Studies on Gravitational Spreading Currents Citation Chen, Jing-Chang (1980) Studies on Gravitational Spreading Currents. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/16RG-AZ72. https://resolver.caltech.edu/CaltechTHESIS:09302010-090407312 Abstract The objective of this investigation is to examine the buoyancy-driven gravitational spreading currents, especially as applied to ocean disposal of wastewater and the accidental release of hazardous fluids. A series of asymptotic solutions are used to describe the displacement of a gravitationally driven spreading front during an inertial phase of motion and the subsequent viscous phase. Solutions are derived by a force scale analysis and a self-similar technique for flows in stagnant, homogeneous, or linearly density-stratified environments. The self-similar solutions for inertial-buoyancy currents are found using an analogy to the well-known shallow-water wave propagation equations and also to those applicable to a blast wave in gasdynamics. For the viscous-buoyancy currents the analogy is to the viscous long wave approximation to a nonlinear diffusive wave, or thermal wave propagation. Other similarity solutions describing the initial stage of motion of the flow formed by the collapse of a finite volume fluid are developed by analogy to the expansion of a gas cloud into a vacuum. For the case of a continuous discharge there is initially a starting jet flow followed by the buoyancy-driven spreading flow. The jet mixing zone in such flows is described using Prandtl's mixing length theory. Dimensional analysis is used to derive the relevant scaling factors describing these flows. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Civil Engineering) ; gravity currents, buoyancy driven flows in both inertial and viscous phases, force scale analysis, similarity solutions, gravity current flows in stagnant homogeneous or linearly density-stratified environments Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): List, E. John Thesis Committee: Unknown, Unknown Defense Date: 17 March 1980 Funders: Funding Agency Grant Number NSF GK-35774X NSF ENG75-02985 NSF ENG77-27398 Record Number: CaltechTHESIS:09302010-090407312 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09302010-090407312 DOI: 10.7907/16RG-AZ72 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6077 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 30 Sep 2010 18:34 Last Modified: 17 Dec 2024 20:09 Thesis Files Preview PDF - Final Version See Usage Policy. 12MB Repository Staff Only: item control page

STUDIES ON GRAVITATIONAL SPREADING CURRENTS Thesis by Jing-Chang Chen In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1980 (Submitted March 17, 1980) © i980 Jing-Chang Chen All·Rights Reserved ii ACKNOWLEDGMENTS The writer wishes to express his deepest gratitude to his thesis advisor. Professor E. J. List. who offered much guidance, encouragement, and valuable criticism throughout this investigation. Several talks with Professors Vito A. Vanoni and Norman H. Brooks always gave a long lasting comfortable feeling, which is especially needed in the life of a graduate student at Caltech. His officemate. Dr. Greg Gartrell, Jr., helped in many phases in accomplishing this research and shared many ups and downs of the research life. The writer would also like to thank Mr. Elton F. Daly, supervisor of the Shop and Laboratory, and Mr. Joe Fontana for their assistance in designing and constructing the experimental set up and their patience in teaching him how to play cribbage during lunch time. Appreciation is also due Mr. David Byrum, who helped in setting ·up equipment and taking photographs; Mrs. Adelaide R. Massengale, who typed the manuscript; and Mrs. Joan Mathews, who typed some parts of the earlier draft. For financial aid of his graduate studies at Caltech, the writer is grateful to The Li Foundation Inc., New York for a two-year (1974-76) Fellowship and to the California Institute of Technology for Graduate Laboratory Assistantship (1974-75), and Graduate Research Assistantship (1975-present). The financial support for this research work from the National iii Science Foundation through Grant Nos. GK-35774X, ENG75-02985, and ENG77-27398 is greatly appreciated. Finally, the author wishes to thank his parents and every member of hls family for their continuous encouragement for a higher education over the past years, and his wife, Yuan-Lin, for her sacrifice in putting up with most of her time alone, days and nights. Without the strong spiritual support from each member of the family, completing this research work would have been impossible. Grandfather. This thesis is dedicated to his beloved iv ABSTRACT The objective of this investigation is to examine the buoyancy-driven gravitational spreading currents, especially as applied to ocean disposal of wastewater and the accidental release of hazardous fluids. A series of asymptotic solutions are used to describe the displacement of a gravitationally driven spreading front during an inertial phase of motion and the subsequent viscous phase. Solutions are derived by a force scale analysis and a selfsimilar technique for flows in stagnant, homogeneous, or linearly density-stratified environments. The self-similar solutions for inertial-buoyancy currents are found using an analogy to the well-known shallow-water wave propagation equations and also to those applicable to a blast wave in gasdynamics. For the viscous- buoyancy currents the analogy is to the viscous long wave approximation to a nonlinear diffusive wave, or thermal wave propagation. Other similarity solutions describing the initial stage of motion of the flow formed by the collapse of a finite volume fluid are developed by analogy to the expansion of a gas cloud into a vacuum. For the case of a continuous discharge there is initially a starting jet flow followed by the buoyancydriven spreading flow. The jet mixing zone in such flows is described using Prandtl's mixing length theory. Dimensional analysis is used to derive the relevant scaling factors describing these flows. v TABLE OF CONTENTS ACKNOWLEDGMENTS ii ABSTRACT iv TABLE OF CONTENTS v LIST OF FIGURES x LIST OF TABLES xxvii NOMENCLATURE xxx CHAPTER 1 INTRODUCTION 1 CHAPTER 2 REVIEW OF PREVIOUS STUDIES 8 2.1 Wastewater Field from Submarine Outfall Discharges 8 Maximum Height or Rise and Spreading Level Observations of Spreading Flows Unsteady Density Spread in a Stagnant Environment 8 8 2.1.1 2.1. 2 2.1.3 2.2 Surface Wastewater Discharge in a Cross Current 2.2.1 2.2.2 2.3 Gravity-Driven Starting Flows 2.3.1 2.3.2 2.3.3 2.3.4 2.4 Formation of Gravity-Driven Flows Lock Exchange Flows Plane Surface Starting Buoyant Jets Von Karman, Prandtl, and Benjamin Theories 26 26 31 38 38 39 47 56 Previous Analysis of Density Front Propagation 69 Spreading in a Homogeneous Environment Spreading in a Linearly-Stratified Environment 69 2.4.1 2.4.2 2.5 Non-Buoyant Models Surface Density Spread Models 16 Further Mixing of Surface Spreading Flow 2.5.1 2.5.2 Description of Spreading Flow Surface Buoyant Jet Models 72 74 74 74 vi CHAPTER 3 THEORETICAL ANALYSIS 78 3.0 Introductory Note 78 3.1 Force Scale Analysis for Density Currents 78 3.1.1 3.1.2 3.1. 3 3.2 Free Surface Spreading Currents Bottom Spreading Currents Internal Density Currents Similarity Solutions for Inertial-Buoyancy Density Currents 3.2.1 3.2.2 3.2.3 Longitudinal Internal Shallow-Water Wave Equations Gas Dynamics Analogy Derivation of Self-Similar Formulae 3.2.3.1 3.2.3.2 3.2.4 Evaluation of Self-Similar Solutions 3.2.4.1 3.2.4.2 3.2.5 3.2.5.2 3.2.5.3 3.2.6.2 3.2.6.3 94 95 98 101 101 105 107 107 112 115 Plane and Radial Surface Density Spread from a Finite Volume Release 116 Starting Two-Dimensional Stratified Flow 120 Surface Radial Starting Plume 122 Asymptotic Solutions for Internal Density Spreading 3.2.6.1 3.2.7 General Derivation Shock Conditions at Front Asymptotic Solutions for Surface Density Spreading 3.2.5.1 3.2.6 Method of Separation of Variables Taylor's Method 78 88 90 124 Plane and Radial Submerged Density Spread from a Finite Volume Release 125 Internal Plane Starting Layer (Continuous Flow) 129 Internal Radial Starting Layer (Continuous Flow) 131 Initial Stage of Density Spread 133 vii 3.3 Similarity Solutions of Viscous-Buoyancy Density Currents 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.4 145 147 150 157 165 167 Introductory Note Turbulent Surface Jet Mixing 167 172 Dimensional Analysis 3.5.1 3.5.2 3.5.3 3.5.4 CHAPTER 4 144 Further Mixing of Surface Spreading Flow 3.4.0 3.4.1 3.5 Nonlinear Long Wave Equations Derivation of Self-Similar Formula by the Method of Separation of Variables Derivation of General Self-Similar Solutions Solutions for Spreading from a Finite Volume Release Plane Viscous Spreading Layer from a Continuous Discharge Radial Viscous Spreading Layer from a Continuous Discharge 143 Radial Surface Buoyant Jet Surface and Submerged Horizontal Buoyant Jets Dynamic Collapse Radial Surface Buoyant Jet in a Cross Current 185 185 194 196 199 EXPERIMENTAL STUDIES AND APPARATUS 203 4.0 Introduction 203 4.1 Surface Buoyant Jet 203 4.1.1 Experimental Set-Up 4.1. 2 Density and Dilution Measurements 4.1.3 Photographic Equipment 203 204 206 Radial Surface Heated and Buoyant Jets 208 4.2.1 4.2.2 4.2.3 208 209 216 4.2 General Experimental Set-Up Temperature Measurements Overhead Photographs viii CHAPTER 5 EXPERIMENTAL RESULTS 217 5.1 217 Half-Radial Surface Buoyant Jet 5.1.1 5.1.2 5.1.3 CHAPTER 6 Radial Heated Surface Jet 242 5.3 Radial Surface Buoyant Jet in a Cross Current 250 COMPARISONS WITH PREVIOUS STUDIES 264 6.1 264 Spreading Currents in Homogeneous Environment 6.1.4 6.1. 5 6.2 Plane Current from a Finite Volume Release 264 Radial Current from a Finite Volume Release 277 Plane Current from Continuous Steady Discharge 284 Radial Current from Continuous Steady Discharge 295 Initial Stage of Finite Volume Release 308 Spreading Currents in Linearly Stratified Environment 6.2.1 6.2.2 6.2.3 CHAPTER 8 220 231 242 5.2 6.1.1 6.1.2 6.1. 3 CHAPTER 7 Starting Half-Radial Surface Buoyant Jet Density Measurements Periodic Feature Plane Submerged Current from a Finite Volume Release Plane Submerged Current from Continuous Steady Discharge Radial Submerged Current 314 314 337 357 ENGINEERING APPLICATIONS 360 7.1 Ocean Outfall Discharge 360 7.2 Surface Buoyant Discharge 366 7.3 Barge Disposal of Dredged Spoil and Sludge 372 7.4 Base Surge Flow 374 DISCUSSION AND CONCLUSIONS 377 8.1 On the Similarity Solutions for Buoyant Spreading Currents 377 ix 8.2 On the Initial Stage of Flows 380 8.3 On the Memory of Spreading Currents 381 8.4 On the Thickness Profile of Spreading Currents 382 8.5 On Dimensional Analysis and Spreading Currents 383 APPENDIX A DERIVATION OF LONGITUDINAL INTERNAL SHALLOW-WATER WAVE EQUATIONS 384 A.O Basic Equations and Assumptions 384 A.l Free Surface Density Spreading Flows 386 A.2 Bottom Surface Density Spreading Flows 389 A.3 Submerged Density Spreading Flows 392 APPENDIX B DERIVATION OF SIMILARITY SOLUTIONS FOR INERTIALBUOYANCY SPREAD 397 B.l Surface Density Spreading Flows 397 B.2 Submerged Density Spreading Flows 399 APPENDIX C DERIVATION OF BOUNDARY CONDITIONS AT THE FRONT 402 C.O Introduction 402 C.l Surface Density Spread 403 C.2 Submerged Density Spread 405 APPENDIX D SEDOV'S (1957) LINEAR SIMILARITY VELOCITY FUNCTION 408 APPENDIX E DERIVATION OF VISCOUS LONG WAVE EQUATIONS FOR DENSITY SPREAD FLOWS 411 E.O Basic Equations and Assumptions 411 E.1 Viscous Surface Density Spread 413 E.2 Viscous Submerged Density Spread 417 LIST OF REFERENCES 421 x LIST OF FIGURES Figure 1.l(a) 1.l(b) 1. 2(a) 1. 2 (b) 1.3 1. 4(a) 1. 4(b) 1. 4 (c) 1.5 1.6 Surface density spread field from a horizontal single-port outfall discharge into a uniform environment. 3 Submerged density spread field from a horizontal single-port outfall discharge into a densitystratified environment. 3 Surface density spreading field due to a multiport diffuser discharge into a uniform environment. 3 Submerged density spreading field due to a multi-port diffuser discharge into a densitystratified environment. 3 Submerged density field from a single-port outfall discharge spreading between two homogeneous layers of densities Pu and Pa (e.g. epilimnion and hypolimnion). 4 Bottom turbidity current from a barged sludge disposal. 4 Spreading turbidity current at the stably stratified interface from barged sludge disposal. 4 Spreading turbidity current at an intermediate level of density stratified ocean from barged sludge disposal. 4 Dense bottom currents from a negatively buoyant discharge. 4 A wide surface density spreading field persisting for a long distance before oceanic turbulent diffusion takes over, aerial photograph taken by Foxworthy and Kneeling (1969) at Ventural Outfall on November 11, 1966. 6 xi Figure A series of radial oceanic fronts in a fjord observed by McClimans (1978). The photograph from which this reproduction was made was taken by T. A. McClimans, who graciously supplied a copy to the author. 12 Radially horizontal spreading cooling tower plume in a stably stratified environment. (Courtesy of Kramer and Seymour, 1976) 15 2.1. 2 (b) A series of discontinuous radially spreading fronts. (Courtesy of Kramer and Seymour, 1976) 15 2.1. 3(a) Flow regions resulting from a positive submerged buoyant round or plant jet discharging vertically upwards into a homogeneous environment. 17 Flow regions resulting from a negative buoyant round or plane jet discharging vertically downwards into a homogeneous environment. 17 Free surface spreading front due to a submerged horizontal buoyant round jet discharge. 18 Bottom surface spreading front due to a horizontal buoyant round jet discharge. 18 Unsteady spreading tip position at time t due to a buoyant pipe flow discharge, after Sharp (1971). 18 Simulation of submerged density spreading by a horizontal jet discharging at its own density level in a linearly density-stratified environment, after Maxworthy (1972). 22 Growth history of the interflowing wedge front due to a horizontal slot jet discharging at a neutral buoyant level into a linearly densitystratified ambient, data from Zuluaga-Angel et aZ. (1972). 23 Three stages of plane submerged density front spread due to a finite volume of homogeneous fluid collapsing at a neutral level in a linearly density-stratified environment, after Wu (1965). 25 2.1.1 2.1.2(a) 2.1. 3(b) 2.1. 4 (a) 2.1. 4(b) 2.1. 5 2.1. 6 2.1. 7 2.1.8(a) xii figure Sequential cross s-ections of plane submerged density front spread at various times, after Wu (1965). 25 A non-buoyant and non-diffusive source in a uniform cross current. 28 Schematic diagram for diffusive field from line-like diffuser in an ocean current, after Brooks (1960). 28 Superposition of a series of circular diffusion patches in a horizontal turbulent current. 30 Superposition of a series of slices of diffusion elements in a horizontal turbulent current. 30 Top view of surface density spread field in a horizontal current. 34 Central cross section along x direction of surface density spread field in a horizontal current. 34 Side-spread model for the plume boundary of a surface wastewater field. 35 Radial spread model for the plume boundary of surface wastewater field. 37 2.3.1 Lock exchange flow. 40 2.3.2(a) Lock and channel before removing gate. 43 2.3.2(b) Advancing lock exchange flow. 43 2.3.2(c) Finite-length exchange flow. 43 2.3.3 Idealized shape of saline front from finite-length lock exchange flow, after Keulegan (1957). 43 2.3.4(a) Forced free surface density currents. 49 2.3.4 (b) Forced bottom surface density currents. 49 2.3.5 Surface density currents and discharging box used by Barr (1959). 51 2.1. 8(b) 2.2.1 2.2.2 2.2.3(a) 2.2.3(b) 2.2.4 (a) 2.2.4(b) 2.2.5 2.2.6 xiii Figure 2.3.6 2.3.7 2.3.8(a) 2.3.8(b) 2.3.9 2.3.l0(a) 2.3.l0(b) 2.3.ll(a) 2.3.ll(b) 2.3.12 3.1.1 3.1. 2 3.1.3 Growth history of starting free surface gravity currents, reproduced from Chen (1975). Experimental data from Almquist (1973) . 53 Advancing wedge on the bottom of a channel, after Keulegan (1946). 55 Configuration of a forced plane gravity current as viewed in a fixed reference frame to the ground. 57 Configuration of a forced plane density wedge as viewed in a moving reference frame with the front, after von Karman (1940). 57 Internally spreading currents between a stably stratified interface. 59 Advancing mass of cold air; actual flow relative to the ground, after Prandtl (1952). 61 Advancing mass of cold air; idealized flow relative to the front, after Prandtl (1952). 61 Cavity front moving at a constant velocity along a two~dimensional box. 64 Arrested cavity front as viewed in a moving frame with the same velocity of front. 64 Comparison of theoretical solution (solid line) of Benjamin (1968) with the experimental results (open circles) of emptying cavity flow in an expansion gallery of surge tank by Ellis and Al-Khairulla (1974). 67 Idealized sketch of free surface spreading density currents. 79 Idealized sketch of bottom spreading density currents. 89 A homogeneous layer intruding at a neutral buoyant level into the linearly densitystratified ambient fluid. 89 xiv Figure 3.2.1(a) Coordinate system for free surface spreading flows. 96 3.2.1(b) Coordinate systems for bottom spreading flows. 96 3.2.1(c) Coordinate system for submerged density flows. 96 3.2.2 Self-similar velocity and thickness distributions of the plane and the radial inertial density wave front propagation due to a finite volume release of buoyant fluid. 119 Velocity and thickness distributions of a starting internal radial ring. 123 Similarity profiles of the half-thickness for a plane (n = 0) and radial (n = 1) interf10wing layer resulting from a collapsing homogeneous mixed region at a neutral density level in a linearly stratified environment. 128 Similarity profiles of the average velocity and the half-thickness of interf10wing radial ring due to a continuous release. 132 Similarity profiles for the mean horizontal velocity and the spreading layer thickness for the initial stage of surface density spreading in a homogeneous environment (i = 0) • 140 Similarity profiles for the average velocity and the half-thickness of interf10wing submerged density spreading layer due to a finite volume release in a linearly densitystratified environment (i = 1) . 140 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.3.1 3.3.2 Self-similar thickness distribution of plane and radial viscous surface spread layer due to a finite vo1umn release (i = 0, m = 0, n = 0, and 1) . 152 Self-similar thickness distribution of plane and radial viscous submerged density spread layer due to a finite volume release (i = 1, m =~O~, n = 0, and 1). 153 xv Figure 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.4.l(a) 3.4.i(b) Self-similar distribution of plane and radial viscous surface (i = 0) or submerged (i = 1) spreading fluid velocity due to a finite volume release (n = 0, 1, and m = 0) • 158 Self-similar thickness distribution of plane viscous surface spread layer due to a constant rate of discharge into a homogeneous environment (i= 0, m= 1, n= 0). Solid line denotes the first approximation, and open circles denote the third approximation. 161 Self-similar half-thickness distribution of plane viscous submerged density spread layer due to a constant rate of discharge into a linearly density-stratified environment (i = 1, m = 1, n = 0) • Solid line denotes the first approximation and open circles denote the third approximation. 162 Self-similar thickness distribution of radial viscous surface spread layer due to a constant rate of discharge into a homogeneous environment (i = 0, m = 1, n = 1). Solid line denotes the third term approximation, open circles, the first term approximation, and triangles, the second term approximation. 168 Self-similar half-thickness distribution of radial viscous submerged density spread layer due to a constant rate of discharge into a linearly density-stratified environment (i = 1, m = 1, n = 1). Solid line denotes the third term approximation, open circles, the first term approximation, and triangles, the second term approximation. 169 Flow regimes of an outfall discharge (soon after startup): I: Submerged buoyant jet flow, II: Surface transition, III: Surface buoyant jet flow. 173 A surface buoyant jet flow model with finite dimension of discharge (soon after startup). 173 xvi Figure 3.4.2 Point source surface jet nomenclature. 174 3.4.3 Two self-similar profiles of mean horizontal velocity, F' (0 = u(r,z) /t1m(r) , and mean tracer when Ku = Kc , e (0 = c(r, z) / cm(r) . Dash line indicates the solution derived from the Prandtl-Tollmien assumption and solid line from Prandt1-Gort1er assumption. 186 3.5.1 Sketch of a radial surface buoyant jet. 187 3.5.2 Three regimes of starting surface or submerged buoyant jet flow. 197 3.5.3 Three stages of dynamic collapse flows. 198 4.1.1 Cross section of the half-radial discharging box. 205 4.1.2 Overall experimental set-up for the ha1fradial surface buoyant jet. 205 4.1.3 Photograph of a conductivity probe. 207 4.1.4 Typical calibration curve of conductivity probe. 207 4.2.1 Overall experimental set-up for radial surface heated jet. 210 Photograph of a rake of five fast-response thermistors. 212 Fast-response thermistor circuit designed by Gartrell (1979). 212 Buffer circuit for A/D converter from Gartrell (1979). 214 Typical calibration curve for fast-response thermistor. 215 Cross sections of half-circular front at various time t after initiation of discharge. (a) Run No. 10/5/77, Fr = 5.25, Re = 7100. 221 Run No. 11/16/77, Fr = 10.0, Re = 8840. 222 4.2.2 4.2.3 4.2.4 4.2.5 S.1.l(a) S.1.l(b) xvii Figure 5.l.l(c) Run No. 11/4/77. Fr = 15.0, Re = 8840. 223 5.1. 2 Growth history of starting radial surface turbulent buoyant jet flow. Closed symbols indicate the photographs are shown in Figure 5.1.1. 224 Normalized growth history of starting radial surface turbulent buoyant jet flow. Closed symbols indicate the photographs are shown in Figure 5.1.1. 224 Relation of starting jet coefficient to jet Reynolds number. 226 Relation of starting jet coefficient to radial jet Froude number. 226 Relation of radial starting plume coefficient to (a) jet Reynolds number, (b) Froude number, (c) Rl/Q. QM • 228 Similarity profiles of viscous radial surface spreading currents at time t (71.4 seconds for No. 9/21/77, 101.7 for 9/28/77, and 96.6 for 10/5/77) after discharge. 230 Radial stratified flow due to a half-radial surface buoyant jet discharge (a) Run No. 10/5/77 (b) Run No. 11/16/77 (c) Run No. 11/4/77. 232 233 234 Density measurements of a half-radial surface buoyant jet (Run No. 10/27/77) with Fr = 10.00 at a radial distance r = 32.39 cm (or r/Q.MB = 0.726). 235 Distribution of relative density difference respectively at a distance r = 17.15, 32.39, and 67.95 cm (or r/~ = 0.384, 0.726, and 1.523) for a radial surface flow field (Run No. 10/27/77). 237 Self-similar distribution of relative density deficiency. Dotted line denotes PrandtlTollmien's solution. 238 5.1. 3 5.1. 4 (a) 5.1. 4(b) 5.1. 5 5.1.6 5.1. 7 5.1. 8 5.1. 9 5.1.10 xviii Figure Relation of the maximum value of relative density deficiency to the radial distance. Symbols are defined in Figure 5.1.10. 239 5.1.12 Relation of the half-thickness to the radial distance. Symbols are defined in Figure 5.1.10. 239 5.1.13 Self-similar distribution of relative density deficiency (Run No. 10/10/77, Fr = 5.26). 240 5.1.14 Relation of the maximum value of relative density deficiency to the radial distance. Symbols are defined in Figure 5.1.13. 241 Relation of the half-thickness to the radial distance. Symbols are defined in Figure 5.1.13. 241 Comparison of the observed period with the predicted value T given by Eq. (3.2.87). 243 Thermal expansion coefficient of water as a function of temperature after Kotsovinos (1975). 247 5.2.2 Kinematic viscosity of water as a function of temperature. 247 5.2.3 Relation of the maximum value of relative mean temperature excess to the relative radial distance r/r .. 248 Relation of the maximum value of relative mean temperature excess to the normalized radial distance r/!/,MB' 249 Relation of the normalized half-thickness to the normalized radial distance. Symbols are defined in Figure 5.2.4. 249 Self-similar relative mean temperature excess above the ambient temperature. Solid line denotes the Prandtl-Tollmien solution given in Chapter 3. (a) Run No. 10/19/78, Fr = 6.36. (b) Run No. 10/20/78, Fr = 6.28. (c) Run No. 10/18/78, Fr=lO.lO. Cd) Run No. 10/17/78, Fr= 9.28. 251 251 252 252 5.1.11 5.1.15 5.1.16 5.2.1 J 5.2.4 5.2.5 5.2.6 xix Figure 5.2.7 5.3.l(a,b) 5.3.1(c,d) 5.3.2 5.3.3 5.3.4 6.1.1 6.1. 2 Profiles of the intensity of turbulent temperature fluctuations for radial surface heated jet (Run No. 10/17/78) versus depth and relative radial distance. 253 Photographs of the development of a surface flow due to a radial surface buoyant jet in a cross current (Run No. 4/10/77). (a) t = 35, (b) t = 80 seconds after initiation of discharge. 256 Photographs of the development of a surface flow due to a radial surface buoyant jet in a cross current (Run No. 4/10/77). (c) t = 110, (d) t = 485 seconds after initiation of discharge. 257 Growth histories of the downstream edge of an initiating surface field. 258 Upstream edge position of surface buoyant field. Symbols are defined in Table 5.3.2. 261 Growth of a surface buoyant field measured from upstream edge. 263 Growth histories for plane surface spreading currents due to a finite volume release. Experimental data are from Almquist (1973). Symbols are defined in Table 6.1.1. 267 Normalized growth histories of plane surface spreading currents due to a finite volume release. Symbols are defined in Table 6.1.1. R 6.1.3 6.1.4 1 = 5 )1/7 and (~ v Coefficients of inertial-buoyancy spreading currents ~xpressed as a function of Reynolds number g~h h /v. o 0 Coefficients of viscous-buoyancy spreading currents jxpressed as a function of Reynolds number g~h h /v. o 0 268 270 270 xx Figure 6.1. 5 6.1. 6 6.1.7(a) 6.1. 7 (b) 6.1.8 6.1.9(a) 6.l.9(b) 6.1.9(c) 6.1.10(a) Frontal velocity decay of density currents from a finite-length lock exchange flow, experimental data from Figure 3 of Keu1egan (1957). H=S.8 em, La = 41. 7 cm, and width of channel = 2.54 cm. 273 Frontal velocity decay of turbidity currents from a finite-length lock exchange flow, experimental data from Figure 10 of Middleton (1966). H = 20.3 em, La = 28.3 em, width of channel = 13.83 em, and average settling velocity of suspension = 0.9 em/sec. 274 Frontal trajectory for an inertial-buoyancy oil slick spreading in a channel, data from Run No. 1 of Liang (1971). V = 131. 2 cm 2 , !;, = 0.136. 276 "Wave gage" measurements (in solid curve) of inertial oil slick spreading thickness by photocell equipment located at three locations (60, 85, 110 em from end wall of channel for Station no. 1,2,3). Dotted curves indicate prediction by similarity solutions, data from Liang (1971). 278 Dynamic collapse of a homogeneous region at a stably stratified interface. Numerical calculation data from Figure 7 of Meng and Thomson (1978). 279 Oil spreadin:g sections at four times after release of 1500 cubic meters of oil (1200 tons). Data from Abbott (1961). 281 Growth history of inertial-buoyancy radial oil spreading front. Data from Figure 6.1.9(a). 282 Comparison of oil spreading layer thickness given in Figure 6.1. 9 (a) with similarity profiles (n = 0 denotes plane case, Eq. (6.1.10) and n = 1 denotes radial case, Eq. (6.1.11». 282 Radial gravity currents produced from circular lock exchange flow. The aspect ratio Ar = 2 and the initial radius Ro = 3.45 em, data from Table 1 of Martin and Moyce (1952, Part V). 285 xxi Figure 6.l.l0(b) 6.1.11 6.l.l2(a) 6.l.l2(b) 6.1.13 6.1.14 6.l.l5(a) 6.l.l5(b) 6.l.l5(c) 6.1.16 6.1.17 Radial gravity currents produced from circular lock exchange flow. The initial relative density difference D, = 0.428, data from Table 2 of Martin and Moyce (1952, Part V). 286 Growth history of a volume of syrup spreading on the shining face of aluminum foil. 287 Plane surface spreading currents from a continuous steady discharge, data from Barr (1959). 289 Plane surface spreading currents from a continuous steady discharge, data from Almquist (1973). 290 Plane bottom spreading currents from a steady continuous discharge. 291 Unsteady surface spreading currents in a channel due to a buoyant pipe flow discharge, data from Sharp (1971). 293 Radial surface stratified flow formed after the diluted effluent reached the free surface (77.9 and 155.5 seconds later). Due to a submerged buoyant round jet (dj = 0.586 cm, D, = 0.0170, Qj = 40.8 cm 3 /sec) discharged vertically into a uniform stagnant environment (H = 11.72 cm). 297 A band of radial surface stratified flow formed after the diluted effluent reached the free surface (70.5 and 166.5 seconds later). Due to a submerged buoyant round jet (dj = 0.586 cm, D, = 0.0142, Qj = 18.9 cro 3 / sec) discharged vertically into a uniform stagnant environment (H=6.l9 cm). 298 A series of outward travelling circular internal waves formed after the diluted effluent reached the free surface (79.1 and 121.6 seconds later). Due to a submerged buoyant round jet (dj = 0.586 cm, D, = 0.1092, Qj = 4.5 cm 3 / sec) discharged vertically into a uniform stagnant environment (H = 23.41 cm). 299 Growth histories of radial surface spreading fronts shown in Figures 6.l.l5(a), (b), and (c). 301 Growth histories of bottom radial gravity currents, data from Britter (1979). 302 xxii Figure 6.1.18 6.1.19 6.1.20 6.1.21 6.l.22(a) 6.l.22(b) 6.1. 23 6.1. 24 6.1. 25 6.1. 26 Growth history of radial surface spreading fronts due to a submerged buoyant round jet discharging horzontally into a uniform stagnant environment, data from Sharp (1969a). 304 Growth history of radial bottom gravity currents due to a submerged buoyant round jet discharge horizontally into a uniform stagnant environment, data from Sharp (1969b). 305 Radial spreading rate of crude oil on ice. The kinematic viscosity v of crude oil is assumed as 60 cm2 /sec for open symbols and as 23.1 cm 2 /sec for solid symbols. Data are from MCMinn and Golden (1973). 306 Growth history of surface spreading front at centerline of boundary buoyant surface discharge, data from Barr (1959) and Hayashi and Shuto (1967). 307 Initial collapse of a rectangular shape column, line sketch of photograph from Martin and Moyce (1952, part IV). 310 Calculated self-similar profiles of initial collapse shown in Figure 6.l.22(a). 310 Growth histo'ry of plane front during initial collapse of a liquid column with rectangular section (hI 3ho /2 in Eq. (6.1.39». Data from Tables 1, 2, and 3 of Martin and Moyce (1952, part IV). 311 Growth history of plane front during initial collapse of a liquid column with semi-circular section (hI = 3 1T ho in Eq. (6.1.39» (Ro = 2 inches), data 8 from Table 4 of Martin and Moyce (1952, part IV). 312 Growth history of radial front during initial collapse of a liquid column with vertical circular cylinder (hl=2ho in Eq. (6.1.40», data from Table 5 of Martin and Moyce (1952, part IV). 313 Comparison of self-similar profile with numerical calculation results by Harlow and Welch (1965) and Fox and Goodwin (1952) for the collapse of a plane symmetrical column in vacuo, initially rectangular at rest, at various relative distance n = R(t) /R . 315 o xxiii Figure 6.1.27 Comparison of self-similar profile with numerical calculation results by the method of characteristics by Penney and Thornhill (1952) for the collapse of an initially semi-cylindrical section column at rest at specific time t ~ 6.1. 28 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 t = 0.8 or n == R(t)/R == 1. 78. o o 316 Comparison of self-similar profile with numerical calculation results by the method of characteristics by Penney and Thornhill (1952) for the collapse of an initially semi-elliptic column at rest at various time t/gh o IRo == 0.5, 1.0, 1.5, and 2.0 or various relative distance n=R(t)/Ro= 1.44,2.08,2.75, and 3.50. 317 Three spreading stages of a plane mixed region collapsing in a linearly density-stratified environment. 320 Comparison of theoretical solutions (the initial and the final collapse stages) for submerged density spreading front with experimental results by van de Watering (1966) on a plane mixed region collapsing in a linearly density-stratified environment. 324 Eleven cases of growth history for submerged density spread front due to a plane mixed region collapsing in a linearly density-stratified environment. Data are taken from Figures 10, 11, and 12 or Wu (1965). Details of experimental parameters for each case are shown in Table 6.2.1. 325 A plane mixed region collapsing in a linearly density-stratified environment. Comparison of theoretical solutions (the principal and the final collapse stages) for submerged density spreading front with experimental results by Wu (1965). 328 A plane mixed region collapsing in a linearly density-stratified environment. Comparison of theoretical solutions for submerged density spreading front (the inertial-buoyancy and the viscousbuoyancy regions) with experimental results by Wu (1965). 329 xxiv :Figure 6.2.6 6.2.7 6.2.8 6.2.9 6.2.10 6.2.11 6.2.12 6.2.13 6.2.14 6.2.1S Coefficient of the principal collapse stage (the inertial-buoyancy submerged spread) as a function of Reynolds number ReN and the initial radius of semi-circular mixed region. 330 Coefficient of the final collapse stage (the viscous-buoyancy submerged spread) as a function of Reynolds number ReN and the initial size of semi-circular or circular mixed region. 331 Plane mixed region in a linearly density-stratified environment: comparison of theoretical solution with experimental results by Wu (196S), for an interflowing front in the final collapse stage. Detail of recalculation is shown in :Figure 6.2.2. 333 Comparison of theoretical solutions with previous numerical calculations by several investigators as shown in Table 6.2.3. 336 Growth history of an inertial starting plume observed by Manins (1976a). Details of experimental parameters are given in Table 6.2.6. 342 Growth history of viscous plume observed by Maxworthy (1972). Details of experimental parameters are given in Table 6.2.6. 344 Growth history for submerged plane horizontal starting buoyant jets observed by Zuluaga-Angel et aZ. (1972). Details of experimental parameters are given in Table 6.2.6. 347 Intrusion front velocity as a function of buoyancy flux Bl = NQ. Experimental data are taken from Manins (1976a),0,N 2 =3.S; 0, N2 =7.0; 11, N2 =1.40. 348 Self-similar distribution of intrusion layer halfthickness near the source. Experimental data are taken from Manins (1976a), 0, NL = 3.S; 0, U2 = 7.0; 11, N2 = 14.0. 348 The plane submerged front velocity due to a starting horizontal slot buoyant jet discharge at its neutral level in a linearly density-stratified environment. The relevant experimental parameters are given in Tables 6.2.5 and 6.2.6. 3S0 xxv Figure 6.2.16 6.2.17 6.2.18 6.2.19(a) 6.2.19(b) 6.2.20 7.1.1(a) 7.1.1(b) 7.1.1(c) 7.1.1(d) 7.1.2(a) The coefficient of submerged plane viscous starting plume (or submerged viscous-buoyancy spread) expressed as a function of discharged Richardson number Ri j and viscous plume Reynolds number Rey. Experimental parameters given in Table 6.2.6. 351 The coefficient for a submerged plane viscous starting plume expressed as a function of R1/~QM' Symbols are defined in Table 6.2.7. 353 The coefficient of spreading for a submerged plane starting jet expressed as a function of discharge Richardson number Rij' data from Table 6.2.7. 355 Viscous intrusion layer profiles at each time t. Data are taken from experiment No. R-11 in Table 4 of Zu1uaga-Ange1 et aZ. (1972). 356 Self-similar thickness distribution of viscous intrusion layer. Solid curve is the viscous long wave solution (3rd order) derived in Subsection 3.3.5. 356 Line sketch for a series of radial ring discontinuities in a horizontal spreading cooling tower plume in atmosphere. 359 Buoyant surface spreading wastewater field from submerged single port (diameter 78 inches) discharge, average discharge rate Q = 120 mgd (185.6 cfs), submerged depth H = 55 ft, /::, = 0.025. Photograph from Figure 21 of Allan Hancock Foundation (1964). 361 Photograph from Figure 22 of Allan Hancock Foundation (1964). 362 Photograph from Figure 20 of Allan Hancock Foundation (1964). 363 Buoyant surface spreading wastewater field from submerged single port (diameter 36 inches) discharge. Average discharge rate Q = 2.20 mgd (3.40 cfs), submerged depth H = 35 feet. Photograph from Figure 4 of Foxworthy and Kneeling (1969). 364 Growth of buoyant surface wastewater field shown in Figures 7.1.1(a), (b), (c), and (d). 365 xxvi Figure 7.1.2(b) 7.1.3 7.2.l(a) 7.2.l(b) 7.2.l(c) 7.2.2 7.3.1 7.3.2 7.4.1 A.3.l Dimensionless plotting of growth of buoyant surface wastewater field shown in Figure 7.l.2(a). 365 Growth of heated (a,b,c) and nonheated (d,e) surface plume field due to a submerged outfall discharge. Data from Table 1 of Sonnichsen (1971). (a) Q = 75,000 cfs, (b) 110,000 cfs, (c) 145,000 cfs, (d) 50,000 cfs, and (e) 100,000 cfs. 367 Line sketch of the thermal image of a discontinuous radial thermal plume field at Sheboygan Power Station, Wisconsin, on September 15, 1971; aerial photograph taken at a flying height 5000 feet. Photograph was shown in Figure 19 of Scarpace and Green (1973). 368 Line sketch of thermal image of a discontinuous radial thermal plume at Sheboygan Edgewater Plant. Photograph was shown in Figure 8a of Green et aZ. (1972). 369 Line sketch of thermal image of a discontinuous radial thermal plume at Port Washington harbor on September 1971. Photograph was shown in Figure 8b of Green et al. (1972). 370 Comparison of oscillating period of inertial radial surface thermal plume with theoretical value given by Eq. (3.2.87). 371 The time an axially symmetric bottom turbidity current arrived at four fixed locations at a radial distance rs from the dump site of dredge spoil; data from Figure 2 of Gordon (1974). 373 "Wave-gage type" record of the spreading layer thickness of the bottom turbid cloud from three different dump operations measured at a radial distance rs = 30 meters from the dump site; data from Figure 5 of Gordon {1974). 375 Propagating mist-laden air forming the base surge following an underwater atomic bomb explosion at Bikini Atoll. 376 Coordinate system of submerged density spread flows. 393 xxvii LIST OF TABLES Table Review of previous experiments on the buoyant round (n = 1) or plane (n = 0) jet rise into a stagnant linearly density-stratified environment. 9 2.3.1 Experimental studies on lock exchange flows. 41 2.3.2 Experimental studies on plane gravity currents due to a constant rate of discharge at one end of channel. 48 Summary of previous theoretical studies of a plane (n = 0) or a radial (n = 1) submerged density front spreading in a linearly densitystratified environment due to a finite volume release (m = 0) or a continuous discharge (m = 1) at a neutral level. 73 3.1.1 Asymptotic formulae for starting surface plumes (Chen, 1975). 84 3.1. 2 Length and time scales for surface density spreading. 86 Asymptotic formulae for surface density spreading front velocity as a function of frontal distance. 87 Asymptotic formulae for submerged density spreading of an intrusion layer in a linearly densitystratified environment. 92 Length and time scales of submerged density spread for normalization. 93 The original characteristic thickness scale hI for the initial stage of surface density spread in a homogeneous environment (i = 0) expressed as a function of the initial shape of liquid volume before spreading. 138 The original characteristic thickness scale hI expressed as a function of initial cross sect10ns for the initial submerged density spread in a linearly density-stratified environment (i = 1) • 139 2.1.1 2.4.1 3.1.3 3.1.4 3.1.5 3.2.1 3.2.2 xxviii Table The value of the integration constant 1o for surface and submerged spreading fro~on a finite volume release (m = 0) . 155 The frontal trajectory functions for the viscousbuoyant surface or submerged spreading due to a finite volume release (m = 0). 156 Integration values 1°10 for plane viscousbuoyancy spreading cijrrents due to a continuous steady discharge plume. 163 Coefficients bOLO for the plane viscousbuoyancy spreaa~ng currents to a continuous steady discharge plume. 164 Integration values 1011 for the radial viscousbuoyancy spreading cijrrents due to a continuous steady discharge plume. 170 Coefficients bOll for the radial viscousbuoyancy spreaa~ng currents due to a continuous steady discharge plume. 171 Prandtl-Tollmien similarity solutions of turbulent surface jet flows. 181 Length, time, and velocity scales of a radial surface buoyant jet. 190 Definitions of volume, momentum, and buoyancy fluxes for a horizontal surface or submerged buoyant plane or radial jet. 195 3.5.3 The ratio of R:sv to Rl and tBv to tl when i = O. 200 3.5.4 The ratio of R:sv to Rl and tBv to tl when i = 1. 201 4.2.1 Fast-response thermistor characteristics. 209 5.1.1 Summary of data for starting half-radial surface buoyant jet experiments, ro = 9.962 cm, ho = 0.3175 cm, and h = 75.0 cm. J J 218 5.1.2 Summary of calculated data for starting halfradial surface buoyant jet experiments. 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.4.1 3.5.1 3.5.2 219 xxix Page Table 244 5.1.3 Summary of observed and predicted periods. 5.2.1 Summary of experiments on radial surface heated jet in a stagnant environment. 245 Summary of experimental data for a radial surface buoyant jet in a cross current. 255 5.3.2 Calculation of upstream edge position. 258 6.1.1 Summary of experiments performed by Almquist (1973) on the plane surface currents due to a finite volume release. 265 Experimental parameters related to Figures 6.2.3, 6.2.4, 6.2.5, 6.2.6, and 6.2.7. 327 Re-calculation of experimental data for the final collapse stage of a plane mixed region in a linearly density-stratified environment. Data are taken from the upper tangent points of principal collapse stage in Tables 1 and 2 of Wu (1965). 334 Parameters of numerical calculations related to Figure 6.2.9. 335 Four flow patterns for a plane intrusion layer due to different combinations of three dimensionless numbers. 339 Summary of experimental parameters for submerged plane intrusion layer propagation in a linearly density-stratified environment due to a horizontal slot buoyant jet discharge at its neutral level 341 Experimental data for plane intrusive layer spreading related to Figures 6.2.10, 6.2.11, 6.2.12, 6.2.15, 6.2.16, and 6.2.17. 343 Calculation of the coefficient of displacement for a submerged plane viscous starting plume as shown in Figure 6.2.17. 354 5.3.1 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.2.7 8.1 Summary of asymptotic formulae of frontal trajectory functions derived by inertial (shallow-water wave) and viscous long wave theories. 379 xxx NOMENCLATURE dimensional constant A h Ar o R aspect ratio of a fluid volume o dimensionless constant a a j , j = 1,2,-- calibration constants for thermistor probe or coefficients in polynomial expansion series B kinematic buoyant flux or channel width B(x) half-width of surface spreading field B. kinematic buoyancy flux, Bo = gllQ for i = 0 and B1 = NQ for i = 1 B initial half-width of surface spreading field 1 o initial width of diffusive wastewater field b b. coefficient in asymptotic time function b (r) jet width based on velocity u distribution b (r) jet width based on tracer c distribution C(x,y) concentration distribution at (x,y) C initial concentration c sound speed or relative density or temperature difference or inertial starting plume coefficient 1IDU u c o c c coefficient related mixing length scale t u coefficient related mixing length tu to r c (r) m maximum tracer concentration at z = 0 c. coefficient in asymptotic time functions of inertial starting plume c. initial tracer value of jet f1uit D channel depth c lmn J c to r xxxi D. diameter of pipe d thickness of box or starting turbulent jet coefficient J diameter of submerged round jet or width of plane jet elevated height of density head trough height near density head crest height of density head total energy released during explosion F initial Froude number of lock exchange flow, U/hg(P2 - P I )H/(P2 -+ PI) inwards horizontal pressure force at front outwards horizontal pressure force at front unbalanced total pressure force for surface or submerged spreading, F S - FA inertial force retarding spreading viscous force due to boundary layer shear in the ambient fluid viscous force due to boundary layer shear in spreading fluid FA ' inwards horizontal pressure force at front for bottom gravity currents FS ' outward horizontal pressure force at front for bottom gravity currents Fp ' unbalanced total pressure force for bottom gravity currents, FS' - FA' hydrodynamic drag force resisting spreading interfacial shear force resisting spreading initial densimetric Froude number of horizontal spreading flow xxxii discharge Froude number of buoyant jet, ~MB/£QM Fr Fr. inlet Froude number, uj/lg~hj Fr Froude number of em/tYing cavity flow based on cavity depth h2' V/ gh 2 J l Froude number of e~tying cavity flow based on total depth d, V/lgd Froude number of emptying cavity flow based on flow depth d-h , V/Ig(d-h ) 2 2 local densimetric Froude number u/lg~h Froude number at far downstream of plane forced gravity currents, v/lg(p s - p a )hex> /p a Froude number at far field of finite length lock exchange flow, V/hg~Pfdl/(PI + P2) Froude number at far field of lock exchange flow, v/hg~Pfd2/(PI +P2) f self-similar fluid velocity distribution function of (i+ 4)th power function of H(.;) for viscous spreading flow g gravitational acceleration, 980 cm/sec 2 g' effective gravitational acceleration, g(p a - P.) /p a 1 H ambient water depth H(F,;) similarity thickness distribution length scale associated with buoyant plume height of rise length scale associated with momentum jet height of rise HA (F,;) approximation function of H(F,;) 1\ (.;) lower bound of H(F,;) Hu(F,;) upper boune of R(F,;) HI (.;) dimensional self-similar thickness scale xxxiii h(r,t) spreading layer thickness at position r and time t h , h(t) average thickness of spreading field or spreading layer h.~ initial surface spreading layer thickness due to a submerged buoyant jet discharge thicknes-s (i = 0) or half-thickness (i = 1) of buoyant jet discharge head loss in emptying cavity flow height of density nose h o initial thickness of a spreading volume at symmetric point thickness scale for initial spreading stage listed in Tables 3.2.1 and 3.2.2 thickness of cavity in emptying cavity flow in a channel h co plane gravity current thickness at far downstream from head 1. integration constant for the case of i, m, n i environment parameter denotes uniform environment with i = 0 and linearly density-stratified environment with i = 1 J dimensional number involving momentum flux integration j index integer number 0,1,2,--- ~mn K u K c k unknown coefficient related to eddy viscosity s u unknown coefficient related to eddy diffusivity E c unknown integer k. shape factor for the case of i and n L(t) frontal position at time t L length of finite length lock exchange flow ~n o lobe length of three-dimensional feature of density current or unknown integer xxxiv length scale defined by M and B length scale defined by Q and M length scale defined by Q and B mixing length scale of velocity u mixing length scale of tracer c M kinematic momentum flux of'buoyant jet m source parameter denotes continuous steady discharge with m = 1, the finite volume release with m = N Brunt-Vaisala frequency of ambient stratification n geometric parameter, denotes plane symmetric flow with n = 0, and radially symmetric flow with n = 1 ° total flow force at downstream of emptying cavity flow Prandtl number p p pressure at the nose observed from a coordinate system travelling with front s total flow force at upstream of emptying cavity flow u p 00 pressure force at far downstream observed from a coordinate system travelling with front p gas pressure Q volume flux of buoyant jet computed flow rate of a plane spreading current in a linearly stratified environment calculated volume flux rate at the source of spreading discharge flow rate of submerged buoyant jet frontal distance or radius at time t Re general symbol for Reynolds number Re. buoyant jet Reynolds number, ~B~MB/Vj or Q/vs Re initial Reynolds number of internal spreading in a linearly stratified environment due to finite volume release, Nh 2/v or NR 2/v J N o s 0 s xxxv Re plume Reynolds number p (NQ)1/2 h j ReV::; --v-_:Ls Reynolds number of plane jet discharging into a linearly stratified environment Reynolds number of density head based on trough height d and average viscosity vm l Ud Re 2 v 3 Reynolds number of density head based on crest height d and average viscosity vm m Ud Re 2 2 2 Reynolds number of density head based on crest height and spreading fluid viscosity v s = -V s frontal Richardson number of plane current in stratified ambient Ri = Ri(r,t) local Richardson number 1 gt:,. ( -~21 )1+J h(r, Rif(t) t) = frontal Richardson number 1 L\ )i+l gR/ h(R, t) ( Nh. Ri. J ---l u. J initial plane Richardson number of slot jet discharging into a linearly stratified environment length scale defined by M and B length scale defined by B and viscous spreading R o initial radial distance and half-length of base transition length scale from inertial-buoyancy spreading to viscous-buoyancy spreading as defined in Tables 3.1.2 and 3.1.5 space variable r r. radius of radial surface buoyant jet r measuring location from the end of wall J s xxxvi S(y,m,n) coefficient in shock wave front propagation S channel slope S c o average distribution at free surface due to submerged discharge T measured temperature or theoretical oscillating period of inertial radial plume T temperature of ambient fluid T. temperature of jet fluid Tob observed oscillating period in density measurement T' fluctuation from mean temperature t time after initiation of discharge a J time scale defined by M and B time scale defined by B and viscous spreading t o dimensionless time scale from the initial collapse stage to the principal collapse stage, Ro/lg~ihol+1 transition time scale from inertial-buoyancy spreading to viscous-buoyancy spreading as defined in Tables 3.1.2 and 3.1.5 u ambient current velocity or frontal velocity density front velocity used by Larsen and Sorensen to estimate spreading field in a current averaEe fluid velocity in emptying cavity flow ~B velocity scale defined by M and B velocity scale defined by Q and M velocity scale defined by Q and B average fluid velocity at time t and location r u(r,z) fluid velocity in horizontal direction fluid velocity just behind the front initial buoyant jet velocity xxxvii u (r) horizontal fluid velocity at free surface z = 0 u' fluctuation value of horizontal velocity v (t) ,V volume of spreading fluid or measured voltage m total volume supplied at time t v.0 initial total volume of spreading fluid W(r) turbulent tracer flux at r w width of channel or fluid velocity in z direction WI fluctuation value of vertical velocity x downstream distance in the direction of current or x = I - l; space variable downstream edge position of initiation of surface field xe upstream edge position of surface field x.]. any independent variable number i y(x) half-width of surface field z vertical coordinate z maximum height of rise m zI/2(r) thickness at 50% value of the maximum density deficiency unknown constant aCT) thermal expansion coefficient at temperature T c inclined angle of frontal interface or some constant initial frontal angle Gamma function of x y specific gas constant relative density difference, II ll(r,z) relative deficiency at (r,z) xxxviii relative density difference b. with i = 0 and densitystratified ambient slope with i = 1 b. (r) m maximum value of density deficiency super-elevation of surface spreading layer above free surface ~aYleigh's unsteady laminar boundary layer thickness, v t a effective eddy viscosity or ambient density gradient, ';':1 dp a (z) dz e: u e: £ eddy viscosity c eddy diffusivity of tracer c o initial eddy diffusivity at origin of wastewater field n relative frontal displacement R(t) to the initial length or radius Ro , n = R(t)/R 0 e angle between the ambient flow direction and the tangent at the boundary of wastewater field ea potential temperature of ambient air es potential temperature of spreading air A decay coeffiCient, A = RR/R2 dimensionless number relative to front velocity to average or head spreading layer thickness kinematic viscosity of spreading fluid (0.01 cm 2 /sec) v a kinematic viscosity of ambient fluid vm average value of kinematic viscosity of two fluids of lock exchange flow v kinematic viscosity of spreading fluid s xxxix dimensionless horizontal coordinate ~ = r/R(t) relative to frontal length or ~ = az/r dimensionless variable 1f 3.14159 P gas density, fluid density mass density of ambient fluid calculated diluted fluid density at source density of discharge fluid mass density of spreading fluid mass density of upper ambient fluid layer of stably stratified fluid mass density of lock exchange flow P= (P l + P 2 ) /2 average mass density of two fluids in lock exchange flow density difference at density head salinity difference between the frontal and the ambient fluids summatio~ of series cr standard deviation of concentration distribution T o ° 1 ess d ~mens~on T zr T(t) ° t~me, t J gD~ Aohoi+i~R 1 ~ fro turbulent shear stress at (r,z) spreading layer thickness scale arbitrary dimensionless function, dependent variable similarity distribution of average velocity dimensional self-similar velocity scale dimensionless function mixing length coefficient related to u mixing length coefficient related to c xl stream function mean velocity scale SPECIAL NOTATION proportional to equal to approximately equal to o differentiation with respect to t prime (') differentiation with respect to ~ or fluctuating part [ ] jump value of the term inside brackets 00 infinity mean SUBSCRIPTS + limit from above limit from below * normalized value 1 CHAPTER 1 INTRODUCTION It is a common practice to dispose of treated wastewater, or cooling water from power plants, by discharging it into a large body of receiving water, such as the ocean, lakes, or rivers. In order to prevent the possible contamination of the ambient water, pollution control agencies usually establish requirements specifying the discharge concentration levels permissible within a given distance from the point of discharge. These permissible concentration levels can then be used to calculate the dilution of the discharge that is necessary. To gain the required dilution two types of submerged discharge facilities are frequently used: one is the single port discharge at the open end of an outfall pipeline and the other is a multi-port diffuser. The usual goal is rapid mixing to reduce the undesirable attributes of the discharge to a tolerable concentration before the discharge is fully diffused into a larger environmental scale by ambient turbulence. The minimum dilution requirement for submerged outfall discharges usually considers only the dilution induced by the discharge momentum and buoyancy without regard for dilution subsequent to the discharge reaching a free surface or becoming trapped by ambient density-stratification. Minimum surface dilution is sometimes predicted on the basis of the dilution at the same elevation as if the discharge were in an infinitely 2 deep receiving water. However, the outfall effluent, which initially behaves like a positive buoyant jet, may reach the free surface and become a free surface horizontal flow. The residual buoyancy and the flow momentum then induce a horizontal flow producing further dilution regardless of whether the ambient fluid is in motion. Since oceans and lakes are usually density-stratified, especially in the summer season, diluted effluent may be trapped at some submerged level of neutral buoyancy and then may spread laterally. Such a submerged wastewater field is favored for avoiding contamination of surface waters. Figures l.l(a) and (b) show the configurations of the surface spreading flow and submerged spreading flow due to a horizontal single-port outfall discharge; Figures 1.2(a) and (b) depict those due to a multi-port diffuser discharge. The same internal spreading layers are also found in thermally stratified lakes as shown in Figure 1.3. A similar, though inverted, situation occurs with marine barge sludge disposal, which can sink as a negatively buoyant plume and spread as a subsurface turbidity current, at the sea floor or some intermediate level as shown in Figure 1.4. Such bottom currents are also formed, for example, by turbid or cold rivers flowing into a reservoir or lake, or by the discharge of a relatively dense liquid, as shown in Figure 1.5. Surface density spread f-ield from a horizontal single-port outfall discharge into a uniform environment. Submerged density spread field from a horizontal single-port outfall discharge into a densitystratified environment. Figure l.l(a) Figure l.l(b) Surface density spreading field due to a multi-port diffuser discharge into a uniform environment. Submerged density spreading field due to a multi-port diffuser discharge into a density-stratified environment. Figure 1.2(a) Figure 1.2(b) w Figure 1.5 Figure 1. 3 IPL..NION . . . . . Dense bottom currents from a negatively buoyant discharge. . ~ Submerged density field from a single-port outfall discharge spreading between two homogeneous layers of densities Pu and P a (e.g. epi1imnion and hypolimnion). Pu \l "---l \ T ._\7 Bottom turbidity current from a barged sludge disposal. Figure 1.4(c) Figure 1.4(b) ~ Spreading turbidity current at an intermediate level of density stratified ocean from barged sludge disposal. Spreading turbidity current at the stably stratified interface from barged sludge disposal. 77J77777777,777777777777777.T77 --~.:~-- Figure 1.4(a) ~·~·u.-,,·:.~ t;J 77;//Tr7T777777777777777777717 "--....J r---1 5 An analogous situation is found in the surface channel discharge of cooling water from power plants and also from large scale river flows into the ocean. When there is a prevailing current the diluted discharge forms a wide downstream field, which may be found at the surface as shown in Figure 1.6, or lie below the surface for discharges trapped in a density stratification. Preventing the surface or subsurface wastewater field from reaching the shoreline involves predicting the size of this field. On the other hand, for cooling water discharges from nearshore power plants it may be preferable to have a wide surface field to accelerate the heat transfer to the atmosphere. The object of the present investigation is to examine surface and subsurface buoyant spreading mechanisms for the wastewater fields due to submarine ocean outfalls discharging wastewater or cooling water, or the ocean barged disposal of sludge or dredge spoil. A review of previous studies of such gravity currents is given in Chapter 2. In Chapter 3, theoretical analyses are presented for unsteady gravity-driven surface and subsurface spreading flows and nearfield surface buoyant jet mixing. A force scale analysis is used to derive a series of functions to describe both surface and submerged density spreading in a stagnant environment. In each asymptotic formula for frontal position there is one empirical coefficient. Similarity solutions of inviscid and viscous long 6 SOJRCE-_ __ DENSITY SPREAD FIELD LAUNCH Ua j TURBULENT DIFFUSION FIELD J SCALE ! 250 o FEET Figure 1.6 A wide surface densit y spreading f i eld pe r sisting fOT a long distance befo r e oceanic turbul ent diffusion takes over, aerial pho to graph taken by Foxworthy and Kneeling (1969) a t Ven t ura Outfall on November 11, 1966. 250 7 wave approximation equations will be used to determine the unknown coefficients. The fluid velocity, the spreading layer thickness for the inertial and the viscous regimes of both surface and subsurface spreading flows will be described. Surface jet mixing of the nearfield spreading flow will be modelled using Prandtl's assumptions. Dimensional analysis is used to show the effect of initial conditions on the spreading currents. Experimental studies of starting half radial surface buoyant jet flows, the full radial surface buoyant jet flow in a crosscurrent and nearfield surface jet mixing are described in Chapter 4. Two sets of experimental measurements on surface buoyant jet mixing were made: one for a full radial heated jet and the other for a half radial buoyant jet. Experimental results are presented in Chapter 5. In Chapter 6, many previously published data of importance to the horizontal spreading problem are compared with present theoretical results. Examples of engineering applications will be presented in Chapter 7. is given in Chapter 8. A summary of conclusions and discussion 8 CHAPTER 2 REVIEW OF PREVIOUS STUDIES 2.1 Wastewater Field from Submarine Outfall Discharges 2.1.1 Maximum Height of Rise and Spreading Level When the receiving body of water is density-stratified, a diluted buoyant field can sometimes reach the free surface and spread outward to become a surface field, but it is more usually trapped at some neutrally buoyant level below the surface. If it is trapped, it then spreads laterally at that level as an intrusion layer and forms a submerged field. To determine whether the diluted water will either become a spreading field at the free surface or at some submerged level, it is necessary to determine the maximum height of rise and the submerged spreading level for a submerged buoyant jet discharging into a density-stratified environment. Several previous investigations have studied the maximum lise problem both theoretically and experimentally and the terminal height of rise can be predicted reasonably well in terms of the parameters of the jet and the environment. These previous results are summarized in Table 2.1.1 in terms of the characteristic lengths appropriate to a turbulent jet or plume. 2.1.2 Observations of Spreading Flows A diluted effluent, upon reaching the free surface, or some intermediate neutral buoyancy level, is deflected from 3.94 4.75 Fan (1967) Fox (1970) Note.. I 3 I H = Mn+3( )- n+3 H Bn+3 ()- 2(n+3) Bg£ 'M g£ ? ? ? ? I I Top of smoke plume by tempera- Top of smoke plume 3.46 3.16 Maximum penetration of dyed salt solution Maximum penetration of dyed ethyl alcohol solution 2.54 0.10 1 1 Maximum penetration of dyed salt solution Maximum penetration of dyed salt solution Top of the fire plume Vehrencamp, Carpenter and Ramie (1955) ? ? ? 1 1 1 Temperature discontinuity height Maximum penetration of dyed salt solution Density gradient changing height ture profile measurement ? ? ? ? 30.22 5.28 ? 1 1 0 1 Maximum Height Definition 3.75 ? ? 1.71 1.80 ? 0.198 ~ ~ 2.40 to n Sneck and Brown (1974) ? 4.38 101.4 ? 324.0 56.3 ? 0.0412 ., ., 0.849 From B MigE 2.77 38.2 19.1 ? 3.41 ., ., ., ., 2.43 17.5 to 7.94 From 8.20 4.76 ? MS/4 Fr - QBI/2 Morton, Taylor and Turner (1956) ? 3.76 Davies (1959) Hayashi and Ito (1974) 3.18 4.43 3.40) 4.11 13.07 \3.70 5.20 4.47 Crawford and Leonsrd (1962) Bardey (1977) to From z m zm/Hs (or ~) ---- - ---- Field photographic record of black smoke from the domestic diesel burning in the desert Lab. temperature profile measurement of air smoke in thermal stratified wood-frame air chamber Lab. photograph of downward dyed salt solution injection Lab. photograph of upward dyed ethyl alcohol solution injection Lab. photograph of downward dyed jet injection Lab. photograph of downward dyed jet injection Field photograph of oil fire plume in stratified atmosphere Schlieren optical observation of upward discharging warm air in an ice skating rink - Lab. photograph of downward dyed jet injection Lab. density measurement along upward fresh water jet flow by conductivity probe Observation Technique Review of previous experiments on the buoyant round (n = 1) or plane en = 0) jet rise into a stagnant linearly density-stratified environment. Abraham and Eysink (1969) Investigstor Table 2.1.1 I I i , ~ 10 vertical flow to nearly-horizontal flow, whereupon it spreads out uniformly. Descriptions on this spreading flow were reported by Rawn and Palmer (1929) on the basis of their laboratory measurements and field observations of four singleport ocean outfalls around southern California. They noted that the diameter of a rising column, when it impinges the free surface, is around one-third of the path length traveled, for both vertical and horizontal single-port discharging flows. The speed of the initial fronts generated by the submerged buoyant discharge were measured by recording the time at which the dyed discharge front passed a designated location. They found that the diameter of the front grew in proportion to the two-thirds power of time. In the subsequent motion, the dyed discharge frequently moved in a series of concentric waves, or annular rings, as described by Brooks (1952) in an unpublished report of his observations of seasonal ocean outfall operations at White's Point and Point Lorna (see also Rawn, Bowerman, and Brooks, 1960). Each wave or ring was followed by an interval of apparently clear water. The waves continued for some distance from the rising column, and gradually merged to form a field of uniform color which remained well stratified. Brooks (1952) did, however, sometimes also observe a steady radial surface flow. This periodic unsteadiness in flow is apparently typical of such flows, for a similar series of the radial surface fronts were observed by Green et at. (1972) and Scarpace and Green (1973) in 11 their aerial photographs of the thermal plume from the surface thermal water discharge at Point Beach Power Plant on Lake Michigan and at three other power plants located at Port Washington, Sheboygan and Madison, Wisconsin. Some field measurements of temperature and velocity of heated discharges were conducted by Frigo et al. (1974) at the Point Beach, J. H. Campbell and Waukegan Power Plants located on Lake Michigan. Unknown coherent oscillating fluctuations in water temperature and velocity were observed with periods ranging from 2 to 20 minutes. It was believed that the long period of fluctuation might be due to the development of "radial thermal fronts" structure as observed by Scarpace and Green (1973). The same band structure was observed on a larger scale by McClimans (1978) in a radial oceanic front in a fjord at Trondheim, Norway, as shown in Figure 2.1.1. According to the early observations by Rawn and Palmer (1929) the interface between a surface horizontal flow and the underlying fluid is not clearly defined close to the surface impingement point. The intensive interfacial mixing of the discharged water with the ambient water, and the periodic concentric wave behavior made it very difficult to measure the thickness of the spreading field. For both vertical and horizontal buoyant round jet discharge, the overall average of observations indicated that the thickness of the two-layer stratified flow was about one-twelfth of the length of travel of the effluent before it reached the free surface. Some Figure 2 . 1.1 A series of radial oceanic fronts in a fjord observed by McClimans (1978). The photograph from which this reproduction was made was taken by T. A. McClimans, who graciously supplied a copy to the author. N !-' 13 laboratory observations of Frankel and Cumming (1965) indicated that the thickness of the stratified layer generated by a horizontal buoyant round jet discharge was around one-third of the total water depth. The same layer thickness to depth ratio checks fairly well with that found by Hart (1961) for a vertical buoyant round jet discharge in an almost stratified fluid. More recent laboratory experiments by Liseth (1970) and Liu (1976) showed that the two-dimensional stratified layer formed by a downward, or an upward, multi-port buoyant discharge into stagnant ambient water, never exceeded 30% of the total depth. Other experiments by Buhler (1974) showed that this two-dimensional bottom stratified layer occupied the lower 40% of the water depth. Roberts' (1977) results for a vertical line plume discharge show a ratio around 30%, which is close to the value predicted by Koh (1976), i.e., approximately one-third of the total ambient water depth. When there is a slow prevailing ocean current the spreading flow has a well-defined upstream edge. According to Brooks' (1952) observations at White's Point and Point Loma, the stronger is the current, the closer the upstream interface is to the point of discharge. The boundary between the discharge and the ambient water also becomes sharper. The ambient current apparently rides under the sewage layer provided that the flowing layer is substantially thicker than the sewage layer. Downstream from the point of discharge the sewage moves as if it were part of the 14 overall current, but spreads laterally. Further mixing, due to the natural turbulence found in the ocean proceeds slowly both vertically and horizontally, and the diluted discharge ultimately becomes an almost non-buoyant tracer. The variation of lateral surface spreading width before turbulent diffusion takes over was found by Foxworthy et al. (1966) to vary as the O.78th power of downstream distance measured from the surface impingement point. An excellent photograph of the transition from gravitational spreading flow to the turbulent diffusion regime is shown in Figure 4 of Foxworthy and Kneeling (1969) . (See Figure 1. 6) Very few field observations have been made examining submerged spreading flow and the subsequent dilution and diffusion. This is possibly because the diluted effluent is trapped beneath the sea surface and no immediate "visible" nuisance is produced. However, in a stably stratified atmosphere industrial discharges are often seen spreading at a neutral density level. In Figure2.l.2(a) is a horizontal spreading plume due to the cooling tower discharge at John E. Amos Power Plant observed by Kramer and Seymour (1976). A series of discontinuous radially spreading fronts can also be seen in Figure 2.l.2(b). Submerged spreading of a turbid cloud at the ocean bottom and at a strongly stratified interface due to ocean dumping of barged sludge was reported by Beyer (1955). No details of measurements (a) Radially horizontal spreading cooling tower plume in a stably stratified atmosphere. (Courtesy of Kramer and Seymour, 1976) Figure 2. i. 2 A series of discontinuous radially spreading front . (Courtesy of Kramer and Seymour, 1976) (b) VI ~ 16 were made. Ocean dump.ing of dredge spoil was surveyed by Gordon (1974) in one of the few published field studies of bottom turbidity currents. He found that after the sinking turbid plume reached the ocean bottom, it spread axisymmetrically over the sea floor and increased in diameter proportionally to the square root of time. The thickness of the turbid cloud at certain locations was observed to decrease as a function of time after initiation of dumping. 2.1.3 Unsteady Density Spread in a Stagnant Environment As has been suggested by several investigators, such as Frankel and Cumming (1965), Jirka and Harleman (1973) and Lee et ale (1974), a starting or unsteady flow has three possible regions of flow: the submerged buoyant jet, the subsequent surface impingement and transition, and the horizontal spreading, as shown in Figures 2.1.3(a) and (b). As noted previously, the surface spreading front on a motionless water surface, and due to a single port submarine outfall discharge, was observed by Rawn and Palmer (1929) in both the field and the laboratory and they found that the diameter of a floating dye plume grew as the two-thirds power of time. Sharp (1969a,b) conducted similar experiments for both upward and downward horizontal buoyant round jets, as shown in Figures 2.l.4(a) and 2.1.4(b). Experimental data we~e plotted 17 SURFACE IMPINGEMENT AND TRANSITION START ING SURFACE B.JOYANT JET SUBMERGED BUOt'ANT JET Figure 2.1.3(a) Flow regions resulting from a positive submerged buoyant round or plane jet discharging vertically upwards into a homogeneous environment. SUBSURFACE IMPINGEMENT AND TRANSITION SUBv1ERGED BUOYANT JET SLBSURFACE JET / Figure 2.1.3(b) Flow regions resulting from a negative buoyant round or plane jet discharging vertically downwards into a homogeneous environment. 18 I- - 7777777777777777777777777/ Figure 2.l.4(a) Free surface spreading front due to a submerged horizontal buoyant round jet discharge. / Figure 2.l.4(b) Bottom surface spreading front due to a horizontal round jet discharge. buoyan~ 14 f? b II Q'fl D·I - - - - - - - Figure 2.1.5 un J -I v Unsteady spreading tip position at time t due to a buoyant pipe flow discharge, after Sharp (1971). 19 using three dimensionless parameters in the form: q, (Lg 5 tg 1l / Qi 2/5 ' l3 / 5 (2.1.1) Q,1/5 1. where L is the surface travel length of the spreading fluid on time t, Q is the volume flux at source of spread, which is i calculated by multiplying the discharge volume flux Q. with J the average dilutions from the graphical solutions by Fan and Brooks (1969), g' is the reduced gravitational acceleration, g I Pa-Pil P ,Pi is the calculated diluted fluid density at source of a spread, v is the kinematic viscosity of the spreading fluid and q, is an unknown function. tory experiments. Hart (1961) also performed small scale labora- However, both failed to show clearly the different possible regions of flow that can occur. Sharp (1971) extended his studies to surface spreading front due to a horizontal buoyant round jet discharge on the surface of a wider channel, as shown in Figure 2.1.5. A further dimensionless variable involving the properties of a discharge from a round pipe was added to those considered previously (2.1.1) to obtain: tg ,3 / 5 Q,1/5 J o (2.1.2) 20 where L is the spreading length from the discharging pipe on time t, D and Q are the diameter and the volume flux of the j j pipe discharge. The added dimensionless variable in Eq. (2.1.2), n g,1/5 J 2/5 , i s related to the discharge Richardson number or o Qj Froude number of the pipe flow. Experiments on radial spreading plumes due to a submerged vertical buoyant round jet discharge were performed by Chen and List (1976). After a very brief initial period the radially symmetric front increased in diameter with the three-quarters power of time and then broke sharply to a function of one-half power of time. (1972). The second regime was also confirmed by Anwar Both regimes were confirmed recently by Britter (1979) in the laboratory experiments on radially spreading gravity currents. Roberts (1977) observed a similar two-stage spreading front from a submerged line plume discharge; the distance from discharge to front increased as a linear function of time t and then changed to t 4 / 5 . This implied constant velocity of the plane spreading front, which depends only on the discharge buoyancy flux, was also found earlier by Liseth (1970) and Buhler (1974) for the flow from a multiport diffuser discharge with dense fluid. Different time functions were noted by Chen and List (1976) for the radial surface spreading front produced by dumping a finite volume of liquid onto the free surface of a denser miscible liquid. The diameter of the spill grew as tl/2 21 initially and then changed to t I 18. Different time functions were also found by Almquist (1973) for a plane surface spreading front produced by the release of a finite volume of buoyant fluid: the distance grew initially as t 2 / 3 and then changed to t l / 5 . In order to study spreading rate for a plane front in a linearly density-stratified environment Maxworthy (1972) examined a neutrally buoyant horizontal slot jet into a density stratified fluid, as shown in Figure 2.1.6. After intensive mixing with ambient fluid occurred in the neighborhood of exit, the interflowing front distance increased as t S / 6 for the case of low source flow rate. Similar experiments were also performed by Zuluaga-Angel et aZ. (1972) and Manins (1976a), but different growth rates of the interflowing wedge were found. Manins found the intrusion layer length to be a linear function of time. found a three-quarters power of time. Zu1uaga-Angel et aZ. However, one of the experiments of Zu1uaga-Angel et aZ. did show a wedge increasing its length in direct proportion to the time t and then changing to t S / 6 before reaching the end of the channel, as plotted in Figure 2.1.7. Thus no conclusive result has appeared for the submerged density spread due to a submerged buoyant round or plane jet discharge. 22 Ps=Po(o) ·············:········k :,' • '.' •• ,'.' I I •••••• " . . . I ":,,::1 J.] Figure 2.1.6 Simulation of submerged density spreading by a horizontal jet discharging at its own density level in a linearly density-stratified environment, after Maxworthy (1972). 23 -E u --0:: 10 ~----~~__~~~~L-_ _~L-~~~-L~~ 10 2 4 10 t Figure 2.1.7 (sec) Growth history of the interf10wing wedge front due to a horizontal slot jet discharging at a neutral buoyant level into a linearly density-stratified ambient, data from Zu1uaga-Ange1 et al. (1972). 24 Wu (1965,1969) found different results for the submerged density spread due to a finite volume of homogeneous fluid collapsing at a neutral density level in a linearly densitystratified fluid, as shown in Figures 2.l.8(a) and (b). In a series of experiments three stages of interf10wing wedge front were observed: an initial stage with Nt < 2.5, a principal stage with 3 < Nt < 25, and the final stage with Nt> 25, where t is the time after initiating collapse and N is the Brunt-Vaisa1a frequency of the stratified ambient fluid. In the initial stage the interf10wing wedge moved almost at a constant velocity, while in the later two stages the wedge front was decelerating during the spreading. Wu gave an empirical formula describing the front propagation in the initial stage as a function to time, R(t) 1 + O.29(Nt)l.08 (2.1.3) = 1.03(Nt)O.55±O.02 (2.1.4) ,R o and in the principal stage: R(t) R o where R is the initial radius of the cylindrical section of the o two-dimensional homogeneous fluid, R(t) is the wedge length at time t. In the final stage, the distance to the front increased at a much slower rate than in the principal stage. The results 25 10 1 SLOPE': j :.--:I kD I INITIAL COLLAPSE E ISTAr I I V ~I ~ ... "-" .....", - 7--i®.-Y-; ~ ~ COLLAPSE "2:.J" FINALSTAGE J7 ~~INJp~~AC~~LAPSE 10' Nt Figure 2.1.8 (a) Three stages of plane submerged den~ity front spread due to a finite volume of homogeneous fluid collapsing at a neutral level ~n a linearly density-stratified environment, after Wu (1965). Figure 2.1. S(b) Sequential cross sections of plane submerged density front spread at various times, after Wu (1965). 26 given by Wu appear inconclusive. However, the front distances does appear asymptotically to grow in proportion to t l/6 (see Figure 4 of Wu (1969) and Figure 2.1.8(a». Similar experiments were performed by van de Watering (1966) with a different apparatus. The dyed wedge was found to move in the initial stage as R(t) R o 1 + O.16(Nt)O.94 (2.1.5) = 1.02(Nt)O.34 (2.1.6) and as R(t) R o for the principal stage. A simple force scale analysis and similarity solutions of long wave equations will be 'used later to derive these different power laws of time for both the surface and the submerged density spread. 2.2 Surface Wastewater Discharge in a Cross Current 2.2.1 Non-Buoyant Models Non-buoyant wastewater field models are reviewed here. The models were developed by considering only the source strengths and the diffusion mechanisms and by neglecting any buoyancy effects. The solution for a non-buoyant source in a uniform current is well-known in hydrodynamics (see for example, Milne-Thomson 27 (1969), pp. 480-481). The induced velocity field, as shown in Figure 2.2.1, is asymptotic far downstream, to a constant width, ~ rrU Q ,where Q is the source strength, U is the current velocity. The upstream edge where the velocity vanishes, is located at a distance, t~ , upstream from the source. No further information about the tracer distribution is given. The turbulent diffusion field from a line non-buoyant tracer source was developed by Brooks (1960) by neglecting the vertical and the longitudinal diffusion. A turbulent diffusion field is assumed to form as soon as the diluted effluent reaches the free surface. The downstream concentration is found to decay in both the transverse and stream directions. An overall sketch of this diffusion waste field is shown in Figure 2.2.2. The lateral spread of the waste field is found to be dependent on the variation of the scale of,eddy diffusivity E along the current direction. The three types of growing field widths that are found are: 24E o~) 1/2 + __ Db b (2.2.1) l2E L o.! - = l+ _ _ Ub b b (2.2.2) and )3/2 b 24Eo x ( 1 + 3Ub (2.2.3) 28 y 2.jQ/TrU I u ----------------------~ x Figure 2.2.1 A non-buoyant and non-diffusive source in a uniform cross current. u x Figure 2.2.2 Schematic diagram for diffusive field from line-like diffuser in an ocean current, after Brooks (1960). 29 each corresponding to one of three types of eddy diffusivity functions assumed: (2.2.1) a constant, (2.2.2) a linear variation of the eddy diffusivity with the diffusion width, and (2.2.3) a four-thirds law of eddy diffusivity, where L is the nominal diffusion width defined as L=2fia (2.2.4) and a is the standard deviation of the concentration distribution function at a given station downstream X, E o is the initial eddy diffusivity value at the origin ,where the initial width of the water field is assumed as b. For the diffusion field from a single-port outfall discharge, mathematical models of the diffusion field, developed to describe the material distribution of an emission from a smoke stack in the atmosphere, are applicable. Frenkie1 (1953) suggested that if the longitudinal dispersion is negligible, a theoretical steady plume model could be assembled by the superposition of an infinite number of overlapping circular patches, as shown in Figure 2.2.3(a), or one-dimensional slice elements, as shown in Figure 2.2.3(b). However, some field density and coliform concentration measurements by Foxworthy et aZ. (1966) and Foxworthy and Kneeling (1969), in the vicinity of submarine outfalls operated by Orange County Sanitation Districts, indicate turbulent diffusion differing from Gaussian profiles. 30 y u x Figure 2.2.3(a) Superposition of a series of circular diffusion patches in a horizontal turbulent current. y u' C(X,y) x U t Figure 2.2.3(b) -J Superposition of a series of slices of diffusion elements in a horizontal turbulent current. 31 It is believed that this is due to the buoyancy forces and would indicate that the details of intermediate scale density spread dynamics are important to the study of large scale diffusion problems. 2.2.2 Surface Density Spread Models To determine a non-diffusive surface wastewater field several environmental factors must be considered in addition to the discharge variables. Those include ocean currents, tides, wind, local climate, and bottom topography. all these factors together is difficult. Inclusion of Only the effect of a uniform prevailing current will be considered here. A density spread model was given by Larsen and Sorensen (1968), Cederwall (1968), and Hyden and Larsen (1975). Harremoes (1967) used a similar model to consider the surface field due to a multiport diffuser discharge. Larsen and Sorensen assumed that there is a uniform thickness h of surface wastewater field at a location x, downstream from the release point, and that the density front propagates with a velocity (2.2.5) The boundary of the surface field will then be defined by the relationship (2.2.6) 32 as shown in Figures 2.2.4(a) and (b), where the coefficient A is chosen as 1.0 by Larsen and Sorensen (1968) and Harremoes (1967), and as 1.0 to 1.4 by Hyden and Larsen (1975). The angle e was related to the half-width of the surface field by (2.2.7) so that for small angles 8 U dB = A/gllh (2.2.8) dx The continuity relationship implies that 2 B(x) h U = S o Q (2.2.9) so that one gets the half-width equation in differential form as (2.2.10) where II is the dimensionless density difference of the surface field is related to the value of the discharge fluid as (2.2.11) 33 y 8 u x Figure 2.2.4(a) Top view of surface density spread field in a horizontal current. Figure 2.2.4(b) Central cross section along x direction of surface density spread field in a horizontal current. 34 Equation (2.2.12) after integration becomes B(x) -B-= o ( 1 + i :0 2/3 (2.2.12) ) in which B A2 gll S Q = ____o_=___ o 2U 3 (2.2.13) The model described here is valid only when the current velocity does not exceed some maximum value since B will o vanish. In the case of low current speeds the longitudinal spreading velocity is no longer negligible, and Eq. (2.2.8) is not a good approximation of Eqs.(2.2.6) and (2.2.7). The width of the field is mainly dependent on the velocity of the density front, which contains an unknown coefficient A. Harremoes (1967) argued that a constant value of A is doubtful since the front edge is observed to propagate with a vanishing thickness, which implies that the front velocity must vanish in Eq. (2.2.5). To overcome such deficiencies, other models are necessary. Rawn and Palmer (1929) proposed the concept of a one-dimensional transverse spreading field superimposed on a uniform current as shown in Figure 2.2.5. Howland (1934) suggested that the experimental results of lock exchange flow by O'Brien and Cherno (1932) 35 u x Figure 2.2.5 Side-spread model for the plume boundary of a surface wastewater field. 36 may be used to predict this side-spread velocity. The same concept was used by Koh (1976) and Roberts (1977) to determine the surface spreading from a finite length diffuser discharge. The same idea was also used by Lee (1971) to predict the surface oil slick due to an oil tanker accident. According to this model the downstream width of the surface spreading field can be approximated by a Galilean transformation of an unsteady time function describing the plane density front motion in a stagnant ambient fluid. In other words, the sharp boundary of the spreading field can be described by just changing the time term t to a space term x/U in the solution for an instantaneous finite volume release. The details of these unsteady functions will be given in Section 3.1. The above approach is a good approximation only when the longitudinal spreading velocity is negligible compared to the current velocity. Otherwise, a radially spreading front is a better approximation, as shown in Figure 2.2.6. The sharp density boundary of a surface wastewater field is in this case similar to the shock wave curve formed by a moving source. There is no upstream wedge in the critical or the supercritica1 flow case, when the current velocity is equal to or higher than the shock front velocity. In the subcritical flow case, there will be an upstream edge formed by the interaction of the current and the spreading flow. 37 y u x x=U 1 Figure 2.2.6 j Radial spread model for the plume boundary of surface wastewater field. 38 2.3 Gravity-Driven Starting Flows 2.3.1 Formation of Gravity-Driven Flows Gravity-driven flows occur both in nature and in engineering practice. They result from unsteady and non-uniform free-boundary motion caused by a density difference which may be due to: (1) A difference in concentration of dissolved solute, such as in saline wedges, or lock exchange flows as observed by O'Brien and Cherno (1932). (2) A difference in temperature, such as atmospheric cold fronts, land breezes after sundown, and katabatic winds in mountain valleys as observed by Margu1es (1906) and Schmidt (1910); sinking melted ice water advancing on the sea floor, (Sandstrom, 1908), the surface thermal plume of power plant cooling water discharge from a channel discharge (Scarpace and Green, 1973), and submerged outfall discharges (Sonnichsen, 1971). (3) Suspended solids, such as in an atmospheric dust storm (a "black blizzard"), turbidity currents of muddy river flows into reservoirs and lakes (Bell, 1942) or from ocean-barge dredge spoil disposal (Gordon, 1974), avalanches of powder snow (Seligman, 1962), and the base surge of misty air from an atomic bomb explosion in a shallow ocean (U.S. Atomic Energy Commission, 1950). 39 (4) Different constituent fluids such as damp gas motion in mining tunnels (Georgeson, 1942), oil slicks spreading on the sea surface (Smith, 1968), and dam break flows into a river channel (Schoklitsch, 1917). (5) Combinations of the above items, such as muddy river plumes after rain storm (Bell, 1942), oceanic and estuarine fronts (Garvine, 1974 and McClimans, 1978), and vaporized liquified natural gas (Drake and Reid, 1977). Thus, although there have been many investigations made of such unsteady flow problems, there is, as yet, no universal theory that can explain the results obtained. The above examples are considered in further detail. 2.3.2 Lock Exchange Flows A lock exchange flow is produced by the removal of a vertical barrier separating two sections of a channel filled with fluids with a slight density difference, as shown in Figure 2.3.1. According to Yih's (1947) laboratory observations, two wedge-shape surge flows are found moving at the same speed but opposite direction. The average value of the initial U frontal velocity U, in terms of a Froude number, F = --r===~======== (P - P ) 2 1 ----:--- H P 2 + PI is around 0.47, as shown in Table 2.3.1. The experimental value, F = 0.47, is 6% lower than the one predicted theoretically by Yih (1947), or Keu1egan (1934), or by O'Brien and Cherno (1932). 40 t LOCK GATE / Figure 2.3.1 Lock exchange flow. u.. ••u Yib (1947) S.lth (1965) (1972) Slap.on (1973) Shvaru fit al. (1932) Cberno O' Brien' (1966) Kiddleton leulelln (1957) (1963) larr , Barr (1967) lov•• til'tor 304.8 1219.2 150 609.60 30.48 ]8.10 3D 38.10 lS.24 60.96 5]. ]4 30 24].84 38.10 50 15.4 500 743.71 1 45.7 22.9 11.3 5.2 2.54 1 42 Depth D(ca) 25.4 10.48 0.64 lS2 Width 8 (ell) WaUl' Tank 7.62 .... 49.05 18.29 6, 12, 24 1 12.19'\,36.58 20"'30.2 5.8'" 52 1 1 H(ca) Water Depth 152.40 ? 20 ? 0.0005"" 0.0{)40 0.002 '\, 0.00; 0.0015 '\. 0.04% 0.012S'\.0.1478 0.015'" 0.101 D.09S'\. 0.21,2 0.005'\0 0.150 ----- 0.05""0.32 ? 0.3 .... 1.2 O.OS'\.lO 0.13"'3.08 0.71 '" 1.07 28.3 9.75'\0 54.56 0.07 '\. 0.30 ? 0 0 ° O· '\.16· 0 0 . ? ? ---- - Uh n 300 < - v - < 10.000 ? ? ? 3200, ~ ,175,000 .. 0 0.07 '" O.~6 ° ? 0 1 ? 1 So Relevant aeynold, Number ". HILo Channel Slope Initial Denatty Disparity ~ A.pect Ratio 41.7",750 1 1 Lo (CII) Initial Length Experimental studies on lock exchange flows. 45.72 10.48 3.81 579.12 487.68 121.92 8750 L( . .) Length Table 2.3.1 Initial Froude Nu ..ber average 0.47 0.50"'0.60 ? 0.25 .... 0. S 0.46"'0.59 average 0.44 0.404 '\. 0, 51 average 0.462 0.42'\0 O. SO ? ? 02 + PI {z:~2-Pl 2.--H F .. ---~-- Concentration difference in •• Unity Concentrat Lon difference in salinity Concentration difference in .alinity and sugar Concentration difference in .aUnity or turbidity Concentration difference in •• Unity turbidity Concentration difference in Concentration difference in I,Unity and sugar ? 1 Due to Density Difference Initial Front Velocity I , , , I I ---- overflow and undedlow overflow underflow underflow underflow underflow underflow overt low and underflow overt low and underflow obeervationsl t-' .po. 42 Keu1egan (1957) performed several similar experiments, but with a relatively shorter lock length as shown in Figures 2.3.2(a) and (b). He found the saline front advancing at a constant velocity which depends only upon the initial water depth and the density difference of two fluids. Froude number, F, varied from 0.42 and 0.50. The frontal The average value was around 0.452, which is very close to Yih's result. O'Brien and Cherno (1932) also found the underf10wing wedge front moved at a frontal Froude number that varied from 0.46 to 0.59. Smith (1965) found a variation from 0.50 to 0.60. For longer travel, as shown in Figure 2.3.2(c), the underflowing front presented a head shape with a crest and trough. The ratio of the two heights, d /d , (see Figure 2.3.3) was 2 1 found to be almost a constant, 2.0, independent of the initial lock size, salinity difference, viscosity of the fluids, and the length of travel. The density head did not propagate with constant speed and in fact quickly decelerated. In other words, the lock exchange flow laws were not applicable. Keu1egan (1957) used another Froude number 43 LOCK GATE T SALT WATER H FRESI-t WATER ~L-___~_2____ L -_ _ _ _ _ _ _ _ _ _p._1_________ ~ Lo Figure 2.3.2(a) ---{ Lock and channel before removing gate. T 1 t u FRESH WATER H fi R U I-- Lo ---l L( t) I- Figure 2.3.2(b) Advancing lock exchange flow. t T FRESH R H 1 I" WATER I l SALT WATER R. .1 un U L----_---1...--_~2 _ - - L - - - - _ Figure 2.3.2(c) Lo 0 ' ·1 Finite-length exchange flow. FRESH WATER R SALT WATER ~ Figure 2.3.3 Idealized shape of saline front from finite-length lock exchange flow, after Keu1egan (1957). 44 to describe the decaying frontal velocity, where ~Pf is the local density difference at the front and d l is the trough height near Udl the front. When the frontal Reynolds number, Re = - - was 1 '\.lm ' greater than 400, the Froude number Fl was constant (1.07). '\.lm is the average value of the initial kinematic viscosity of the two fluids. But when the frontal Reynolds number ReI was less than 400, the Froude number Fl was a function of the local Reynolds number ReI (2.3.1) The decelerating velocity was also observed by O'Brien and Cherno (1932) for a saline front in a finite-length lock exchange flow. The surge front from a finite-length turbid water lock exchange flow was obseryed by Middleton (1966). He found a constant velocity initially with the frontal Froude number, F, varying from 0.404 to 0.51 with an average value of 0.44. Similar rapid deceleration of the head at a longer travel distance was also noted by Middleton. Shwartz et aZ. (1973) found that, for both saline and the turbidity flows from a finite-length lock on a horizontal bottom, and also with a mild bottom slope, the asymptotic decay of the frontal velocity U with distance traveled x was (2.3.2) 45 There is no clear explanation why this rapid deceleration occurs. Previous investigators have speculated that this may be due to viscous bottom friction, or side wall friction or the exhaustion of the supply of sediment caused by deposition of the suspended material. A three-dimensional structure to the finite-length lock exchange flow was observed by Simpson (1972). A series of cleft and the lobe structures were seen to develop in the transverse direction to the "nose". The elevated height of the nose, dn , was found as a function of frontal Reynolds number, Ud2 Re = - 3 v s (2.3.3) where v s is the kinematIc viscosity of the moving fluid. The lobe length between two clefts, i, was found as a function of Reynolds number, i d 2 _ 7 4 Re - 0 • 3 9 ± O. 2 -. 3 (2.3.4) Similar experiments for lock exchange flow connected to a wider channel, were also performed by Keu1egan (1958). He found that a saline front intruding into a narrow channel also moved at a constant speed. The observed Froude number F was a function of the width ratio of the two channels. In the limiting case of a 46 river channel connected to an open sea, the saline front moved initially at a faster constant speed than that for a constant width channel. F was 0.57. The average value of the frontal Froude number The frontal head characteristics were also found to be different with the head being flatter and slightly longer. The ratio of the crest height, d ' to trough height, d 1 , was Z 2.16. The Froude number, F , based on the head velocity and the l trough height, d , was 1.04 when the frontal Reynolds number, l Udl ReI = --- , was greater than 550. For a frontal Reynolds "m number, ReI less than 550 0.206 Re l 1 / 4 (2.3.5) Similarly, it was found that (2.3.6) when the frontal Reynolds number Re 2 was greater than 1200 and (2.3.7) when the frontal Reynolds number Re 2 was less than 1200. 47 2.3.3 Plane Surface Starting Buoyant Jets Experimental studies on plane gravity currents have been performed by many investigators, as shown in Table 2.3.2. As shown in Figures 2.3.4(a) and (b), most of them produced a current by discharging horizontally a fluid of density p. onto J the free surface or bottom surface at one end of channel filled with a homogeneous fluid of density p . a This type of density current is found to move at approximately a constant speed, even with viscosity drag effects. The earliest experimental observations were made by meteorologists modeling atmospheric cold front propagation. Schmidt (1910) performed model experiments by discharging a salt solution onto a channel floor. An advancing saline front formed a nose, whose speed of advance depended on the salinity difference between the frontal and the ambient fluids, and the height of nose, h. n ~p n , Schmidt (1911) proposed that the frontal velocity, U, be expressed as (2.3.8) U A is a dimensional constant. Margules A similar results was derived by (1906) for modeling atmospheric cold front propagation by equating the change of potential energy to the change of kinetic energy. (t..) Wood (1966) 182.88 36.57.6 WUlr.ll'leOa (1910) 500 181 , 2286 3048 Sc'-tclt (1910) (1966) "feldl.toa Willock (l96S) ..... , 1-.1. . . . (1946) Ceo!"I··D (1942) 1 1 :1 , "'.'" Iraucher (l950) !IUaoa , Turner (U59) I Bell (l942) 152.4 304.8 451.2 518.161 luI' (1959) ? 487.71 (1913) (elll) Fluid HDe~!:) I 4~.12 ? 30.48 .58.42 15.4 60.96 152.4 10 1> 11.43 182.88 '" '0 45.12 55.88 60.96 31 I'" 1121.92 I 5.,2125.4 25.4 25.4 38.1 86.4 45.12 12.1 ? 0O 20.57 1 ,, 1 ,, 182.88 I I ? Rate pu I 0.00]6 .... 0.015 0.29 'J ,_ !e. R.latlve Den.ity Excell 1>.33 I 141.6 .... 1144 0.0023"'0.0120 0.0557 0.0263 0.091'\0 0.0196 0.00)) ..... 0.1160 0.0032"'0.146 0.0022'\00.1605 O.OO296'\o0.14R O.OO197'\oO.OIiR0 I 11.5.88 ... .58.441 49.24'1.1111.92 I 115.51'1. 53.61 0.18 10.4] ].41 .... 64.12 0.0020'\0 0.01l4 0.0021'" 0.0131 0.0020'1. 0.0106 0.0001'" O.OOSO 19 . .51'\0 61.94 2.61 ... 4.30 2.35'\04.22 2.35 'I. 4.37 I 4.40 .... 12.3910.00247 ... 0.00210 0.945"''',)] ? (C. 2 /lec) Q Unlt Wldt.h II ...,.n:\ Oloch"•. , , , , , Width (elll) Depth , 9.91 11.27 1 2.61 0.56 .... 4 . .55 I""'""O 0.64'" 1.06 :t 0.01 0.0092'\0 0.01115 :to.Ol :to.Ol :to.Ol I 1 14,100'\0 114,400 1560'\. 5840 "" 4924'\.11.900 1550 ... 5360 261 '\04]0 235"'422 2]5'" 431 440 ... 1240 1 I '0- '0' 0- 0' ,.- O..... 0 . .541 O. '\00.040 1/10 O· '\090· 0.010 0.015 0.005 0.251'1. 8.01 , ........ I , I ·Y· 1.06t 0.10 1.24 t 0.08 1.51 t 0.11 1.53 t 0.18 1.1UO.14 1.04.5 1: 0.041 ConcMltratlOf1 dtfh,rence In aallnity ConClntntlon difference In . .Unity Concent... tion difference in •• Unity CoDeentr.tlon dlfferenc:e in •• Unity difference of hlnh ".t.r T~rnure diffe...nc. in .aUnit, 1 eo..c..uattoa 0.100 DUferftlt con.tituHt of I ••• fired..., and h,drolell fro- .11' Coacntratloe difference ln •• Unity CoDceauet:lOD Utferece to ..Unit,. ad fluor.an- Coac_tnt1oe diffu_ce la auapaneled ••d1_nta or a",ar OT adt or aodi_ thioaulhte aM: tellllPerature dtfhnnce r..,.... tut. dlff.tenu of fr •• h ... tet difteunci in •• Unity COQcentrattOD DUt.nnt conatitulnt I I .,.d Sequential photoluph of dy" •• It front (underflow) It varloul twel Sequlnti.l photo.uph of d,ed .. line hont (undf'fflov) It .,.rioul ttael Dred . . Un. front profile ob •• rvatlonl (underfloW) Sequent ta 1 ,hoto.uphy of dy" •• 11n. f .. ont at .,.rlou. ttae., veloclty and thickll." _ •• ure· _nt. or !,InHon flow behind front Dr" .... ted: h •• h ",.u.. f ..ont (m'.l'flov) Yelocity oblervuionl. velocity and hy.1' thickn . . . ... w .....nt. of unUon flow behind front SaUa. front ( ..... ) y.locity _a ... r~ta in at ill fnah . . t......It .at.r dhchat'Jed . .rtlca11, upward. (underrIow) U red.., or hydrOietl la. front advanc lnl upwa .. d. doni the inclined 1.11 ... y or the roof of the airv., by thin, p.a. the d .. lp.ted 10cationa .......1'. _an vdoclty of in.,erted Dy" fnah _taT front oy.rflova on .. Une _tel' aurrace. dyed .dt .at.r froat underfl~ down on lacUnad f ....h . . ur box lequnr:ld photoanph of underflowi", dyed front and v.locity and thicluteea "'~nlentl of unifonl flow khlnd the front Dred dndry c:un_U ullll.rflon on dopina channel Sequntul ,hotolnph of .... t.d front (OyurIow) .t yarlou. tbtu ded.ut.d locatlona ts. Htod.tna of dyed fr •• h front (O'l'uUov) p. . t the . . tel' It...,. .,dotit, •••ur..nt. of p .... fUn oU front (overflow) Ipr •• dlD1 on fl'elh •• ter Deadty Diff.rencil Cr."lty Currnta Due to Flow Obaervltiona I )0.644",O.,H ....1'.,. 0.55"'0.81 1.06 '\01.18 0.12 ... 0.85 0.16'1.0.96 0.11 '\0 1.10 1.10'" 1.50 0 c s :t: 0.01 "'0.64 94.5 .... 4)) ? "j ae.'s- U BIll I 5.lS ? ~j(c.2/ •• c) Channa I Botto. Slope 1.23 ? -~ j~ Inlet Reynold. N!,nd,er ~O.Ol Fr Fluid of DJacharged J(ln __ tlr; Vhcodty ,, ? 'j (e.-) Inlet. Fro,"t. N.... ber Inht ~pth Experimental studies on plane gravity currents due to a constant rate of discharge at one end of channel. ICh~" I Llqth L .U~uht Abbot.t (1961) Inveatl.ator Table 2.3.2 ~ 00 49 Figure 2.3.4(a) Forced free surface density currents. U·J ~__-J~______- J_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _~~ Figure 2.3.4(b) Forced bottom surface density currents. Q 50 Barr (1959) produced so-called plane "forced surface density currents" on the free surface of a flume, as in Figure 2.3.5. He found that sometimes a roller type front developed initially after the discharge and sometimes a faster moving and pointed wedge type front occurred, either initially or after some travel. No clear explanation was given for this behavior although surface tension effects were suspected. Lean and Willock (1965) also performed similar experiments and found the front moved out initially with a nearly constant speed, with an intensive mixing region behind the front. The thickness of the surface layer increased away from the entry point, but downstream the surface layer changed to a uniform depth and there was little further mixing with the underlying cold water. The turbulence in the interfacial layer damped out smoothly to form a sharp interface. Lean and Willock found that a higher inlet Froude number flow always lead to a thicker uniform surface layer and that for an inlet Froude number less than about 1.0, the thickening of surface layer was slight. The u local densimetric Froude numbers, F~ = ---- , were 0.69 to 1.14, /gllh which generally agrees with the results of Ellison and Turner (1959) for the case of negligible mixing. Almquist (1973) also observed surface density front motion and the details of the experimental parameters are shown in Table 2.3.2, and in Koh (1976). The wedge-shape front grew Figure 2.3.5 H • • Po --~--.~7:..::_:::~_=_..- ~_=_.~~.=_:._=_~_=_.~_~.=~__:_:_ -=-.~7":-:..~ , / . -. -. ,. __ . _/ /~:::s R I· _______ -_~:----- _--' -7L Surface density currents and discharging box used by Barr (1959). h-+-lll Q VI f-' 52 1;irs-t with time t, and then changed to t4/5, as shown in Figure 2.3.6. The density fronts therefore propagated initially with a constant speed and then decelerated, which was consistent with the results of Barr (1959) and Lean and Willock (1965). Density currents on the bottom were investigated by Bell (1942), who used saline flows to model the turbid river flow entering a reservoir. Middleton (1966) performed a series of experiments to investigate the sediment transport mechanism by the turbidity currents. A suspended sediment density current "head" was formed and after an initial acceleration this flowed at an approximately constant velocity down the sloping floor of a channel. The steeper the slope is, the longer the acceleration stage will be. Eventually the flow was decelerated to a nearly constant velocity and the head shape was preserved. Figure 12 of Middleton (1966) shows nearly uniform distributions of velocity and layer thickness in the flow behind the current head. Keulegan's (1957) results for density currents were also checked by Middleton. The Froude U number, F2 = --~=========== , was found to vary from about 0.7 211p g p, + P d 2 J a for the low slope case (S = 0.005) to about 0.8 to 0.9 for the c high slope case (S = 0.04) • c Similar experiments on a milder sloping floor were also performed by Braucher (1950). The front propagated with a constant velocity proportional to the cubic root of the buoyancy flux discharged, B. U ' coe ff"~c~ent---Th e va1 ue 0 f t h ~s B1 / 3 Figure 2.3.6 a:: -- u E ..-- 2 / 10 SLOPE 1 t ( sec) 102 , I Growth history of starting free surface gravity currents, reproduced from Chen (1975). Experimental data from Almquist (1973). 4 10 10 VI W 54 as shown in Column 9 of Table 2.3.2, is found to vary slightly with channel slope. Experimental studies of plane forced density current for a range of bottom slopes, by Wood (1965). (0°, 6°, 28°, 50°, 90°), were conducted All experimental results shown that after an initial period the magnitude of the constant frontal velocity was proportional to the cubic root of kinematic buoyancy flux with the coefficient a function of slope angle. This coefficient had its maximum value for a 50° slope and for the horizontal bottom the value was close to that observed by Wilkinson (1970). Keulegan (1946) also performed this type of experiment but the discharge arrangement, as shown in Figure 2.3.7, lead to a wedge shape front instead of the head or the cap shape observed by Middleton (1966), Wood (1965), and Wilkinson (1970). Although a constant frontal velocity was also observed, the ratio of frontal velocity to the cubic root of buoyancy flux had an average value of around 0.700. Georgeson (1942) studied frontal motion as a model of gas spreading in mining tunnels and found that the propagating velocity of the frontal edge was proportional to the cubic root of the rate of inflow and varied little with the travel distance along a sloping gallery roof. The ratios of the frontal velocity to the cubic root of the buoyancy flux have been re-calculated, as shown in Table 2.3.2, and vary from 1.06 to 1.18 for methane gas and from 0.55 to 0.81 for hydrogen. 55 u L----~------,t Figure 2.3.7 Advancing wedge on the bottom of a channel, after Keulegan (1946). 56 2.3.4 Von Karman, Prandt1, and Benjamin Theories Von Karman (1940) made a pioneering theoretical study of bottom density currents propagating in an ambient fluid of infinite extent. The essential ideas are repeated in Yih (1965). Figure 2.3.8(a) shows the nature of a dense current advancing into a less dense fluid as viewed in a reference frame fixed relative to the ground. Figure 2.3.8(b) shows the current as viewed in a moving frame traveling with the front velocity U. In this moving frame the front becomes a stagnation point and the layer behind the front becomes a stationary layer under a uniform and steady flow with velocity U. Applying the Bernoulli equation along a streamline in the ambient fluid in the neighborhood of the discontinuous interface between stagnation point X (subscripts)and the far downstream point Y(subscript oo ) gives (2.3.9) where P s and Poo are pressures, P is the ambient fluid density, a hw is the thickness of the uniform layer at the far downstream point Y and U is the horizontal current velocity. Since the current is assumed steady, the application of Bernoulli's equation along another streamline on the other side of interface implies P s = P00 + g Ps hw (2.3.10) 57 Figure 2.3.8(a) Configuration of a forced plane gravity current as viewed in a fixed reference frame to the ground. Po Figure 2.3.8(b) Configuration of a forced plane density wedge as viewed in a moving reference frame with the front, after von Karman (1940). 58 where p s is the fluid density of the arrested current. By equating the two pressure differences in Eqs. (2.3.9) and (2.3.10) one obtains (2.3.11) or Fco Using free streamline theory, von Karman (1940) also showed that the inclination angle of the frontal interface, at 60° to the horizontal. B, must be A similar angle was also derived by Stewart (1945) and Jeffreys (1952) using inviscid potential flow theory in polar coordinates. Rannie (1975) also derived the same angle, 60°, assuming an approximate profile in an infinitely deep ambient fluid. Von Karman (1940) also suggested that there must be a head at the front which is thicker than the downstream layer. The head shape of the discontinuity surface has not yet been determined. Ambient stratification effects on such currents were discussed by Kao (1976,1977). According to Kao's study, the velocity of an intruding uniform current at a level of neutral density in a stratified environment will be (2.3.12) 59 where the ambient stratification E: = 1 dp (z) -=a and Po dz = p (0) p 0 a which is to say that the Richardson number Ri = Ig£ he, = 1 (2.3.13) U An advancing density current propagating along a sharp interface between two infinitely deep homogeneous fluids, as shown in Figure 2.3.9, was studied by Enge1und and Christensen (1969), Moshagen (1973), Thorpe (1973), and Kao (1977). On the basis of the Bernoulli theorem the front should move at a speed U, (2.3.14) Figure 2.3.9 Internally spreading currents between a stably stratified interface. 60 The limiting case of a surface spreading current when the density p u of the upper ambient fluid is small, i.e., p /p and p /p + o. u a u s Then (2.3.15) Prandtl (1952) studied the rate of advance of atmospheric cold front, as shown in Figure 2.3.l0(a). In a ~ference system of coordinates traveling at the same velocity as the front, the moving front is a stagnation point. He reasoned that pressure must be the same on both sides of the stagnation point so by neglecting the hydrostatic pressure terms, and equating the dynamic pressure, he found 1:.2 p s (u f - U) 2 = -21 P a U 2 (2.3.16) which gives (2.3.17) where U and u f are the velocities respectively of the front and the fluid particle just behind the front, pa and p s are the densities of the ambient air and the cold air, as defined in Figure 2.3.l0(b). 61 Figure 2.3.l0(a) Advancing mass of cold air; actual flow relative to the ground, after Prandtl (1952). Figure 2.3.10(b) Advancing mass of cold air; idealized flow relative to the front, after Prandtl (1952). 62 Prandt1's theory implies that the cold front must advance at a velocity of about one half the velocity of a fluid particle behind it, since the density of the frontal air usually differs little from that of the ambient air. Thus, a roller-type stream is expected to be formed when the fluid behind the front is deflected upward at the front in a manner that is similar to the development of a starting jet flow. For the case of an avalanche, in which the density of the moving mixture of snow and cold air is considerably greater than that of the ambient air, the frontal velocity should become nearly equal to the fluid particle velocity. Prandtl claimed that the results of field observations by Koschmieder (1936) supported his Eq. (2.3.17). These ratios of the front velocity to the fluid velocity were also recalculated by Berson (1958) and the mean values varied from 0.50 to 0.62. However, this simple theory does not agree with the laboratory experiment results obtained by Keulegan (1957), Ellison and Turner (1959), Wood (1965), and Middleton (1966). Neither does it agree with other field measurements of cold fronts by Clarke (1961). Wood (1965) suggested that Prandtl's neglect of the hydrostatic pressure terms in the steady layer behind the front accounts for the error in the result. Benjamin (1968) also argued that Prandtl's theory can only apply to the transient problem immediately following the release of a stream of dense fluid into a deep expanse of lighter fluid. 63 Benjamin (1968) studied the flow produced by a cavity emptying a two-dimensional box, as shown in Figure 2.3.11(a). In a reference frame moving with the same velocity as that of cavity tip, Benjamin considered the steady flow past a cavity, (as shown in Figure 2.3.ll(b)). Benjamin balanced the total flow force, that is, the momentum flux plus pressure force, between the approaching and the receding streams and in this moving coordinate system the cavity tip becomes a stagnation point where the pressure can be chosen to be zero. Applying Bernoulli's theorem along the free surface streamline downstream one obtains (2.3.18) for the plane cavity flow, in which U is the mean fluid flow 2 speed downstream where the mean thickness is (d - h 2 ), d is the total thickness of the box and h~ is the unknown head loss between the approaching and the receding streams. The total flow force P u for upstream of the stagnation point is equal to the sum of the hydrodynamic pressure, - t PaU2d , the hydrostatic pressure, 1 P gd 2 and the momentum flux, 2 a ' P u = 12 Pa U2d + 12 Pa gd 2 (2.3.19) 64 d-h 2 77777777777777777777777)77 Figure 2.3.ll(a) u Cavity front moving at a constant velocity along a two-dimensional box. Ps « Po 7777777777777777777777777 Figure 2.3.ll(b) Arrested cavity front as viewed in a moving frame with the same velocity of front. 65 And the total flow force P d far downstream is equal to the sum of the hydrostatic pressure force, t ag (d - h 2) P 2, plus the momentum flux, P U 2 (d- h 2 ), i.e. a 2 (2.3.20) Equating the flow force terms and using the continuity relationship, Ud = U (d - h 2 ) 2 (2.3.21) g h (d - h ) (2d - h ) 2 2 2 d(d + h ) 2 (2.3.22) gives U2 and g d h2 (2d - h 2 ) 'U 2 = 2 (d - h )(d + h 2 ) 2 (2.3.23) Using Eqs. (2.3.22) and (2.3.23) one has the relevant Froude numbers U =--= I gh (2.3.24) 2 (2.3.25) 66 and . (2.3.26) Equating (2.3.18) and (2.3.23) gives the head loss term hi between the approaching and the receding streams as h 2 (d- 2h ) 2 2 (2.3.27) (d 2 - h22) This rela tionship gives h2 = d/2 when hi = 0, i. e., no head loss, and h2 < d/2 when there is non-negative head loss. Without an energy source the case h2> d/2 is clearly impossible. Benjamin (1968) recommended that all the equations derived here were still applicable to gravity currents by replacing g by g', where g' = g (P a p-a Ps). Benj amin found that the equations so derived checked fairly well with the experimental measurements of lock exchange flows made by Keulegan (1958) and with measurements of forced gravity currents on a sloping floor by Braucher (1950), but not with the turbidity flow measurements on sloping bottoms made by Middleton (1966). The energy loss Eq. (2.3.27) indicates that the possible range of the cavity flow thickness h2 is from 0 to one-half of the box thickness and the possible ranges of Froude numbers, are from 1:2 to ~ 1:2 for Fr , from 0 to l i for Fr 2 , and from 0 to 1:2 67 h2 for Fr , depending on the depth ratio, dl ' of the receding 3 streams. A fairly good comparison with Benjamin's theory for a plane propagating cavity flow was given by Ellis and Al-Khairulla (1974), as shown in Figure 2.3.12. Benjamin (1968) extended his analysis to cavity flows propagating in a circular tube. In the limiting case without any surface tension and viscosity effects, the theory was confirmed fairly well by the experimental results of bubble motion in a circular tube by Zukoski (1966). Using a hodograph transformation, Benjamin (1968) was also able to derive the actual shape of a free boundary for a twodimensional arrested cavity. The inclined angle at the tip was found to be 60 0 just as von Karman found, but no prominent "head" shape, as von Karman suggested, was found. Benjamin (1968) argued that there must be some separation and turbulence behind the crest point of the density head in order to balance the hydrodynamic force between the tip and a point far downstream. In the limit of infinite depth of ambient fluid, h /d-+ 0, in Eq. (2.3.24), Benjamin has ~ 2 y'g h2 which was also derived by von Karman earlier as U Ig '1100 = 12. 12, Benjamin claimed that although von ~rman overlooked the energy loss, he obtained the right value with wrong reasoning. Recently, many experimental measurements were made by Britter and Simpson (1978) and Simpson and Britter (1979) on an arrested density wedge produced by a continuous release of saline water into 68 0..6 Eq. (2.3.25) 0..5 0 0..4 8 :>~ 0.3 0 0 0 0 8 000 II ~N lL.. U 0.2 ~ d t 0.1 0..0. L - -_ _....l.-_ _--L.._ _ _L -_ _-L-_ _---l 0.0. 0.1 0.2 0.3 0.4 0.5 ~ d Figure 2.3.12 Comparison of theoretical solution (solid line) of Benjamin (1968) with the experimental results (open circles) of emptying cavity flow in an expansion gallery of surge tank by Ellis and A1-Khairu11a (1974). 69 uniformly flowing fresh water. The wedge angle was found to be around 40 ± So, which is smaller than the 60 0 predicted by von Karman and others. Much higher values of the Froude numbers than those predicted by Benjamin's theory were found for the range of the relative depth, 0.0 < h2/d < 0.3. The trend of the experimental results is that the Froude number will go to a h2 large value for the case of vanishing relative depth, - = 0 d rather than the value, 12, predicted both by von Karman and Benjamin. The difference between the experimental results by Britter and Simpson (1978) and the theoretical results by Benjamin is probably due to the downstream source condition. 2.4 Previous Analysis of Density Front Propagation 2.4.1 Spreading in a Homogeneous Environment In Section 2.3, most of the density fronts discussed belonged to the class of lock exchange flows or to the plane forced surface density currents. Most were observed to propagate with constant speed and any deceleration was believed due to the effect of viscosity. The influence of other aspects of these problems, such as the source condition (a finite volume release or a continuous steady discharge), or radial motion, requires further discussion. A force scale analysis was used by Fay (1969,1971) to study plane and radial spreading of an oil slick on the sea surface. He concluded that an oil front spreading from a finite volume 70 oil spill will be first driven by an unbalanced horizontal pressure force due to the buoyancy and then later by surface tension. The spreading is retarded at first by inertial forces and then by the viscous force associated with an unsteady boundary layer growth in the ambient water. A similar force balance method was previously used by Langmuir (1933) to examine the extent of an oil film lens. Chen and List (1976) used the same methods to analyze the frontal motion due to a steady submerged buoyant jet discharge and also that resulting from a finite volume surface release of miscible buoyant liquid. Two regimes were found for both cases, each with different time histories corresponding to a buoyancyinertial spread and a buoyancy-viscous spread. In the finite volume release case the inertial density front grows at the same rate as that for a sprea,ding oil slick, but the results are different in the viscous regime. The saMe force argument can be used to predict oil front motion resulting from a constant discharge spill and the results are quite different from the case of a finite volume release, (Waldman et al. (1972) and Chen (1975». In order to solve these problems for more than just the frontal position function, several direct numerical calculations of the Navier-Stokes equations have been used. Daly and Pracht (1968) and Vasiliev (1975) considered lock exchange flows, Kao et at. (1977,1978) plane forced surface density currents, 71 and Harlow and Welch (1965) and Sokolov (1972) some miscellaneous problems. Another approach, potential flow solution, was used by Penney and Thornhill (1952), to determine the inertial phase of flow of a squatting liquid column which was a different density from that of ambient fluid. Similar method was also used by Pohle (1952), for the dry-bed dam-break flow. Numerical calculations were performed by Meng and Thomson (1978) and Meng (1978) to study a mixed region collapsing in a stably densitystratified interface between two homogeneous fluids. Nonlinear internal shallow water wave equations were developed by Penney and Thornhill (1952) to study the base surge flow due to an atomic bomb explosion in a shallow ocean. These long wave approximation equations were also used by Abbott (1961,1968) and Hou1t (1972) to study inertial spreading of oil slicks. Numerical computations of the equations using the method of characteristics were performed by Penney and Thornhill (1952), Abbott (1961,1968) and Reid (as reported by Ichiye, 1972), but problems were found in specifying the source conditions and the initial and the boundary conditions at front. Other se1f- similar solutions of the long wave equations were used by Hou1t and Suchon (1970), Fay (see Liang, 1971), and Fanne10p and Waldman (1972) for finite volume oil spills. Recently, Mei (1966) and Buckmaster (1977) considered the spreading flow of a sheet of viscous liquid down an inclined plane. The special case involving spread over a horizontal plane 72 was considered by S. H. Smith (1969b). He obtained similarity solutions to appropriate to an instantaneous point source and found that a liquid front moves over a horizontal bed with a decelerating velocity proportional to t- 4 / S for plane flow and proportional to t- 7 / 8 for radial flow. More general solutions including varying source conditions are presented in the next chapter. 2.4.2 Spreading in a Linearly-Stratified Environment Numerical studies for interf10wing density fronts were made by Wessel (1969), Padmanabhan et aZ. (1970), Young and Hirt (1972), Vasiliev et al. (1972), Bell and Dugan (1974), Dugan et aZ. (1976), and Cerasoli (1978), as listed in Table 2.4.1. For the so-called "principal stage" region, Manins (1976b) and Kao (1977) showed that a plane submerged density front resulting from a volume of homogeneous fluid collapsing at a neutrally buoyant level in a linearly stratified environment will increase its horizontal length scale in proportion to tl/2. However, the viscous long wave approximation by Padmanabhan et aZ. (1970) showed that it will increase as tl/6 when time is very large. By an inverse analogy to selective withdrawal theory, Imberger et aZ. (1976) obtained different power laws for the plane intrusion layer. They found that in the absence of molecular diffusion, the interf10wing wedge due to a horizontal slot jet discharge at a neutral buoyancy level in a linearly density-stratified environment will grow in proportion to the time t in the inertial-buoyancy 73 Table 2.4.1 Autbor Summary of previous theoretical studies of a plane (n = 0) or a radial (n = 1) submerged density front spreading in a linearly density-stratified environment due to a finite volume release (m = 0) or a continuous discharge (m = 1) at a neutral level. . Bell and Ilu&ao (1974) D Bu.er1cal Solution 0 - Nt- 0 1 - "'~totlc ADalyt.ical Solution fana of Il(t)/R o fo ~~~:;) f t~!:·)" It ( " 2 2 ':\0 1+ N t de 4 _ O(Nlttlt) R .t-2" de '" 1 +- N26t2 _ O(trt lt ) V.Ud Dca.in Initial Stage Initial Stage Rate leynolds Humber MR' v. ~_.-2 10 rather large Cer •• oli (1978) 0 0 Dugan .t al.. (1976) Hartaan and t.vt. (1972) - . . .er1cal Calculation Finite 0 0 Difference 0 0 - labereer et a!. (1976) 1 0 - liuaerical Calculation - - lavera. Withdrawal ? '\, 1+T- N2 t 2 _11 Tiae .... 2 _ £. _ 2(1- £.)J~~t) s.&11 '" t Theory 5 6 '\. t'3/'" 0 0 - !.ill _ {i (Nt+c)" Ito e • iDtegration conat.nt Time .... t 1 ,O'+ '" t / Eao (1974) ! 1- • Inertlal- Inflow buoyancy .omentUlll ill _all Vi.COU8buoyancy V1ac:ous- diffusion Principal ... (Nt)" Stage - 1.0{C{- 0.8 Eao (1976) 0 0 - !.ill_ 1 + N' t ' +O(N"t") Ro Nt! 2.05 8 [ !.ill_ ~ (Nt - 0.57)" '" (Nt)" Ro Manina (1976.) 0 0 - h~~:~) - (1 + 2Fr Nt)" !.ill _ Ro (t Nt)" J! Nt~2.05 Principal ASSUlDe Stage elliptical section and Fr- R(~) .1 Nb(o.t) Manius (1976b) 1 0 R(t) - f (.Q. v Nt •~) H N"o\ • ~ R(t) '" N"o\ '\, t S/6 Kaxworthy (1972) 1 0 - R(t) ..... £1/6 Q2/3 tS/6 Hd (1969) 0 0 - R(t) - (1+ h'(o.t)t')" 0 0 Padmanabhan .t at. (1970) Weasel (1969) Young and Hirt (1972) Zulua.a-Angel at al. (1972) 0 0 1 0 0 0 Finite Difference - - Viscous Long Wave Approximation Numerical Calculation - MAC Method (Mark.er and Call) - Neglect diffusion IniUal Stage ..... t o . 33 Inviecid • Nt'! 20 ..... to. 33 Viscous . Nt ~ 20 l16 Final Stage t » 1 '" t 8 InerHaIbuoyancy Assume an elliptical initial section ..... 1+O.122(Nt)1.70 O~Nt~2.75 Ito is one-half ..... O.907(Nt)o.S8 2.75~Nt"25 the side of the initial square of mixed fluid - 1 Initial and Principal ll(t)"~(Nt)3/' ..... t3/~ lfsl6 "llll B Sta~e 74 region, and then in proportion to t 5 / 6 in the viscous-buoyancy region. A linear function of time was also obtained by Manins (1976a) using a different approach. Mei (1969) found by using internal long wave approximations that a finite volume of homogeneous fluid collapsing initially in a linearly stratified ambient fluid will propagate asymptotically in proportion to t. Although Manins (l976a,b) obtained two different power laws for interf10wing fronts for two different source conditions, he (l976b) suggested that there must be some dynamic similitude between them. However, there is not, as yet, a unified theory which can show how the wide variety of power laws for interf10wing fronts are related. A simple order of magnitude analysis will serve this role in the next chapter. General self-similar solutions of long wave approximation are also presented for the inertial and the viscous phases of interflowing fronts, including the initial stage, the principal stage, and the final stage. 2.5 Further Mixing of Surface Spreading Flow 2.5.1 Description of Spreading Flow When a round or slot buoyant jet flow reaches the free surface or the bottom, it may still have residual buoyancy and horizontal momentum which will cause it to spread out as a horizontal buoyant jet. For instance, in the two-dimensional case, Jirka and Harleman (1973) found that a two-dimensional 75 surface spreading flow moved out from the transition zone at an initial densimetric Froude Number F. around 2.6, where 1 F. 1 u ~ Pa -po1 g Pi hi i and hi are respectively the mean velocity and the thickness of the spreading flow at the boundary of the transition zone, and the density of the spreading fluid Any residual horizontal momentum flux will drive the diluted water away from the surface intersection point and produces a secondary entrainment of ambient fluid during horizontal spreading. By contrast, residual buoyancy will gradually erode the capacity of the spreading fluid to mix with the ambient fluid and will force the less dense fluid into a thinner floating layer. Eventually, interfacial mixing will be reduced to a negligible amount and the interface will be sharpened. A two-layered stratified flow will be formed. The initial region of the surface spreading flow was found by Jirka and Harleman to have rapidly increasing depth while the local densimetric Froude number decreased to a lower value. Intensive interfacial mixing with the ambient fluid occurred inside this region. Jirka and Harleman (1973) and Lee et aZ. (1974) suggested that this region formed an internal hydraulic jump. List (1977) preferred to call this increasing depth zone surface jet mixing rather than an internal hydraulic jump, since although the Froude number of the spreading flow is 76 reduced by an interfacial mixing process in the absence of a downstream control there can be no stably hydraulic jump. Some velocity measurements were made by Murota and Muraoka (1967) and Anwar (1972) and excess-density or temperature measurements by Anwar (1972) and Pryputniewicz and Bowley (1975). Similar flow structures have been observed in other experimental studies of surface buoyant jet discharges, such as by Wood (1965,1967), Wilkinson and Wood (1971), Wilkinson (1970), Stefan (1972) in the plane jet case and Jirka et aZ. (1977) in the radial case. An intensive interfacial mixing zone was followed by a two-layered stratified flow where vertical entrainment was almost negligible. The plane jet mixing was found to be sensitive to changes in the discharge condition as well as in the downstream control. An internal hydraulic jump does occur in this jet mixing zone when the entrainment length is controlled by a broad-crested weir downstream (Wilkinson and Wood, 1971). 2.5.2 Surface Buoyant Jet Models A radial surface jet model by Chao (1975) simulates the initial mixing zone of the surface spreading flow created by a vertical single port discharge. Following the wall jet solution by Glauert (1956), Chao derived the mean velocity field of the radial surface jet for both laminar and the turbulent flows. He found that there are similarity profiles for the radial velocity, and that the maximum mean surface 77 velocity decays in inverse proportion to the radial distance from the stagnation point, while the surface jet thickness increases linearly with the radial distance. Similar results for radial surface jet flow and surface buoyant jets were also derived earlier by Koh and Fan (1970) using an entrainment theory approach. However, no information about the surface jet dilution was presented. An analytical solution for surface jet mixing, for both plane and radial surface jets, will be presented in the next chapter. 78 CHAPTER 3 THEORETICAL ANALYSIS 3.0 Introductory Note In this chapter theoretical solutions describing gravity- driven surface and subsurface spreading currents, in both homogeneous and linearly density-stratified environments, due to a variable rate source will be presented. An order-of-magnitude analysis by a force-balance method is used to derive a series of asymptotic time functions for both the inertial and the viscous spreading regimes. Self-similar solutions of non-linear long wave equations are also used to examine these asymptotic formulae. For inertial spreading the solutions analogous to blast wave propagation and to gas cloud expansion are derived for the inertial-buoyancy spreading region and the initial stage. For viscous spreading a solution analogous to the "power-law" diffusion mechanism is derived. Solutions of surface jet mixing in the momentum jet regime are also derived on the basis of Prandt1's assumptions. Dimensional analysis will be used to consider the initial condition effects on the spreading mechanisms. 3.1 Force Scale Analysis for Density Currents 3.1.1 Free Surface Spreading Currents Suppose there is a pool of liquid with uniform density p s floating on a liquid of infinite depth with a uniform density 79 p , (p < p ), where the dimensionless' density difference ratio 11 a s a is (p - p )/p ,as shown in Figure 3.1.1. ass At time t after Q R(t)i '\l --.....p' ~ Figure 3.1.1 1 / J~ !\ Fs Ps --.l ~h(t) Jt) ~ Idealized sketch of free surface spreading density currents. starting the discharge Q, the average thickness of the spreading pool is h = h(t) and the pool radius R = R(t). Suppose the vertical acceleration is very small in comparison to the horizontal one, so that the pressure distribution will be hydrostatic. When the floating layer is in hydrostatic equilibrium in the vertical direction, the average super-elevation of the floating pool over the free surface is I1h(t) , which can be simply derived from the hydrostatic assumption. Assume that there is no molecular diffusion and a negligible amount of mixing through the interface, then the total volume of the surface layer, which is 80 proportional to (1 + f,) hRn+1, is fixed by the discharge. I f the integrated source discharge is Qtm, then the volume conservation equation is simply (3.1.1) in which n is a parameter that denotes the plane (n = 0) and the radial (n= 1) cases, Q is the discharge rate, and m is a source parameter that denotes an instantaneous finite volume release (m = 0) or a continuous steady source respectively (m = 1) . Q represents a finite volume V when m= 0, and a constant discharge rate Q when m = 1. If the surface tension force is negligible, the driving force, F ' at the plume front is the resultant of two horizontal p pressure forces FS and FA' which are directed outwards and inwards at the front. These are (3.1.2) (3.1.3) and (3.1.4) 81 There are two forces retarding the spreading plume: one is the inertial Fr which is proportional to the product of the plume mass with the acceleration (3.1. 5) and the other is the viscous force. The viscous force may arise from two different mechanisms of interfacial shear. For example, an oil slick is usually very thin and its kinematic viscosity \l s is much higher than that of the ambient fluid \l a underneath the oil layer, thus the viscous force FVA for a spreading oil slick can be approximated by the boundary layer shear in the ambient fluid, and , (3.1.6) where 01 is the Rayleigh's unsteady laminar boundary layer thickness;v-t. a For miscible fluids, the viscous force is dominated by the fully developed boundary layer in the plume fluid itself and we write n 2 R + P \l s s It h (3.1.7) 82 From the comparison of the two possible retarding forces two different asymptotic regimes for the starting plume can be distinguished. In one, the inertial force dominates the viscous force and in the second, the viscous force is predominant. By balancing the driving force with either of the aforementioned retarding forces and introducing the average thickness h from the continuity relationship (3.1.1) the possible growth rates are derived. For a buoyancy-inertial plume m+2 R(t) ~ (g~Q)n+3 t n + 3 1 (3.1.8) with an average thickness 2 . n+3 Q h (t ) ~ --n-+"""'1~2"""(:--n--m+-l:"':")(gll) n+3 t (3.1.9) n+3 For a miscible buoyancy-viscous plume R(t) and its average thickness ~ t 3m+l 3n+5 (3.1.10) 83 2 Q3n+S h (t) (3.1.11) IV - - - - ' - " - - - - - - n+l n-2m+l (~) 3n+S t 3n+S The inertial-buoyancy spread regime is therefore identical with that of a spreading oil slick. However, a different growth history is found in viscous-buoyancy spreading regime due to the different mechanisms of interfacial shear. The equations equivalent toEqs. (3.1.10) and (3.1.11) are, in this regime, 1 2(n+2) R(t) IV gilQ2 4m+3 t 4 (n+2) ( \I 1/2 ) (3.1.12) a and h(t) IV (3.1.13) n+1 3n-4m+3 ( ~) 4(n+2) t 4(n+2) More details of spreading oil slicks are given in Fay (1969,1971), Hou1t (1972), and Buckmaster (1973). It is interesting to note that, from Eqs. (3.1.8) and (3.1.10), the starting surface plume does not necessarily propagate with a constant velocity except when m = n + 1 (with no viscous effect) and when m= (3n+4)/3 (with a viscous effect). Table 3.1.1 gives the surface starting plume formulae for two conventional discharge 84 Table 3.1.1 ~ Case ;Plane n=O Asymptotic formulae for starting surface plumes. (Chen, 1975). Source Inertial Viscous condition plume plume Instantaneous finite volume release R( t) 'U (gllV) 1/3 t 2/3 R(t) 'U ge: R(t) 'U (gllQ)1/3 t 6Q'r/5 t 4/5 R(t) 'U (~ v s R(t) 'U (gllV) 1/4 tl/2 R(t) 'U -gv s ( 'r 115 t m=O Steady continuous discharge m=l Instantaneous finite volume release Radial m=O n=l Steady continuous discharge m=l R(t) 'U (gllQ)1/4 t 3/4 (6v'rlS t 1/8 ( ,(S t R(t) 'U geg 1/2 85 conditions, corresponding to m = 0 and m = 1 for both plane and axisymmetric cases. The relevant length and time scales for normalizing experimental data as derived from Eqs. (3.1.8) and (3.1.10), are shown in Table 3.1.2. The frontal propagation velocities can be simply derived by differentiation. For the o inertial regime the front velocity of the plume R decays along with frontal travel distance according to ~ m-n-l R(R) ~ (gbQ)m+2 R m+2 (3.1.14) In the viscous regime o R(R) ~ ( g~~3 1 3m+l 3m-3n-4 R 3m+l ) (3.1.15) These are shown in Table 3.1. 3 for the cases in which n = 0, 1 and m= 0, 1. An alternative approach to the above, used by Koh and Fan (1970), employed the hydrodynamic drag force F , D (3.1.16) and the interfacial shear force F IS F ~ .£h (dR) (Rn+l _ R n+l) IS dt 0 (3.1.17) 86 Table 3.1.2 Length and time scales for surface density spreading. ~ Case Plane n=O condition Length Scale Rl Instantaneous finite volume release (g~."25) 1/7 Source Time Scale tl ( (gt.)2 v· v 3 s t/7 m=O Steady continuous discharge g2 g2 (gt.Q) 173 v s (gt.Q) 2f3 Vs ;T (_v )1/3 gt.v m=l Instantaneous finite volume release Radial n=l (gv6 /12 s m=O Steady continuous discharge m=l ( Q5 gt.v s 3 (8 ( Q gt.vs r 87 Table 3.1.3 Asymptotic formulae for surface density spreading front velocity as a function of frontal distance. ~ Case Plane n=O Source Inertial Viscous condition plume plume Instantaneous finite volume release R(R) tV (gAV) 1/2 R- 1/2 3 4 R(R) tV (ge: )R- R(R) tV (gAQ) 1/3 R(R) tV m=O Steady continuous discharge • (g6vQ't/4 R-1/4 -- s m=l Instantaneous finite volume release Radial n=l R(R) tV (gAV)1/2 R- l R(R) tV (ge R(R) tV (gAQ)l/3 R-l/3 R(R) '\i V3 )R- 7 m=O Steady continuous discharge m=l ( 'r' ge~ R- l 88 to express the two forces resisting spreading, Here € is an effective viscosity coefficient and R =R(O). o Since the dimensions of these two resisting forces, in Eqs. (3.1.16) and (3.1.17) are homogeneous respectively with those of the previously derived inertial force FI and viscous force FVS' the same results for frontal motion, as shown in Table 3.1.1, can be derived from this approach (see Koh (1976». 3.1.2 Bottom Spreading Currents A bottom starting plume can be viewed as an inverted version of the free surface plume without super-elevation ~h. The two retarding forces are the same as for the free surface case since the inertial force is still involved and the interfacial shear dominates the wall shear. As shown in Figure 3.1.2 the driving force Fp' for a starting bottom current is the resultant of two pressure forces FS' and FA', (3.1.18) (3.1.19) and Fp' =FS' -FA' so that:F p " has the same form as Eq. (3.1.4): (3.1.20) and the free surface results may be applied directly. 89 Q R( t) -1 T r r F.A - ........;-.- F.S Figure 3.1.2 Idealized sketch of bottom spreading density currents. z \ \ \ \ -~~---2h(t) \ r R(t) Figure 3.1.3 A homogeneous layer intruding at a neutral buoyant level into the linearly density-stratified ambient fluid. 90 3.1.3 Internal Density Currents The driving force for submerged density currents is the unbalanced horizontal pressure force Fp. It is assumed that the internal current is homogeneous and is spreading at a neutral density level in a linearly stratified environment, as shown in Figure 3.1.3; the density difference between the current and the ambient stratified fluid is a linear function of depth. The total pressure force Fp in this case is proportional to the cubic power of the average thickness of the current rather than the square power as for surface and bottom spreading currents, i.e. (3.1. 21) In the two dimensional case (for which n = 0), the total pressure 1 force is 12 g d p (z) d: . h 3 , as derived by Kao (1974,1976). The retarding forces for submerged density currents are the same as for free surface and bottom currents, i. e. the inertial force Fr and the viscous force FVS as defined in Eqs. (3.1.5) and (3.1.7). Equating the pressure force Fp with the inertial force Fr and cancelling the average thickness term h with the aid of the continuity relationship (3.1.1) enables the inertial spreading front to be expressed as a function of time t as 91 1 m+1 R(t) 'V (NQ)n+2 t n+ 2 since N = /gE. (3.1.22) Equating the pressure force Fp to the viscous force F ' gives the radius of the viscous spreading front VS 1 R( t) 'V 4m+1 N2 Q4 ) 2 (2n+3) t 2 (2n+3) ( \) (3.1.23) S Table 3.1.4 gives specific results for the various values of m and n. The average thickness h(t) for the inertial spreading wedge can be written as 1 Qn+2 h (t) 'V -n+-1--=:"--::-3m+n--+-1 Nn+ 2 t n+2 (3.1.24) and for the viscous spreading wedge as 1 Q2n+3 h (t) 'V --~'-n-+':-:1;----n-:-+71--2;:-m( (3.1.25) ~:) 2 (2n+3) t2(2n+3) The length and the time scales for normalizing experimental data are given in Table 3.1.5. 92 Table 3.1.4 ~ Case Plane n=O Asymptotic formulae for submerged density spreading of an intrusion layer in a linearly density-stratified environment Source Condition Finite Volume Release m=O Steady Continuous Discharge m=l Radial n= 1 Finite Volume Release m=O Steady Continuous Discharge m=O Viscous Submerged Spread Inertial Submerged Spread R(t) ~ (NV)1/2tl/2 R(t) ~ R(t) ~ (NQ)1/2 t R(t) ~ R(t) ~ (NV) 1/3 t l/3 R(t) ~ R(t) ~ (NQ)1/3 t 2/3 R(t) ~ (t (t N~4 N~~4 (2 Nv: 4 t 1/ 6 6 t 5/ 6 flO t 1 / 1O (t N~~4 6 10 t1/2 93 Table 3.1.5 h Case Length and time scales of submerged density spread for normalization. Source Length Scale Time Scale Condition Rl tl Instantaneous finite volume release Plane m=O n=O Steady continuous discharge NV 3 v s ( f'4 Q3/2 Nl / 2v m=l Instantaneous finite volume release Radial m=O n=l Steady continuous discharge m=l s (N~s r _QN v s 1/7 ( f/7 (NO:: ,) ( Q3r'S (N4vQ2 3 NV 3 v s Nv 2 s s r s 94 3.2 Similarity Solutions :for Inertial-Buoyancy Density Currents The force scale analysis presented in Section 3.1 provides order of magnitude estimates of frontal trajectories for unsteady and non-uniform flows. However, in each power law growth rate there is an unknown coefficient that must be determined from experimental studies. Moreover, all of the fluid was assumed to move at the frontal velocity and the spreading layer thicknesses were assumed to be uniform. In this section two generalized shallow-water equations will be used to describe the inertialbuoyancy force balance for surface and the submerged density spreading flows in more detail. Taylor's (1950) self-similar technique for describing blast wave propagation is extended and used to find the unknown coefficients and the non-uniform features of the flows. The series of asymptotic formulae that were derived by the force scale analysis will be determined directly from the equations of motion. The effects of finite dimensions will be considered for both the surface and the submerged spreading flow produced by a finite volume release, and the self-similar solutions are found to describe the early stage of flow before "shock wave" type propagation develops. This solution is similar to the expansion of a suddenly released finite volume of gas. 95 Theoretical results developed in this section will be compared with previously published numerical and experimental data in Chapter 6. 3.2.1 Longitudinal Internal Shallow~Water Wave Equations We consider the two flow configurations as shDwn in Figures 3.2.l(a) and (b). At time t, there is a spreading pool with a horizontal extent R(t) and layer thickness h(r,t) at a radial distance r. If the horizontal length scale is much larger than the maximum vertical length scale, then the vertical acceleration and velocity will be smaller than the corresponding horizontal values. The pressure distribution is therefore hydrostatic and the horizontal velocity u(r,t) is independent of the vertical direction z. At hydrostatic equilibrium the thickness of the surface pool above the water surface is 6(r,t) ~ (3.2.1) h(r,t) where the dimensionless density difference ~ = (p ~p )/p . ass For a submerged spreading flow in a linearly density~stratified medium, as shown in Figure 3.2.l(c), the flows are assumed symmetrical. The coordinate system is chosen such that the origin is at the point of symmetry and h(r,t) represents only one~half the thickness of the submerged spreading layer. Using a long~wave approximation, two generalized shallowwater wave equations can be derived for these spreading flows (see Appendix A). First, the continuity equation is 96 8< r, t) r z Figure 3.2.l(a) Coordinate system for free surface spreading flows. z r Figure 3.2.l{b) Coordinate systems for bottom spreading flows. z I Po(o)=R r \ Figure 3.2.l(c) Coordinate system for submerged density flows. 97 n Clh +...!... (r uh) = ah + a (uh) + nuh = 0 Clt n Clr at Clr r (3.2.2) r and the horizontal momentum equation is ~ + u au + 11 hi dh = 0 at ar g i ar (3.2.3) where the parameter n denotes the case of plane spreading flow for n = 0 and radial spreading flow when n = 1. The parameter i = 0 denotes surface and bottom spreading flows in a uniform environment. Submerged spreading flow in a linearly density- stratified environment is denoted by i = 1. By adding Eq. (3.2.2) multiplied by u and Eq. (3.2.3) multiplied by h the momentum equation can be rewritten as (3.2.4) Equation (3.2.4) is actually an energy equation for shallowwater waves. 98 Equations (3.2.2) and (3.2.3), with i= 0 and n= 1, were derived by Penney and Thornhill (1952) to study the base surge mist-air flow formed by an atomic bomb explosion in shallow water. Equations (3.2.2) and (3.2.3), with i= 0 and n= 0, are the general form of dam-break flow equations without friction effects, in which case the relative density difference ~ is unity. The same shallow water wave equations, with i = 0, were used by Hou1t (1972) to study inertial spreading of an oil slick on the sea surface. Equations (3.2.2) and (3.2.3), with i = 1 and n = 0, were used by Mei (1969) to study the initial stage of collapse of a plane mixed region in a linearly density-stratified fluid. 3.2.2 Gas Dynamics Analogy It is well-known (see Courant and Friedrichs (1948), pp. 32-35) that the two shallow-water wave equations (3.2.2) and (3.2.3), when i = 0, are similar to the equations of one-dimensional unsteady isentropic gas-dynamics. (3.2.5) (3.2.6) when the specific gas constant y is equal to 2, in which the sound speed c= Idp/dp and the parameter n denotes the plane, cylindrical or spherical flow respectively when n = 0, 1, and 2. 99 ~ollowing the study of Penney and Thornhill (1952), the characteristics equations for density spreading flows' can therefore be written as ~I '+1 nuygbihl. ± -- r =0 (3.2.7) The Riemann invariants along the characteristics curves, dr = u ± dt - ~gbihi+l (3.2.8) are 2 u + - i+l ~+ g i - n It u~gbihi+l r dt = constant (3.2.9) or 2 ~ gbih i+l ± n u + - i+l r u ~ gbih i + l dr u± ~ gbihl.'+1 -= r constant With suitable initial and boundary conditions these equations may be solved numerically except when a jump discontinuity develops due to the non-linearity. (3.2.10) 100 Similarity solutions for the blast wave propagation produced by an atomic bomb explosion were introduced independently by Taylor (1950), and Sedov (1957). Later, this self-similar technique was extended by several investigators, (see Rogers (1958), Freeman (1968), Dabora (1972), Pitkin (1977», to solve the more general problem of the shock wave produced by a variable rate of energy release. Taylor and Sedov found that shock wave front propagated with a velocity that depends only on the specific energy release. The trajectory of the front is described as R(t) = S(y,m,n) ( _1_ m+2 ~p) n+3 t n+ 3 (3.2.11) where the proportionality coefficient S(y,m,n) is found from an energy conservation condition. The similarity distributions for particle velocity, density and pressure were solved analytically and numerically. Sedov's self-similar technique was used by Hoult and Suchon (1970) to study the inertial phase of spreading of a plane oil slick formed by the release of a finite volume of oil. However, their computed results are not in agreement with their experimental results. Later, Hoult (1972) revised the technique to use a different independent self-similar variable but problems were still met in specifying the boundary conditions and the similarity profile for the spreading layer thickness was not given for the radial spreading case. Similar difficulties were 101 also encountered by F.ay (see Liang, 1971) and Fanne10p and Waldman (1972) • Here, we use Taylor's self-similar technique, as extended by Rogers (1958), to resolve the general problems of a variable flow rate buoyant-discharge. However, first we must derive the correct Taylor's self-similar formulae and the necessary boundary conditions at the front. 3.2.3 Derivation of Self-Similar Formulae Two methods of derivation will be given here. The first one is the method of separation of variables and the second one is the Taylor's method used for the blast wave propagation. 3.2.3.1 Method of Separation of Variables Assume that the two unknowns, u(r,t) and h(r,t) in the shallow-water wav,e equations, have solutions in the following forms: u(r,t) = net) ~(~) (3.2.12) h(r,t) = T(t) H(~) (3.2.13) where the dimensionless variable ~ is defined as r/R(t), and R(t) is the frontal position at time t, net) and T(t), which will be determined later, are two unknown scale functions for mean velocity and spreading layer thickness, respectively. 102 Substituting the above into the shallow-water wave equations (3.2.2) and (3.2.3), one obtains Q Q RT H _! I;H' + <pH' + <p'H + n4>H = 0 nT n I; (3.2.14) and (3.2.15) in which 11011 denotes the differentiation of a function with respect to time t and I, ," to the dimensionless variable 1;. In order to have the assumed form of solutions given in Eqs. (3.2.12) and (3.2.13), one must let four unknown functions, R/n, RQ/~2, RT/~T and g~. Ti+l/~2, be dimensionless. Choose ~ o unity for the first one R/n. Then, the velocity scale will be the front velocity and the similarity assumption for mean velocity will become o u(r,t) = ~(t)4>(I;) = R(t)4>(I;) (3.2.16) o The second function R~/~2 will therefore automatically become a dimensionless number A that we call the decay coefficient, R~ A =- ~2 00 = R(t) R(t) iF(t) (3.2.17) 103 There are two kinds of self-similar solutions to choose for the spreading layer thickness. The first one is to set the fourth function g~. Ti+l/~2 to unity. That is to say that ~ the length scale for the spreading layer thickness will be 1 --- 1 --- T(t) ::: (~) i+1 = (R2(t) ) i+l g~. (3.2.18) g~. ~ ~ o In this case the third function RT/QT will automatically become a constant value that is related only to the decay coefficient, i.e. o o~ RT 2 RR 2 = --- = -- A QT i+ 1 R2 i+ 1 (3.2.19) and the similarity assumption, Eq. (3.2.13) will become h(r,t) (3.2.20) o The second solution is to set the third function RT/QT to a constant value a, i.e. 000 RT RT nT -=-0-="'--= a nT RT nT where the dimensionless frontal position n = R(t) /R o ,R0 is the initial frontal length R(O). From Eq. (3.2.21) one has the length scale for the spreading layer thickness (3.2.21) 104 (3.2.22) where T(O) is the initial length scale of the spreading layer thickness. By choosing T (0) = hI' then the second solution for the spreading layer thickness becomes (3.2.23) in which ex is an unknown constant and hI is an initial layer scale thickness. For this case the fourth functjon, g~.Ti+l/~2, 1. will be a dimensionless number that is related only to the decay coefficient A. The first set of similarity solutions, Eqs. (3.2.16) and (3.2.20), indicate that .only the frontal velocity plays the role in defining the spreading layer thickness. In the second set,Eqs. (3.2.16) and (3.2.23), only the relative frontal length n is important. The first set thus is used to describe the far-field gravitational spreading problem where the front is so displaced from the origin of motion that the initial dimension is no longer important but not so far that viscous effects are negligible. The second set of solutions is appropriate for the near-field problem and will be considered in Section 3.2.7. 105 The first set of similarity solutions can also be derived by Taylor's (1950) approach to solving the blast wave propagation as extended by Latter (1955). 3.2.3.2 Taylor's Method Assume that two unknowns, u(r,t) and h(r,t), have similarity solutions of the following forms: (3.2.24) (3.2.25) where k and ~ are two unknown integers that will be determined later, ~l ahd HI are two unknown functions that are also to be determined later. Substituting above two assumptions in the shallow-water wave equations (3.2.2) and (3.2.3), one gets (3.2.26) and • (3.2.27) In order to homogenize both Eqs. (3.2.26) and (3.2.27) into dimensionless forms, one has to set 106 (3.2.28) (i+l),!I,-2k==0 (3.2.29) and assume (3.2.30) 1 i+1 2 HI (0 = (:1:::. ) H(O (3.2.31) i in which A is a dimensional constant, and ~(s) and H(s) are two dimensionless similarity functions. From Eqs. (3.2.24), (3.2.25), (3.2.28), (3.2.29), (3.2.30), and (3.2.31) two similarity solutions are derived as. (see Appendix B ~or details of derivation) o u(r,t) = R(t)~(~) (3.2.16) and h(r,t) = (it 1 i+1 2 ( t) ) gl:::.. H (s) (3.2.20) ~ The two similarity functions ~(s) and H(s) are the dimensionless distributions of the relative mean particle velocity and the local Richardson number behind the front. For the uniform 107 environment case where i = 0, the local Richardson number is clearly defined as Ri(r, t) H(E;) = gf.h(r.t) (3.2.32) R2(t) and for the linearly stratified case where i = l, it is defined as Nh(r,t) R(t;) = Ri(r,t) = o (3.2.33) --'-~:'- R(t) in which N is the Brunt-Vaisa1a frequency of the ambient stratification. 3.2.4 Evaluation of Self-Similar Solutions 3.2.4.1 General Derivation The similarity assumptions (3.2.16) and (3.2.20) when substituted into the continuity and the momentum equations (3.2.2) and (3.2.3) reduce the two partial differential equations to two ordinary differential equations: (<I> - ~) H' + i! 1 A + <p' + ~4> H o (3.2.34) (<1>- 04>' +Acj>+ R~' = 0 in which ' denotes differentiation with respect to ~, i.e. 00 0 (3.2.35) d/d~. The decay coefficient A=RR/R2 in Eqs. (3.2.34) and (3.2.35) can be derived from the frontal function R(t). 108 Now suppose a buoyancy source supplies a time-varying volume V.(t) of fluid that may be positively, negatively, or neutrally ] buoyant. Then we have V. (t) = Q t m (3.2.36) ] where m = 1 denotes the case of a steady, constant rate discharge Q, and m = 0 denotes the case of an instantaneous release of a finite volume Vj = V. Assuming that the amount of interfacial mixing between the two fluids is negligible and also that free surface evaporation loss or bot~om floor seepage is negligible, then the total volume Vet) of the spreading layer at time t must equal the amount that has been supplied V.(t), i.e. ] Vet) (3.2.37) V. (t) J The total volume of a spreading layer at time t is 2 Ret) Vet) = f o k. ~n h(r,t)rndr = kin limn R(t)n+l R(t)i+l 1 (g~.)i+l ~ (3.2.38 ) 109 in which the integration constant in each case will be (3.2.39) and the shape factor k. ~n is defined as (i+ 1) { n=O (3.2.40) (i+l)21T n=l When the relative density difference b is rather small, i.e. for the case of free surface spreading of sewage or thermal water discharge, Eq. (3.2.38) is still a good approximation with an error to the order of b, but for a spreading oil slick, the relative density difference is not very small and the shape factor has to be multiplied for a factor (1 + b) for correction, i.e. (I + b) kon = { (1+b)21T n=O (3.2.41) n= 1 From the similarity assumptions (3.2.19) and the condition of volume conservation, one obtains the differential equation for the trajectory function, no i+1 -2- (i+1) (n+1) 2 Ret) m(i+1) dR(t) dt = k. 1.n t I. 2 ,(3.2.42) 1.mn for both surface and submerged spreading flows. Integrating Eq. (3.2.42) from R = R(O) = R to R = R( t) gives the frontal function, o R(t) = 2 (i+l) {n+l)+2 i+l { (i+l) (n+1)+2 + (i+l)(n+I)+2 R 2 o m(i+l)+2 Z.~ Bi. ) 1.n 2 t mCi +;)+2 } l.mn (3.2.43) in which the generalized definition of kinematic or specific buoyancy flux B. is 1. 1 B. = (g~.)i+l Q = 1. 1. { g~Q i= 0 NQ i =I {3.2.44) When the initial size of the spreading pool is very small, i.e. (i+l)(n+l>+j R fR{t)« 0(1) or R 2 i+l B.2 0 0 1 . asymptotic form for R(t) is m(i+l)+2 t 2 « 0(1), an 111 R(t) c imn i+l m(i+l)+2 B.(i+1) (n+l)+2 t(i+l)(n+1)+2 ~ in which the coefficient in the asymptotic form c. ~mn c. = ~mn ("+1)(2 +1)+2j{ n { m(i+1)+2 is i+1 } (i+1) (n+1)+2 f (i+1) (n+1)+2} 1: l (3.2.45) kin limn (3.2.46) The power law is a general form for describing inertial-buoyancy motion for both surface and submerged spreading currents due to a variable source of buoyant discharge. The decay coefficient A becomes A = 'R(t)°R'(t) = (i+1) (m-n-1) R2(t) m(i+1)+2 (3.2.47) The two non-linear ordinary differential equations for H and ~ become np(~-~) + (1-i)(m-n-1) 2 (m-n-1) ~ m(i+1)+2 ~ - m(i+1)+2 ~ -- = --~------~~~~~------~~~~-i 1 H H + _ (~_~)2 HI (3.2.48) 112 !!i) (i+1) (m-n-1) H<P-f;) _ (2(m-n-1) + Hi +1 <P' = ---:m::. (::. .::i::. .;+-=l:.!. .)+...:..2=---_ _ _::-:-::_:....:m::..(::...::i::...;+..=1:L)+...:..2=---~f;z........:._ __ Hi +1 _ (<p-f;)2 (3.2.49) To solve these two differential equations for the similarity profiles two boundary conditions are needed and they are found from the shock discontinuities at the front. 3.2.4.2 Shock Conditions at Front From continuity of mass, Eq. (3.2.2), and momentum Eq. (3.2.3) or energy Eq. (3.2.4), three shock conditions at the front can be derived: U[h] [uh] (3.2.50) (3.2.51) U[uh] = [ u2h + g.ilohi+2] ~+2 (3.2.52) where [ ] denotes the jump value of the term inside brackets between the right-hand side (+ side) and the left-hand side o (- side) of the discontinuity and U = R( t) is the frontal velocity. The derivation details for these three jump conditions are given in Appendix C. 113 Now, assume that the ambient fluid is stagnant, i.e. ua = u+ = u(R+, t) = 0, and the spreading layer thickness there vanishes, Le. h+=h(R+,t) =0. From Eq. (3.2.50), one has o U = R(t) (3.2.53 ) L e. t) = u(R_,t) = 1 cj>( 1) = u(R, o o R(t) (3.2.54) R(t) From Eqs. (3.2.51) and (3.2.53), one has i 1 g6. t4+1 g6. i h - + i 1 2 1.2 u2- + 0 - "2 u+ i+1 i+l U = R(t) = u- -u.r 1 g6. i h= "2 u_ + i+l u_ i+1 1 (3.2.55) i.e. H(l) (i;9 1 i+l (3.2.56) Similarly, from Eqs. (3.2.52) and (3.2.53), one has (3.2.57) 114 i.e. 0.2.58) Two kinds of front are therefore possible depending on whether momentum or energy is conserved: one is the wave front with a constant frontal Richardson number, 0.2.59) and the other is the front with a vanishing frontal Richardson number, Ri f = H(l) o Two kinds of moving surface fronts for the case i = 0 were also found by Abbott and Torbe (1963) using ehe method of characteristics: one was called the wave front (or moving hydraulic jump or bore) propagating with a finite frontal Richardson number, the other was called the Saint Venant front propagating with a vanishing frontal Richardson number. According to their theoretical arguments, the front moving with (3.2.60) 115 a vanishing frontal thickness occurs at the coincidence of two characteristics curves and the wave front moving with a finite frontal Richardson number occurs when one of the characteristics becomes time stationary. In the case of one fluid spreading over or under another, a wave front must always be formed as long as the velocities in both fluids are in the same direction. A spreading front formed by a shallow fluid layer flowing over or under another fluid of much greater depth will eventually develop into a moving wave front since it does not induce appreciable motion in ambient fluid. The vanishing submerged frontal thickness in Eq. (3.2.58) when i = land n = m == 0 was also derived by Mei (1969) for the initial stage of dynamic collapse of a mixed region in a linearly density-stratified environment. 3.2.5 ASymptotic Solutions for Surface Density Spreading The asymptotic form of the frontal trajectory function in this category, i = 0, has the same time-power function as for blast wave propagation produced by a variable rate of energy release, 1 m+2 B n+3 t n + 3 R(t) = c omn o In this case the kinematic buoyancy flux, B = B = g6Q, or the o specific buoyant force per unit mass, B = B = g6V, is analogous o (3.2.61) 116 to the specific power or energy release E/p in Eq. (3.2.11). The coefficient c omn ' 2 c omn = {(m+ 2) n:C-k_3--::I~_}n+3 on (3.2.62) omn can be determined from the integration constant I omn of the self- similar thickness profiles H(~) for each case n and discharge conditions m. The similarity distributions for spreading layer thickness and mean velocity can be solved analytically or numerically from the two reduced ordinary differential equations, m-n-l (~_ 20 + nH~ - t;) m+2 ~ $' m-n-l H~ _ m+2 n _ [2(m-n-l) +!!1] H m+2 ~ = ~~-------------:--~~------~~-- H - (~- t;)2 (3.2.63) (3.2.64) using two Rankine-Hugoniot type frontal boundary conditions, H(l) ="21 and HI) = 1. 3.2.5.1 Plane and Radial Surface Density Spread from a Finite Volume Release The ordinary differential equations in this case (n =0, 1, and m= 0) become 117 (<1>-1;) H' +H(-n-1 + n<l> + <1>') = I; ° (3.2.65) and (<1>_1;) <1>' - (n;l) <I> + HI = ° (3.2.66) Corresponding to the boundary conditions <I> (1) = land H(l) not specified, these equations have the solutions (3.2.67) and H(O = (n1l) (1;2 - 1) + H(l) (3.2.68) The Saint Venant front condition, H(l) = 0, gives a negative value for H(~) except at the front. The wave front condition, 1 H(l) = 2 ' gives the thickness distribution as (3.2.69) Thus the integration constant I oon = r1 I;n H (E;) d = -:--~l:..,.---::--=- 1o (n+l) (n+3) (3.2.70) 118 and the spreading layer coefficient is = (cn+1) (n+3) c oon 4 k 3) 1 n+3 (3.2.71) on The plane inertial spreading front (n = 0) due to a finite volume release therefore grows asymptotically according to (3.2.72) and the radial inertial spreading front (n = 1) grows asymptotically as R(t) = 1.502(g~V)1/4 tl/2 (3.2.73) The density wave front from a finite volume release therefore grows with the greatest thickness at the front and a continually decreasing thickness behind the front. No "density head" or "density nose" is given by this blast wave analogy solution. The whole spreading fluid acts as a stretching layer between the front and the origin. The front pulls the spreading fluid with the same velocity as the front at the frontal wall and a linear distribution of fluid particle velocity, proportional to its dimensionless location, ~, occurs behind the front. details of this spreading flow are shown in Figure 3.2.2. The 119 1.0 0.8 0.6 04 0.2 0.0 0.0 02 04 06 o.s 1.0 r R( t) Figure 3.2,2 Self-similar velocity and thickness distributions of the plane and the radial inertial density wave front propagation due to a finite volume release of buoyant fluid. 120 This strong shock wave solution is applicable only when the density fronts are so far away from the origin point that the initial dimensions are negligible in comparison to the frontal distance. The initial dimension plays no further role in defining the length scale to the problem. In the case where the initial dimensions are important, the initial flow problem is totally different and will be discussed in Subsection 3.2.7. 3.2.5.2 Starting Two-Dimensional Stratified Flow In this case n = 0 and m = 1 so that providing that (¢ - 0 2 - H(O " 0, the two similarity equations in this case are = ° (3.2.74) H' (0 = ° (3.2.75) cj>' (0 and The non-trivial wave front solutions are cl>(t;) = 1 (3.2.76) = 2"1 (3.2.77) and H(t;) Hence the integration constant (3.2.78) 121 and the coefficient for the two-dimensional inertial starting flow is 1.260 (3.2.79) i.e., the starting free surface inertial spreading flow grows as Ret) 1. 260 (gllQ) 1/3 t where Q is the volume flux of the discharge. (3.2.80) In the absence of any viscosity effect the two-dimensional starting flow therefore grows with a constant velocity which is proportional to the cubic root of the discharge buoyancy flux B = gllQ. The velocity and the thickness of a two-dimensional starting flow are therefore distributed in such a way that the local Richardson number Ri(r,t) is uniformly distributed with a constant value 0.5 over the entire spreading length, i.e. Ri(r,t) gllh(r,t) u 2 (r, t) This is fully analogous to a two-dimensional propagating strong shock front driven by a continuous steady energy release, as derived by Rogers (1958). The shock wave front also propagates with a uniformly distributed velocity, density (pressure also) and Mach number. (3.2.81) 122 3.2.5.3 Surface Radial Starting Plume In this case n = 1, m = 1, and the two ordinary differential equations describing the similarity solution are 4>'(~) = 1. H4>-~) + H(i_~) 3 ~ 3 (3.2.82) (c/1- ~)2 - H and H{ H' (~) = t (4) - 20~ - H4> - O} (3.2.83) ~{(<p_~)2-H} These two differential equations were solved numerically at Caltech's Booth Computing Center by a program called "MODDEQ/ Differential Equation Solver" using frontal boundary conditions HI) = land H(l) = 1/2. The numerical calculation is based on the method of Runge-Kutta-Gill with automatic error control. Two similarity profiles are shown in Figure 3.2.3. It is peculiar to see that there is an infinitely large fluid velocity inside the plume at the location ~ = 0.753. At this point the thickness vanishes and another internal ring forms after the former ring is dispatched. The integration constant 1011 as calculated by Simpson's rule is = (1 10.753 ~H(~)d~ = 0.09645 (3.2.84) 123 1.5 1.4 1.3 1.2 1.1 1.0 05 0.4 0.3 0.2 0.1 0.0 0.7 0.9 0.8 r { =----:R( t ) Fig. 3.2.3 Velocity and thickness distributions of a starting internal radial ring. 1.0 124 Thus, the coefficient for the radial starting plume is (3.2.85) The frontal bore therefore propagates outwards according to R(t) = 1.309 (g~Q)1/4 t 3/ 4 (3.2.86) Suppose that there is a stationary observer located at a distance r from the surface buoyant source point. He will experience a series of internal ring fronts at a period T, which is 4 1 T =----(1.309)4/3 = 0.699 1/3 (r ) B (3.2.87) Velocity measurements at the same location r from the source point will have an oscillation in magnitude, which is coherent with an oscillation in density, at the same period T. 3.2.6 Asymptotic Solutions for Internal Density Spreading The asymptotic Eq. (3.2.45) of the frontal trajectory function in this category, i = 1, becomes 1 R(t) :=; c lmn B n+2 I m+l t n+2 (3.2.88) 125 in which the kinematic buoyancy flux Bl = NQ for the steady continuous discharge case, m=l, and the specific buoyant force per unit mass Bl =NV for the finite volume case, m= O. The coefficient c mn ' l 1 c n+2 lmn - { (m+ 1) k. ~n I }n+2 (3.2.89) lmn can be determined from the integration of the self-similar thickness profile H(~) for each case n and discharge conditions m. This self-similar thickness distribution H(~) can be solved analytically or numerically from two ordinary differential equations, n<j>{cp - ~) ~ m-n-l </>(<1> _ 0 </>' = m+l - m-n-l ~ m+l _ (m-n-l + m+l H2 _ (</>-0 2 (3.2.90) ¥) H2 ~ (3.2.91) with the boundary conditions H(l) = 1 and </>(1) = 1. 3.2.6.1 Plane and Radial Submerged Density Spread from a Finite Volume Release The ordinary differential equations in this case (n= O. 1, and m= 0) become 126 HI (</> _~) + H(</> I + ncjl - n - 1) = F; ° (3.2.92) and cjl'(cjl-~) + HH' - (n+1)cjl = ° (3.2.93) Corresponding to the boundary conditions cjl (1) = 1 and H(l) = I, these equations have the solutions (3.2.94) and (3.2.95) H(O = I{n+ 1);2 - n The half-thickness will be seen as a linear function of the dimensionless variable ~ for the plane flow, n= 0, (3.2.96) H{O = F; and for the radial flow, n= 1, will be H(~) = h~2 - 1 where the value exists only for 1/1:2 ~ F; ~ 1. structure must be formed for this radial surface spreading. cas~ (3.2.97) A circular internal ring which is different from that of Details of self-s'imilar half-thickness 127 profiles are given in Figure 3.2.4. Therefore, the integration constant I Ion becomes 1 lIon = (n+ 1) (n+ 2) and the coefficient c l on (3.2.98) becomes (3.2.99) A plane inertial spreading front (n = 0) due to a finite volume release therefore grows asymptotically according to R(t) = (2NV)1/2 tl/2 (3.2.100) and the radial spreading front (n = 1) grows asymptotically as R(t) (3.2.101) For the special case of an initially cylindrical section of radius R , (n = 0), the frontal position function will be o (3.2.102) 128 1.0 0.8 0.6 H(e) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 r 0.8 1.0 e = R( t) Figure 3.2.4 Similarity profiles of the half-thickness for a plane (n = 0) and radial (n = 1) interflowing layer resulting from a collapsing homogeneous mixed region at a neutral density level in a linearly stratified environment. 129 :For an initially spherical section of radius R , (n = 1), the o frontal position will be specified by (3.2.103) 3.2.6.2 Internal Plane Starting Layer (Continuous Flow) Providing that t;) 2 - H2 (t;) :f 0, the two (<I> - similarity solutions in this case (n = 0 and m = 1) are =0 (3.2.104) <1>'(0 = 0 (3.2.105) H'(t;) and The non-trivial wave front solutions for the half-thickness and the mean velocity of the intrusive layer are: H(t;) = Nh(r,t) 0 1 (3.2.106) 1 (3.2.107) R(t) and <I> (t;) u(r,t) 0 R(t) 130 The integration constant in this case is (3.2.108) and the coefficient for the plane intrusive layer is (3.2.109) An intrusion layer front resulting from a discharge at a neutral density level, with a discharge Richardson number Ri. J Nh· =~ Uj of unity, will increase its displacement according to R(t) = !22 (NQ)1/2 t (3.2.110) where h. is the half-thickness of the slot jet opening, u. is J J the mean jet velocity and Q is the volume rate of discharge, When the jet Richardson number is less than Nh· unity, i.e. Ri. = ~ < 1 J U· J ' there will be a region of mixing. This jet mixing will reduce the mean velocity to a lower value such that the flow will move at a Richardson number of unity. However, the total volume of the interf10wing layer will be larger than the amount discharged. In this case the constant . of proportionality ~n Eq. (3.2.110), R(t) , will be greater (NQ)1/2 t than 1/12. 131 3.2.6.3 Internal Radial Starting Layer (Continuous Flow) In this case n = 1 and m = I and the decay coefficient A 1 - 2. Two reduced differential equations are obtained: R' = t Rep + H (ep - ~ ) ( t-t) (3.2.111) H2 - (ep-02 and (3.2.112) With the two boundary conditions, HI) = 1 and R(l) = 1, these two differential equations were solved numerically using the subroutine "MODDEQ/Differential Equation Solver" at Caltech's Booth Computing Center. The subroutine is based on the method of Runge-Kutta-Gill with automatic control of error truncation. The two similarity profiles for mean velocity and half-thickness obtained for the spreading layer are shown in Figure 3.2.5. It is peculiar to see that there is also an infinitely large fluid velocity inside the plume at the location where the half-thickness vanishes. .; = 0.6915 Since the neutral density source is continuously discharged at the origin, a series of such internal rings should form after the first ring is dispatched. 132 1.4 1.2 1.0 0.8 H(e) 0.6 H(e) 0.4 0.2 0.0 __ 0.5 0.6 ~ ....L..-_ _""""""_ _- - ' -_ _- - J ._ _- - - ' 0.7 0.8 0.9 1.0 r e= R(t) Figure 3.2.5 Similarity profiles of the average velocity and the half-thickness of interflowing radial ring due to a continuous release. 133 The coefficient of integration is computed using Simpson's rule as rl I;H(I;) dl; 0.231 (3.2.113) ) 113 = 0.802 (3.2.114) )0.6915 and the radial front coefficient becomes = ( 3 2 x 41T X 0.231 An internal ring front will therefore propagate according to R(t) = 0.802 (NQ)1/3 t 2/ 3 (3.2.115) Suppose there is a stationary observer located at a distance r from a submerged neutral density source discharging with a flow rate Q in linearly density-stratified environment for which the Brunt-Vaisala frequency is N. The observer will see a steady series of fronts passing at a period T 3)1/2 T = 1.391 ( ~Q 3.2.7 (3.2.116) Initial Stage of Density Spread A spreading density front generated by release of a finite volume of liquid has an initial stage formed by the collapse of the volume. Subsequently the front formed develops 134 into the decelerating shock wave type propagation that was considered in the previous Subsections 3.2.5 and 3.2.6. As was discussed in Subsection 3.2.3.1 to solve the two shallowwater wave equations (3.2.2) and (3.2.3) in this case, similarity solutions for velocity and thickness were developed in the form o u(r,t) = R(t)~(~) (3.2.16) and (3.2.23) where ~ is the dimensionless variable r/R(t) as before, n is the frontal position relative to the original radial length R , o Le. n=R(t)/R ' hI is the original characteristic thickness o before spreading, and a is an unknown number. ~(~) and H(~) are the similarity profiles for the mean velocity and the spreading layer thickness. Substitution of these two similarity solutions into the internal shallow water-wave Eqs. (3.2.2) and (3.2.3) gives (<t> - ~) H' + H ( <t>' + n~<t> + a) = 0 (3.2.117) and , (3.2.118) 135 where the decay coefficient ~ is equal to RR/R 2 as before and i=O for surface spreading and i=l for internal spreading. To homogenize the above equation (3.2.118), one must have that (3.2.119) for SOme constant S, so that Eq. (3.2.118) becomes (~ - ~ H' + !I.~ + SH~' = 0 (3.2.120) Equations (3.2.117) and (3.2.120) have solutions (see Appendix D) (3.2.121) 1 H(O (1- ~2) i+1 (3.2.122) satisfying =1 (3.2.54) H(l) = 0 (3.2.58) </>(1) provided that constants a and S are chosen 136 CL = -en + 1) (3.2.123) (i+1)A (3.2.124) 2 The characteristic thickness hI can be derived from a conservation of volume. The total volume of spreading fluid Vet) must be equal to the total volume V at the start of spreading. o The relationship is Vet) V (3.2.125) o where R(O) V I o o h(r,o) k. l.n (3.2.126) and RCt) Vet) = I h(r,t) k. lon o (3.2.127) It thus becomes possible to find the unknown length scale, hI' IRo h(r,o)rn dr V r 0 hI = k. R0 l.n n+1 E;~(E;)dE; 0 0 = R n+1 0 r 1 E;n (1 - E;2) i+l dE; 0 (3.2.128) 137 The values of the original characteristic thicknes.s scale hI are listed in Table 3.2.1 for spreading in a homogeneous environment (i = 0) and in Table 3.2.2 for spreading in a linearly density-stratified environment (i = 1). Thus, the unsteady and non-uniform thickness distribution h (r, t) can be expressed explicitly as h(r,t) hI (n+l) • (3.2.129) = R(O These two similarity profiles for spreading fluid thickness are plotted in Figure 3.2.6 for i = 0 and in Figure 3.2.7 for i = 1. The thickness profile of an initial spreading layer in a homogeneous environment (i= 0) is not an elliptical section as Longuet-Riggins (1972) expected. However, the thickness profile in a linear density- stratified environment (i= 1) is an elliptical section which was also assumed by Mei (1969) .. The frontal contact angle S with the horizontal bed (or the neutral water level) can be found from the following relationship, S = arctan {ah~!, t)} = arctan {-2G~) (R:~) r} (3.2.130) for i= 0 and a ~ = arctan {ah(R,t) } __ arctan {~ } = 2 1T ar (3.2.131) 1 ~ Ro~ ~____ _ _ _ _ ._._ 2 ho n=1 L......-_ _ _ _ _ _ _ _- - L __ I hemi-sphere 80 37T h semi-cylindrical column j.Ro~ ~Ro....J t semi-ellipsoid 80 37T h semi-elliptic column '3 ho '3 ho 3" ho 2 4 4 triangular cone circular cylinder radial 1 h triangular column j-.Ro~ 40 1. h rectangular column . I I ii .9° fT~T~. 1 i ~~o LLJ~o r- i ~ho I 20 ~~~~~~n Initial The original characteristic thickness scale hI for the initial stage of surface density spread in a homogeneous environment (i=O) expressed as a function of the initial shape of liquid volume before spreading. n= 0 plane symmetry h Table 3.2.1 I I 00 UJ f-' lh 2 a lh 2 a a n=l 1T ~h double cone 0 rhombic column cylinder 1T ~h rectangular column Ro~ radial n=O plane I ho _-!--_I---.L 0 h a sphere h cylindrical column a h a ellipsoid h elliptic column The original characteristic thickness scale hl expressed as a function of initial cross-sections for the initial submerged density spread in a linearly densitystratified enviromnent (i = 1) . Initial Table 3.2.2 I-' VJ 1.0 0.4 r 0.6 0.8 )! 0.4 0.6 0.0 Similarity profiles for the mean horizontal velocity and the spreading layer thickness for the initial stage of surface density spreading in a homogeneous environment (i = 0). ( = R(t) 1.0 0.0 Figure 3.2.7 0.0 0.2 4>(!) H( () 0.8 1.0 0.0 K " 0.2 sc::c:: 0.2 0.4 0.6 I Figure 3.2.6 4>( 0 H(() 0.8 I. 0 0.6 ( = R[tl 0.4 0.8 1.0 Similarity profiles for the average velocity and the half-thickness of interflowing submerged density spreading layer due to a finite volume release in a linearly densitystratified environment (i = 1) . 0.2 H(( ) = ( 1- (2)"2 I ~ t-' o 141 for i = 1. This means that the internal wedge angle at the front is always perpendicular to the propagation direction at the neutral level for a stably stratified environment (i = 1) . But more interesting, the frontal contact angle of inertial spreading in the homogeneous environment (i = 0) is a function of: (:1) the relative frontal distance n = R(t) /R ; (ii) the o initial shape of liquid volume; and (iii) the initial aspect ratio (or h1/Ro) and also the case considered n. For a very long reservoir, h1/Ro -+ 0, the initial value of f3 defined by B0 : arctan {- (:~) } (3.2.132) will be asymptotically zero, which coincides with the dry-bed break flow result obtained by Ritter (1892). h R (R~) (R(~») For a long time, n+2 -+ 0, so the frontal contact angle will be asymptotic to a constant value, 0, for a reservoir with finite aspect ratio. The frontal trajectory function is determined by integration of the equation, 00 1 di 2 R(t) = 2" dR 2 ~ h i+l = i+ I giRl o -[(i+l) (n+l)+l] (:) 0 (3.2.133) 142 Equation (3.2.133) can be rewritten as 4 -[ (i+ 1) (n+ 1)+1] d . (dd;)2 • = i+ 1 n dn (3.2.134) where neT) = R(t)/Ro R(t)/R(O) (3.2.135) (3.2.l36) This equation is solved with initial conditions nCO) 1 (3.2.137) dn(O) =0 (3.2.138) dT to give T = (i ; 1) (n + 1) 1/2 rn _____d-'n_ _----,-_ _ ),l [1- n-(i+l)(n+l) ] 1/2 (3.2.139) The normalized frontal velocity for large time is clearly o R(t) 2 _~b.h.i+i 11 l. l. (i+l) (n+l) 1/2 ~;;======;=;:,.=---"---- (3~2.l40) 143 For plane surface spreading in a homogeneous environment n = 0 and i=O so that Eq. (3.2.139) becomes (3.2.141) while for radial surface spreading in a homogeneous environment (i= 0 and n= 1) Eq. (3.2.139) becomes (3.2.142) For density stratified environments and internal spreading (i= 1 and n = 0) the plane spreading result is L = (3.2.143) and while for radial spreading the integral cannot be evaluated explicitly. In each of the above cases the characteristic length scale hI must be evaulated from the given initial shape of the spreading volume. 3.3 Similarity Solutions of Viscous-Buoyancy Density Currents In this section the asymptotic functions describing both surface and submerged density currents in the viscous-buoyancy regime, which were derived by force scale analysis in Section 3.1, will be evaluated by a self-similar analysis. Exact solutions 144 analogous to point source solutions for nonlinear diffusion will be given for the first four cases. These correspond to a plane (n = 0) or a radial (n = 1) finite volume release (m = 0) in a uniform (i = 0) or a linearly density-stratified (i = 1) environment. Series expansion solutions will be given for the continuous discharge cases (m = 1) . Self-similar distributions of thickness and velocity and the unknown numerical coefficients in the power law descriptions for the viscous spreading front will be found. A comparison of the results obtained with available experimental data will be presented in Chapter 6. 3.3.1 Nonlinear Long Wave Equations The relevant long wave approximation equation for the viscous-buoyancy region of surface and submerged density spread is the "( i + 4)" power nonlinear diffusion equation, i 4 1 d (n dh + ) at = 3(i+ 4)v rn r dh g~i ar ar (3.3.1) where h = h(r, t) is the thickness of the spreading layer (for i = 0), and is the half-thickness of an interflowing layer in a linearly density-stratified environment (for i = 1), n is a parameter that denotes the radial case for n = 1 and the plane case for n = 0, v is the kinematic viscosity of spreading fluid, g~. 1. denotes the reduced gravitational constant, g (p - p ) /p , for s a s i = 0, and the square of Brunt-VaisaHi frequency of ambient 145 dp (z) density stratification, N2 = gE =.::.& a ,for i = 1, in p dz s which p a and p s are mass densities of the ambient and the spreading fluid. The coordinate system of this spreading flow is that shown in Figure 3.2.1. (See Appendix E for the derivation of viscous long wave approximation equation.) 3.3.2 Derivation of Self-Similar Formula by the Method of Separation of Variables Assume that the unknown spreading layer thickness h(r,t) in the nonlinear long wave equation has solutions of the following form: h(r,t) = T(t) H(~) (3.3.2) where the dimensionless variable ~ is defined as r/R(t), R(t) is the frontal position at time t, T(t) is the unknown scale function for the spreading layer thickness and H(~) is the similarity function to be determined. Substituti~g Eq. (3.3.2) into Eq. (3.3.1) gives the ordinary differential equation 3dv gll.T i +3 1 o 3RRv gll.Ti+3 1 (.+4) ~H' =..!.- ~ ~n dH ~n d~ 1 d~ (3.3.3) 146 in which 11011 denotes differentiation of a function with respect to time t and II-I II differentiation with respect to the dimensionless variab1e~. In order to have a self-similar solution of Eq. (.3.~.2), one mus.t have the two terms. RRv/gll.r i +3 1 o i+4 and R2vT/g~.T ,in the left-hand side of Eq. (3.3.3), be l. dimensionless and constant. For simplicity, choose the first term RRv/g~.Ti+3 equal to unity, so that the scale function l. T(t) is 1 T(t) : : ; (Riv)i+3 (3.3.4) g~. l. Then the second term will automatically be a dimensionless number, (i! ) where the decay coefficient A RT° = 0 ::::; RT 1+ A i+ 3 (3.3.5) 00 O is defined as A::::; RRi'R 2 and can be determined from the frontal trajectory functiop R(t). Therefore, Eq. (3.3.3) can be rewritten as 3(l+A) H - 3~H'::::; (i + 3) ( ~ n i 4 dH + ) d~ (3.3.6) 147 Thus, the similarity solution for surface and submerged spreading layer thickness h(r,t) has the form 1 ~v gU 0 h(r,t) = ( ) '+3 H(s) 1 (3.3.7) i The same self-similar formula Eq. (3.3.7) can also be derived straightforwardly by substituting the Taylor's (1950) similarity assumption Eq. (3.2.25) into the nonlinear wave Eq. (3.3.1) (see Appendix E for details of derivation). 3.3.3 Derivation of General Self-Similar Solutions If interfacial mixing and diffusion, and surface evaporation were negligible, then the total spreading volume v (t) at time t supplied V.(t). J must be equal to the volume that has been Since the total spreading volume at time t is Ret) Vet) = f h(r,t)k. 1n o rndr (3.3.8) where k.1n is a shape function of i and n and is defined in Eqs. (3.2.40) and (3.2.41). The total volume supplied is assumed to be V.(t) J so that one obtains = Q tm (3.3.9) 148 R(n+l) (i+3)+1 R= a g~.Qi+3 1 t(i+3)m (3.3.10) v 1 where a , and the integration constant I. is 1mn (k. I. )i+3 1n 1mn defined as = II ~n H(~)d~ I. 1mn (3.3.11) o which is a function of i, m, and n (since the similarity distribution H(~) is a function of i, m, and n). Carrying out integration of Eq. (3.3.10) from R= R to R= R(t) and t = 0 o to t = t gives the explicit form of frontal traj ectory functions: 1 (n+1)(i+3)+2 ag~iQ(i+3) m(i+3)+1 } m(i+3)+1 ' --=--V-- Ro ~n+l) (i+3)+2 (n+1) (i+3)+2 (3.3.12) When the spreading distance R(t) is much larger than the initial length R , i.e. R(t»>R o 0 or equivalently when the time t» 1 VRo (n+1) (i+3)+2 }m(i+3)+1 { g~.Qi+3 1 spreading becomes ' the asymptotic form of viscous-buoyancy 149 1 R(t) ~_.~_i_+_3 ) (n+1) (i+3)+2 b iron ( _g_D. m(i+3)+1 t(n+1) (i+3)+2 (3.3.13) in which the coefficient b b""'n = ~ imn is i 3 {(n+1~ (i+3)+2} (n+1) (l + )+2 I{k. (n+1~ ti!3)+2 m(~+3)+1 / . ~n ~n (3.3.14) 1. } This function has exactly the same form as that derived by the force scale analysis in Section 3.1. The unknown coefficient mainly depends on the spreading layer thickness distribution which can be derived by integrating Eq. (3.3.6) with suitable boundary conditions. From Eq. (3.3.13) the decay coefficient now l.S 00 RR 11.= -O = R2 (i+3) (m-n-1) - 1 m(i+3) + 1 (3.3.15) so that the ordinary differential equation for a viscous layer spreading thickness (Eq. (3.3.6» 3(2m-n-1) H - 3~H' m(i+3) + 1 .,. = becomes (3.3.16) 150 3.3.4 Solutions for Spreading from a Finite Volume Release In this cas-e (m;:; 0) the ordinary differential Eq. (3.3.16) for surface and submerged spreading layer thickness is - 3(n+ l)H - 3~H' = i 4 1 d (~n dH + ) (i + 4)Sn d~ d~ (3.3.17) or i+4 )' = 0 3 (i + 4) (~n+1 H)' + ( ~n d~~ (3.3.18) Integrating twice from ~ = 1 to ~ = ~ and using the boundary condition H(l) o (3.3.19) and assuming o (3.3.20) gives the similarity distribution for spreading layer thickness as (3.3.21) 151 which is consistent with Eq. (3.3.20). Thus the self-similar profile for the surface and submerged spreading layer thickness becomes 1 h(r,t) = (Riv )i+3 {3(i+3) g~. 2 1. The shape of this thickness 1 (1- ~2) i+3 (3.3.22) } profile is shown in Figure 3.3.1 for the case of spreading in a homogeneous environment (i = 0) and in Figure 3.3.2 for the case of spreading in a linearly density-stratified environment (i = 1) • The integration constant lion is (3.3.23) The special case n = 1, gives ~ = {3(i +3) } 1.+3 1.1.01 2 i ~(1_~2)i+3 l 1 1 = {3(i+3) }i+3 2 3 2(i+4) (3.3.24) and for n= 0, 152 2.0 \.0 -II w :::r: 0.0 0.0 . 0.2 0.4 0.6 0.8 1.0 ! = R(t) Figure 3.3.1 Self-similar thickness distribution of plane and radial viscous surface spread layer due to a finite volume release (i= 0, m= 0, n= and 1). ° 153 2.0 r----,-----.,...---..-----.,.-----, --.. t0:: ~ ~ ~ 1&1 01 -.-' 1.0 -:r II v...P 0.0 0.0 Figure 3.3.2 0.2 0.4 0.6 0.8 1.0 Self-similar thickness distribution of plane and radial viscous submerged density spread layer due to a finite volume release (i= 1, m= 0, n= 0 and 1). 154 1 i+5 cosi+3 ada = { 3(~+3) } i+3 r(i)r (m) 3i+11 ) 2r ( 2(i+3) (3.3.25) where r(x) is the Gamma function of x. The approximate value of the integration constant I.~on is listed in Table 3.3.1 for each case of i and n. The approximate value of the coefficient b.~on in viscous-buoyancy spread Eq. (3.3.13) is then calculated and the viscous-buoyant trajectory function for each case i and n is listed in Table 3.3.2. The frontal contact angle S defined by e = arctan {ah~~' t)} = ~ (3.3.26) for all time during viscous~buoyant spreading. As derived in Appendix E, the horizontal velocity of a fluid particle will be given by gt..h i +2 u(r,z,t) = - ~ 2v ah ar (3.3.27) or (3.3.28) 155 Table 3.3.1 The value of the integration constant lion for surface and submerged spreading from a finite volume release (m = 0) . ~ In Homogeneous Environment i= 0 In Linearly Density-Stratified Environment i = 1 Plane Spreading Radial Spreading n=O n=l cos 5 / 38d8 \f{2 18 11" VI L2 = 1. 389 11" ¥L2 = ° = 0.619 cos 3/ 2 ed8 ¥o i 1.386 = 0.626 156 Table 3.3.2 The frontal trajectory functions for the viscousbuoyant surface or submerged spreading due to a finite volume release (m = 0). ~ In Homogeneous Environment Plane Spreading Radial Spreading n=O n=l ,)'/5 tl/5 0.779 gtl: ( 24) I/e tI/G 0.552 N: 1.133 (gtl: i = 0 In Linearly Density-Stratified Environment i= 1 9.689 N : ( 'f' ( 2 tI/8 .Y/IO tl/lO 157 as shown in Figure 3.3.3. The peak fluid particle velocity will therefore be 50% faster than the frontal velocity. The cross-sectional average fluid particle velocity is, however, continuous with the frontal velocity and still a linear function of location, since u(r,t) = o R(t) 3.3.5 1 her, t) h (r,t) I u(r,z,t) dz = ~ (3.3.29) R(t) 0 Plane Viscous Spreading Layer from a Continuous Discharge Equation (3.3.16) in this case (m = 1 and n == 0) is (Hi+4)II = 3H - 3(i+4)~H' (3.3.30) According to the mathematical study by Gilding and Peletier (1976), there is no exact weak solution to Eq. (3.3.30). the solution was shown to exist and be unique. However, Moreover, it is bounded between a lower bound solution ~, (1- ~2) } 1 i+3 (3.3.31) and an upper bound solution H , u Hu (~) = {[3(i+3)+~ (3.3.32) z Figure 3.3.3 h(r,ll 0.0 !=O.l 0.2 0.3 0.5 R( f ) o u( r,z,f) 1.0 r != R(f) 1.5 Self-similar distribution of plane and radial viscous surface (i= 0) or submerged (i = 1) spreading fluid velocity due to a finite volume release (n = 0,1 and m = 0). 0.0 0.5 1.0 VI CP ~ 159 An approximate solution HA between two bounds, 1 HA(~) = {3(i+3)(1-~)} i+3 (3.3.33) was given by Grundy (1979) with the method of phase plane analysis. The approximate solution in Eq. (3.3.33) is actually a solution near the front as ~ -+ 1. It can also be derived easily by integrating the simplified Eq. (3.3.30) twice near the front, (Hi+4)" ~ -3(i+ 4)H' (3.3.34) with two boundary conditions, H(l) = 0 and Hi +3 (1)H' (1) = O. The lower bound solution is the exact weak solution given in Eq. (3.3.21), which was derived in the Subsection 3.3.4. Equation (3.3.30), can be rewritten in new form, 1 __1_ i+4 i 4 3f + + 3 (i + 4) (1- x) df dx i 4 where f (x) = H + (~) and x = 1 - ~. (3.3.35) Assume that the solution of Eq. (3.3.35) has the following series expansion form near the front, i+4 f(x) = [3(i+ 3)x]i+3 {l + f: j=l j a.x } J (3.3.36) 160 Substituting the series solution in Eq. (3.3.35) one can find the coefficients a . j Two values, a l and a , are found, 2 -1 al (3.3.37) 2(i+4)(i+3) and 2(i+4)3 - 2(i+4)(i+3) - (i+3) [2(i+4)2 + 4 (3i+10) (i+3) (i+4) + (i+3)] [2(i+4)2(i+3)] (3.3.38) The solution of Eq. (3.3.30) can be expressed up to the third order as 1 R(O [3(i+3)(1- O]i+3 {l (1- 0 2(i+ 4) (i+ 3) + 1 2(i+4)3 - 2(i+4)(i+3) - (i+3) (1- 0 2 [2(i+4)2 + 4 (3i+10) (i+3) (i+4) + (i+3)] [2(i+4)2(i+3)] l i+4 + ... J (3.3.39) Two self-similar profiles of plane spreading layer thickness are shown in Figure 3.3.4 for the homogeneous environment (i = 0) case and in Figure 3.3.5 for the linearly density-stratified environment (i = 1) case. The integration values lilO are given in Table 3.3.3 and the coefficients b ilO in the asymptotic form of frontal trajectory functions are given in Table 3.3.4. 161 2.0 -I.., -1.0 -II 4A.P J: 0. 0 L - -_ _-i...-_ _--'--_ _-..l...._ _---I._ _-...J 0.0 0.2 0.4 ~= Figure 3.3.4 0.6 o.s 1.0 r R( t ) Self-similar thickness distribution of plane viscous surface spread layer due to a constant rate of discharge into a homogeneous environment (i = 0, m = 1, n = 0). Solid line denotes the first approximation, and open circles denote the third approximation. 162 2.0 r - - - - - - - , - - - - - - - r - - - . . . . , . . - - - - , - - - - - - - , II -:c v..,p 0.0 GO 0.2 0..4 0..6 0.8 1.0. ! = Rtf) Figure 3.3.5 Self-similar half-thickness distribution of plane viscous submerged density spread layer due to a constant rate of discharge into a linearly densitystratified environment (i = 1, m = 1, n = 0). Solid line denotes the first approximation and open circles denote the third approximation. 163 Table 3.3.3 Integration values lila for plane viscousbuoyancy spreading currents due to a continuous steady discharge plume. i ~ , derived from Upper bound of solution Uniform Environment Linearly DensityStratified Environment i=O i =1 1.6078 1.5140 1.3890 1.3679 1.5600 1.4889 1. 5507 1.4848 1.5510 1.4849 u (0 H Lower bound of solution ~(O 1st term approximate solution of H( 1;) 2nd term approximate solution 0 f H(O 3rd term approximate solution of H(t;;) 164 Table 3.3.4 Coefficients b i10 for the plane viscousbuoyancy spreading currents to a continuous steady discharge plume. i b ilO derived from Upper bound of solution Uniform Environment Linearly DensityStratified Environment i=O i=l 0.7864 0.4925 0.8585 0.5270 0.8008 0.4980 0.8036 0.4990 0.8036 0.4989 R (0 u Lower bound of solution ~(O 1st term approximate solution of R(O 2nd term approximate solution of R(O 3rd term approximate solution of R(O 165 The series solutions are convergent very fast over all the domain between 0 and 1. Therefore, the solutions derived from the third term approximation are good enough for comparison. The frontal trajectory functions can be expressed as R(t) t 4/5 (3.3.40) for the spreading currents in the homogeneous environment (i = 0) and as R(t) 24)1/6 0.4989 ( N vQ 5/6 t (3.3.41) for the spreading currents in the linearly density-stratified environment (i = 1) . 3.3.6 Radial Viscous Spreading Layer from a Continuous Discharge Equation (3.3.16) in this case (m = 1 and n = 1) is (3.3.42) The simplified Eq. (3.3.42) near the front can be expressed in the same form given in Eq. (3.3.34). Thus, the approximate solution near the front is also of the same form in Eq. (3.3.33). Rewrite Eq. (3.3.42) in the form, 166 1 d 2f df df i + 4 (l-x) - - = 3(i+4)(1-x)2dx dx i+4 where f (x) = H . CO and x = 1 - t;. (3.3.43) Assume that the solution of Eq. (3.3.43) can also be expressed in the series expansion form about the solution near the front, i.e. Eq. (3.3.36). Substituting the power series expansion Eq. (3.3.36) in Eq. (3.3.43), one can find the coefficients a . j Two values, a 1 and a , are found, 2 (i + 2) a 1 = 2(i + 3) (3.3.44) and (i+2) {_~(i::;.;+-=2~)..!.,;( +:....::5~)--....:.+--=2:,..:,(-...:.4=-i+.:...:.1::..::3:..<.)~(=-i+.:...:.3~)....,:,(-=i+_4-,-,),---} 2(i+3) 2(i+4)2 + (i+3) + 4 (3i+10) (i+3) (i+4) l.::;,.' 2(i+4)2 + (3.3.45) {2(i+4)2 + (i+3) +4(3i+10)(i+3)(i+4)} Thus, the self-similar function H(t;) can be expressed up to the third order as 1 H(t;) 1 {3 (i + 3)(1 - t;) } i+3 {1 + 2([:11) (1 - 1';) + a 2 (1 - 1';) 2 + .•. } i+4 (3.3.46) 167 Two self-similar profiles of radial spreading layer thickness are shown in Figure 3.3.6 for the homogeneous environment (i = 0) case and in Figure 3.3.7 for the linearly density-stratified environment (i = 1) case. The integration values Iill are given in Table 3.3.5 and the coefficients bill in the asymptotic form of frontal trajectory functions are given in Table 3.3.6. The series solutions are also convergent rather fast all over the domain between 0 and 1. Therefore, the frontal trajectory function can be expressed as R(t) t 1/2 (3.3.47) for the radial spreading currents in the homogeneous environment (i = 0) and as 2 4) 1/10 R(t) = 0.45 ( ~ 'V 1/2 t (3.3.48) for the radial spreading currents in the linearly densitystratified environment (i = 1) . 3.4 Further Mixing of Surface Spreading Flow 3.4.0 Introductory Note After a discharged effluent reaches a free surface or the bottom, the momentum flux causes an increase in pressure which together with any buoyancy flux causes the diluted effluent to 168 o o -- .. oa::a::---<J I. 0 It) ..:. ~ -- ;::,C'I II - W I 0.0 0.0 Figure 3.3.6 0.2 0.4 0.6 0.8 1.0 Self-similar thickness distribution of radial viscous surface spread layer due to a constant rate of discharge into a homogeneous environment (i = 0, m = 1, n = 1). Solid line denotes the third term approximation, open circles, the first term approximation, and triangles, the second term approximation. 169 2.0 --I" ..: cr: N _ ocr: z .s:: 1.0 ~ -w II I 0.0 0.0 0.2 0.4 ~ = Figure 3.3.7 0.6 o.s 1.0 r R( t ) Self-similar half-thickness distribution of radial viscous submerged density spread layer due to a constant rate of discharge into a linearly densitystratified environment (i=l, m=l, n=l). Solid line denotes the third term approximation, open circles, the first term approximation, and triangles, the second term approximation. 170 Table 3.3.5 Integration values Ii1l for the radial viscous-buoyancy spreaaing currents due to a continuous steady discharge plume. i lill derived from 1st term approximation of Uniform Environment Linearly DensityStratified Environment i=O i=l 0.6686 0.6618 0.6895 0.6795 0.6977 0.6858 H(~) 2nd term approximation of H(t;) 3rd term approximation of H(~) 171 Table 3.3.6 Coefficients bill for the radial viscousbuoyancy spreading currents due to a continuous steady discharge plume. i bill derived from 1st term approximation of Uniform Environment Linearly DensityStratified Environment i=O i=l 0.6366 0.4593 0.6293 0.4545 0.6265 0.4528 H(~) 2nd term approximation of H(O 3rd term approximation of H(O 172 flow horizontally. The pressure gradient induced by the momentum flux makes the initial regime of horizontal flow behave as a surface jet and therefore causes a considerable amount of mixing with the ambient fluid beneath it, as shown in Figure 3.4.l(a). Further away from the impact point the buoyancy effects due to any small density difference become more important. The buoyancy gradually reduces the vertical mixing to a negligible amount and eventually produces a two-layer stratified flow. The behavior is therefore quite different from a vertical submerged buoyant jet. The mixing due to both plane and radial buoyant jets, as shown in Figure 3.4.l(b), will be discussed in this section. 3.4.1 Turbulent Surface Jet Mixing Suppose there is a radial or plane momentum source located at the origin of an r-z coordinate system in the free surface, as shown in Figure 3.4.2. The steady jet flow produced is assumed turbulent so that the molecular transport quantities are neglected. With the usual boundary layer assumptions the continuity, horizontal momentum, and mass conservation equations for mean jet flow can be written as n ~ d(r u) r n dr + dW = 0 dZ (3.4.1) 173 Figure 3.4.1(a) Flow regimes of an outfall discharge (soon after startup): I: Submerged buoyant jet flow, II: Surface transition, III: Surface buoyant jet flow. Figure 3.4.1(b) A surface buoyant jet flow model with finite dimension of discharge (soon after startup). 174 r W( r) ( r, U) M( r) ( z,w) Figure 3.4.2 Point source surface jet nomenclature, M(r) W(r) = r r pk k on on n r u 2 dz, M(r + dr) n r cwdz, W(r+dr) = M(r) + elMer) W(r) + dW(r). 175 (3.4.2) u ~ + w ~ = Cl (W) Clr Clz Clz (3.4.3) where u and ware the time-averaged velocities in the horizontal and the vertical direction, respectively; c is the relative mean temperature or mean density difference; L zr is the turbulent shear stress and W is the turbulent flux of c at horizontal position r and depth z, n=O denotes the plane surface jet flow, and n = 1 the radial surface jet flow. The ambient fluid is assumed infinitely deep. Assume that the time-averaged horizontal velocity u(r,z) has a similarity solution of the form u(r,z) (3.4.4) u (r)f(O m where the mean velocity at free surface u(r,O) = u (r) and the m dimensionless similarity variable ~ = az/r. The exact form of the maximum free surface velocity um(r) can be derived from the condition of horizontal momentum conservation, namely that, l~ o n+1u 2 m p.k r J on a r' f2(~)d~ =p.M J Jo ,(3.4.5) 176 where the initial kinematic momentum jet flux M= k on r. n h. u. 2 J J J and kOO = 1 for n = 0 and kOl = 21T for n = 1 and the subscript j implies the initial jet values. J = -;----:-::7""'7-::(n+l)/2 u ( r) m 00 where J = {aM/kon 1 Thus, (3.4.6) r f2(~)d~ } 1/2 or simply in terms of the o discharge jet dimensions and jet velocity u : j u (r) a m ---= u (3.4.7) j A Stokes' stream function can be derived from the Eq. (3.4.1), u(r,z) = J:.. ~ n dZ (3.4.8) _ J:..~ (3.4.9) r w(r,z) r n dr to give n+l u (r ) r m 1jJ(r,z) = - - - - F(~) a (3.4.10) 177 F(~) = J~~ f(~)d~. where Therefore the vertical mean velocity can be deduced as u (r) w(r, z) m a {~F' (0 - n+l (3.4.11) 2 and the vertical gradient of the mean horizontal velocity as Clu(r,z) Clz a u (r) _....:.m=--_ F" ( ~) (3.4.12) r where the prime denotes differentiation with respect to ~. Two unknown values of turbulent quantities in Eqs. (3.4.2) and (3.4.3), L zr /p and W~ are assumed as L zr P - u'w' = £ u Clu Clz W = c'w' (3.4.13) Clc = - £ C Clz by introducing two eddy viscosity terms, denote fluctuations from the mean values. £ (3.4.14) u and £ c , value primes Following Prandtl- Tollmien's two assumptions are made for the two eddy viscosity terms: 178 e: e: where R, u and R, c (3.4.15) u R, c u R, c I~I oZ (3.4.16) are Prandtl's mixing length for the momentum and the tracer transfer, R, R, u c = cu r (3.4.17) c r (3.4.18) c in which c u and c c are two unknown coefficients. In short, Prandtl-Tollmien assumptions for two eddy viscosity terms are e: u (3.4.19) (3.4.20) where two unknown numbers K and K are related to K = c 2 u c u u and Kc = c u c. c An alternate approach is, following Prandtl-GBrtler (1942), to assume that 179 E E u = X'u b u (r) u m(r) (3.4.21) X b (r) u (r) (3.4.22) c C c m where b (r) and b (r) are two mixing thickness scales that are u c defined as b (r) u k b (r) k c u c r (3.4.23) r (3.4.24) implying that the eddy viscosity terms are expressed as E u = Ku r um(r) (3.4.25) (3.4.26) where two unknown numbers K and K are related to K = X k u c u u u and K = X k in Prandtl-Gortler assumption. c c c Substituting the Prandtl-Tollmien assumptions, Eqs. (3.4.13) and (3.4.19), into the integral form of horizontal momentum equation, 180 1 uw + r -d n dr [Z T n r u 2 dz - ~ = 0 (3.4.27) p given the ordinary differential equation, (3.4.28) where a = Vn+l 2Ku from derivation of Eq. (3.4.28). With boundary conditions at the free surface of u(r,O)=u (r), w(r,O)=O and m at dep th u (r ,co) = W (r ,co) = 0, 1. e. F' (0) = 1 and F (0) = F (co) = F' (co) = 0, the similarity profile for mean velocity is obtained by integrating Eq. (3.4.28) as shown in Table 3.4.1. The mean tracer distribution of the turbulent jet flow is found from the self-similar distribution assumed to be of the form c (r, z) c (r)e(!;) m (3.4.29) where the longitudinal tracer distribution at the free surface c(r,O) = cm(r). The total amount of tracer loss through the free surface is assumed very small so that the total integral amount of tracer distribution should be conserved at each radial location r, i.e. 181 Table 3.A.l Prandtl-Tollmien similarity solutions of turbulent surface jet flows. aw(r,z) = um (r) ~F' (0 _ n+l F(O 2 ~ u(r,z) = F' (q u (r) m F(~) Plane Surface Jet n=O Radial Surface Jet n=l 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 1.000 0.979 0.940 0.897 0.842 0.782 0.721 0.660 0.604 0.538 0.474 0.411 0.357 0.300 0.249 0.200 0.165 0.125 0.095 0.067 0.046 0.030 0.020 0.009 0.000 0.000 0.100 0.200 0.287 0.370 0.450 0.520 0.590 0.650 0.710 0.760 0.810 0.860 0.880 0.910 0.930 0.950 0.960 0.965 0.970 0.980 0.985 0.990 0.995 1.000 0.000 0.048 0.088 0.126 0.152 0.166 0.173 0.167 0.158 0.129 0.094 0.047 -0.002 -0.050 -0.106 -0.165 -0.211 -0.268 -0.312 -0.358 -0.398 -0.430 -0.451 -0.477 -0.500 0.000 -0.002 -0.012 -0.018 -0.033 -0.059 -0.087 -0.128 -0.167 -0.226 -0.286 -0.358 -0.432 -0.490 -0.561 -0.630 -0.686 -0.748 -0.794 -0.843 -0.888 -0.922 -0.946 -0.974 -1.000 182 k on r n+1 c (r)u (r) m m a [0 f(~)e(~)d~ = c.Q J (3.4.30) where c j is the initial jet tracer concentration and Q is the jet discharge rate, Q = k n on r. h.u .. J J J Therefore, the maximum tracer distribution is decaying according to c (r) _m__ = _ _ _ _ _ __ • c. .~.J f;ft J r J n+1 (3.4.31) Substituting the Prandt1-To11mien assumption Eq. (3.4.14) with Eq. (3.4.20) in the integral form of tracer equation, 1 cw+ -Cl- r n Clr [Z rn cu dz - W = 0 (3.4.32) gives K ~ F" (~) e ' (0 - F (0 e (0 K =0 u By the aid of Eq. (3.4.28), Eq. (3.4.33) can be written as (3.4.33) 183 F (F;) F"(O F"(O F'(O (3.4.34) Integrating Eq. (3.4.34) providing with boundary conditions e(o) = 1 and F' (0) = 1 from c (r) = c(r,O) and u (r) = u(r,O) m m gives the self-similar distribution 8(F;) for mean tracer as a power function of mean velocity distribution, K /K e(F;) =[F'(~n c K /K u = £ c u(F;) (3.4.35) Thus, the mean tracer distribution is expressed as c(r,z) = c. J a I: (Od~ M K /K f c u(O £2 I: f(08(Od~ J r n+l 2 J . (3.4.36) Similarly, using the Prandtl-Gortler assumptions Eqs. (3.4.13) and (3.4.14) with Eqs. (3.4.25) and (3.4.26) in the integral form of horizontal momentum Eq. (3.4.27), gives 2F(OF' (F;) + F"(O o (3.4.37) 184 with ~~ a = ~ ~. The tracer equation (3.4.32) becomes K U e'(~) + 2 K F(~)e(~) c (3.4.38) ° Integrating Eq. (3.4.37) twice with boundary condition F' (0) = 1 and F(O) = 0, gives F(O = tanh t;, (3.4.39) The similarity distribution of a mean horizontal velocity in dimensionless form is therefore u(r,z) u (r) m F'(O (3.4.40) From Eqs. (3.4.37) and (3.4.38) a' (P = _ 2 Ku F a(t;,) K c Ku F" (0 =KF'(t;,) (3.4.41) c Integration of Eq. (3.4.41) with boundary conditions a(O)=F'(O)=l gives K /K a(t;,) = F' u c(t;,) = K /K [sech 2 (t;,)] u c (3.4.42) 185 Thus, the mean tracer distribution is expressed as c(r,z) c. J ~a r f 2 (1;)d, f: f(oe(Od~ __ __ _I r . n h . 2K /K '~~l ~J~ sech u c (~) n+l r .(3.4.43) 2 The two mean tracer distributions, Eqs. (3.4.35) and (3.4.42), derived from the Prandtl-Tollmien and Prandtl-Gortler assumptions, therefore do not necessarily coincide with the mean horizontal velocity distribution f = F' (0 Ku and Kc are equal. except when two mixed coefficients As shown in Figure 3.4.3 in this case the profiles of mean horizontal velocity and mean tracer distribution derived from the two assumptions give values close to each other. 3.5 Dimensional Analysis The advantage of using kinematic fluxes instead of the actual physical dimensions to describe the general time-mean properties of a submerged buoyant jet has been demonstrated by List and Imberger (1973). The same dimensional methods may be used to solve the problem of a horizontal turbulent surface buoyant jet. 3.5.1 Radial Surface Buoyant Jet As shown in Figure 3.5.1 a radial surface buoyant jet discharges a fluid of uniform density p. and temperature J Tj at the free surface of a large body of water of density Pa 186 0.0 0.0 0.5 1.0 1.0 2.0 3.0 Figure 3.4.3 Two self-similar profiles of mean horizontal velocity, F'(O = u(r,z)/um(r), and mean tracer when Ku=Kc, 8(0 = c(r,z)/cm(r). Dash line indicates the solution derived from the Prandtl-Tollmien assumption and solid line from the Prandtl-Gortler assumption. 187 r u( r. z ) B( r. z) H z Figure 3.5.1 Sketch of a radial surface buoyant jet. u 188 and temperature T. a The jet originates from a circular slot of radius r. and thickness h .• J ambient fluid For the case of stationary J and when surface tension is neglected, there are (8 + i) independent variables to describe this buoyant jet flow so that any dependent variable ~ is given by (3.5.1) where H is the depth of the ambient fluid, x.1 are the space or the time independent variables which can be r,z,t, or rand z, or only t, v is the kinematic viscosity of the discharging fluid. It is assumed that the density difference between the discharged fluid and the ambient fluid is sufficiently small in comparison to some reference density, say Pj , that the density difference is only important in the body force term of the jet flow. Now, four independent variables r. , h. , u ' and gLl = g (p - p . ) / p . j a J J J J are transformed into three kinematic flux variables, Q, M, B, and one physical dimension of the jet, either r . or h., so that J ~(X. ,Q,M,B,r. J 1 J or h. ,H,V) J (3.5.2) in which the volume flux Q = 21T r. h. u., the momentum flux M= Q u. The dinensions corresponding J J J J and the buoyancy flux B = gLlQ. to each flux are respectively as L3/T, L4/T2 and L4/T3. There are three length scales associated with any combination of two 189 flux variables, as given in Table 3.5.1. has its own meaning. Each length scale For example, £MB is the length scale located where the momentum flux produced by the buoyancy will begin to dominate the initial discharge momentum flux M. Thus, two regimes of flow, which are dominated respectively by the momentum flux M and by the buoyancy flux B, are roughly separated at a distance from the origin of order £MB. Other time scales and velocity scales can also be derived from the similar combinations of any two fluxes, as given in Table 3.5.1. The densimetric Froude number Fr (or the inverse square root of Richardson number Ri) of the buoyant jet flow is defined as the ratio of two length scales, £MB and £QM' i.e. M5 / 4 QB 1/ 2 u. (3.5.3) = _;:::::::::J==== J gt:J2Tfr.J h.J 1 Similarly, it is found that the ratio of any other two length scales, time scales, or velocity scales also forms a power of Froude number Fr. To characterize the turbulent flow two Reynolds numbers are defined. The first is the Reynolds number of the jet flow Re., J Re. J u. hTfr .h. J J J 'V (3.5.4) 190 Table 3.5.1 ~ Length~ time, and velocity scales of a radial surface buoyant jet. Combinations of Item ~ Length scale Q and M M and B Q and B £QM = --L M1 /2 M3/4 £ =-MB Bl/2 £QM =~ Bl/S (21Tr.h.)1/4 u. = (2u.h.)1/2 = J J Time scale = (21Tr.h.)1/2 J J u. (21Tr.h.u.)1/S u. =J (gb)3/S Bl/2 M Q = - ~ J J J = gb J QM Q4/S -tQB =B3/S M ~ =B" M3/2 u (21Tr.h.u.)2/S J J J = (gb)l/S J (gb)1/2 =~ tQM J J =-- u M1 / 4 B2/S QB =-- Ql/S Velocity scale = u. J = (gb)1/2 (21Tr.h.)1/4 J J = (gb)2/S (21Tr.h.u.)1/S J J J 191 which indicates the turbulence level in the jet flow regime and the second is the Reynolds number of the plume flow Re , p Re = uQB tQB = Q2/5 Bl/5 p v \) = (3.5.5) which indicates the turbulence level in the plume region. The ratio of these two Reynolds numbers is also a function of the Froude number, (3.5.6) This implies that Froude number and one or the other Reynolds number is necessary to model the whole regime of buoyant jet flow. Different levels of turbulence in the plume flow downstream may result depending on the jet Reynolds number, which is generally neglected in jet flows since it is very large. From Buckingham 1T-theorem (7 + i) - 2 = (5 + i) dimensionless terms can be formed to describe the flow problem given in Eq. (3.5.2). The first two 1T-terms are the densimetric Froude number Fr and the jet or the plume Reynolds number. There is no difference in choosing r. or h. to be normalized by the J J length scale t QM , since the resultant dimensionless terms are reciprocal to each other, i.e. 192 = ~Jr; ,,~ (3.5.7) and (3.5.8) and describe the aspect ratio r./h. of the jet. J J One more TI-term The other (1 + i) is the relative depth of ambient fluid R/h.. J terms can be formulated by suitable normalization with the length, time, or velocity scales an appropriate choice depending on the problem being considered. Thus, Eq. (3.5.2) can be rewritten in dimensionless form, ~* = ~*(x.*,Fr,Re,r./h.,R/h.) ~ J J J (3.5.9) in which x.* and ~* denote the dimensionless independent and ~ dependent variables, the Reynolds number Re can be Re. or Re • J p Consider the initiation flow for a radial buoyant surface jet. The independent variable x. in this case is time t and the -~ dependent variable is R(t), the growth history of the radial tip. The dimensionless equation becomes R*(t*) ~(t*,Fr,Re,r./h.,R/h.) J J J (3.5.10) 193 where R* and t* are suitable normalized values of R(t) and t. When the dimensionless length R(t)/~MB or the dimensionless time t/tMB is much less than unity, there is a region of starting jet flow, i.e. R(t) = M1 / 4 t1/2 'I'~(Fr , Re ~ r./h J j~ Hth.) J (3.5.11) Similarly, there is a region of inertial plume flow, R(t) B1 / 4 t 3 / 4 ~(Fr,Re,r./h.,H/h.) J when tMB < t < tBv or ~ < R(t) < RBv • J (3.5.12) J Subsequently there is a region of viscous plume flow ( ) _- (~) Rt 1/8 V when tBv < t. t 1/2 ~ ( Fr,Re,r./h.,H / h. ) J J J (3.5.13) All the coefficients for starting jets, inertial plumes, and viscous plume flows are functions of the densimetric Froude number, a Reynolds number, the aspect ratio relative depth of ambient fluid. and the The limiting values for two starting plume coefficients were given in Sections 3.2 and 3.3 respectively as c 011 and bOll' The time scale tBV and the length scale RBv are proportional to the values t1 and Rl given in Table 3.1. 2. 194 3.5.2 Surface and Submerged Horizontal Buoyant Jets The derivations of the dimensionless parameter in Subsection 3.5.1 can be extended to the case of a plane surface buoyant jet and to plane or radial submerged buoyant jets. The latter two cases involve the discharging of a fluid into a linearly density-stratified environment at a neutral density level. In these cases the general definitions for the volume flux Q, the momentum flux M, and the buoyancy flux B are given in Table 3.5.2. The general form of Eq. (3.5.2) can be written as <t>(x. ,Q,H, B, v,H,nr.) ]. J (3.5.14) In the plane jet case there is one less independent variable r., J since the coordinate system can be shifted to eliminate the jet length r .. J Just as found in the previous section, there are three possible starting flow regimes for the initiation flow, viz. a starting jet flow, 1 2 n R(t) ~ M +3 t n+3 a starting inertial plume flow, (3.5.15) 195 Table 3.5.2 Definitions of volume, momentum, and buoyancy fluxes for a horizontal surface or submerged buoyant plane or radial jet. ~ Definition Dimensions 2i u.h. (2'ITr.) n Ln+ 2 T 2i u~h. (2'ITr . ) n -- Fluxes Volume flux Q J J Momentum flux J J M Buoyancy flux B J -1(gll. ) ~ i+l Ln+ 3 J T2 i n 2 u. h. (2'ITr . ) J J J (n+l) (i+l)+2 i+l L i+3 Ti + l 196 1 i+3 R(t) ~ B(n+l) (i+l)+2 t(n+l)(i+l)+2 (3.5.16) and a starting viscous plume flow, gt.iQ R(t) Two time scales, ~ ~B ( 1 i+3) (n+l) (i+3)+2 t v and t Bv i+ 4 (n+l) (i+3)+2 , and two length scales, ~ • (3.5.17) and R , can be derived from the above three equations (3.5.15), Bv (3.5.16), and (3.5.17) to distinguish the three regimes of starting flow and steady flow, as shown in Figure 3.5.2. However, not all three regimes necessarily exist in a starting flow or in a steady buoyant jet flow. For example, when the ratio of tBv1tMB is very small, there is no starting inertial plume regime. Similarly, there is no inertial plume regime, when RBv/~ is very small. 3.5.3 Dynamic Collapse The dynamic collapse of a homogeneous mass in a linearly density-stratified environment (i = 1) or for the squatting of a liquid column in a uniform environment (i = 0) may be similarly described. sho~m in Figure 3.5.3. There are three flow regimes as There is an initial stage, a principal flow stage, and a final spreading stage. Two time scales, 197 Rev t - - - - - - - - - - - - - - - - / (i+ 4) ( n + I )( i + 3)+ 2 log R ( i + 3) (n+l)( i+I)+ 2 STARTING JET INERTIAL STARTING A...UM VISCOUS STARTING PLU\1E log Figure 3.5.2 Three regimes of starting surface or submerged buoyant jet flow. 198 R~~------------------------~ (n+l)( i+3)+2 log R SLOPE 2 (n+l)( i+\)+ 2 1 - - - - - INERTIAL-BUOYANCY V1SCOUS-Bl.K)YANCY (INITIAL STAGE) (PRINCI~L STtrGE) (FINAL STAGE) log t Figure 3.5.3 Three stages of dynamic collapse flows. 199 t o I J 0" 0 ;:: R b . h. i +1 g ]. and t Bv • and tWQ length scales Ro and RBv ' can be developed to distinguish the three regimes although as before, not all three regimes necessarily exist. For example, when the ratio of ~v to Ro (or tBV to to Rol ~ g~ihoi+l) is rather small, there is no principal stage. The relationship of RBv and tBV to Rl and t l , which were given in Tables 3.1.2 and 3.1.5, are given in Table 3.5.3 for i = 0 and in Table 3.5.4 for i = 1. The limiting values of two ratios, ~v/Rl and tBvlt l , from the long wave solutions are also given. 3.5.4 Radial Surface Buoyant Jet in a Cross Current The above analysis can be extended to describe spreading in a crossflow. If U is the magnitude of horizontal current velocity, and ~ is the downstream distance from the source then substituting t for x/U in Eqs. (3.5.11) and (3.5.12) gives the width of (inertial) surface wastewater field, y(x), M1 I 4 ( X )1/2 U (3.5.18) y(x) tV B1 I 4 ( x) 3/4 U (3.5.19) y(x) tV and The location of the upstream edge from the center of a radial buoyant jet~e' can be derived from balancing the 200 Table 3.5.3 The ratio of ~-v to Rl and tB-v to tl when i = O• ~ ~-v -- . n Plane n = 0 Radial Instantaneous release m = 0 ( bOOO 10 ) ' " Continuous discharge m= 1 bOlO Instantaneous release m = 0 currents n = 1 R1 m currents Continuous discharge m = 1 tB-v 3 tl = 0.9l0 cOOO 5 - - = 0.133 4 cOlO eo" 4 ) 1/' cOOl b~ll = 0.626 - - = 0,143 c2 011 (~) cooo 15/7 = (~)' cOlO (~ )"" cOOl (~) cOlI 0.334 = 0.106 = 0.174 4 = 0.0523 201 Table 3.5.4 The ratio of RBv to R1 and tBv to t1 when i= 1. ~ n Plane R Bv t1 R1 m Instantaneous release m= 0 tBV -- Ciao) 1/2 = 0.481 CIOO blOO - - = 0.487 cIOO currents n = 0 Radial Continuous discharge m= 1 Instantaneous release m= 0 b~lO - - = 0.0874 c 5llO 10 C:Ol y/7 = 0.407 c IOI = 0.124 e~)' CllO C )"/7 I 01 c IOI = 0.0469 currents n = 1 Continuous discharge m= 1 b4 -III - - 0.0816 c 3lll = 0.0325 (~)' c lll 202 starting flow velocity with the ambient current velocity, to give x e 1 2 =- d 2 Ml/2 -U (3.5.20) and (3.5.21) in which d is the starting jet coefficient R(t)/M 1 / 4 t 1 / 2 in Eq. (3.5.11) and c is the inertial starting plume coefficient R(t)/B 1 / 4 t 3 / 4 in Eq. (3.5.12). From Eqs. (3.5.15), (3.5.16), and (3.5.17) similar relations can also be derived for the internal spreading of a submerged wastewater field (i = 1) . 203 CHAPTER 4 EXPERIMENTAL STUDIES AND APPARATUS 4.0 Introduction Many experimental studies have been made o~ the unsteady phase of plane gravitation-driven spreading flows, and a systematic review of previous studies on these subjects was presented in Chapter 2. The theoretical results developed in Chapter 3 will be compared with these previous experimental results in Chapter 6. In this chapter experiments on radial surface buoyant jets are described. Apparently no previous experiments of this type have be~n performed, at least none seem to be described in the research literature. The experiments to be described were performed to dete~ine the effect of initial jet momentum on the jet mixing by evaluating the density distributions. Conductivity measurements will be described in which a periodic radial spreading motion develops, as predicted by the analysis in Chapter 3. Surface buoyant jet flows were also performed in which the buoyant surface flow field prevailing in a steady cross current was measured. 4.1 Surface Buoyant Jet 4.1.1 Experimental Set-Up The experiments were conducted in a square tank (4.0 meters by 4.0 meters) one meter deep filled with tap 204 water to a depth of 75 centimeters. A half-radial buoyant jet flow was generated by discharging blue-dyed buoyant water from a half-cylindrical chamber combining flow-straightening rings a half-circular slot 9.962 cm in radius and 0.3175 em wide (see Figure 4.1.1 for details). A constant head tank was used to provide the pressure necessary to drive the radial jet and a calibrated precision bore flow meter (tube number B6-27-10/27 manufactured by Fischer & Porter Corporation) was used to regulate the flow rate with an accuracy of 1%. Another flow meter was connected to a drain pump and was set to a flow rate equal to the discharge simultaneously in order to keep the free surface at a constant depth during the experiment. The overall experimental set-up is shown in Figure 4.1.2. Before running each experiment, the effluent fluid was mixed well and cooled to a temperature close to that of the tank water. Both temperatures of the discharging flui~ and tank water were checked with a thermometer (graduated in O.l°C intervals and read to ±O.02°C). 4.1.2 Density and Dilution Measurements The discharged fluid was a mixture of tap water, methanol, sodium chloride (Food grade 999 salt supplied by Morton Salt Company) and a small amount of blue dye (blue concentrate supplied by 7-K Color Corp) for flow visualization. The sodium chloride was the tracer. The concentration change due to mixing was measured by a calibrated conductivity probe 205 i- fOf =-'"'~ -tt~! J t~! .....,1 t.... t 6 -jti----- 2 ~ I i Figure 4.1.1 l Tl 3i*AL.L SCALES IN INCH Cross section of the half-radial discharging box. CONSTANT t£AO TANK TO BRIDGE CIRCUIT AND SANBORN RECORDER DYED MIXTIJRE Figure 4.1.2 PlM' I· Overall experimental set-up for the half-radial surface buoyant jet. DRAIN 206 (shown in Figure 4.1.3), forming part of a Wheatstone bridge circuit. The response of concentration change was measured as a conductivity change, which was recorded on a strip chart via a Sanborn analog recorder (Model 150). The conductivity probe was mounted on a carriage, and could be moved vertically with an accuracy to 0.01 em and radially to 0.10 em. The probe electrodes were cleaned regularly and p1atinized using a platinum chloride solution. The probe was immersed in distilled water when not in use. Before experiments were performed, the values of specific gravity of both the effluent and the tank water were measured by a specific gravity hydrometer (specified as ASTM 151H) in a graduated plastic cylinder (1000 c.c.). The accuracy of the specific gravity measured by hydrometer is within ±0.0002. A calibration curve was plotted for the conductivity probe by diluting the effluent fluid with the tap water in the tank (in a series of proportion of 1/2, 1/4, 1/8, 1/16, 1/32 ..• ). A typical calibration curve for the conductivity probe is shown in Figure 4.1.4. 4.1.3 Photographic Equipment Side-view photographs of the discharge flow were taken with a Nikon F camera equipped with a 17 mm lens and mounted on a tripod in front of glass-side wall of tank. camera was equipped with a motor drive which was connected The via a cable to the timer of the Sanborn recorder, so that the strip 207 ~ ..... ;.' ...~~ ""I )\ I , . " c "I I IJ .. #, ), r ~ I, I I I' \ , I, , " ,. '1 ) Figure 4.1.3 Photograph of a conductivity probe. ATTENUATION 7 'V . 6 1 0 5 z I 0 X2 0 X5 c:, XIO 4 Cl.:' Q...-' XI Q:..~ 3 2 TOTAl Figure 4 . 1.4 READING IN eM Ty pical ca libration curve of conductivit y probe (Run No. 10/ 27/77). 208 chart was marked as each photograph was taken. The photographs were scales from a projected image of the film negative using the known dimensions of the discharge box. 4.2 Radial Surface Heated and Buoyant Jets 4.2.1 General Experimental Set-Up Two series of radial jet experiments were performed in a rectangular recirculating flow basin. The first set was for a heated radial surface jet in a stagnant environment. The second was set for a buoyant radial surface .i et in a uniform steady current. The test basin (36 feet long, 20 feet wide, and about 16 inches deep) is described in detail in Koh et aZ. (1974). The bottom of the basin is a uniform white sand (median diameter 0.8 rom) and for the experiments' it was leveled to within ±l rom of horizontal by use of a grader rolling on steel rails at each side of the tank. The recirculation system consists of two pumps which allow the simulation of a wide range of current speeds. The average current speed was measured by timing cross-shaped lucite drogues (1 inch high and 2 inches long) while traveling a known distance in the tank. Five manifolds in both ends of basin were adjusted to guarantee the horizontal uniformity of the cross current. Two swimming pool skimmers were used to clean the free surface film from the water in the basin. 209 The general set-up for the radial surface heated jet experiments is shown in Figure 4.2.1. The flow injection system is essentially similar to that for the half-radial surface buoyant jet experiments shown in Figure 4.1.2, except an electrical heater was included to provide hot water at a constant temperature at the circular slot jet (7.62 cm radius and 0.3175 cm thick) generating the radial flow. Two precision bore flowmeters (Tube number FP-1-35-G-IO/27 manufactured by Fischer & Porter Corp.) were used. In the observations of a radial surface buoyant jet in a cross current the discharge was a mixture of tap water with methanol and a small amount of blue dye for flow visualization. The overall experimental set-up is similar to that shown in Figure 4.1.2. 4.2.2 Temperature Measurements Six fast-response thermistors were used for temperature measurements in the heated jet experiments and their characteristics are given in Table 4.2.1. Table 4.2.1 Each thermistor was insulated and Fast-response thermistor characteristics Type Bead Diameter Dissipation Constant-Still Water at 25°C Time Constant in Moving Water Small head Veco thermistors type 52A26 0.35 rom 0.75 mw;oC 27 or 45 milliseconds Resistance at 25°C ~200 KQ ~---- 6" I" 8 I" Detail of Circular Discharge Box 4 -------- Constant Head Tank 12 I" 8 Nikon F Camera I" -11 4 ' I" 104 N I-' o Valve Heater T Drain Thermostat Figure 4.2.1 Overall experimental set-up for radial surface heated jet. 211 mounted in a stainless steel tube. One thermistor was installed inside the discharging chamber to record the initial temperature of the jet fluid. The other five formed an equally spaced vertical rake as shown in Figure 4.2.2. The rake was mounted on a carriage so that it could be moved in the radial direction (accurate to ±0.10 cm)and vertically perpendicular to the free surface (accurate to ±0.01 cm). The thermistor response to temperature was measured with a bridge circuit designed by Gartrell (1979), as shown in Figure 4.2.3. A dual operational amplifier (Texas Instruments, SN558) was used to buffer, amplify and low-pass filter the signal from the bridge. All resistors used in the circuit were precision metal-film or wire-wound resistors. The electrical current passing through the thermistor was approximately 4.0 x 10- 6 amperes so that the ohmic dissipation of the thermistor was approximately 2 x 10- 6 watts. Hence the self-heating of thermistor was about 0.003°C by using the dissipation constant given in Table 4.2.1. The amplified signal response was about 1.0 volt/oC, and the output of the circuit could be offset so that the zero-level output would correspond to a convenient temperature level. Six bridge circuits were connected in parallel to a DC power supply (±15 volts) and were housed in a grounded aluminum box. The drift of the thermistor circuit was less than 10 mV over an extended period, giving the thermistor an accuracy of better than 212 .. ... '';- " Llq ... ,:- 11.1 .. ) / Figure 4.2.2 Photograph of a rake of five fast-response thermistors. 2.2M 2.2M .005 j-Lf 1.2 k t--V'o.l'v-_...c;.5OK Figure 4.2.3 Fast-response thermistor circuit Gartrell (1979). 3v . designed by 213 O.Ol°C. A buffer circuit made from an integrated circuit operational amplified (Texas Instruments SN741), as shown in Figure 4.2.4, followed the thermistor circuit and was used to increase the input impedance of the analog-to-digital converter system to about 1 megohm. Each analog output voltage was fed into one channel of an eight channel ana1og-to-digita1 data acquisition system (manufactured by Digital Data System) which digitized the information and packed it on magnetic tape. Data written on the magnetic type were retrieved and processed at Ca1tech's Booth Computer Center with the IBM 370/168 high speed digital computer. The sample time of the A/D system was set at 150 seconds and the sample rate at 10 samples per second. The total length of the measurement records was 2 minutes for each location. Each thermistor was individually calibrated with the same bridge circuit and same cable as in experiment. A third-order polynomial was fitted, in a least-square sense, to the calibrated data so that (4.2.1) where T is the temperature in °c, V the analog output from the bridge in volts, and a., j J = 0,1,2,3, are regression coefficients. Typical calibration curves along with calibration data are shown in Figure 4.2.5. 214 r----< -15v IOkn CHANNEL INPUT Figure 4.2.4 ">-__- -..... OUTPUT 10 6 AID CONVERTER Buffer circuit for AID converter from Gartrell (1979). 30 u 0 .... w cr 25 :::> ~ cr ~ I / -1 ~ W .... 20 -10 -5 o +5 VOLTAGE Figure 4.2.5 Typical calibration curve for fast-response thermistor. +10 N I-' V1 216 4.2.3 Overhead Photographs Overhead photographs of the dyed buoyant surface flow field were taken with a Nikon F camera equipped with a wide angle lens (17 mm). The camera was equipped also with motor drive so that it could be triggered remotely. The scale of flow field was read from the negative image by comparing the known spacing between two fiducial marks on the sand floor of the basin. The upstream edge position was also checked with direct measurement with a steel scale. Elapsed time photographs were also taken for some experiments involving initiation of flow in a steady cross current. 217 CHAPTER 5 EXPERIMENTAL RESULTS 5.1 Ha1f-~adia1 Surface Buoyant Jet A total of thirty experiments were performed using the apparatus described in the previous chapter. data are given in Table 5.1.1. The relevant These experiments covered the range of initial jet conditions as shown in Table 5.1.2. The densimetric Froude number Fr ranged from 2.80 to 25.0, the jet Reynolds number Re = Re. from 5040 to 13,700, and the ratio of J two length scales, R1/~QM' from 31.64 to 72.28 (or equivalently, Re/Fr2 from 29.5 to 645.6). These dimensionless numbers were defined in Section 3.5 as (5.1.1) (5.1.2) (5.1.3) (5.1.4) Table 5.1.1 Summary of data for starting half-radial surface buoyant jet experiments, rj = 9.962 em, h j = 0.3175 em and H = 75.0 em. -- - j Pa Pj f:J. u j Q M (OC) (OC) (gm/ cm 3) (gm/cm 3) (10- 3) (cm/sec) (cm 3/ sec) (cm" /sec 2) (cm 3/sec") Density Measurements 0.9970 0.9988 0.9971 0.9985 0.9984 2.0 1.6 2.3 0.5 0.6 30.70 20.13 28.38 28.38 28.38 610 400 564 564 564 18,724 8,052 16,000 16,000 16,000 1,199.2 627.2 1,272.0 276.6 332.0 No No Yes Yes Yes T Run No. a T B 8/23/77 24 26 28 29 - - 25.05 25.90 25.40 25.05 27.35 25.40 0.9990 0.9994 0.9994 0.9990 0.9990 30 31 9/01/77 06 19 25.95 25.60 25.30 24.80 24.75 25.70 25.80 25.45 25.80 24.70 0.9990 0.9990 0.9990 0.99905 0.9991 0.9985 0.9986 0.9987 0.9975 0.9958 0.5 0.4 0.3 1.65 3.3 21.34 15.90 17.21 18.84 17.61 424 316 342 374.4 350 9,046 5,024 5,886 7,054 6,164 208.0 124.0 100.6 605.9 1,132.9 Yes Yes Yes Yes No 20 21 22 27 28 24.60 24.40 24.65 23.50 24.60 23.90 24.45 25.00 24.70 24.60 0.9991 0.99915 0.99915 0.9996 0.9990 0.99635 0.99505 0.9954 0.9972 0.00715 2.75 4.1 3.75 2.4 1.85 16.35 15.92 11.32 11. 32 11.75 320 316.4 225 225 233.6 5,315 5,038 2,548 2,548 2,746 876.7 1,272.4 827.6 529.4 424.0 No No Yes No Yes 10/05/77 10 14 17 22 23.70 23.50 23.90 24.30 23.10 24.70 24.60 24.40 24.70 24.20 0.9992 0.99945 0.9995 0.9995 0.9992 0.0071 0.00735 9.9984 0.9984 0.9980 2.1 2.1 1.1 1.1 1.2 15.92 15.92 15.92 15.92 16.71 316.4 316.4 316.4 316.4 332.0 5,038 5,038 5,038 5,038 5,547 651. 7 651.7 341.2 341.2 459.9 Yes Yes Yes Yes Yes 24 27 29 11/01/77 04 23.60 23.40 22.90 22.70 22.70 24.60 25.00 24.20 23.60 23.80 0.9995 0.9995 0.9995 0.9995 0.9995 0.9980 0.9984 0.9988 0.9994 0.9991 1.5 1.1 0.7 0.1 0.4 20.50 21.94 17.51 16.53 19.83 407.4 436.0 348.0 328.4 394.1 8,352 9,566 6,094 5,429 7,817 599.2 470.2 238.8 32.2 154.6 Yes Yes Yes Yes Yes 13 16 20 29 12/12/77 21.90 22.35 23.00 23.85 0.9995 0.9998 0.9997 0.9998 1.0000 0.9991 0.9989 0.9988 0.9974 0.9980 0.4 0.9 0.9 2.4 2.0 26.44 19.83 19.83 17.00 15.55 525.3 393.9 394.1 337.3 309.0 13,891 7,811 7,816 5,741 4,804 206.0 347.5 347.7 794.4 605.6 No Yes Yes No Yes - - - - 21. 90 20.80 23.20 22.60 -- ----- - N I-' 00 I I Table 5.1.2 Summary of calculated data for starting half-radial surface buoyant jet experiments. R1 l'.QM R(t) Ml/4 tl/2 127.3 154.8 158.0 34.4 41. 3 63.83 50.44 59.73 72.28 70.65 1.05 9,511 7,089 7.672 8,399 7,851 45.7 49.1 34.0 170.7 365.3 60.48 51. 75 56.37 48.20 42.38 1.04 4.71 3.76 2.80 3.49 4.13 7,290 7,097 5,047 5,047 5,240 317.8 502.0 645.6 413.0 306.8 41.55 38.72 31.64 33.46 35.38 1.40 1.32 1.53 1.43 1.30 1.00 0.84 0.77 0.725 0.85 0.777 7.73 7.73 14.76 14.76 12.06 5.25 5.26 7.26 7.26 6.72 7,097 7,097 7,097 7,097 7,447 257.1 257.1 134.6 134.6 164.8 42.10 42.10 45.65 45.65 45.59 1.25 1.15 1.12 1.05 1.18 0.80 0.75 0.713 35.69 44.60 44.63 111.44 66.87 13.94 20.34 25.51 168.57 50.57 8.00 10.00 10.01 25.00 15.00 9,139 9,780 7,806 7,368 8,842 142.6 97.7 77 .9 11.8 39.3 51.43 55.78 51.26 63.06 59.43 1.07 1.02 1.07 0.95 1.04 0.71 0.70 0.81 0.88 0.89 89.15 44.57 44.57 23.40 23.45 67.43 22.48 22.48 7.23 7.93 20.00 10.00 10.00 5.26 5.26 11,785 8,838 8,841 7,577 6,931 29.5 88.4 88.4 274.8 250.6 71.13 53.69 53.70 43.14 41. 74 1.08 1.07 1.05 1.30 0.88 0.77 0.78 0.85 - - Rm tMB (em) (sec) 8/23/77 24 26 28 29 46.22 33.94 39.90 85.56 30.75 15.61 12.84 12.58 57.86 48.22 10.37 7.61 8.95 19.19 17.52 13,684 8,973 12,652 12,652 12,652 30 31 9/01/77 06 19 64.32 53.60 66.98 31.27 20.67 43.50 40.52 58.48 11.64 5.44 14.43 12.02 15.03 7.01 4.64 20 21 22 27 28 21.02 16.76 12.46 15.58 18.42 6.06 3.96 3.08 4.81 6.48 10/05/77 10 14 17 22 23.42 23.42 32.37 32.37 29.97 24 27 29 11/01/77 04 13 16 20 29 12/12/77 Run No. -- - Fr '- Re Re/Fr 2 1.04 1.03 1.11 - 1.06 1.07 1.30 R(t) BI/4 t3/4 - - R(t) (g~Q3 t8tJ/2 - - - - - 1.04 - 0.93 - 0.70 0.78 - - - - - - - - N ...... \0 220 in which the kinematic viscosity of the jet fluid, v, is assumed a constant at 0.01 cm2 /sec in all of the calculations. In these experiments the aspect ratio of jet dimensions, hj/rj' was fixed at 0.03187 while the relative depth of ambient fluid, H/h., was also fixed at value of 236.2, corresponding to J relatively deep water. 5.1.1 Starting Half-Radial Surface Buoyant Jet Typical side-view photographs of a starting half-radial surface buoyant jet are shown in Figure S.l.l(a), (b), and (c). Figure S.l.l(a) shows a starting buoyant jet at an initial Froude number, Fr = 5.25. Figure S.l.l(b) shows that with a Froude number Fr = 10.0 and Figure S.l.l(c) is for Fr = 15.0. These photographs show the typical behavior of such jets. The jet is initiated with a frontal head that is thicker with larger Froude number. Just behind the front there is a vortex growing and traveling with the front until at some point the vortex collapses, as shown in Figure S.l.l(c). The growth histories of the radial fronts, as determined by the photographic methods described in the previous chapter are shown in Figure 5.1.2. This graph is a logarithmic plot of front radius R(t) as a function of time from release. It is clear that the data fall into two categories corresponding to a slope of 1/2 or a slope of 3/4. However, some of the data show an initial slope of 1/2 followed by a slope of 3/4 leading t ( sec) 41.2 54.9 80.5 96.6 Figure 5.1.1 Cross sections of half-circular front at various time t after initiation of discharge (a) Run No. 10/5/77, Fr = 5.25, Re = 7100. N N '-' 222 ¢ rn o o ..;t co co II QJ p:: t (sec) 104.2 155.2 N N W 170.4 Figure 5.1.1(c) Run No. 11/4/77, Fr 15.0, Re 8840. 224 5 R( t) 100 SYMBOL RUN NO. 0 9/21/77 I>. 9/2B/77 0 10/5177 v IV4177 0 IVI6177 SLOPE ! (em) 10 100 500 t (see) Figure 5.1.2 Growth history of starting radial surface turbulent buoyant jet flow. Closed symbols indicate the photographs are shown in Figure 5.1.1. 10 SYMBOL RUN NO. 0 9/21/77 '" 9I2B177 0 1015177 ..B.L!.l v 1114177 l"a 0 11116177 Figure 5.1.3 Normalized growth history of starting radial surface turbulent buoyant jet flow. Closed symbols indicate the photographs are shown in Figure 5.1.1. 225 to a further slope of 1/2. According to the discussion in Chapter 3 these can be categorized, respectively, a starting jet, a starting inertial plume, and a starting viscous plume. The data in Figure 5.1.2 are replotted in Figure 5.1.3 in a normalized form, where length scales have been made relative to tMB and the time scale by~. These normalized data do not show a common curve for all three segments, as is to be expected, since the coefficients in each relation are not necessarily constant. From the dimensional analysis given in Eq. (3.5.11) the coefficients for starting turbulent surface jet flows are a function of initial Froude number and Reynolds number, ----.:;R~(...::.t!...)_ = <t>( F r , Re) Ml/4 tl/2 (5.1.5) The relationship of the jet coefficient to Reynolds number is shown in Figure 5.l.4(a) and to the densimetric Froude number in Figure 5.l.4(b). The value of starting jet coefficient decreases with increasing Froude number and it begins to approach to an asymptotic value around 1.05 when the Froude number is greater than 10. That means that the buoyancy enhances the initiation jet flow when the Froude number is less than 10. At a Froude number higher than 10 the buoyancy plays a less important role in initiation of flow 80 that starting turbulent buoyant jet flows can be described by the asymptotic formula, 226 1.5 0 0 0 0 .L ~ 0 R( t ) .L 0 M4 t 2 0 0 0 o 0 (foro 00 1.0 0 8 0 0 0 0 5 10 Re Figure 5.l.4(a) 15 20 10' ) Relation of starting jet coefficient to jet Reynolds number. 1.5 R( t) M'*" t + 1.0 0.0 5 10 15 20 Fr Figure 5.l.4(b) Relation of starting jet coefficient to radial jet Froude number. 25 227 R(t) (5.1.6) From Eq. (3.5.12) the starting radial plume coefficient will also be a function of initial Froude number and Reynolds number, i. e. R(t) Bl / 4 = q,(Fr ,Re) (5.1.7) t 3/ 4 As shown in Figure 5.1.5(a) there is no systematic dependence of the starting plume coefficient on Reynolds number. Figure 5.1. 5 (b) and (c) shown the relation of starting plume coefficient to Froude number and R1/~QM = (Re 3 • Fr2) 1/8 All the values of the starting plume coefficients are between 1.04 and 0.70, which is lower than theoretical results 1.31, given in Subsection 3.2.6. Momentum and viscosity effects may be the possible reasons. Only three values of viscous starting plume coefficient are found (0.713, 0.725, 0.777) as listed in the final column of Table 5.1.2. These values are about 15% higher than the theoretical result, 0.63 obtained from viscous long wave solution. There are insufficient data to show the dependence on the initial Froude and Reynolds numbers, i.e. R(t) ( gllvQ3 ) 1/8 t l / 2 <I>(Fr,Re) (5.1.8) 228 R( t ) ...L 8 4 tt 1.0 0 0 0 0 0 o 00 80 0 00 0 0.5 0 0 a 5 10 15 ( 10"3) Re (a) o 1.0 " 0..._-- ~~~~ 0 C30-..o o 20 0 ___ - - - 0 g--~---..Jj.------00 0 Fr (b) " R(f)= 1.309 at l 10 o I (c) Figure 5.1.5 R, 2 3 T - - = (Fr ·Re ) lQf1. Relation of radial starting plume coefficient to (a) jet Reynolds number, (b) Froude number, (c) Rl/tQM" 229 Without any initial effects, such as momentum jet mixing, ~ will approach to the theoretical value, 0.63, as the dimensionless ratio Rl/~QM tends to be small. Three similarity profiles for viscous spreading currents are compared with viscous long wave solution, as shown in Figure 5.1.6. The experimental profiles are all thicker than the theoretical profile and there is a prominent head at the front and some internal waves behind the front. The greater profile thickness may be due to jet mixing and possibly some error due in the kinematic viscosity value. The spreading currents thickness scale T(t) in this case is independent of time t, i.e., T(t) RiV)1/3 = (<P2)1/3(v)1/4 ~ = ( -gfl 2 gfl (5.1.9) in which ~ is the viscou's spreading plume coefficient given in Eq. (5.1.8) and v is assumed as 0.01 cm 2 /sec. This thickness scale was also found independent of time by the force scale analysis (see Eq. (3.1.11)). Although the thickness scale for two-layered radial stratified flow is independent of time, in an infinite extent of ambient water, it does not mean that the thickness of flow is also uniformly distributed in GIl space (see Figure 5.1.6). 6.0 i o 9/21/77 6 9/28/77 o o o 10/5/77 0 (Rg~V) I 0/(/ T / 2.0 \ cf ,P'--o~ d0\ / ° A 000 /\ A h( r, t) ~ 00 o ° 0A o 4.0 o X ..... JA A A A IS AA A 0 __0- -0-0 \ P t;. I A t;. 0 0 OL\. '0-0 -0 1 I ~, 0 \ I if>. rtl , '0/ \ I o 0 o.o~, __ 00 6 ~ __ ~ __ 0.2 ~ __ ~ __ ~L- 04 __L-__L-__~__~_ _ 0 1.0 06 as r ! = R( t ) Figure 5.1.6 Similarity profiles of viscous radial surface spreading currents at time t (71.4 seconds for No. 9/21/77, 101.7 for 9/28/77, and 96.6 for 10/5/77) after discharge. N W o 231 5.1.2 Density Measurements Figure 5.l.7(a) shows a two-layered radial stratified flow being formed from a radial surface buoyant jet discharge at Fr = 5.25. The first photograph (t = 200.5 seconds) in Figure 5.l.7(a) was taken when the circular front just touched two side-walls (see Figure 5.l.l(a) for reference). The surface jet thickness became constant at a distance which was approximately the length scale tMB as indicated. It then collapsed to produce circular interfacial waves which were damped out quickly. After the circular front touched all the walls of the tank, the surface layer thickness kept increasing. The circular surface jet zone then looked like a circular internal hydraulic jump followed by a series of undular internal waves with a circular plan form. Figure 5.1. 7 (b) shows the surface flow field produced by the half-radial surface buoyant jet with Fr = 10.0 and Figure 5.l.7(c) with Fr = 15.0. began In those two cases the circular jet mixing also to collapse at a distance which is roughly equal to t MB . However, more violent jet mixing and a propagating wavy interface made it difficult to locate the exact point of the jet collapse. Typical records of density measurements as recorded by a conductivity probe are shown in Figure 5.1.8. The turbulent fluctuation was found to decrease as the depth of measurement increased at any radial distance. Mean values of density at a given radius were obtained by dividing these depth-density curves into two regions having equal area. The distribution of fluid N W N t = 200.5 262. 6 1MB = 23.42 em 472.0 seconds Figure 5.1.7 Radial stratified flow due to a half-radial surface buoyant jet discharge (a) Run No . 10/5/77. 23 3 en "§ () ~ <t- -:r<ioi =NN NNN II E u t = 247.0 286.9 seconds N W "'" r---------~-----JI Figure S.1.7(e) Run No. 11/4/77 2MB = 66.87 em, t = 330 .3 344.1 seconds 235 Figure 5.1.8 Density measurements of a half-radial surface buoyant jet (Run No. 10/27/77) with Fr = 10.00 at a radial distance r = 32.39 cm (or r/t = 0.726). MB 236 density difference on depth at three radial locations are shown in Figure 5.1.9, in which circles denote the mean density difference and a bar denotes the maximum and the minimum values. In Figure 5.1.10 the relative mean density difference, ~(r,z)/~m(r) plotted against normalized depth z/zl/2(r), is compared with the similarity tracer distribution given by the radial surface jet solution in Section 3.4. The half-thickness zl/2(r) is defined as the thickness where the relative density difference is half of the maximum value ~ (r) at the free surface. m In this case the discharge Froude number is Fr = 10.0. maximum values of mean density difference, ~ The (r), are found to m agree fairly well with the radial jet solution, i.e. the density deficiency decays inversely proportional to the radial distance r, as shown in Figure 5.1.11. However, the half-thickness zl/2(r) is a linear function of radial distance r for the region where the buoyancy effects do not become important, i.e., where r/£MB < 1, as shown in Figure 5.1.12. Similar results are shown in Figures 5.1.13, 5.1.14, and 5.1.15 for a radial surface buoyant jet with a lower densimetric Froude number, Fr = 5.26. In this case it is hard to find a linear growth function of half-thickness zl/2(r). It approaches to a constant thickness without any sharp collapse from the jet regime. 237 o 5 100 5 ,.. 0 I I I 0 r-----o---; I ~ I 0 I ~ I 0 ~ I " Depth z I 5 - 2 -v- - (em) "" - I- " - "- - ~ 4 - - ,.. - ~ 5 - -o----r ~ 6 Figure 5.1.9 I I... Distributions of relative density difference respectively at a distance r = 17.15, 32.39, and 67.95 cm (or r/£ B = 0.384, 0.726, and 1.523) for a radial sur~ace flow field (Run No. 10/27/77). 238 ~(r,z) ~m( r) 0.0 0.5 0.0r---~--~----~--~--~~--'---~----~--~~~ <J 0.5 [> Z ZI 2" o 1.0 1.5 o SYMBOl RUN NO. t:,. 10/27177 ~ " [> " Cl 10/29/77 0 " 0 11/16/77 0 11120177 <J " 2.0 2.5 '-------'------1____...L....-_ _----L____L -_ _-1-_ _--L____L -__....L__---l Figure 5.1.10 Self-similar distribution of relative density deficiency. Dotted line denotes Prandt1-To11mien's solution. 239 o 0.1 10 0.1 Figure 5.1.11 Relation of the maximum value of relative density deficiency to the radial distance. Symbols are defined in Figure 5.1.10. r (em) Z..L 2 (em) 5~------------~------------~--------------~--~ Figure 5.1.12 Relation of the half-thickness to the radial distance. Symbols are defined in Figure 5.1.10. 240 ~( r,z) ~m( r) Z Z.L 2 Figure 5.1.13 SYMBOL- rilMB 0 1.789 0 1.355 ~ 0.705 '1 0.488 Self-similar distribution of relative density deficiency (Run No. 10/10/77, Fr = 5.26). 241 Figure 5.1.14 Relation of the maximum value of relative density defiency to the radial distance. Symbols are defined in Figure 5.1.13. r (em) o ~ 70 O~---~I,---~I---~I---~I---~I--~I--~ .~ Z-'- Ir- 2 ( em) 2f- "~ ........ - -0---0------- - 31~--~'----~1--~1----L-,--~1----~1--~ Figure 5.1.15 Relation of the half-thickness to the radial distance. Symbols are defined in Figure 5.1.13. 242 5.1.3 Periodic Feature The analysis in Chapter 3 predicted the existence of a periodic wave feature in the buoyancy-inertial regime of flow. However, only a weakly periodic radial surface plume was observed in these experiments. Since there is intensive surface jet mixing in the upstream and strong viscous effects in the downstream, most of the periodic feature can only be observed in the position close to the interface. The period record as shown in the last record of Figure 5.1.8 results from breaking internal waves which form travelling mixing billows. The billows then collapse into a series of circular internal waves. In Figure 5.1.16 all the observed periods Tob are compared reasonablywellwith the predicted value T as given by Eq. (3.2.87). The details of the measurements are presented in Table 5.1.3. 5.2 Radial Heated Surface Jet Four radial heated surface jet experiments were performed as listed in Table 5.2.1. In these experiments the density variations are produced by the temperature variations specified by the equation, 1 dp (T) p (T) dT = aCT) (5.2.1) where peT) is the fluid density at temperature TOC, aCT) is the thermal expansion coefficient (11°C) at temperature TOC. 100 i iA I - Tob = T = 0.699 ( ~4) "3 0 Q) en .0 ~O .. "'0 Q) > Q) en .0 0 10 N ~ w . "'0 0 Q) Q.. 2 10 Period Figure 5.1.16 Predicted 100 T (sec) Comparison of the observed period with the predicted value T given by Eq. (3.2.87). 244 Table 5.1.3 Summary of observed and predicted periods. Run B No. (cm'+/sec 3 ) Location of Measurement r r/~MB z Tob T (sec) (sec) (em) (em) 8/26/77 1272 29.21 0.732 2.30 5.8 5.80 8/28/77 276.6 44.45 0.520 5.46 13.5 16.90 " " 29.21 0.340 1.96 8.0 9.65 9/1/77 100.7 39.37 0.588 3.96 11.8 20.10 9/6/77 605.9 29.21 0.934 2.40 9.0 7.40 " " 59.69 1.909 3.90 17.0 19.30 10/10/77 651.5 41. 91 1. 789 2.50 10.6 11.70 10/14/77 341.2 17.78 0.549 2.11 7.2 4.64 " 22.86 0.706 2.50 9.1 6.49 " " " 38.10 1.177 2.50 13.8 12.80 " " 53.34 1.648 4.00 14.4 20.10 10/17/77 341.2 27.94 0.863 2.50 8.0 8.48 " " " 43.18 1.334 3.50 13 .2 15.20 " 68.58 2.119 3.50 25.6 28.10 10/22/77 459.9 17.15 1.453 1.20 4.0 4.00 " " 22.23 1.883 2.20 7.0 5.66 " " 62.87 5.328 2.70 16.4 22.60 10/24/77 599.2 66.04 1.850 2.00 4.3 5.18 " " 52.71 1.477 4.00 14.0 16.40 10/27/77 470.2 32.39 0.726 3.50 7.9 9.28 10/29/77 238.9 52.71 1.181 4.37 22.7 22.30 11/16/77 347.5 134.62 3.020 4.68 53.5 68.60 11/20/77 347.7 114.30 2.560 6.05 45.0 55.10 " " 93.98 2.110 5.05 35.53 42.50 11/29/77 794.4 93.98 4.020 3.43 35.50 32.30 12/12/77 605.6 69.85 2.979 2.66 23.40 23.80 " " 83.82 3.574 2.66 31.20 30.30 Table 5.2.1 Summary of experiments on radial surface heated jet in a stagnant environment. Ta T j "j a(T j ) tJ. B R.MB (OC) (OC) (10- 2 em2 / sec) (10-1t) (10- 3) (cm lt /see 3) (em) RUN NO. Fr = Re j '" Re /Fr2 • j R /.t = 1 QM MS /4 MI/2 ~ QB 1 /2 "j " j M2 ("j~Q2B')'" 10/17 /78 21.5 26.4 0.7955 2.748 1.346 422.2 36.19 9.28 10320 119.8 55.85 10/18/78 21.0 25.3 0.8138 2.646 1.138 356.8 39.36 10.10 10090 98.94 56.55 10/19/78 20.9 30.2 0.7378 3.081 2.865 898.5 24.81 6.36 11120 274.8 52.27 10/20/78 21.4 30.8 0.729 3.131 2.943 923.0 24.47 6.28 11250 285.7 52.33 rj - 3 in. (7.62 em), h j - 1/8 in. (0.3175 em), .tQM = 12nr j h j = 3.899 em N Note: Q - 320 em 3/see Uj - Q/2nrjhj - 21.05 em/sec M - QU j - 6736.3 em4 /see 3 a(T j ) - (-0.773 + 0.19 Tj - 0.0027 Tj 2 + 0.000021 Tj3) x 10- 4 .pVI 246 For pure water an algebraic expression for the thermal expansion is given by Batchelor (1970) as a(T) (-0.773 + 0.19 T - 0.0027 T2 + 0.000021 T 3 ) x 10- 4 (5.2.2) which is plotted in Figure 5.2.1 as a solid curve. The values of kinematic viscosity of the hot water discharged are given in Figure 5.2.2, v. (T) J (1. 745 - 0.0468 T+ 0.0005823 T3 - 0.2614 x 10- 5 T 3) x 10 2 cm 2 /sec, (5.2.3) which is from the data also given by Batchelor (1970). In these experiments the ambient water depth H was kept at 17.78 cm (7 inches). The relative depth of ambient water, H/h., was then fixed at 56 and the aspect ratio of the jet, J hj/rj' was fixed at 1/24. The maximum value of the temperature excess at radial distance r, defined by I:::.T m I:::.T. J T(r,O) - T a =-----T. - T J a was found to decay proportionally with the inverse of r, as shown by the data plotted in Figures 5.2.3 and 5.2.4. However, 247 5.0. 4.0. ~.-- I+~ 30. lj~ -I~ 2.0. ~tt: !t!~ 1.0. C(~ ~u O. 0.0. 10. 20. TEMPERATURE Figure 5.2.1 40 30. 50 (OC) Thermal expansion coefficient of water as a function of temperature after Kotsovinos (1975). 1.5 ] VI N....... E u 0 b >- 1.0. ~ (i) 0 ~ :> u ~ 0..5 ::E ~ ~ 0..0. L--L--..L--...l.--....l--~_--J....-~---'-----'----;-;ICO 0. TEMPERATURE figure 5.2.2 (OC) Kinematic viscosity of water as a function of temperature. 248 1.0 ~n-------r----"""'T"'----'T"--r----:r--"""'T"'-r--""--., 0.1 DATE Fr o 10/20/78 6.28 6.36 10.10 9.28 10/19/78 6 10/18/78 'V 10/17/78 Cl Re ~ 11250 0.002943 11120 0.002865 10090 0.001138 10320 0.001346 0.05 1.0 5.0 r/r· J Figure 5.2.3 Relation of the maximum value of relative mean temperature excess to the relative radial distance r/r .. J 10 249 Fr o o 6. 'V 6.28 6.36 1010 9.28 SLOPE -I 0.1 4 0.1 Figure 5.2.4 Relation of the maximum value of relative mean temperature excess to the normalized radial distance r / ,t~fB • r Irj T-__-T____~4____~5____~6____,7____~8____~9__--.10 0~~-T____ 2 ZIII hj 6 8 10 12~--~----~--~----~--~----~--~----~--~--~ Figure 5.2.5 Relation of the normalized half-thickness to the normalized radial distance. Symbols are defined in Figure 5.2.4. 250 while the decay exponent of -1 is clearly independent of Froude number the coefficient of proportionality is not. The half-thickness zl/2 (r) of the time averaged mean temperature excess above the ambient temperature is not a linear function of the radial distance r. In Figure 5.2.5, it is clear that the relative half-thickness, zl/2(r)/h j is a concave upwards curve and is deflected and the deflection is more rapid for a lower Froude number which implies it is due to a buoyancy effect. The time averaged mean temperature excess is found to have a self-similar distribution, as shown in Figure 3.2.6(a), (b), (c), and (d). It is very close to the Prandtl-Tollmein's solution obtained when the two coefficients, K and K , are u c equal as discussed in Chapter 3. One profile of the intensity of turbulent temperature fluctuations is shown in Figure 5.2.7. The maximum value of turbulent intensity, ~T'Z/~T seems to increase with increasing m' radial distance r/r. and the location of the maximum turbulent J intensity deepens with increasing radial distance. 5.3 Radial Surface Buoyant Jet in a Cross Current Experiments were performed to evaluate the performance of a buoyant radial surface jet in a cross current. The primary purpose of the experiments was to determine the relative importance of the jet momentum and buoyancy fluxes. 0, O. ~ 1.0 v / lor tit v r 7J TMB 0.34 0.37 0.43 0.49 0.55 0.61 0.77 0.92 • <l 0 r 1.12 0 1.2 A 1.4 v 1.6 <l 1.8 I> 2.0 . <> 2.5 3.0 0 r 2.01- 41'> ..L 0 A 0 Z 112 A v <l 2.0 t> • r r 2.30 2.70 4.00 6.00 8.00 10.00 072 0.84 1.25 1.87 2.49 3.11 rr1.30 0.41 1MB 0 3.0 3.0 o 0.5 LlT(r,Z) LlTm(r) Figure 5.2.6 1.0 o 0.5 Ll T( r, Z) LlTm(r) Self-similar relative mean temperature excess above the ambient temperature. Solid line denotes the Prandtl-Tollmien solution given in Chapter 3. (a) Run No. 10/19/78, Fr = 6.36. (b) Run No. 10/20/78, Fr = 6.28. 1.0 N V1 I-' 0, ~ 1.0, o~ Z ZI/Z 1 r r 1.15 1.30 1.50 1.70 2.25 2.75 3.50 5.00 9.00 1MB 0.22 0.25 0.29 0.33 0.44 0.53 0.68 097 1.74 7J 0 0 t:. 2.01- 0, Ar V <J t> 0 vo/ •+ ] 1.0 Z r r; ZI/2 0 0 <J t:. V <J 0 2.0 t> 0 V •x t> + <J 1.1 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 6.0 r ~B 0.23 0.25 0.30 0.34 0.38 0.42 0.53 0.63 0.84 1.26 t:. -1 3.0r 0 0.5 1.0 6T( r, Z) 6Tm( r) (c) Run No. 10/18/78. Fr ~ 10.10. 3.0~ V 0 0.5 6T(r,Z) 6Tm( r) (d) Run No. 10/17/78. Fr ~ 9.28. 1.0 N V1 N 253 0.2 ch\7 <l [> 0 0.3 etl x CD 0 - 0 6Q[>e 0 ~ - 0 z Ll:J (em) CTJ ~ IP " x • ~ r J 1MB 0.23 0.25 0.30 0.34 0.38 0.42 0.53 0.63 0.84 1.26 0 0 2 [> ex 0 + r· 0 0 0 0 1.1 1.2 6 1.4 'V 1.6 <l 1.8 [> 2.0 2.5 e 3.0 X 4.0 + 6.0 <> 3 Figure 5.2.7 f I Profiles of the intensity of turbulent temperature fluctuations for radial surface heated jet (Run No. 10/17/78) versus depth and relative radial distance. 254 As listed in Table 5.3.1 seventeen experiments were performed and in all the tests the ambient water depth was maintained at H = 19.81 cm with a circular slot jet thickness of h. = 0.3175 cm J (1/8 inch). H/h. J The relative depth of water was therefore constant at 62.4 for all experiments. A typical development of initiation flow pattern to a well- developed flow field due to a radial surface buoyant jet discharge in a cross current is shown in Figure 5.3.1. The downstream edge position Xd in the current direction can be simply predicted by the superposition of a radial surface buoyant jet front and a uniform current. Therefore, the downstream edge grows as a starting jet flow followed by a starting plume flow, but with the velocity relative to the free stream, i.e. (5.3.1) R(t) where U is the current speed. When the downstream displacement is plotted against time as in Figure 5.3.2 it is clearly seen that two different spreading regimes are distinguishable. One is the starting jet flow which specifies a jet coefficient defined by R(t) x d - Ut (5.3.2) In the four tests where this is appropriate the coefficient varies from 1.22 to 1.45. The other regime is the inertial starting Table 5.3.1 Summary of experimental data for a radial surface buoyant jet in a cross current. Fr Re 128.7 17.4 8400 " 386.3 10.1 " " " 482.9 9.0 " 325.3 21.40 6970 287.3 11.5 8340 10.010 328.0 21.58 7080 321.8 11.0 8400 0.9988 7.008 " " " 225.3 13.2 " 0.9998 0.9989 7.010 202.7 13.33 2700 179.0 7.2 5200 0.949 0.9999 0.9985 14.523 328.0 21.58 7080 466.8 9.2 8400 3/08/78 0.933 0.9999 0.9989 10.011 202.7 13.33 2700 198.8 6.8 5200 3/09/78 0.933 1.000 0.9993 7.005 328.0 21.58 7080 225.2 13.2 8400 3/21/78 1.00 0.9996 0.9995 1.001 315.4 20.75 6540 30.9 33.5 8090 3/22/78 " 0.9995 0.9991 4.004 " " " 123.8 16.8 " 3/22/78a " " 0.9990 5.005 " " " 154.7 15.0 " 3/27/78 " 0.9997 0.9995 2.001 320 21.05 6740 62.8 24.1 8200 . 3/28/78 " 0.9994 0.9992 1.501 376 24.73 9300 55.3 32.7 9640 3/29/78 " 0.9997 0.9973 24.568 320 21.05 6740 770.4 6.9 8200 4/10/78 " 0.9992 8.006 376 9300 295.0 14.1 9640 Q uj M B (10~4) (cm 3/sec) (em/sec) (cm 4 /sec 2 ) (cm 4 /sec 3) 0.9987 4.005 328.0 21.58 7080 0.9997 0.9985 12.018 " " 0.962 1.000 0.9985 15.020 " 3/05/78 " 0.9997 0.9988 9.011 3/06/78 " 1.000 0.9990 3/06/78a " 0.9995 3/07/78 0.926 3/07/78a Run No. U Pa P~ A (em/sec) (gm/cm 3) (gm/cm 3) 2/23/78 0.920 0.9991 2/27/78 " 3/02/78 N VI VI i I I I 1.000 '----- - 24.73 _ _L ----- - -~ - 256 (a) (b) Figures 5.3.l(a,b) Photographs of the development of a surface flow due to a radial surface buoyant jet in a cross current (Run No. 4/10/77). (a) t = 35, (b) t = 80 seconds after initiation of discharge. 257 (c) (d) Figures 5.3.l(c,d) Photographs of the development of a surface flow due to a radial surface buoyant jet .in a cross . current (Run No. 4/10/77). (c) t = llO, (d) t = 485 seconds after initiation of discharge. 50 ~ I E - 10 t- 0 ..- :::> ~ SYMBOL 1RUN NO. V 3/5/78 • 3/9/78 ~ 13/22/780 o 14/10/78 ~ SLOPE -=- ~ "0 X 10 100 t (sec) Figure 5.3.2 Growth histories of the downstream edge of an initiating surface field. N V1 00 259 plume (slope 3/4) for which the starting plume coefficients, defined by R(t) x d - Ut (5.3.3) vary from 1.24 to 1.44. In general, the location of the upstream edge of radial buoyant jet in a crossflow is specified by x e --= ¢ ( 2 U ,Re, Fr, R/h.) B/Ml/2 J (5.3.4) Where M and B are specific or kinematic fluxes of momentum and buoyancy. It was shown in Section 3.5.4 that the asymptotic solutions corresponding to a radial momentum jet is (5.3.5) and (5.3.6) for a radial surface plume. d and c are respectively the coefficients of spreading defined by jets and plumes in the absence of a crossflow that d R(t) (5.3.7) 260 and (5.3.8) Figure 5.3.3 is a plot of the upstream position x e normalized by B/U 3 and it shows how the upstream position varies from a jet flow to a plume flow. The upstream displacement was actually found to fluctuate between a maximum and a minimum value which is denoted by bars on the figure. The straight lines are shown for d = 1.05 and c = 0.75. Figure 5.3.4 is a logarithmic plot of the surface half-width produced by a radial buoyant jet. The surface field also shows the transition from a jet flow to a plume flow. For Run No. 3/22/78a, the appropriate dimensionless numbers for each range of half-width of the surface field are found to be (5.3.9) y _ _ ;::; O. 76 _ _--:t... B1 / 4 (x/U)3/4 (5.3.10) and (5.3.10) The value of kinematic viscosity v was assumed to be 0.01 cm 2 /sec. 3 , PLUME + JET Xe 8/U 3 SLOPE 1 N 0'\ f-' 0.1 0.1 U 5 2 B/FM Figure 5.3.3 Upstream edge position of surface buoyant field. in Table 5.3.2. Symbols are defined 262 Table 5.3.2 Symbol Calculation of upstream edge position. Run No. x - e- U2 B/Ml/2 B/U 3 Minimum Median Maximum 0 2/23/78 0.553 - 0.216 - 0 2/27/78 0.184 - 0.152 - 6 3/02/78 0.161 - 0.112 0.145 \l 3/05/78 0.269 - 0.142 - <J 3/06/78 0.242 0.168 - 0.197 r> 3/06/78a 0.345 0.150 - 0.170 0 3/07/78 0.249 0.101 0.130 0.158 + 3/07/78a 0.162 - 0.100 0.112 3/08/78 0.228 0.093.5 - 0.0987 3/09/78 0.325 0.0963 0.110 0.119 3/21/78 2.616 0.970 1.099 1.293 • • • • 3/22/78 0.654 0.299 0.347 0.388 ~ 3/22/78a 0.523 0.181 0.194 0.2l3 ~ 3/27/78 1.308 0.590 0.669 0.701 )( 3/28/78 1. 743 - 0.687 - <:> 3/29/78 0.106 0.117 - 0.151 • 4/10/78 0.327 0.l36 0.153 0.207 (em) Y 3 I Figure 5.3.4 10 10 2 10 10 x (em) Growth of a surface buoyant field measured from upstream edge. .. I SLOPE "2 3/9/78 0 o 3/22/780 RUN NO. 0 SYMOO... II II Vol N 0\ 264 CHAPTER 6 COMPARISONS WITH PREVIOUS STUDIES In this chapter theoretical solutions derived in Chapter 3 will be compared with the results of previous experimental studies. The first section discusses spreading currents in a homogeneous environment (i = 0) and in the second section those in a linearly density-stratified environment (i = 1) . 6.1 Spreading Currents in Homogeneous Environment 6.1.1 Plane Current from a Finite Volume Release From the theoretical results previously developed for this case a plane current (after a brief initial pnriod) will advance asymptotically according to the regime described by (6.1.1) for an inertial-buoyancy balanced flow and according to R(t) ( = 1.133 g~v 3)115 t 1 J5 (6.1.2) for a viscous-buoyancy balanced flow without any mass diffusion. Table 6.1.1 lists twenty experiments performed by Almquist (1973). In these experiments a tilting partition plate separated 265 Table 6.1.1 Summary of experiments performed by Almquist (1973) on the plane surface currents due to a finite volume release. {gllh p. p. p. - Ps p. V R h (gm/em 3 ) (gm/em 3 ) (10- 3 ) (1n2) (in) (in) 1 1.018 1.014 3.945 4.5 2.12 4.25 2 " " " 8.0 2.83 3 " " " 4.0 0 4 " " " • 5 " 1.009 t> 6 " 7 Sy.bo1 R(t) 3 115 (gllvV ) tl/S 7.51 1.580 1. 720 5.65 11.51 1. 750 1.610 2.00 4.00 6.86 1.571 1.630 2.0 1.414 2.83 4.08 1.493 1. 810 8.920 3.0 1. 732 3.46 8.30 1.564 1.660 " " 1.0 1.00 2.00 3.65 1.430 1.881 " 1.015 2.956 1.0 1.00 " 2.10 1.471 1.897 8 " " 2.956 3.0 1. 732 3.46 4.78 1.455 1.670 110. 6 'V • • A • 0 • •e ho (gllv)I/3 t 213 Run 0 0 o v R(t) 9 " " " 9.0 3.00 6.00 10.91 1.695 1. 730 10 " 1.009 8.920 " " " 18.95 1.696 1. 710 1B 1.020 1.004 15.936 4.5 2.12 4.25 15.10 1.820 1.730 2B " " " " 9.0 3.00 6.00 25.33 1.808 1. 726 3B " " 18.0 4.24 8.49 42.63 1.835 - 1.012 7.905 4.5 2.12 4.25 10.63 1. 762 1. 700 Q 4B . 0 5B " " " 18.0 4.24 8.49 30.02 1. 760 - + 6B " " " 9.0 3.00 6.00 17.84 1.840 1. 730 <3 7B 1.0217 1. 0177 3.930 2.0 1.414 2.83 4.07 1.377 1.700 <> 8B " " " 4.5 2.12 4.25 7.50 1. 550 1. 760 0 9B " " " 9.0 3.00 6.00 12.58 1.666 1.635 )( lOB " " " 18.0 4.24 8.49 21.17 1.740 1.540 Note: Initial aspect ratio h /R o 0 = 2.00. 266 a right-triangular section of volume V per unit width from the ambient fluid. The triangular section had an initial horizontal length R on one side and an initial vertical thickness h o the other side. 0 on The plate was removed and the time when the dyed front passed through several fixed locations was recorded. The growth histories of twenty fronts are plotted in Figure 6.1.1. Two slopes are apparent: one is 2/3, which is appropriate for inertial-buoyancy spreading, and the other is 1/5 which is appropriate for the viscous-buoyancy spreading. A normalized plot of the twenty experiments is shown in Figure 6.1.2, however, all of the experimental points do not collapse into a single curve. Equation (6.1.1) is found to be an upper limit for inertial-buoyancy spreading. Equation (6.1.2) gives a low estimate for the viscous-buoyancy spreading. (The value of the kinematic viscosity of spreading fluid was assumed to be v=O.Ol cm 2 /sec for all calculations). An understanding of the variations of the coefficients in Eqs. (6.1.1) and (6.1.2) is apparent from a dimensional analysis. The growth histories of a frontal trajectory can be described by = 4> (%- ' :1 , :0) (6.1.3) 100 in which the time and the length scales t1 = ( Rl = (g~~5) )/7 are as listed in Table 3.1.2. 4 V )1/7 and (gll)2 v 3 The second R(t) 5 10 e Figure 6.1.1 ( in) 2 10 i TIME t (sec) 10 • 102 I 5" Growth histories for plane surface spreading currents due to a finite volume release. Experimental data are from Almquist (1973). Symbols are defined in Table 6.1.1. • e 3 SLOPE ..L , ( " N 0- At, ) I -R-- Ie· Figure 6.1.2 0.1 I 0..01 5 o • I 2 0..1 TT At , ) = 1.890. (g.W) , /'\1 'I -,- ,t <1 1 v R :::; ( gbV5)1/7 and t =( V'+ )1/7. 3 1 (g6)2v Normalized growth histories of plane surface spreading currents due to a finite volume release. Symbols are defined in Table 6.1.1. + )( . +~ At, ) = 1.133 (g~V3 ) T I ~.~ ~ 0 I I .. N C1' 00 269 dimensionless number, R /R ' represents the viscosity effect in 1 o terms of ratio of the two length scales, R1 and Ro. The ratio is R1 -= R o (g~V25 ) 1/7 h02)117 R o which contains a Reynolds number Ig~h of the initial geometry h /R. o 0 experiments was kept at 2.0. -+00 (6.1.4) R 2 o o h /v 0 and an aspect ratio The aspect ratio h /R 0 0 in these Therefore, when the length scale , an inviscid buoyant spreading current will result. In this case the coefficients which have the following form (6.1.5) will approach the limiting value c and as given in Eq. (6.1.1). 000 as shown in Figure 6.1.3 Similarly, when the length scale ratio R1/Ro -+ 0, the flow will begin with a viscous current. The coefficient for a viscous-buoyancy spreading current, (6.1.6) 270 5 I I I I 2 (g6Vl"'3 t 3 0.5 - --•-I / R( t } = 1.890 ( 9 6V ) 3 t 3 - e - I I I I 10 figure 6.1. 3 I - ~~"'IOX.O ~.A.- I .1. .L R( t ) I 2 10 Coefficients of inertial-buoyancy spreading currents expressed as a function of Reynolds number Ig~h o h 0 Iv. R( t ) ~ho 1/ figure 6.1. 4 Coefficients of viscous-buoyancy spreading currents expressed as a function of Reynolds number Ig~h o h 0 Iv. 271 will approach to the limiting value b OOO given in Eq. (6.1.2). The fact that the data in Figure 6.1.4 lie above the data specified by Eq. (6.1.6) may be due to the error of assuming a constant value of kinematic viscosity of spreading fluid v = 0.01 cm 2 /sec for all computations, although this is unlikely because of the fact that the experimental data are all very consistent. A similar set of experiments were performed by Keulegan (1957) although the details of the initial flow are somewhat different. His experiments involved the density front produced by a finite length lock exchanging fluid with a semi-infinite channel. This flow eventually produces a rapidly decelerating velocity due both to the viscous effects and the fact that there is a finite volume of fluid spreading. Keulegan made observations of the frontal velocity at different locations rather than observing the frontal travel as a function of time. In this case it is best to rewrite Eq. (6.1.2) in the form (see also Eq. (3.1.15) with n=m=O). o U(R) = R(R) 0.373 (g/:lV43 ) (6.1. 7) vR which may be used to describe the decelerating flow. Equation (6.1.7) rewritten in terms of lock variables, as defined in Figure 2.3.2, is 272 (6.1. 7a) in which R(t) = L(t) + Land H is the initial lock fluid depth. o Figure 6.1.5 shows that the frontal velocity decays with travel distance as the distance to the -4 power anticipated by Eqs. (6.1.7) and (6.1.7a). Equation (6.1.7a) compares very well with one set of experiments (denoted by circles). The initial Reynolds number Ig~H H/v is approximately equal to that given by Keulegan (1957), Ig~H H/v , where v is the mean value of m m kinematic viscosity of both ambient and saline fluids and the dimensionless distance R/H is approximately equal to L/H when R(t) »L • o Middleton (1966) performed similar experiments using turbidity currents, however, Eq. (6.1.7a) overestimates the velocity of the currents produced. The theoretical value, U(R)/lg~H, (denoted by dotted line in Figure 6.1.6) is about ten times the value actually observed at the same location L/H. The simplest explanation for the discrepancy is the rapid settling of suspension which reduces the relative density difference to a value about 1% of the initial value. From Subsection 3.2.4 we know that Eq. (6.1.1) is valid only when the initial relative density difference is much smaller than the order of 0.1. Otherwise, the coefficient in Eq. (6.1.1) 273 6P .J'96H H SYMBo.L 6 = - U(R) Uo 0.1 P.. v'" 0 0..0.11 1810. 0 0..0.19 270.0. " 0..0.38 370.0. 0 0..0.75 50.0.0. + 0..143 650.0. SLo.PE -4 0.0.1 2 10. 10. L 1=1= Figure 6.1.5 R -Lo H Frontal velocity decay of density currents from a finite-length lock exchange flow, experimental data from Figure 3 of Keu1egan (1957). H = 5.8 em, L = 41.7 em, and width of channel = 2.54 em. a 274 \ \ \ o~ 60 \ \ initial \ 6p 0 ( gmlcm 3 ) 0.128 6 0.242 \ u .j 96H 0.1 - - - for the case of 0 \ \ \ \ \ \ \ \ \ 10 40 L H Figure 6.1.6 Frontal velocity decay of turbidity currents from a finite-length lock exchange flow, experimental data from Figure 10 of Middleton (1966). H = 20.3 em, L = 28.3 em, width of channel = 13.83 em, and average sgttling velocity of suspension = 0.9 em/sec. 275 needs correcting to the form R(t) 1.890 (g~V)1/3 t2/3 (1+~)1/3 (6.1.8) For example, in the case of a spreading oil slick the relative density difference is ~= 0.136 for an experiment by Liang (1971). In this case Eq. (6.1.8) will be corrected to (6.1.9) However, as shown in Figure 6.l.7(a) the theoretical value given by Eq. (6.1.9) is still slightly higher than experimental observation. The reason, for this is not clear. Measurements of the spreading layer thickness h(r,t) are of two types. One is a "wave-gage-type" measurement by fixing the measurement location and varying observation time. The other is a "photographic typel' measurement performed by fixing the time and observing the spatial location of the front. Results for plane currents obtained by using the first technique will be discussed here. current case obtained Results for the radial with the second method will be discussed in the next subsection, 6.1.2. The similarity profile of inertial buoyancy spreading layer thickness due to a finite volume release was derived in Section 3.2.5 as h(r,t) (6.1.10) 276 I 2 R{ t ) = 1.811 ( gil V) 3" t '3 R( t ) (em) 100 o 30 I 10 t (see) Figure 6.l.7(a) Frontal trajectory for an inertial-buoyancy oil slick spreading in a channel, data from Run No. 1 of Liang (1971). V = 131. 2 cm 2 , /). = 0.136. 50 277 for which the frontal displacement function R(t) is in Eq. (6.1.8). For a spreading oil slick the inertial layer thickness measured by photocell equipment located at r thickness h + 8 = s is actually the total (1 + ll)h so that the thickness as measured by Liang (1971) is of the form h(r ,t) s (6.1.11) The value c can be computed from c = R(t)/(g~V)1/3 t2/3. Thus, the spreading layer thickness at one fixed location, r s , will decay proportionally to t2/3. Figure 6.1.7(b) shows that the agreement of the similarity solution (6.1.11) with three experimental observations by Liang (1971) is fairly good except for a damping internal wave train behind the steep front. The collapse of a mixed region at a stably stratified interface can be viewed as a combination of two surface spreading currents. As shown in Figure 6.1.8 the numerical calculation performed by Meng and Thomson (1978) supports this assumption. The density front displacement grows proportionally to t 2 / 3 rather as a linear time function. 6.1.2 Radial Current from a Finite Volume Release After an initial stage the radial spreading current produced by a finite volume release grows asymptotically according to R(t) (6.1.12) 278 3.5 I I n 30 S'TIITION 1\ I 1\ 1\ I, : 1\ 25 \ 1\ \ \ I I 20 \ 1.5 \ \ \ 10 ,, .... .... .... ....... ,-- 05 --- ---- -------- 00 25 I II 1\ 20 STATION 1\ 2 1\ I \ I \ I 1.5 I I I ( I + 1\) h (em) \ ' I I I I I 10 05 "- ''--.... .... I I --- -- ---------- I I 0.0 20 ~ 1\ I , I \ 15 I 3 STATION \ \ \ , "- 10 ................... 05 - -- ---------- 00 0 6 8 TIME figure 6.1.7(b) 10 12 14 16 "Me) "Wave gage" measurements (in solid curve) of inertial oil slick spreading thickness by photocell equipment located at three locations (60, 85,110 em from end wall of channel for Station no. 1,2,3). Dotted curves indicate prediction by similarity $olutions, data from Liang (1971). 279 10 I- R( t )--j P,~ R( t ) ~ 0.1 10 t jg~ 21TRo Figure 6.1.8 Dynamic collapse of a homogeneous region at a stably stratified interface. Numerical calculation data from Figure 7 of Meng and Thomson (1978). 280 for the inertial-buoyancy stage and then according to R(t) = 0.779 ( g~V for the viscous-buoyancy stage. and List (1976) 3)1/8 t 1/ 8 (6.1.13) Experimental results by Chen showed that two such regions of flow do exist but they found higher coefficients in both equations which can be ascribed to mixing. As previously discussed for an oil slick, spreading in inertial stage, Eq. (6.1.12) needs an adjusted coefficient to give R(t) (6.1.14) For the experiments by Abbott (1961) shown in Figure 6.1.9(a), ~ = 0.250 and Eq. (6.1.14) becomes R(t) (6.1.15) Figure 6.1.9(b) shows that the agreement of Eq. (6.1.15) with the observations made by Abbott (1961) is very good. Figure 6.1.9(c) shows the comparison of similarity profiles developed in Subsection 3.2.5 with observations given in Figure 6.1.9(a). The similarity profile for thickness of a radial spreading layer was computed as 281 0 20 40: I 60 Radial distance (meters) I I 10 I I I I I 20 18 seconds after release 0 20 E 40 - 60 - 10 u 26.2 seconds (f) (f) Q) c .:s:. .~ ..c - 0 0 20 40 60: I - I 10 I 20 40 37.2 seconds 60 10 49.7 seconds Figure 6.1.9(a) Oil spreading sections at four times after release of 1500 cubic meters of oil (1200 tons). Data from Abbott (1961). 282 100 E 10 100 10 I (sec) Figure 6.l.9(b) ~68!r,l) 0 Growth history of inertial-buoyancy radial oil spreading front. Data from Figure 6.1.9(a). 0.1 2 R(I) 0.0 0.1 SYMBOL I ( sec) 0.2 2t.h{r,l) Rof o IB.O c 262 V 37.2 ll. 497 0.3 0.4 0.5 0.0 02 04 0.6 {= Figure 6.l.9(c) 08 1.0 1.2 1.4 R[n Comparison of oil spreading layer thickness given in Figure 6.l.9(a) with similarity profiles (n= a denotes plane case, Eq. (6.1.10) and n= 1 denotes radial case, Eq. (6.1.17». 283 h(r,t) for each time t. (6.1.16) In dimensionless form, this may be written as 2 (6.1.17) gt.h(r,t) 1{ r } cB 1 / 4 t- 1 / 2 ) 2 ="2 R(t) in which the total buoyant force per unit mass B = gt.V and c = R(t)/B 1 / 4 tl/2 for each instant time t after release. The supere1evation o(r,t) over free surface is equal to t.h(r,t). Although the frontal Richardson number, gt.h(R,t)/R 2 (t) is found to agree with the theoretical value of 0.5, the spreading layer thickness follows the plane front solution (n = 0 in Figure 6.1. 9 (c» ratheI: than the radial solution (n = 1) given by Eq. (6.1.16). It is not clear how this result comes about. In order to consider the results of Martin and Moyce (1952) for the radial spreading problem, Eqs. (6.1.12) and (6.1.13) are rewritten as R(R) Igllh Ro (6.1.18) = 2.00 R(t) o and R(R) Igt.h o = 0.526 Re • Ar • {R:~) J1 7 (6.1.19) 284 in which Rand h o 0 are the initial radii and the initial height of circular column before spreading. Re = Ig~h o h 0 /v The Reynolds number and Ar is the aspect ratio h 0 /R 0 , v is the kinematic viscosity of the spreading fluid and is assumed to be 0.01 cm2 /sec in the calculations. Figures 6.l.l0(a) and (b) show that Eq. (6.1.18) is an upper limit for inertial spreading and Eq. (6.1.19) is a lower limit for viscous spreading. Similar experiments were also performed by Roge and Brooks (1951), however, there is insufficient information on the experiments to perform a comparison. For the viscous spreading front, as described by Eq. (6.1.13), the frontal velocity decelerates proportionally to t- 7 / S , which agrees with a similarity solution found by Smith (1969b). One experiment performed using syrup on a flat aluminum surface appears to confirm this basic result, as shown in Figure 6.1.11. 6.1.3 Plane Current from Continuous Steady Discharge In this case the density front advances at an asymptotically constant velocity according to R(t) 1.260 (g~Q)1/3 t (6.1.20) in the inertial-buoyancy stage and according to 3 R(t) = 0.804 (g~vQ) 1/5 t 4/ 5 (6.1.21) 285 x 9 B l:.. V ~ I <> f • o R(R) SYMBOL 0 0 l:.. V 0.1 <I t> <> • • + X II X 0.429 0.324 0.259 0.123 0.069 0.040 0.012 0.0058 0.0011 0.0006 0.0001 •<> o Ro -;:::R==0.526'Re'Ar{R( ~ 91lh o 7 t)} <> for the case of <> 0.02 ------.J---L-..l-..--l...---L~--'-~10 LI R( t ) Ro Figure 6.l.l0(a) Radial gravity currents produced from circular lock exchange flow. The aspect ratio Ar = 2 and the initial radius R = 3.45 em, data from Table 1 of Martin and Moyce ~1952, Part V). 286 Ro ;:::::;:::::;=- = 2.00 t:. e 0.5 0 0 0 • t i t t -0 .() -0 Ar = hoI Ro 2 3 Ro (em) 2.2 0 2.82 0 3.45 t:. • • • R( t ) ~ ~ 4 0- -0 ~ -0 -it Q-0 ~ 0 t::. • • A. 7 0 .; 9Aho =0.526·Re 'Ar{R~~ y R o for the ease of 0 0.1 '--_ _ _ _ _ _- L -_ _ _--L_ _--L---!..----1_---l._....l---l I 4 2 3 6 7 8 5 R( t ) Ro Figure 6.1.10(b) Radial gravity currents produced from circular lock exchange flow. The initial relative density difference ~ = 0.428, data from Table 2 of Martin and Moyce (1952, Part V). 3 10 --I 4 1 I I I 10 I -0 TIME ,.. t (sec) ~ 0 I ---0-= --0 \ 102 I SLOPE _1- 8 , , Growth history of a volume of syrup spreading on the shining face of aluminum foil. r-o I Figure 6.1.11 «...J «00::: ~ lj 0 0 <t ~ LLJ LLJ ..... 0::: 0 0 - E - 30 1 N 00 ...., 288 in the viscous-buoyancy stage. These results apply after a short period in a starting jet regime. Figure 6.1.12 (a) shows the agreement of these results with experimental results by Barr (1959) (Experimental apparatus was shown in Figure 2.3.5). There appears to be some initial starting jet flows before approaching the plume flows. Figure 6.1.12(b) also shows agreement with another set of experimental results by Almquist (1973) who used a different discharge apparatus from Barr's (1959) (see Koh (1976». Figure 6.1.13 also shows the agreement with Eqs. (6.1.20) and (6.1.21) of plane density currents propagating on a flat bottom as measured by Wood (1965) and Wilkinson (1970). There is insufficient information given by above three investigators to determine the dependency of the coefficients on the discharge Froude number and Reynolds number as discussed previously. _ ---"R"-"-C..::."t)<--- R ( t) or or (gt.Q) 1/3 t M1/3 t 2 /3 _ _ _.....:R:.:..(~t:..!..)_ __ ( cp(Fr, Re 3 ) 1/5 ~ v (6.1.22) where Fr = u./lgt.h. , Re = u.h./v J J J J = Fr 2 / 3 • Re. Plane surface currents, which were produced by a horizontal buoyant surface discharge from a circular pipe into a channel of motionless ambient water as illustrated in Figure 2.1.5 were observed by Sharp (1971). Four dimensionless numbers, Figure 6.1.12(a) RI R( t) .. o fa = 0.804 (g~Q3)5 t ; Plane surface spreading currents from a continuous steady discharge, data from Barr (1959). 0.1 R( t ) I 290 11)2 A III • 3.736 7.BSIl 6.809 6.513 lij.926 1ij,ij2B iij.ij2B + ••" 101 (0.001) 3.932 3.539 It z wQ Q (cm 3/sec) 12,5 26.5 (cm2/sec) 0.984 2.087 12.B 25.6 51.5 12.0 23.3 ij7.B 1.008 2.016 4.055 0.945 1.835 3.921 55.0 4.331 WIDTH OF CHANNEL w = 5 INCHES R( t) -R-,- 100 1 R( t)= 1.260(9AQ)~ t 10- 1 Figure 6.1.12 (b) Plane surface spreading current s from a continuous steady discharge, data from Almquist (1973). 6 , 000053 0.0022 I Wilkinsal (1970) 0.0138 0.0236 000104 0.0114 00182 0.0071 0.0574 00033 0.0061 0 \l 0 0 0.0437 0.0032 0.0141 0.121 I Wood (1965) 0.0324 0.0239 0.1160 C> 0.0130 0.157 <J 0 0.0074 0.0420 0.01 / ," 0.1 11 10 =:J INVESTIGATOR 00546 I 0.1160 ,B sec') (ft I Plane bottom spreading currents from a steady continuous discharge. 0.001 Q (ttl/sec ) 6p ~ 4 6 SYMn..1 I ____~__~~~-L~WW____~__-L~~-L~~____~__-L~-L- R(,) = 1.260 (gt,Q)"!", 00011~6wl~--~--~~~~~WW 0.01 Figure 6.1.13 R, ~ 0.1 3 R(')=0.804(g~Q )5,T N \0 I-" 292 cp L(g')1/5 ( , t(g,)3/5 Q.2/5 o Q.1I5 J J (6.1.23) were used to described these currents, in which the definition of each variable is given in Eq. (2.1.2). Sharp (1971) used at least two figures to illustrate Eq. (6.1.23). However, by using the following transformations, = t(g,)3/5 D . (g , ) 1/5 ) 4/3 • --,,-J_ __ ( Q.1/5 J Q.2/5 J (6.1.24) Q. (g,)ll3 )-3/5 J • L(g,)1/5 • Q.2/5 J ( v5/3 (D. (g')1/5 )5/3 -dJ_______ Q2/5 (6.1.25) it is seen that the dimensionless length and time scales R/Rl and t/tl are contained within Eq. 6.1.23. Using these variables, all the experimental data given by Sharp (1971) can be plotted in one figure as shown in Figure 6.1.14. Two slopes, 1 and 4/5, can also be seen to indicate the inertial and the viscous starting plumes. 16 3 L. Figure 6.1.14 10 I -5 10 lli J4 ( OJ) 2 OJ(~=0.58 I - I 1~2 I o ~ 0-. J'8 • • '0. + .~: • ~ t:.~ 0.,7 It,.~ --t . ·8 250 200 150 100 50 •+ • Op! _ t ~ g')5/5 SYMSQ I 10- 1 I v <> SYMOOL Of5 = 1.16 Dj (OOQL) 2 tV{g'-R)215 Q{ g' )"5 Unsteady surface spreading currents in a channel due to a buoyant pipe flow discharge, data from Sharp (1971). LV(g,..01.)·/5 ~~~~~~YI n' ,1/5 lOll N \0 W 294 From Eq. (6.1.20) we know that inertial-buoyancy spreading currents will propagate at a constant velocity which is proportional to the cubic root of kinematic buoyancy flux, i.e. (6.1.26) Such constant horizontal surface spreading velocities were observed by BUhler (1974) and Roberts (1977) when a submerged line plume reached the bottom or a free surface. From Eq. (6.1.26) the theoretical value of R/(2g~.Q.)1/3 (since the submerged buoyancy J J flux is two times that of a one-sided spreading plume) will be o R ~)1/3 ~ 1.0 1.26 ( ~j Q , (6.1.27) j when the buoyancy flux is assumed conserved. The average value of experimental results is 0.66 from Buhler (1974) and 0.68 from Roberts (1977). It is not clear why these values are lower than predicted. For the viscous spreading regime, Eq. (6.1.21) can be rewritten as (6.1.28) 295 From Roberts (1977) the experimental range of (Q./(2g~.Q.)1/3 H)2/S J J J varies from 0.0972 to 0.229. Thus, Eq. (6.1.28) becomes ___R->.(..::,.t<-.)__ = (0.062 IV 0.146) (QQJ. )2/5 (H vt2 )-1/5 (6.1.29) (2g~.Q.)1/3 t J J when the buoyancy flux is assumed conserved, i.e. g~Q = g~jQj. The experimental results for viscous spreading reported by Roberts (1977) give _-..:.R.:;...:><c.::.t£-)_ _ = 0.27 ( H vt2 ) -1/ 5 (2g~.Q.)1/3 t (6.1.30) J J The difference is believed mainly due to the mixing occurring before the plume reaches the free surface, so that Q/Q. is J significantly greater than 1. Numerical calculations of plane overflowing and underflowing currents in a finite depth of ambient fluid were performed by Kao et aZ. (1977,1978) using the full Navier-Stokes and diffusion equations. The range of calculations for surface currents varied from a starting jet flow to a starting plume flow. local Richardson number Ri(r,t) The = g~h(r,t)/u2(r,t) for the surface flows was found to be roughly constant at 1.00. However, no further information is available for comparison with the similarity solutions given above. 6.1.4 Radial Current from Continuous Steady Discharge In this case the spreading front advances asymptotically according to 296 (6.1.31) for the inertial-buoyancy stage and according to R(t) _ 0.63 ( g~VQ3 ) 1/8 t'/2 (6.1.32) for the viscous-buoyancy stage. These two stages of flow for surface currents were investigated by Chen and List (1976). The spreading currents were produced by a submerged buoyant round jet discharging vertically into a uniform stagnant environment. Typical flow patterns produced by radial surface spreading currents are shown in Figures 6.l.l5(a), (b), and (c). Figure 6.1.l5(a) shows the radial two-layered stratified flow formed behind a circular front. Figure 6.l.l5(b) shows a band of radial stratified flow formed behind the circular front and connected to a radial surface jet zone (also note that the circular surface wave formed by the impingement of submerged jet on the free surface). Figure 6.l.l5(c) shows the formation of a series of travelling circular rings behind the outer front. The rings are smoothed out gradually by viscous effects to from a stratified radial surface flow. The ratio of radii of two sequential dark streaks is found to be from 0.70 to 0.77, which is close to the theoretical value 0.753 found in Subsection 3.2.5.3. 297 Figure 6.l,15(a) Radial surface stratified flow formed after the diluted effluent reached the free surface (77.9 and 155.5 seconds later). Due to a submerged buoyant round jet (dj = 0.586 em, ~ = 0.0170, Qj = 40,8 cm 3 /sec) discharged vertically into a uniform stagnant environment (H = 11.72 cm). 298 Figure 6.1.15(b) A band of radial surface stratified flow· formed after the diluted effluent reached the free surface (70.5 and 166.5 seconds later). Due to a submerged buoyant round jet (d j = 0.586 cm, 6 = 0.0142, Qj = 18.9 cm 3/sec) d~schraged vertically into a uniform stagnant environment (H = 6.19 cm). 299 Figure 6.1.15 (c) A series of outward travelling circular internal \vaves formed after the diluted effluent reached the free surface (79.1 and 121.6 seconds later). Due to a submerged buoyant round jet (dj = 0.586 cm, 6 = 0.1092, Qj = 4.5 cm 3 /sec) discharged vertically into a uniform stagnant environment (H = 23.41 cm). 300 The spreading stage of each photograph presented in Figure 6.1.15 is shown in Figure 6.1.16 by solid symbols. Britter (1979) performed experiments on bottom spreading radial gravity currents and Eqs. (6.1.31) and (6.1.32) rewritten as R(t) 1.309 (6.1.33) for the inertial-buoyancy spreading currents, and as R(t) = !t(v g ll Q)1/2j-l/4 -_..>..:...<._0.63 Q (gllQ)1/4 t 3 / 4 (6.1.34) for the viscous-buoyancy spreading currents agree well with his experimental data for the viscous region as shown in Figure 6.1.17. However, the theoretical value of the constant in the inertial spreading stage is greater than that observed. Many experimental data describing radial surface and bottom density currents have been analyzed by Sharp (1969a, b), using following dimensionless form (previously the given in Eq. (2.1.1», (6.1.35) in which Q. is the calculated volume flux at the surface origin l. of spreading. Using the following transformations, 2 Figure 6.1.16 10 2R(t) 10 ( em) 3 6.1.15( a ) ( b) (e) o o 6. t (see) Growth histories of radial surface spreading fronts shown in Figures 6.1.15(a), (b), and (c). 10 FIGURE SYMBOL i i 10 I~--'----'--'-'"TTTT1 I ~r-r-r­ 10 3 I I I I i I f-' w a 3 0.1 J 0.01 1.0 + Figure 6.1.17 (g6Q)"4" t4 I R( t ) 10 ~ = 0.1 I I I 3 .L aT I (g6V)2 t 1.0 I I ! R( t )= 0.63 ( g6QI 10 I , ! I SLOPE -"4 Growth histories of bottom radial gravity currents, data from Britter (1979). + R( t ) I '4 '4 1.309 ( g6Q) t N w a 303 L = (g'Q.)1/4 t 3 / 4 1L(g')I/sl . 1 t(g,)3/S Q.2/S l. l. l. and t(vg'Q.)1/2 l. Qi = Q.l/S 1t(g')3/sl_ Q.1/5 l. r g ·) 1/3 Q l i r/ 4 (6.1.36) -3/10 v S/ 3 (6.1.37) all of Sharp's data can be replotted as in Figure 6.1.18 for the surface currents, and as in Figure 6.1.19 for the bottom currents. Since the experimental frontal position L(t) is not exactly the same value as R(t), there are some deviations, however, the agreement with Eqs. (6.1.33) and (6.1.34) is generally fairly good. It is clear that most of Sharp's experiments were performed in the transition region. McMinn and Golden (1973) studied the spreading of crude oil on ice, as shown in Figure 6.1.20. Equation (6.1.32) also predicts radial spreading of crude oil reasonably well. A surface buoyant discharged from a boundary can be viewed as an inertial plane current in the near field and as a radial viscous current in the far field. The spreading front therefore will eventually increase its distance from the boundary in proportion to tl/2. As shown in Figure 6.1.21 the experimental data from Barr (1959) and Hayashi and Shuto (1967) for such a boundary discharge show this to be the case. 0.001 0.11 r1- I 3 1.0 2.0 4.0 6.0 8.0 10.0 SYMBOL 0 x A 0 + v 0.01 I I II Q.I -L (V9'Qj) 2 Q A 0.1 II Po x )ff<X---=:\""'" II 3..L 1 R( t )=o.63(g~Qj) 8 t 2" II II, Growth history of radial surface spreading fronts due to a submerged buoyant round jet discharging horizontally into a uniform stagnant environment, data from Sharp (1969a). t(9 Qi"5 ,)3/5 R( t )=1.309(9L\Qj)4 t 4 Figure 6.1.18 (g'Qj)'t L 10, w o .po. Figure 6.1. 19 0.1 (g'Qlr t-f L 2 50 40 30 20 10 ~ v A 0 0 J I 01 (vg'Q)T t 3 ~ 'iJ R( t )= 0.63 ( geOr)t t 2 ~ Growth history of radial bottom gravity currents due to a submerged buoyant round jet discharge horizontally into a uniform stagnent environment, data from Sharp (1969b). 0 00 0.11/5 60 0.1 CDO t (g,)3/5 II) t> SYMBOL '6dO 0000 00 1.p~e gc 1 R( t) =1.309 (g60j)4 t 4 3 Vt w o 306 I R( t )=0.63 (g~Q 3)8 t2I 10 0.1 Figure 6.1.20 Radial spreading rate of crude oil on ice, The kinematic viscosity v of crude oil is assumed as 60 cm 2 /sec for open symbols and as 23.1 cm 2 /sec for solid symbols. Data are from MCMinn and Golden (1973). , R( t ) -1.260( 0110) 3' t ¢ I> o 'V I> I> 1>1> ¢ 1>1> DATA FROM BARR (1959) wO ::; 0.0337 w = 0.5 ft o ::.... ¢ I> I> ft}sec ¢ 1>1> Q¢ 400 38.0 31.0 13.5 14.2 (OC) J T o 00 ~ 00 o 0 00b>~t>tt> 0 '51 12.0 170 17.0 6.0 14.0 (DC) To 00 00 DATA FROM HAYASHI ond SHUTO (1967) wO = 0.19 1tsee w = 5.3 em 7.283 5.816 3.437 0.629 0.028 !::. (O.OOt) 0 ~ I 2 (w'v 0 ) 1- 2 (g60)'2 t ..}.. o ~ CJ SLOPE 00 ¢ 'V o o <> I> SYt.9)L ,J Growth history of surface spreading front at centerline of boundary buoyant surface discharge, data from Barr (1959) and Hayashi and Shuto (1967). 01 ~I______~____~~L-~J-~LJ-L______~____L-~__~J-J-~~______~__~. ill 30 0.5 Figure 6.1.21 (~o)f L 5 81 w o -....J 308 6.1.5 Initial Stage of Finite Volume Release The initial collapse of a liquid column in a homogeneous environment was found to have a self-similar thickness profile, previously derived from the shallow-water wave equations, given by h(r,t) n+l n hI 1 - 1;2 (6.1.38) where n = 0 for plane currents and n = 1 for radial currents. The thickness scale hI is dependent on the initial shape of liquid column and examples are listed for reference in Table 3.2.1. The relative frontal position n = R(t)/R , grows as o t/g~hl 1 { 1 2 } n / (n-l)1/2 + £n[n 1 / 2 + (n - 1)1/2] 2 (6.1.39) --.::.= - R o for the plane currents and as (6.1.40) for the radial currents (see Section 3.2.7). 3 As for plane-symmetric flows the thickness scale hI = 2 ho · f or a rectangu 1 ar sectl.on an d h 1 -- 31T 8 h0 f or a seml.-Cl.rcu .. 1 ar section. Martin and Moyce (1952, part IV) made many laboratory observations of the collapse of a liquid column in air (~= 1) . 309 Figure 6.l.22(a) shows a sketch of three typical photographic sections taken during collapse of a liquid column with an initially rectangular section. Figure 6.l.22{b) shows the comparison of the self-similar profile given by Eq. (6.1.38) (with n = 0) with these three observations. The growth history of a plane front formed during collapse of a rectangular column is shown in Figure 6.1.23. It is interesting to note that the theoretical curve overestimates the displacement. The smaller the aspect ratio h /R of the initial rectangular section is, the o 0 closer are the experimental data points to the theoretical curve. Two cases with initial aspect ratio h /R o 0 = 1 begin to show deceleration with a change in slope of the displacement curve to 2/3 (i.e. R{t) ~ t 2 / 3 ) around t~/R o 0 one case with an aspect ratio h /R the theory overestimates In the = 4 the plane front shows a = 20. Figure 6.1.24 also shows o constant velocity up to t~/R o 0 6. 0 the displacement for the collapse of a semi-circular column section. The reason may be due to an insufficient description by the shallow-water wave equations of the initial stage of the collapse since the vertical acceleration is not small in this stage of the flow. The radial front displacement produced by the collapse of a vertical cylindrical column is plotted in Figure 6.1.25. theory also overestimates the observed displacement. The The displace- ment also appears to be dependent on the initial aspect ratio as shown in Figure 6.1.25. R( t ) ~I I : I I .. I I -----i---i I I I I r-------l R( t ) Figure 6.l.22(a) Initial collapse of a rectangular shape column, line sketch of a photograph from Martin and Moyce (1952, Part IV). I· ~-- I.. ~ r-------, I--- R( t )--1 I-- Ro-l D~ 0 NII'0 '--"" (fa: -;:1 ...-- -....,..... 0 0 00 0.2 6. 6. o 0.6 , _ r - R(t) 0.4 6. 6. 0 0 SYMBOL 0.8 4.75 2.27 1.97 Ro R(t) 1 1.0 Figure 6.l.22(b) Calculated self-similar profiles of initial collapse shows in Figure 6.l.22(a). o 0' 0.2 0.4 0.6 0 0.8 .!:..s::. -=:1 .s::. ...-- 6. 0 1.0~ 1.2t" 1.41- o W I--' I '0.1 5 10 Figure 6.1.23 Ro R( t) 100. I ISl 1.0 00 lSl o 0 ~ o <> ~ <> RO t.jghO (p fJ' o OJ 't. V <:Ecffi o 0° <> <> <> 5 <> 0 10 0 _ilPo 0 c:x?;, 00 ~6. OC 6. 00 lSl 0 6. <> QPrtft1 oClJ lSl V aD 1-1/8 ., v !b I-lIB 2 2 4 2-1/4 2-1/4 4-1/2 Ro(jn) INITIAL LENGTH I I INITIAL ASPECT RATIO ho/Ro 0 o o 0° 00 50 history of plane front during initial collapse of a liquid column with rectangular section (hI = 3ho/2 in Eq. (6.1.39)). Data from Tables 1, 2, and 3 of Martin and l'1oyce (1952, part IV). Gro'~th 0.5 --.-- F I 0 v 6. ° 0 SYMBOL p 100 W f-' f-' Figure 6.1.24 Ro R( t ) 10 Ro t~ 00 [rom Table 6 of Martin and Moyce (1952, part IV). with semi-cirrular section (hI:]; ho in Eq. (6.].39)) (Ro=2 inches), data Growth history of r1.ane front during -initial collapse of a liouid column 0.1 o o 00 0/ 10 W I-' N I b.1 5 10 Figure 6.1. 25 Ro R( t) 50 -- 0.5 0 .--~.--- 2 INITIAL ASPECT RATIO hoiRo 0 1.0 I 0 0 0 0 0 0 L-_ _ _ _ _ _ 2-1/4 2-1/4 INITIAL RADIUS RoOn) 0 0 0 0 0 0 0 0 000 00 0 Ro t ./ghO 0 0 00 r 000 5 10 EQUATK>N (6.1.40) 50 Growth history of radial front during initial collapse of a liquid column with vertical circular cylinder (hI ,= 2ho in Eq. (6.1.40», data from Table 5 of Hartin and Noyce (1952, part IV). -.---; 0 0 SYMBOL 100 100,---------~----r-~r__r_,_,_,_r,_--------,---~r--;r--r-,-,-,-r,_--------r----,---,~A~-r-r-r~ W I-' W 314 Although the theoretical frontal trajectory displacement functions Eqs. (6.1.39) and (6.1.40) are only asymptotic limits valid when the initial aspect ratio is small, the self-similar thickness profile compares very well with the results of numerical calculations with finite difference method and the method of characteristics as shown in Figures 6.1.26, 6.1.27, and 6.1.28. 6.2 Spreading Currents in Linearly Stratified Environment In the first subsection the results of experimental studies and numerical calculations for a plane mixed region collapsing in a linearly density-stratified environment will be compared with the theoretical solutions that were developed in Chapter 3. Other experimental results for a plane starting intrusion layer due to a horizontal slot discharge into a linearly densitystratified environment will be examined in Subsection 6.2.2. There are no experimental results available for the radial submerged spreading case. 6.2.1 Plane Submerged Current from a Finite Volume Release In Sections 3.2 and 3.3 a three-stage description was developed for a front advancing at its own density level in a linearly density-stratified environment. case i = 1, n = m = O. It is given by the In particular the front due to collapse of a finite volume of homogeneous fluid is described by: (I) Initial Stage (6.2.1) 315 "7 • • \l 1.83 t:::. 2.55 0 0 3.30 0 4.15 1.4 ~ 1.2 • 0 • • • 1.0 - ~ ~ ~ 0 NWER ICAL CALCl1.ATION SYMBOL Harlow and Welch ( 1965) 2.00 Fox and Goodwin ( 1952) 2.74 0.8 .s:::. -:c I· ~ f 0:0: 0.6 0 ~ Nlr'IJ cl •eJ 0 0.4 0 • 0.2 • 0.0 L---_ _J......-_ _..l--_ _..l..-_ _..l..-_ _.l...----1 0.0 Figure 6.1.26 0.2 0.4 0.6 08 1.0 Comparison of self-similar profile with numerical calculation results by Harlow and Welch (1965) and Fox and Goodwin (1952) for the collapse of a plane symmetrical column in vacuo, initially rectangular at rest, at various relative distance n = R(t) IRo· 316 1.2 o o o 1.0 - 0.8 ~~O 0.6 -.,.. ~To CEil: --- 0.4 CDI~ 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ~=~r_ R(t) Figure 6.1.27 Comparison of self-similar profile with numerical calculation results by the method of characteristics by Penney and Thornhill (1952) for the collapse of an initially semi-cylindrical section column at rest at specific time t ~ ~ = 0.8 or n = R(t) IRo = 1. 78. 317 1.2 6 0 0 1.0 SYMBOL 7] 0 3.50 2.75 2.08 1.44 0 6 --- -- 0.8 " ,:~ ~ 0.6 ,,:.... ~I (fa::0 - COI~ 0.4 0.2 0.0 0.0 Figure 6.1.28 0.2 Q4 0.6 o.s 1.0 Comparison of self-similar profile with numerical calculation results by the method of characteristics by Penney and Thornhill (1952) for the collapse of an initially semi-elliptic column at rest at various t/gh time R 0 = 0.5, 1.0, 1.5, and 2.0 or various o relative distance n and 3.50. = R(t)/R = 1.44, 2.08, 2.75, 0 318 (II) Inertial-Buoyancy Balance Region (6.2.2) (III) Viscous-Buoyancy Balance Region R(t) where Rand h o 0 = 0.69 ( g~: 4 )1/6 t 1/ 6 (6.2.3) are the initial horizontal and vertical length scale, V is the initial half-volume of homogeneous fluid, R(t) is the horizontal extent at time t, v of the spreading fluid. s the kinematic viscosity The thickness scale h1 is dependent on the initial shape of spreading fluid and is listed in Table 3.2.2. For an initially semi-circular cross-section, the value of h1 is equal to h. In this case, the horizontal length scale Rand o 0 the vertical length scale h . 0 are both equal to the initial radius The volume per unit width of 7fRo 2 the semi-circular section is equal to V = --2-- • Formulae for of the semi-circular section. the three stages of spreading flow will, in this case be simplified to: (I) R(t) - - - = { 1 + (Nt)2} R 1/2 (6.2.4) o (II) R(t) --= 7f1/2 (Nt)1/2 R (6.2.5) 0 (III) R(t) --= 0.93 R 0 C'v: . Nt) 1/6 (6.2.6) 319 NR 2 o in which the dimensionless parameter ---- on the right-hand side \is of Eq. (6.2.6) is a Reynolds number Re N• The three stages of submerged spreading flows are illustrated clearly in Figure 6.2.1. For a volume of semi-circular cross- section fluid with a Reynolds number R~ greater than 500, the interflowing front will have three complete stages of flows which have been previously described by Wu (1965) as: the.initial collapse stage, the principal collapse stage, and the final collapse stage. With an initial Reynolds number between 500 and 5, only two stages will exist for the flow (the principal stage is excluded). For an initial Reynolds number less than 5 there is only one final collapse stage. The dimensionless transition time from the initial stage to the principal stage is about Nt = 2.8 and the dimensionless transition length is about R(t)/R o = 3.0. However, the dimensionless transition time and length from the principal stage to the final stage are dependent on the initial Reynolds number R~. The larger the Reynolds number is, the longer the transition time and the transition length are. For an initially rectangular shape of plane mixed region, three stages of collapse can be expressed: (I) Initial Stage R(t) R o (6.2.7) 321 (II) Principal Stage (Inertial-Buoyancy Balance) h )1/2 R(t) - - = 2 R: R ( (6.2.8) (Nt)1/2 o (III) Final Stage (Viscous-Buoyancy Balance) R(t) R o in which Rand h o 0 = h )113(Nh 1.10· 2 V:· ( R: ,(6.2.9) are now respectively the one-half length of the horizontal and the vertical sides of the rectangular section. The ratio of these two length scales, h o /R 0 , is the aspect ratio of the initial shape and the initial Reynolds number now is defined as Re Nh 2 N = __0_ where the velocity scale is Nh and the length Vs 0 scale is h . o For the special case when the aspect ratio h /R o 0 is unity, i.e. the initial shape is square, and the three collapse stages are described by: (6.2.10) (I) (II) R(t) R = 2(Nt)1/2 (6.2.11) o 2 (III) R~:) = 1.10 • (~: • Nt) 1/6 (6.2.12) 322 Consider only the inertial-buoyancy and the viscous-buoyancy regions. A more general way to normalize the experimental data is by using the length scale Rl : t 1 = (..}L) N\l s (~3 ) ' / 1/2 as listed in Table 3.1.5 4 and the time scale Then the intercept of two oblique lines given by Eqs. (6.2.2) and (6.2.3) will be at the point where the vertical ordinate is R/R1 = 0.481 and the horizontal abscissa is tlt1 = 0.487 as listed in Table 3.5.4. Two sets of experimental data are available for comparison: Wu (1965,1969) and van de Watering (1966). Both experiments were originally performed to study the collapse of a turbulent wake in a density-stratified fluid. A semi-circular or circular homogeneous mixed region was prepared to simulate a uniform parcel of turbulent wake. Sequential pictures were taken as soon as the mixed region began to collapse. Thus, a history of horizontal spread was obtained by measuring the horizontal spreading length as a function of time. A comparison by Mei (1969) of the theoretical solution for the initial collapse stage of a circular mixed region with the experimental results by Wu (1965) was found to be fairly good for dimensionless time Nt between 0 and 1. The theoretical curve shown in Figure 2 of Mei (1969) actually is the same curve given by Eq. (6.2.4). When the dimensionless time Nt is greater than 1, the theoretical solution overestimates the experimental results. The reason why the error is introduced is that the data probably lie in the so-called principal collapse stage. 323 In Figure 6.2.2, the comparison of two theoretical solutions with the experimental results is presented for the transition behavior from the initial collapse stage to the final collapse stage. All the experimental data are from Figure 16 of van de Watering (1966). The Reynolds number Re N for each run is calculated to check its effect on submerged density spread. Since the kinematic viscosity of the spreading fluid was not given, it is assumed to be the same as that of pure water, 0.01 cm 2 /sec. A log-log plotting of the data shows that the experimental results follow the theoretical prediction in the initial stage very well. For the final stage, or the viscous-buoyancy balanced spread, a slower growth rate of horizontal spread is found according to the 1/6 power of t as shown. Most of the experimental results by van de Watering (1966) are in the lower Reynolds number Re N range. Therefore, from the theoretical solution shown in Figure 6.2.1 it is known that the horizontal front moves as such a rate that it does not pass through the principal collapse stage. This is probably the reason why a false conclusion (see Eq. (2.1.6» was drawn by van de Watering (1966) in his description of the principal collapse stage. Figure 6.2.3 shows the transition from the principal collapse stage to the final collapse stage for eleven experiments by Wu (1965). The two different stages are noted by the sharp change of slope from 1/2 to 1/6 in the log-log plot of growth history for a horizontally spreading front. All the data are from Ro R( t) 0.1 2 0.836 0.824 2.17 2. " 6 0 ~ ) 2 Nt 2,157 2,127 10 I I 2 NRo ~ 100 IA Comparison of theoretical solutions (the initial and the final collapse stages) for submerged density spreading front with experimental results by van de Watering (1966) on a plane mixed region collapsing in a linearly density-stratified environment. 2 2 1 0.808 2.03 <> Ro = ( 1 + N t 521 1 0.417 0.54 0 R(t) 269 I 0.254 0.20 0 2 11. (10 ·ft ) 164 NR! ( in) (sec-f) -I Ro E -2 I I N SYMBOL Figure 6.2.2 10 100 I UJ N .po ( tt) 01 +I Figure 6.2.3 R( t ) 10 10 t (sec) I ""6 100 SLOPE 6I 500 Eleven cases of growth history for submerged density spread front due to a plane mixed region collapsing in a linearly density-stratified environment. Data are taken from Figures 10, 11, and 12 of Wu (1965). Details of experimental parameters for each case are shown in Table 6.2.1. SLOPE SLOPE N VI W 326 Figures 10, 11, and 12 of Wu (1965). Details of experimental parameters for each run are given in Table 6.2.1. The data shown in Figure 6.2.3 are normalized in two ways, One is shown in Figure 6.2.4 along with Eqs. (6.2.5) and (6.2.6). The other is shown in Figure 6 .. 2 .. 5 with Eqs. (6.2.2) and (6.2.3), Although the theoretical solutions predict very well the power laws of growth for the submerged spreading rate for both initial and final collapse stages, there is some discrepancy in the magnitude of coefficients in the two equations (6.2.2) and (6.2.3). The variations in the coefficient are found to depend on the initial size of the mixed region and the Reynolds number Re N (for data presentation the kinematic viscosity of the spreading fluid is assumed to be 0.01 cm 2 /sec for all the eleven cases). As shown in Figure 6.2.6 the theoretical value of the coefficient (1:2) for inertial-buoyancy spreading is an upper limit for the eleven experimental results. For each initial size of homogeneous mixed region the coefficient for inertial spreading forms a curve which is asymptotic to the upper limit at the lower Reynolds number R~. Although there are not enough data available to show the value of Reynolds number where the intercept to the upper limit occurs, some idea can be gained from Figure 6.2.6. The larger the initial size, the larger the intercept Reynolds number will be. Similar results are also seen in Figure 6.2.7 for the coefficient for the final collapse stage. The only difference is that the theoretical value for the viscous-buoyancy 0.644 0.321 " 1.042 " " 1. 764 1.375 0 <l l> a 0.904 1.764 " 0.875 X 0.511 ).. " ~ 0.706 1.162 " 0 1.115 1/ " \l 1.612 N (sec-I) + 6 (in) 0 R 390 790 1,270 2,150 450 570 870 11,860 16,400 25,900 37.440 s -V 0- NR 2 1.40 1.37 1.26 1.03 1.40 1.29 1.08 1.18 1.14 1.01 0.92 R(t) (NVt)1/2 t (' 1. 21 1.04 0.88 0.60 1.29 1.13 0.97 0.75 0.71 0.59 0.46 (N~:' R(t) 12 10 6 1 30 31 33 4 1 3 9 Run No. 10 11 12 Figure Source from Wu (1965) - Experimental parameters related to Figures 6.2.3. 6.2.4, 6.2.5. 6.2.6, and 6.2.7. 0 Symbol Table 6.2.1 Vol N -...J /, Ro I Nt I I I 100 --Rill = 0.93 (1iB!. f( Nt f'~ Ro II. I 300 Wu (1965). A plane mixed region collapsing in a linearly density-stratified environment. Comparison of theoretical solutions (the principal and the final collapse stages) for submerged density spreading front with experimental results by I R{ t ) = (17" N t) "2 Figure 6.2.4 R:- R{t) 10 30~r--~-r-r-r.-rr------~----'--'--'-'-Tl-'r-------.---~--'-~-'-r'-'T-------'--~ W N 00 o ll. R( t)= (2NV)t tt I I 01 t -t-,- 2 4 -L R( t ) = 069 ( N",V ) 6 t A plane mixed region collapsing in a linearly density-stratified environment. Comparison of theoretical solutions for submerged density spreading front (the inertial-buoyancy and the viscous-buoyancy regions) with experimental results by Wu (1965). 001 o 0.1 L-...:q Figure 6.2.5 ~ R( t ) i I I 10 3rr--------_r----_r--_r--~_r_r_r,_~--------~----~--~~--._._~~--------~----~--~--,__r~~~ W N \0 (Nvt)i R( t} Figure 6.2.6 10 2 I 3 10 I Ro , ,= '~ Re N lis 2 =~o Ro = I. 375 in. ~_ 10 4 I - - - ______ ~, , 0.875 m. - 1- Q- -Ro= 6 in. ~--~ R ( t ) = ( 2 NV t ) 2 5 10 'I Coefficient of the principal collapse stage (the inertial-buoyancy submerged spread) as a function of Reynolds number Re and the initial radius of N semi-circular or circular mixed region. , -------D-:a:: __ +.::: ==t>- _ "<L oj 4, w w o ( i 0.3 2 10 I-- Figure 6.2.7 N2V4 ~ lis ~ R( t} 3, i 3 '- '- ReN = NRo lis 2 - R( t ) 4 10 I Ro= 6 in. -V, ......... 1'1 II \ I (N:: t)(; 4 - 0,, =0.68 -6--...rt I - i 5 10 - I Coefficient of the final collapse stage (the viscous-buoyancy submerged spread) as a function of Reynolds number Re and the initial size of N semi-circular or circular mixed region. 10 I ....... i Ro= 087~n'a 1.375 in. X-'<l. --o..t~::::~ ______ _ i w w ...... 332 spreading coefficient is almost at the middle value of the experimental results. Other experimental data for comparison with the theoretical solutions for the final collapse stage are taken from Tables 1 and 2 of Wu (1965). The value of upper tangent points of the principal collapse stage are re-calculated and plotted in Figure 6.2.8. (Details of the re-calculation are given in Table 6.2.2.) Equation (6.2.6) obviously is an upper limit for all 26 experimental points. Several numerical calculations were performed using various methods by Cerasoli (1978), Dugan et az' (1976), Meng (1978), Padmanabhan et aZ. (1970), Wessel (1969), and Young and Hirt (1972). All investigators claim that their numerical results are in agreement with the experimental results by Wu (1965,1969), although only Young and Hirt (1972) give all the details of the numerical parameters as shown in Table 6.2.3. Most results of the numerical studies predict values lower than the theoretical solution in the inertial-buoyancy spread region, with the exception of the study by Padmanabhan et aZ. (1970), as shown in Figure 6.2.9. There is no way to compare the numerical results with the experimental results obtained by Wu (1965,1969) for the principal or the final collapse stage, since information such as the initial size and the Reynolds number Re N are not given. The inertial-buoyancy spread coefficient calculated by Young and Hirt (1972) is around 0.98 for the Reynolds number ReN = 301 and the 10 10 :5 ~ Figure 6.2.8 R. Rill 2 10 4 10 2 ZI. NRo ·N t t:?6 6 \ 5 10 1.375 0.875 6 0 , 10' J Plane mixed region in a linearly density-stratified environment: comparison of theoretical solution with experimental results by Wu (1965), for an interflowing front in the final collapse stage. Detail of re-calculation is shown in Figure 6.2.2. Ro I 6" R( t) = 0.93 (" ~~o . Nt) 2 6.0 ( in) Ro 0 SYMBOL II w t; 334 Table 6.2.2 Re-ca1cu1ation of experimental data for the final collapse stage of a plane mixed region in a linearly densitystratified environment. Data are taken from the upper tangent points of principal collapse stage in Tables 1 and 2 of Wu (1965). Nr 2 0 Re -=-Fr vs Re Nt Fr (10 3 ) (10 3 ) 17.0 22.0 28.0 24.0 25.0 5.894 8.335 11.86 16.41 20.12 100.1 183.4 332.1 393.8 502.9 6.3 6.6 6.6 6.2 6.2 25.0 25.0 27.0 26.0 21.0 23.20 25.89 28.21 37.44 40.00 580.6 647.3 761.5 973.4 840.0 1.764 1.619 1.458 1.331 1.162 2.7 3.0 2.9 2.9 3.6 10.0 14.5 13.2 14.5 16.0 2.15 1.97 1. 78 1.62 1.42 21.52 28.63 23.47 23.54 22.68 0.0337 0.0254 0.0213 0.0171 0.0129 1.042 0.904 0.828 0.742 0.645 3.1 3.0 3.4 3.4 3.5 9.6 10.5 11.0 11.0 11.0 1.27 1.10 1.01 0.91 0.79 12.20 11.58 11.11 9.96 8.65 0.0043 0.0032 0.372 0.321 3.3 3.2 6.3 7.8 0.45 0.39 2.86 3.05 0.0966 0.0659 0.0419 0.0254 1. 764 1.456 1.162 0.904 3.7 3.6 3.5 3.4 16.7 13.6 12.2 15.0 0.87 0.72 0.57 0.45 14.55 9.78 7.00 6.70 R -1 dP a e=-- dz I-'s N = (ge)~ (in) (ft- I ) (sec-I) 0.00204 0.00401 0.00801 0.01549 0.02334 0.254 0.359 0.511 0.707 0.866 6.7 6.8 7.2 7.0 6.9 0.03101 0.03861 0.04578 0.08070 0.09210 1.000 1.115 1.214 1.612 1. 722 0.0966 0.0814 0.0660 0.0550 0.0419 0 1 R(t) R0 Nt 6.00 1.375 0.875 Mixed Region Produced by Wa11mixer Padd1emixer Padd1emixer (em) (sec-I) 0.006 0.7062 0.5107 0.25 0.990 0.01549 fC 1 0.00810 ft- l ? 0.001 cm- 1 'fiJ • 0 0 ? ? 15.6 ? ? ? ? ? ? 0 0.820 R N 1 <1 ? ? (0.187 °C/cm) 0.0209 ft- l E ? ? 0 ? s 0.8 cm2 /sec 1 1.2 x 10- 5 ft 2/sec \I 301 ? ? ? ? ? ? s -\I 0- NR 2 Figure 1 Figure 1 Table 1 Figure 7 Figure 3 Figure 4 Figure 13 Finite Difference Equations Harker-and-Cell Method Wessel (1969) Young and Hirt (1972) (1970) et at. Primitive Variable Decomposition Method Vortex-in-Cell ! Method I Finite Difference Method Finite Difference Method Numerical Method Padmanabhan Heng (1978) Dugan et at. (1976) Cerasoli (1978) Source Parameters of numerical calculations related to Figure 6.2.9. • 0 ~ Symbol Table 6.2.3 w w U1 R.. Jilll. 10 Figure 6.2.9 03 o 0 li 0 0 0 0 0 ./ li \ ...L = (1r Nt) 2 li 0 0 0 Nt 0<1 • 10 li 'b¢O ./' li 0 0 o o - I 100 ~: ) = 0.93 ( ~Ro. Nt) , 2 Comparison of theoretical solutions with previous numerical calculations by several investigators as shown in Table 6.2.3. • R;"" R( t) 30rl--.--r~-.-r,,--------r----r--.--r~-.-rTl--------r----'--'--'-.-.-',,--------r---~ w W 0\ 337 initial radius of circular section Ro = 15.6 em. It can be seen that it is a little lower than the experimental results by Wu given in Figure 6.2.6. Three cases of transition from the principal collapse stage to the final collapse stage were given by Padmanabhan et al. (1970) and Dugan et at. (1976). Unfortunately, no comparison between theoretical and experimental results in the final collapse stage is possible, since the value of initial size of mixed region was not given. Similarly, there is insufficient information available about the experiments reported by Amen and Maxworthy (1980) to compare their results with the present theories. 6.2.2 Plane Submerged Current from Continuous Steady Discharge From the theoretical results and dimensional arguments of Chapter 3, it is known that in the absence of any diffusion effects there are three possible successive growth rates for a submerged intrusion front due to a horizontal slot buoyant jet discharging into a linearly density-stratified environment. These three stages are: (I) Starting Jet (6.2.13) (II) Inertial Starting Plume (or Inertial-Buoyancy Balanced Spread), B1 F NQ, R(t) = 12 B 1/2 t 2 1 (6.2.14) 338 (III) Viscous Starting Plume (or Viscous-Buoyancy Balanced Spread) 24)1/6 R(t) = 0.499 N ( VS (6.2.15) It should be noted that not all three stages of spreading are necessarily present in any given discharge situation. end wall effects may modify these results. on three dimensionless parameters. number, Ri. J Also, The results depend The first is the Richardson = Nh./u., which is used to classify whether the J J starting flow is a starting jet or a starting plume. is the Reynolds number, Re. J The second = Q/v s , which is used to determine whether the discharge flow is turbulent or laminar initially. The third is also a Reynolds number, R~. = (NQ)1/2 h./v , which v J s is used to determine whether there exists an inertial plume regime or not. Excluding the case of a starting laminar jet, all three stages of spreading front exist only when the initial discharge Richardson number Ri. is rather small and two Reynolds J numbers, Re. and Re , are simultaneously large. J v If the discharge Richardson number, Ri., is large and the two Reynolds numbers are J both small, there will exist only the viscous starting plume stage --.l)1/2 • (or more precisely, when Rl/~QM is very small,Rl/~QM = ( N2h. u.h. J ~). There are two other combinations of these three parameters v for the two other flow patterns. The details are given in Table 6.2.4. 339 Table 6.2.4 Four flow patterns for a plane intrusion layer due to different combinations of three dimensionless numbers. Dimensionless Numbers Condition Flow Patterns Discharge Richardson Number Discharge Reynolds Number Nh. Ri. = -..J.. Re. =..iL v J Three spread stages R(t) '" t 2 / 3 u. J J s Plume Reynolds Number (NQ)1/2 h. Re v = J v s small large large small large large large large large large small small t t s16 Two spread stages R(t) '" t 2 / 3 t S/ 6 Two spread stages R(t) '" t t S/ 6 One spread stage3 R(t) '" t s16 340 These three dimensionless numbers are useful in predicting the four possible different growth histories of the plane intrusion layer. It is easily seen why three kinds of growth rate were observed by Manins (1976a), Maxworthy (1972), Zuluaga-Angel et aZ. (1972) under different combinations of the three parameters. In an analogous way, Imberger et aZ. (1976) showed why two different growth rates occurred with a selective withdrawal theory with no diffusion effects. As shown in Table 6.2.5 three sets of experiments by Manins, Maxworthy, and Zuluaga-Angel et aZ. have almost the same range of volume flux Q, as does the discharge Reynolds number Re. Q/V s , since J the variation of kinematic viscosity of discharge fluid is rather small. However, the discharge slot thickness 2 h. and the J Brunt-Vaisala frequency, N, of the ambient stratification, did vary one order of magni~ude. Richardson number, Ri. J Consequently, the discharge = Nh./u. = 2Nh. 2 /Q, in these three sets J J J of experiments has a wide range of variation and covers three orders of magnitude. As shown in Figure 6.2.10, Manins (1976a) observed that the length of the intrusion layer increased in proportion with time t in most of his experiments. In some cases, such as in experiments A, B, C, G, and H, the layer decelerated to grow with t S / 6 at the larger times. As given in Figure 6.2.11, Maxworthy (1972) reported that the length of intrusion layer increased with t S / 6 , I I luluaga-Angel. Darden and Fisehpr (! 972) 30.48 38.10 (1. 25 ft) (42 ft) (1 ft) 1280.2 J65.76 10.3 183 (12 ft) (em) (em) (1972) Width Length 45.72 20 (em) Depth - _.. _ - - - - - - (4.00 ft) 121.92 0.5 ftl Experimental Tank ------- 100 to (1/32 in) 0.079/. 1. 93 to and 0.712 0.277 0.170 1.479 0.055 and 1.479 1.0';5 0.635 ? (1/4 In) 2.57 3.94 to 0.72 1& 1. 76 (em 2 /.ec) (.<ec- I ) to 3 (em) 2hl Q Frequency Thirkne!;s Volume Flux N BruntVllI s:tlll !;lnt Inlet to 13 (em) Water Depth s 1.100 to 0.942 ? ? 00- 7 cml/sec) v Vt~cnqity Kinematic 233 to 5.5 540 up to 175.5 to v s aJl. 27.0 Re j Number Slot Jet Reynolds I j 3.87 to 0.091 23.6 to 3.39 Q 0.0081 to 0.00051 Ri 2Nh 1 _ ..'::l Slot Jet Richardson Number ? ? 300 Pr Pundtl Number - -_.. _ - - - - - - t 3/4 t S/6 t Law PowerTime Summary of experimental parameters for submerged plane intrusion layer propagation in a linearly density-stratified environment due to a horizontal slot buoyant jet discharge at its neutral level. Haxworthy (1976al Hanins Investigator Table 6.2.5 "'......" w 342 300 ~~JY ;4~~~/ 100 - ~ Jd 0/ ~VNC<S E 0 ~N/a~ --0::: 10 t~ SLOPE 10 100 TIME figure 6.2.10 t (Sec) Growth history of an inertial starting plume observed by Manins (1976a). Details of experimental parameters are given in Table 6.2.6. 343 Table 6.2.6 Symbol or Run No. 426.4 242.1 259.4 296.8 323.6 0.94 0.80 0.81 0.79 0.87 0 2.55 2.70 1.95 1. 76 1. 87 224 255 86 159 208 5.13 4.76 10.2 4.98 4.05 358.7 393.6 251.8 250.8 296.1 0.90 0.93 0.83 1.00 1.01 p 1. 94 257 3.39 334.6 1.03 + 1.48 1.06 1.06 1.48 1.06 233. 231 161 117 117 0.128 0.091 0.132 0.256 0.183 58.9 49.8 41.4 41.7 35.2 1.30 1. 27 1.06 1.06 1.06 57.3 10.8 5.5 0.371 1.98 3.87 24.7 10.7 7.65 - 5 0.325 0.221 0.406 0.473 0.356 175 136 132 129 82 0.000531 0.000506 0.000927 0.00108 0.00135 2.86 2.16 2.85 2.99 2.13 6 7 8 9 10 0.170 0.284 0.519 0.316 0.219 89 82 88 83 29 0.000601 0.00108 0.00197 0.00120 0.00476 1.55 1.90 2.77 2.04 0.986 11 0.712 0.417 28 27 0.00810 0.00475 1.60 1.31 M N a x • 0 • 'V ~ 1 2 3 4 ., • 12 • (NQ)1/2 7.18 11.5 10.0 8.04 6.39 K L It j 3.78 3.51 3.81 3.39 3.94 72 llO Source R 225 101 116 148 181 J e s 0 ReV 3.59 2.58 2.58 2.65 2.57 F ~ \I .~ u 142 180 183 G H I 0 J Ri j 0.70 0.74 0.84 0.87 0.86 E • (sec- 1) Re. .. JL 247.5 294.7 348.8 370.6 402.6 D 0 N 23.6 14.4 12.1 8.48 9.68 A B C 4 I> <I Experimental data for plane intrusive layer spreading related to Figures 6.2.10, 6.2.11, 6.2.12,6.2.15, 6.2.16, and 6.2.17 • -- Place Author Figure 3 Manins (1976al Figure 4 Maxworthy (1972) Tables 1 and 2 Zuluaga-Angel, Darden and Fischer (1972) - - - - - - 1.25 - 1.12 - Figure 6.2.11 -~ ~ I 0 W ( f) - 0.1 ~ /" TIME / 10 t ( MINUTES) ." Growth history of viscous plume observed by }1axworthy (1972). experimental parameters are given in Table 6.2.6. I 10 L /./ 100 Details of 100 J ~ ~ w 345 especially at the lower source flow rate. He found also that it grows proportionally with two-thirds power of discharge rate, i.e. Q2/3. The intrusion layer thickness, after an intensive mixing near the source point, increased as t 1 / 6 , which is predicted by the force scale analysis and also by the viscous long wave theory. In the experiments by Zuluaga-Angel et aZ. (1972), no discharge Richardson number, Ri. was above 0.01. J It is therefore to be expected that the discharge fluid would undergo intensive jet mixing on entering the stratified tank, just as in the case of the higher flow rate experiments by Maxworthy (1972). The intrusion layer length would then have at least two kinds of spreading history before meeting the end wall of tank. That is to say that the intrusion layer would increase with t 2 / 3 at first corresponding to the starting turbulent jet region, and then with t and tS/6 for both the inertial and the viscous plume region. The criterion to determine which flow pattern would occur between the two possibilities in the latter stage is the plume Reynolds number Re , as defined previously. v As shown in Table 6.2.5 the Reynolds numbers R~ in the experiments performed by Zuluaga-Angel et aZ. (1972) are rather small, 0(1) in magnitude, whereas Maxworthy's are 0(10) and Man ins , order 0(100). Clearly, it is not easy for an inertial starting plume region (i.e. an inertial-buoyancy spreading) to exist behind the starting jet region in those experiments by 346 Zuluaga-Angel et ale As shown in Figure 6.2.12, most of their experiments have two stages of spread, i.e. t 2 / 3 and then t 5 / 6 , except for Run No. R-9. The three-quarters power of time reported by Zuluaga-Angel et ale may be due to the fact that the average value of time power between two-thirds and five-sixths was taken in order to cover the two different regions of flow. According to the three experimental parameters listed in Table 6.2.6, the viscous starting plume region should have appeared after the inertial starting plume in Manins' experiment, however, his experiments were limited by the available length of tank. This is apparently the reason why only one inertial plume was observed by Manins in most of his experiments. Although a slightly longer channel was used by Maxworthy, only the larger time record was presented. It can be surmised that there were also two possible flow patterns which existed before the viscous plume region found by Maxworthy. A comparison of the experimental results by Manins with the constant propagation velocity inertial intrusion front similarity solution (n = 0 and m = i = 1) is given in Figure 6.2.13. In Figure 6.2.14 the intrusion layer thickness near the source point is compared with the self-similar solution, i.e. h(r ~ 0, t) o = R~t) H(O) = ( ~ ) 1/2 H(O) , (6.2.16) 10 2 10 Figure 6.2.12 (em) R( t ) 10' 10 TI ME t (see) Growth history for submerged plane horizontal starting buoyant jets observed by Zu1uaga-Ange1 et al. (1972). Details of experimental parameters are given in Table 6.2.6. 2 SLOPE 3" -...J w +-- 348 6 - I R( t ) = ./2 (N Q)2 t 0 CD tn "- e - 2 4 0 00:: 2 o o 1.0 2.0 3.0 4.0 5.0 I (N Qc ) 2" Figure 6.2.13 -e ( em/sec) Intrusion front velocity as a function of buoyancy flux B1 = NQ. Experimental data are taken£rom Manins (1976a),0, N 2 = 3.5; 0, N2 = 7.0; tJ., N2 = 1.40. 3 0 - 2 ~ H(e -0) h( r-O. t) = 0 --.. ~ ~ 0 ...t - ~O~ .&::. LDOc€ 0 0 0 05 1.0 1.5 -L (~) 2 Figure 6.2.14 (em) Self-similar distribution of intrusion layer halfthickness near the source. Experimental data are taken from Manins (1976a)0. N2 =3.5; O. N2 =7.0; tJ., N2 = 14.0. 2.0 349 where R(O) is equal to 1. That is to say, the intrusion nose velocity is proportional to the product of the thickness near the source point and the Brunt-Vaisala frequency of the ambient stratification. The values of volume flux Q used in Figures 6.2.13 and 6.2.14 are the Qc computed by Manins from the actual flow patterns, rather than an initial discharge value Q. The discrepancy shown in the two figures between the experiments and theoretical value may therefore be due to the small amount of jet mixing near the source point. These errors can be seen more clearly from the direct calculation of the coefficients in Eq. (6.2.14). In Figure 6.2.15 is given the inertial starting plume coefficient, R(t)/(NQ)1/2 t , which is expressed as a function of discharge Richardson number Ri .. J Twenty data points are asymptotic to the theoretical value 1:2/2 at the higher value of Richardson number. Significant discrepancies occur at the lower Richardson numbers. This is logical because more jet-like flows at lower Richardson numbers will induce more jet mixing near the discharge point so that larger values of coefficients will result. A similar explanation can be applied to the value of the viscous starting plume coefficient. As shown in Figure 6.2.16, the two groups of data are both asymptotic to a common value around 0.499 at a higher Richardson number. The two groups of data are separated by the dimensionless number R~, which is the product of the Reynolds number Re. and the square root of J I E R(t) A \ =~ (NO)+ t 0.0. lao. Ri.= J 150 J -U'- Nhj 20.0. 25.0 30.0. The plane submerged front velocity due to a starting horizontal slot buoyant jet discharge at its neutral level in a linearly densitystratified environment. The relevant experimental parameters are given in Tables 6.2.5 and 6.2.6. 50 o.alL--------L-_ _ _ _L-.._ _ _---L-_ _ _ _L--_ _ _- l -_ _ _~ 0.5 F K ;----O--i#_ -G-c__ -s- __ O N L' ...... " ...'R ...... Figure 6.2.15 (NO)f R(t) o La ", 1.5 ,...,-----.------,--------r------.------~---__, o w VI -4 Figure 6.2.16 10 0.1' (N~r:1)t/f I R(t ) ~ -3 10 -2 NII~ Ri· = J R( t) = 0.499 ( Z 4 t T 5 Nh· J U· J , r - 10 II• 1 8t h. o 7.65s _,_J s 58.9 II v ~ The coefficient of submerged plane viscous starting plume (or submerged viscousbuoyancy spread) expressed as a function of discharged Richardson number Ri j and viscous plume Reynolds number Re. Experimental parameters given in v Table 6.2.6. 10 I' • 0.986 S-II.-S . ~ 2.99 " 10 IOrr----r--r-.--rrrr'l' , ---r-r---ro-rrrrr---"-'-'-.--rrrrr--'-~---rn-r--'------'-~i i f-' W \JI 352 Richardson number Ri.. They are actually separated by the J Richardson number, since most of the two sets of experiments cover almost the same range of Reynolds number Re .. J The inertial effects on the viscous intrusion layer spreading can be seen more clearly in Figure 6.2.17. In Figure 6.2.18 it is seen the starting jet coefficient decreases with increasing Richardson number. The eleven data points, which are taken from Zuluaga-Angel et al., are asymptotic to a constant value of around 1.90 at a lower Richardson number. Figure 6.3.l9(a) presents the typical viscous spreading layer profile at each time of measurement as recorded by Zuluaga-Angel et al. The data have a self-similar profile as shown in Figure 6.3.l9(b). The normalizing thickness is (v RR/N 2)1/4, and the horizontal distance is normalized by s the frontal length R(t) each time. Since the frontal velocity was not measured by Zuluaga-Ange1 et al., the value is obtained by differentiating the frontal position function, i.e. R(t) = i ( vS24)1/6 c N t- 1 / 6 in which c is the coefficient in Eq. (6.2.15). (6.2.17) Therefore, the time-varying thickness can be computed directly with time t and the unknown discharge and ambient conditions, i.e. 0_5 I • '\J / 10 I I I R( t ) JOM RI • = 24..L 2 2 Q J J 2 h'I.«NQ)~ 10 • Q.- __ ~ ---... ._x-I I I 5 =0.499 (~) 6 tT J I I I -- --+0 II ..0. ~~ /" 6_~ ~ 0 The coefficient for a submerged plane viscous starting plume expressed as a function of R /t . Symbols are defined in Table 6.2.7. 1 QM 0.21 I Figure 6.2.17 (;Q\'-tl R( t) 4 I II /" II V1 w W 354 Table 6.2.7 SYMBOL and RUN NO. + • 0 • • \l ~ [> g2 N (l0-2 cm 2/ sec ) (l/sec) (cm 2 /sec) 1.0 1.48 2.33 0.910 460 1.06 2.33 0.850 544 1.06 1.61 0.615 314 Figure 4 1.48 1.17 0.569 163 of 1.06 1.17 0.573 192 Maxworthy 1.06 0.573 0.553 66 (1972) 1.06 0.108 0.620 5.43 1.06 0.055 0.460 1.95 .. .. .. .. .. .. X R1 Vj . 0 Calculation of the coefficient of displacement for a submerged plane viscous starting plume as shown in Figure 6.2.17. Q R(t) (N~t y/6 S/ 6 t 1QM - 2h v (NQ)I/2 j j Source 1 1.100 0.325 1.93 1.350 5387 2 1.014 0.221 1.38 1.430 4284 <J 3 1.043 0.406 1.38 1.187 3073 <> • 4 1.072 0.473 1.38 1.060 2770 5 1.014 0.356 0.83 0.993 1575 of 0 6 0.995 0.170 0.89 1.226 2578 Zu1uaga- ~ 7 1.014 0.284 0.83 1.057 1763 Angel et al. <» 8 0.942 0.519 0.83 0.936 1404 (1972) Q 9 0.995 0.316 0.83 0.500 1703 () 10 1.014 0.219 0.29 0.523 414.6 e 11 0.995 0.712 0.277 0.507 218.8 12 1.028 0.417 0.277 0.536 276.7 • Table s 1 and 2 I Figure 6.2.18 M3 t 3 R(t) ..L .1. -3 10 J Ri·J = Uj Nh·J -2 10 4 The coefficient of spreading for a submerged plane starting jet expressed as a function of discharge Richardson number Ri., data from Table 6.2.7. 0.4 41 w V1 V1 356 5 4 Eu 3 .: 2 :E' 5600 see 0 0 100 200 400 300 500 r (em) Figure 6.2.l9(a) Viscous intrusion layer profiles at each time t. Data are taken from experiment No. R-ll in Table 4 of Zuluaga-Angel et ale (1972). 6 5 ~ . oct: ~ o~~~ (\oj a:: Z 3 0 ~ tJ. ... I:. ib VtJ. o'tJ tJ. 4 ~ '-- u V Oll;) V 0 2 tJ. V 0 o 0.0 0.5 1.0 ( . Rtf) Figure 6.2.l9(b) Self-similar thickness distribution of viscous intrusion layer. Solid curve is the viscous long wave solution (3rd order) derived in Subsection 3.3.5. 357 (v:~ ) (% 1/4 = C 2) 1/4 (\t Q2 2 )"6 (6.2.18) Using this scheme the four profiles at each time collapse into a self-similar curve. The theoretical profile R(~) calculated from the second-order nonlinear ordinary differential equation in Subsection 3.3.5, 3R-15i;R' (6.2.19) with boundary condition R(l) = 0 is shown by solid curve. The higher values given by the experimental curve may be due to intensive entrainment due to jet mixing near the discharge point which will give a larger value of Q than used in the normalization. 6.3.3 Radial Submerged Current Only one field observation result of a horizontal spreading plume in a stably stratified atmosphere is available for comparison with the theoretical prediction. Two aerial photographs of a cooling tower plume were taken by Kramer and Seymour (1976) at the John E. Amos power plant (shown in Figures 2.l.2(a) and (b) of Chapter 2). The submerged radial spreading ring structure at one instantaneous time, as shown 358 in Figure 2.1.2(b), is reproduced in the line sketch in Figure 6.2.20. If it is assumed that the center of radial spreading plume is located at the point of the maximum height of plume rise, then four subsequent locations of the circular discontinuity can then be scaled from the figure. Three successive ratios for the radii of the discontinuity ring are calculated to be 0.73, 0.72, and 0.71, which are only 6% higher than the theoretical value 0.692 given previously. Experiments performed recently by Zatsepin et al. (reported in Barenb1att and Monin (1979)) show that the radial viscousbuoyancy spreading current advances proportionally to t IllO in a linearly density-stratified environment. Similarity solutions developed independentlY by Barenb1att and Monin (1979) also show that the plane viscous-buoyancy spreading front increases its length in the linearly density-stratified ambient proportionally to t l/6 and the radial front to tl/IO. 359 8.6 6.1 4.4 3.2 em -rc~ ---C C-S-S----=-.:V ---- - ---. Figure 6.2.20 L----C/ :: Line sketch for a series of radial ring discontinuities in a horizontal spreading cooling tower plume in atmosphere. 360 CHAPTER 7 ENGINEERING APPLICATIONS A primary application of the results of the present investigation is intermediate field problems (buoyancy-driven gravitational spreading) associated with ocean outfall discharge of treated waste waters. Barge disposal of sludge and dredged spoil are also important environmental applications. The results of these investigations can also be applied to the analysis of accidental releases of hazardous fluids, such as the spreading of dense poisonous or explosive gases or liquids. The spreading of cooling water from power plants is a further application. 7.1 Ocean Outfall Discharge Figures 7.l.l(a), (b), (c), and (d) show four photographs and line sketches of a -typical buoyant surface spreading wastewater. field under an ambient cross current. The photographs show the situation prevailing before turbulent diffusion becomes dominant and a sharp boundary between the diluted effluent and the ambient sea water can be clearly seen. The width of the sharp boundaries has been plotted in Figure 7.1.2 to determine the lateral growth rate of the plume. The width of the wastewater field as a function of distance from the source is plotted in Figure 7.1.2 for the four cases shown in Figures 7.l.l(a) - Cd). There is no clearly defined growth rate for these field results although a dependence on xl/2 361 FEET 1000 o 1000 U = 0.34 KNOTS 28 JUNE 1963 ORANGE COUNTY OUTFALL . Figure 7. Ll(a) Buoyant surface spreading wastewater fie l d from submerged single port (diameter 78 inches) discharge, average discharge rate Q = 120 mgd (185.6 cfs), _ submerged depth H = 55 ft, 6 = 0.025. Photograph from Figure 21 of Allan Hancock Foundation (1964). 362 FEET I 1000 o 1000 U= 0.32 KNOTS 28 JUNE 1963 ORANGE COUNTY OUTFALL Figure 7. l.l(b) Photograph from Figure 22 of Allan Hancock Foundati6n (1964). 363 FEET 1000 o U = 0.32 KNOTS 27 JUNE 1963 ORANGE COUNTY OUTFALL Figure 7.1.1(c) Photograph from Figure 20 of Allan Hancock Foundation (1964). 1000 364 FEET 250 I I 0 250 U = 0.16 KNOTS II NOVEMBER 1966 VENTURA Figure 7 . 1.1 (d) OUTFALL Buoyant surface spreading waste\vater field from submerged single port (diameter 36 inches) discharge. Average discharge rate Q = 2.20 mgd (3.40 cfs), submerged depth H = 35 feet. Photograph from Figure 4 of Fo~vorthy and Kneeling (1969). 365 3 10 0 FIGURE 7.1.1 (0) 6 ( b) 0 (c) \7 (d) r ILLJ LLJ ~ >- SLOPEt ~.X x (FEET) Figure 7.l.2(a) Growth of buoyant surface wastewater field shown in Figures 7.1.l(a), (b), (c), and (d). o y o 19 0& o o 6 o 6 0 o o 0.1 y I 6 o 19 o l. (g~<t) 8" (x/U) 2 o V o 012 o 0 00 0 6 o 000 o o ______~~__~~~~~ 10 100 0.4~----~--~-L-L~~~ I xl U Figure 7.1. 2 (b) Dimensionless plotting of growth of buoyant surface wastewater field shown in Figure 7.l.2(a). 366 appears reasonable. A growth rate of x 1 / 2 corresponds to buoyancy driven spreading with viscous friction. In Figure 7.l.2(b) one can see the inertial plume coefficient y/B 1 / 4 Cx/U)3/4, is decreasing while the viscous plume coefficient, y/(g~Q3/v)1/8(x/U)1/2, remains at a constant value, which is around the experimental value given in Eq. (5.3.10). Figure 7.1.3 shows a surface thermal field observed by Sonnichsen (1971). The width of the plume grows as x 3 / 4 which is consistent with a buoyancy driven inertial plume. There are again unfortunately insufficient data to establish the buoyancy flux. 7.2 Surface Buoyant Discharge The intermittent surging of a radial discharge as discussed in Section 3.2 has been observed in the field. Figures 7.2.1(a), (b), and (c) show three line sketches of a thermal plume due to cooling water discharged at a low Froude number from the shoreside power plant. The sequential ratio of the position of the radial discontinuities (as noted by circles) is found to vary from 0.79 to 0.63 for Figure 7.2.1(a), from 0.75 to 0.67 for Figure 7.2.l(b), and from 0.79 to 0.72 for Figure 7.2.1Cc). The average value of this ratio is around 0.74 which is quite close to theoretical value 0.753 given in Subsection 3.2.5.3. As shown in Figure 7.2.2 the oscillating period experienced by a stationary observer in measuring the temperature and the velocity of the 10 Figure 7.1.3 N '+- o t- ->- 10 0 x (ft) 10 3 .y ~ In/SLOPE~ ///~ (e) (d) ~ ~ (c) (b) 0 0 (0) Growth of heated (a,b,c) and nonheated (d,e) surface plume field due to a submerged outfall discharge. Data from Table 1 of Sonnichsen (1971). (a) Q = 75,000 cfs, (b) 110,000 cfs, (c) 145,000 cfs, (d) 50,000 cfs, (e) 100,000 cfs. r 2~ o X 4 10 -1 w --..J 0- 368 Figure 7.2.l(a) Line sketch of the thermal image of a discontinuous radial thermal plume field at Sheboygan Power Station, Wisconsin, on September 15, 1971; aerial photograph taken at a flying height 5000 feet. Photograph was shown in Figure 19 of Scarpace and Green (1973). 369 Figure 7.2.1(b) Line sketch of thermal image of a discontinuous radial thermal plume at Sheboygan Edgewater Plant. Photograph was shown in Figure 8a of Green et aZ. (1972). 370 Figure 7.2.l(c) Line sketch of thermal image of a , discontinuous radial thermal plume at Port Washington harbor on September 1971. Photograph was shown in Figure 8b of Green et aZ. (1972). 2 10 I I • I I I SCARPACE S Period Calculated 2 10 I / T (sec) 0-0 / GREEN (l973) FRIGO et 01. ( 1974) Tob =T =0.699 ( ~ )3 POINT BEACH CAMPBELL I POINT BEACH SOURCE Comparison of oscillating period of inertial radial surface thermal plume with theoretical value given by Eq. (3.2.87). 0 ~ 0 ISYMBOL I POWER STATIOJ 10 ~ I I Figure 7.2.2 0 .c V) Q,) "- ~ '"C ~ "i: 0 '"C 1--0 ..c 3 10 t- 3 10 w -....J f-' 372 thermal plume is also compared fairly well with the predicted value given by Eq. (3.2.87). The periodic sharp temperature gradients induced by this surging are probably undesirable environmentally. In order to eliminate these oscillations in the plume zone, one would reduce the discharge flow rate Q (simultaneously increasing the relative density difference ~ since the waste heat rate of a power plant is fixed). Rl ~ This would decrease the length scale (Q5/g~vs)1/8 so that a viscous surface thermal plume with an almost uniform thickness of order (v Q/g~)1/4 would result. s 7.3 Barge Disposal of Dredged Spoil and Sludge Field observations of bottom turbidity clouds produced by the dumping of the noncohesive dredged spoil in the near-shore ocean waters were made by Gordon (1974). After sinking, the negatively buoyant parcel of turbid water impacted the ocean floor, and an outward spreading turbidity current was produced. The turbidity current formed moved as a homogeneous fluid at a speed substantially higher than any ocean current speed. The turbidity current was measured by observing the turbidity and salinity and a sharp distinct interface was observed. Such a radial turbidity current increases its radius proportionally to tl/2, as shown in Figure 7.3.1, and decreases its thickness at a fixed location r s from the dump site according to 373 2 10 -E SLOPE - 2I ~ --II 0:: o 10 --2------~--~--~~~--~~ 3 10 10 t ( sec) Figure 7.3.1 The time an axially symmetric bottom turbidity current arrived at four fixed locations at a radial distance rs from the dump site of dredge spoil; data from Figure 2 of Gordon (1974). 374 r 2 s her ,t) = -.:..-- (7.3.1) S However, there is insufficient information about the relative density difference D., to compare those field data measured by Gordon (1974) with Eq. (7.3.1). However, an oblique line with a slope -2 is found to fit the field data, reasonably well, as shown in Figure 7.3.2. This is at least consistent with Eq. (7.3.1). 7.4 Base Surge Flow According to the report of the U. S. Atomic Energy Commission (1950), at the Bikini shallow ocean atomic bomb test it was observed that as the column of water and spray constituting the plume fell back into the lagoon, there developed a gigantic cloud of radioactive mist on the surface at the base of the column. This mist wave travelled rapidly outward, a short time after the detonation, maintaining an ever-expanding doughnutshaped form. This dense mist-laden air has the property of flowing almost as if it were a homogeneous fluid. As shown in Figure 7.4.1 its radius increases asymptotically as tl/2 at first and then as tl/8 as predicted by the long wave theories for bottom current developed in Chapter 3. 375 10 )( o -E --.. x o x o 0.1 2 3 10 10 t (seC) Figure 7.3.2 "Wave-gage type" record of the spreading layer thickness of the bottom turbid cloud from three different dump operations measured at a radial distance rs = 30 meters from the dump site; data from Figure 5 of Gordon (1974). 4 3XI0 , 10 10 5 Figure 7.4.1 o ::J ..... 0: W <t 0: o :::> (f) 0: o E --- TIME t (sec) 10 , '2 2 I Propagating mist-laden air forming the base surge following an underwater atomic bomb explosion at Bikini Atoll. ,/ ,p'/ ,/J>' /p ./0 ./" I '" "'-I W 16 ! 4Xl051r-------.-----r--,--~IIII~rT--------~--~--~~--r-~~ 377 CHAPTER 8 DISCUSSION AND CONCLUSIONS Gravitational spreading currents are a type of time-dependent, non-linear, non-uniform and unsteady free boundary flow. Given this it is very difficult to have a complete solution which will cover all regimes of flow and include all possible effects from initial and boundary conditions. This investigation is a step forward in this complicated problem which, however, will remain the subject of much further research. The problem has been approached by finding a set of asymptotic solutions appropriate for each region of flow. These were derived by the method of force scale analysis and the development self-similarity solutions for the inertial (shallow-water wave) and viscous long wave equations. Dimensional analysis was used to find the pertinent dimensionless numbers which control the viscous regimes of the spreading currents. 8.1 The following conclusions may be drawn. On the Similarity Solutions for Buoyant Spreading Currents From the self-similar solutions of inviscid shallow-water waves and the viscous long wave equations, the buoyancy-driven spreading currents grow asymptotically according to i+l (i+l) (n+l)+2 R(t) = c.l.mn B1 m(i+l)+2 (i+l) (n+l) +2 t (8.1.1) for the inertial spreading layer with the spreading layer thickness 378 1 given by 02 h(r,t) = ( g~i ) i+1 H(~) (8.1.2) For viscous-buoyancy spreading currents the displacement is given by 1 g~iQ R(t) = b i mn ( v i+3 ) (n+1) (i+3)+2 m(i+3)+1 (n+1) (i+3)+2 t (8.1.3) s with the thickness given by (8.1.4) h(r, t) in whiih the kinematic buoyancy flux in general form Bi is equal to (g~i)i+3 Q • In the above: i = 0 denotes a uniform density environment i = 1 denotes a' linearly density-stratified environment ~ o =~ the density anomaly of the source 1 dp (z) ~1 = E: = - -Ps n = 0 denotes a plane current n = 1 denotes a radial current m m a dz density gradient of stratified environment denotes the source condition with = 0 an finite volume source in which Q is replaced by V the volume of the source m = 1 a continuous source with flow rate Q Equations (8.1.1) and (8.1.3) for each case of i, n, and m are listed in Table 8.1. i .. 1 Environment Stratified Density- Linearly i - 0 Environment Homogeneous i cas Continuous n .. 0 Continuous n .. 1 Continuous n .. 0 Continuous n = 1 m= 1 m= 0 Radial Instantaneous m .. 1 m= 0 Plane Instantaneous m= 1 m=0 Radial Instantaneous m .. 1 m=0 Instantaneous Plane n Source Condition R(t) = 0.802 (NQ)1/3 t 2 /3 R(t) = 1.127 (NV)1/3 t l/3 R{t) = 0.707 (NQ)1/2 t R(t) = 1.414 (NV)1/2 tl/2 R{t) .. 1.309 (g~Q)1/4 t3/4 R{t) = 1.502 (g6V)1/4 tl/2 R{t) K 1.260 (gaQ)1/3 t R(t) = 1.890 (g~V)1/3 t 2/3 Inertial-Buoyancy Regime lIs lIs t l/S e+l s 2 4 R{t)=0.45(~) 1 110 116 lIla R{t) .. 0.552 (N~V4 ) 2 4 t 1/2 t 1110 t s/6 ( 2 4 1/6 N:) tl/6 R{t) .. 0.499 (~ ) R(t) = 0.689 3 lis R(t) = 0.63 (~ ) tl/2 R{t) .. 0.779 (g~V3 ) R{t) .. 0.804 (g~vQ3 ) 3 lIs R{t) .. 1.133 (g~vV) t llS Viscous-Buoyancy Regime Summary of asymptotic formulae of frontal trajectory functions derived by inertial (shallow-water wave) and viscous long wave theories. ~ Table 8.1 I I I I I ! I W \0 -...J 380 The spreading layer thickness scale, T(t), can be derived from Table 8.1. For the inertial-buoyancy currents, (8.1.5) T(t) and for the viscous-buoyancy currents, 1 0 T(t) = vsRR ( gt. )i+3 • (8.1.6) i Similarity profiles H(~) for each case of i, m and n in Equations (8.1.2) and (8.1.4) can be found in Sections 3.2 and 3.3. 8.2 On the Initial Stage of Flows Before spreading currents develop fully into the case discussed in the previous section, there is a short initial stage which is governed by the initial conditions. For a finite volume release an initial spreading current is found to advance with a self-similar mean velocity distribution given by u(r , t) o - - = R(t) <I> (0 = ~ (8.2.1) and a self-similar thickness profile given by h(r , t) (Rit) )n+1 = H(0 h1 (8.2.2) 0 where the initial thickness scale h1 is developed from the initial shape 381 (see Table 3.2.1 for a homogeneous environment (i=O) and Table 3.2.2 for currents in a linearly density-stratified environment (i=l). In this initial stage the trajectory function of frontal displacement is generally described by T - (1;1) (n+l// J n 2 dn [ 1 - n-(i+l) (n+l) ] (8.2.3) 1/2 1 -' i+l IRo and n = R(t) IRo • in which the dimensionless time T = t "gt:.ih l For a continuous buoyant discharge there is initially a starting jet zone followed by a starting plume flow. The starting jet front grows according to 1 n+3 R(t) IV M 2 n+3 (8.2.4) t The coefficient in above equation (8.2.4) is found to depend mainly on the discharge Froude number. For the limiting case where the Froude number is very large, the coefficient appears to be asymptotic to a value of about 1 for a radial surface buoyant jet with small aspect ratio discharging into a homogeneous environment (i=O). The coefficient is about 1.50 for a plane slot jet discharging into a linearly densitystratified environment (i=l) at a neutrally buoyant level. 8.3 On the Memory of Spreading Currents The asymptotic similarity solutions are derived on the basis of assuming that the fluid is either perfectly inviscid with no friction or viscous with laminar friction. A transition length scale is established, one limit of which gives the inviscid solution, another 382 limit gives the viscous solution. However, it is clear that the co- efficients in these limiting solutions are, to some degree, dependent on certain initial dimensionless numbers such as the Froude number, Reynolds number, and other geometrical ratios such as the relative depth, and aspect ratio of the discharge. The assumption on the asymptotic solutions is that these parameters only enter in a secondary role in certain regimes of flow in which one effect alone is dominant, as for example, the buoyancy flux in the inertial-buoyancy regime. It is clear that some of the effect of the initial discharge parameters is to drift the time origin, or geometric origin of the flow. It is not always clear whether the variation between experiments and theory is to be accounted for with a time origin shift or through functional dependence of the coefficients on the initial dimensionless parameters of the discharge. In other words, a memory for the initial conditions appears to exist although not at a primary level of flow influence. 8.4 On the Thickness Profile of Spreading Currents The thickness profile of spreading currents varies significantly from one type of flow situation to another. In general, the transi- tions in thickness from one regime to another are smooth, but the asymptotic solutions developed in this investigation, of course, do not give such smooth transitions. However, it is clear that it is simply not appropriate to assume an arbitrary distribution for the thickness of a spreading layer, as has been frequently done in previous investigations. The choice of the thickness profile is not free as has been often assumed. The profile thickness appears to be strongly 383 influenced by initial conditions, especially momentum-induced entrainment, which results in significant increases in spreading layer thickness. Furthermore, the influence of interfacial friction between the spreading layer and the fluid beneath is controlled to a large degree by the momentum flux of the discharge. With a strong relative buoyancy flux and weaker initial momentum flux the friction is clearly viscous as the consequence of a laminar shear layer production. This appears to be given by the transition length scale Rl (see Tables 3.1 and 3.5). 8.5 On Dimensional Analysis and Spreading Currents Dimensional analysis is an extremely powerful tool for the development of asymptotic solutions where the major effects of discharge parameters can be reduced to a primary variable such as the kinematic buoyancy flux. The development of characteristic length and time scales based on the relevant parameters appears to give an excellent guide to the appropriateness of given solutions for specific regimes of flow. 384 APPENDIX A DERIVATION OF LONGITUDINAL INTERNAL SHALLOW-WATER WAVE EQUATIONS Two sets of shallow water wave equations are derived here for the inertial regime of surface and subsurface density spreading flows. A.O Basic Equations and Assumptions When surface and interfacial tension forces are neglected, the continuity and the momentum equations for plane or axisymmetric, unsteady, inviscid, and incompressible flow in an infinite depth of fluid are 1 n 0 (r u) rn or + ow = a oz (A.a.l) (A. a. 2) (A.a.3) where u and ware horizontal and vertical velocities in the r and z directions as defined respectively in Figures 3.2.I(a), (b), and (c). The sign before gravitational term g in Eq. (A.D.3) depends on the direction of z chosen; positive when downwards and negative when upwards, p s is the mass density of the spreading 385 plume and p=p(r.z,t) is the pressure at point (r,z) when time is equal to t. The complete solutions of these three nonlinear equations seem i.ntractable and some assumptions are necessary to simplify the problem. Consider the case when the horizontal scale of a surface spreading layer is very large in comparison with the vertical scale. Then the vertical acceleration is a negligible fraction of the horizontal acceleration so that (A.O.4) and the unsteady pressure distribution is hydrostatic -..l..~+g=O p s dz- (A.O.5) Thus the equation of motion is simplified to (A.O.6) Assume that there is no diffusion and no mixing through the interface and no evaporation and no breaking at the free surface, then a particle once on the interfacial surface will remain there, i.e. the density on a particle path is constant. 386 A.I Free Surface Density Spreading Flows In this case the two kinematic boundary conditions on the interface and the free surface of a spreading plume are ah(r,t) ah(r,t) at +u(r,h,t) ar (A.I.l) w(r ,-0, t) =_ ao(r,t) _ u(r,-o,t) ao(r,t) at ar (A. I. 2) w(r,h,t) = and hydrostatic pressure distribution is p(r,z,t) = p g (Z to (r, t) s dz = p g[z+o(r,t)] s (A.I.3) where the superelevation o(r,t) above free surface can be derived as a function of the spreading layer thickness h(r,t) below the free surface from the hydrostatic equilibrium: p gh = p g(h+ 0) a (A.I.4) s i.e. o(r,t) lIh(r,t) (A. I. 5) Integrating the continuity Eq. (A.O.l) and the simplified momentum Eq. (A.O.6) from z = -o(r,t) to z = h(r,t) one finds 387 (A. I. 6) h l-0 ~~ dz + lh ~ dz + ~ lh ~ dz = a -0 dr ps • (A. I. 7) -0 This provides the use of the two kinematic boundary conditions (A.L1) and (A.l.2) and by the aid of Leibniz' rule 1 n d (rnu) dr dz= r ld ndr lh ~ n dh dO rudz-u(r,h,t)ar-u(r,-o,t)ar r-u (A. I. 8) h i -0 i h -0 u ~~ . dz = ddt h r J- o udz - u(r,h,t) ~ dz = l ~ lh u 2 dz dr 2 dr -0 1 -2 ~~ dO u(r,-o,t) dt (A.1.9) u 2 (r,h,t) dh - l u 2 (r -0 t) dr 2 ~ "dr (A. LID) 1 lh -0 pS .E.E. d dr z = g (h+ 0) ~ dr (A. loll) 388 The integral continuity equation becomes a at (h+ 0) + - 1 rn aal (rn u)dz r -0 h o (A.1.12) and the integral momentum equation becomes ~ aa lh udz + 1:. lh u 2 dz + g(h+ 0) t -0 2 dr -0 = u(r,h,t) :~ ah ao ah ao at + u(r,-o,t) at + 2I u 2( r,h,t) ar + 2I u 2 (r,-o,t) ar (A.1.13) Since the spreading layer is very thin, a uniform horizontal velocity distribution is a good approximation, i.e. u(r,z,t) ~u (r,t) (A.1.14) Thus, Eqs. (A.I.12) and (A.I.13) are reduced to a set of nonlinear shallow water wave equations in explicit form, o (A.1.IS) and ~ + u au + gf:" ah = 0 at or ar (A.1.16) 389 which were used earlier by Hoult (1972) to solve the inertial oil slick spreading problem. A.2 Bottom Surface Density Spreading ~lows For the case of bottom surface spreading layer as shown in Figure 3.3.2 there is only one kinematic boundary condition at the interface, i.e., Eq. (A.l.l). If the bottom floor is perfectly horizontal and impervious, then = Ha (A. 2 .1) w(r,o,t) - 0 (A.2.2) H (r) a The hydrostatic pressure distribution is then ~dz - - psg (A. 2 .3) or = Pa g[H a -h(r,t)] + p s g[h(r,t) - z] p(r,z,t) (A.2.4) Integrate continuity Eq. (A.O.l) and momentum Eq. (A.O.6) from z = 0 (rather than from z = -0 (r, t) in the previous case) to z = h(r,t), i.e., 1 d (rnu) {h(r, t) dW ~~~ dz + dz = 0 n dr dZ r 0 (A. 2.5) 390 and rh 1o h au dz + r u au dz + ~ rh ~ dz at 10 ar p S 10 ar o • (A. 2.6) With one kinematic boundary condition (A.I.l) at the interface and the impervious condition (A.2.2), and also by the aid of Leibniz 1 rule, we have h ~ Cl(rnu) dz = ~~ lh rnudz - u(r h t) n Clr n ar " lo rr 0 lh o ~~ dz = ;t lh udz - u(r,h,t) ~~ ~hr (A.2.7) 0 (A.2.B) 0 h u au dz =!~ Ih u 2 dz -!2 u 2 (r,h,t) Clh Clr 2 Clr dr Io 0 ~d dr z ( a) gh-drdh P P 1 - - (A 2 9) • • (A.2.l0) s The integral continuity equation becomes dh + -1 -CI Jh (rn u)dz -CIt n Clr r 0 o (A.2.11) 391 and the integral momentum equation - a lh udz + -1 - a lh u 2 dz + g ( 1P - ~ ) hah at 2 ar P ar o s 0 ah 1 2( ah u(r,h,t) at + 2 u r,h,t) dr (A.2.l2) If a uniform horizontal velocity (A.l.13) were also assumed, the same set of shallow water wave equations (A.l.14) and (A. 1. 15) . Ps-P a Pa-P s is also derived but w1th instead of for ~. Ps Ps The same equations were also derived by Penney and Thornhill (1952) for the squatting flow of a liquid column with a density comparable to the ambient medium. Freeman (1948) also derived similar plane shallow water wave equations for atmospheric cold front propagation, except ~ es - ea es ,where ea and es are the potential temperatures ,of the ambient warm air and the spreading cold air respectively. 392 A.3 Submerged Density Spreading Flows Assume that the submerged spreading interflows are radially symmetric. Let the origin of the chosen coordinate system be at the point of symmetry, as shown in Figure A.3.l. The linear ambient density distribution can be expressed as p (z) = p (1 - EZ) a where E = I P1 dpdza I • s (A.3.l) Based on the shallow-water wave approximation, s the two equations of continuity and of motion become (A.3.2) and dU + u ~ + J:.- 1E. = 0 dt dr p dr s (A. 3.3) The hydrostatic pressure assumption, dp(r,z,t) dZ (A. 3.4) gives = p g(h-z) + p g(H _12 EH 2-h+.!2 Eh 2 ) s S s s (A. 3.5) 393 z Ps h( r, t) r Figure A.3.l Coordinate system of submerged density spread flows. 394 The horizontal pressure gradient can be derived directly by the differentiation of Eq. (A.3.5) dp(r,z,t) or ah p g£h -;;S or (A. 3.6) in which h = h(r, t) is the half-thickness of the interflowing layer. Integrating Eqs. (A.3.2) and (A.3.3) from z=- h(r,t) to z = +h Cr, t), i. e. (rnu) dz + Jh -h r i ow dz = 0 h u h ou dz + dz + Jh -h at -h ar p s -h i ~ ~ (A. 3.7) oz ~ dz = 0 (A.3.8) Then, with the aid of relationships from Leibniz' rule: = - 1 - a lh (r n u)dz n or h r - u(r~h,t) Clh ~r 0 oh - u(r,-h,t) or h ou dz = ~ Jh udz - u(r h t) oh ot ot " ot -h -h i (A.3.9) Clh u(r ,-h, t) at ,(A.3.l0) 395 2 ah u (r,-h,t) ar } (A.3.ll) The hydrostatic pressure distribution gives ~d ar Z = 2p s ah ge:h 2 ar (A.3.l2) and the kinematic boundary conditions at interface, namely, w(r ,±h, t) + ah + ( h ) ah - at - u r,±,t ar (A. 3 .13) gives ah 2 at + n1 ara L(hh (rnu) dz = 0 r (A.3.l4) - and a (h ah ah at Lh udz - u(r,h,t) at - u(r,-h,t) at +!~ Jh u 2 dz - ! u 2 (r h t) ~ - ! u 2 (r -h t) ah 2 ar 2" ar 2 "ar -h (A. 3 .15) Again, assuming that the horizontal velocity is uniformly distributed, 396 u(r,z,t) (A.3.l6) u(r,t) gives the continuity equation n (r uh) ar o (A.3.l7) and momentum equation au + u au + gEh ah = 0 at ar ar (A.3.l8) Both shallow-water wave equations CA.3.l7) and CA.3.l8) were used earlier by Mei (1969) to solve the initial collapse stage of a mixed region spreading in a linearly density-stratified environment. 397 APPENDIX B DERIVATION OF SIMILARITY SOLUTIONS FOR INERTIAL-BUOYANCY SPREAD Two sets of similarity solutions: (3.2.16) and (3.2.32) and (3.2.16) and (3.2.33), for the inertial-buoyancy regime of both surface density spreading flow and subsurface density spreading flow will be derived here. The derivation technique was initiated by Taylor (1950) to determine the spherical shock wave propagation produced by an intense explosion. A similar technique was also used by Lin (1954), Latter (1955) and Rouse (1959). An overall review of these self-similar processes in gasdynarnics was given in Chapter VII of the textbook by Ze1'dovich and Raizer (1967). A mathematical discussion on the general type of self-similar solution was presented in a review article by Barenb1att and Ze1'dovich (1972). Other application to gas flows were also given by Sakurai (1965) and Korobeinikov (1971). B.1 Surface Density Spreading Flows In Appendix A two shallow-water wave equations n ah + ~ a(r u h) at n ar =0 (B.1.1) ~ + u ~ + gtl. ah = 0 (B.1.2) r and at ar ar 398 were derived to describe free surface density spreading and an underflowing density spread. We now assume that the two unknowns, u and h, of this unsteady and non-uniform flow have the following form (B.L3) (B.L4) where the dimensionless variable ~ is defined as r/R(t), and R(t) is the frontal position at time t; k and £ are two unknown rational numbers. Substituting the above two relations into the shallow-water wave equations gives o (B.L5) and (B.L6) In order to homogenize both Eqs. (B.l.5) and (B.l.6) into a dimensionless form, one has to set -k 0 R =A (B.L7) £ - 2k = 0 (B.L8) R 399 and assume in which A is a dimensional constant. Thus, from Eqs. (B.l.3), (B.l.4), (B.l.7), (B.l.S), (B.l.9) and (B.l.lO), the similarity solutions are derived as having the following forms o u(r,t) = R(t)~(~) (B .1.11) and (B.1.l2) h(r,t) and the ordinary differential equations (B.l.S) and (B.l.6) are reduced to (B.1.l3) (~ - ~) ~' B.2 + k~ + H' = 0 (B.1.l4) Submerged Density Spreading Flows The two shallow-water wave equations in this case are n ah + ~ a (r u h) = 0 at r n ar (B.2.l) 400 and (B.2.2) in which N is the Brunt-Vaisala frequency for the linearly densitystratified ambient fluid. Assume now that the two unknowns, u and h, of this unsteady and non-uniform flow have the following form (B.2.3) (B.2.4) where ~ = r/R(t), R(t) is the submerged interflowing front position at time t. Substituting these assumed forms into two shallow-water wave Eqs. (B.2.l) and (B.2.2), gives To homogenize both Eqs. (B.2.S) and (B.2.6) in.o a dimensionless form, one has to set -k 0 R R A (B.2.7) 401 t - k (B.2.8) 0 and assume that CPl(~)=ACP(O Hl(~) (B.2.9) A =N H(~) (B.2.l0) in which A is a dimensional constant. Thus, from Eqs. (B.2.3), (B.2.4), (B.2.7), (B.2.8), (B.2.9), and (B.2.l0) the similarity solutions have the following form o u(r,t) = R(t) cp(~) h(r,t) = R~t) H(~) , (B.2.11) o (B.2.l2) and the ordinary differential Eqs. (B.2.S) and (B.2.6) are reduced to (cp - ~) H' + (cp' + t + nCP) H = 0 ~ (<jl - ~) cp' + kCP + HH' o (B.2.13) (B.2.l4) 402 APPENDIX C DERIVATION OF BOUNDARY CONDITIONS AT THE FRONT C.O Introduction To solve the two similarity forms of solution for the shallow-water wave equations, as derived in Appendix B, it is necessary to find appropriate boundary conditions. For surface and bottom spreading we have u(r,t) = (C.O.l) 0 R(t) and H(~) g~h(r,t) (C.0.2) R2(t) For an interflowing layer we have u(r,t) o R(t) (C.0.3) = Nh(r,t) o (C.0.4) R(t) These functions therefore give the ratio of particle velocity at any location to the frontal propagation velocity, and express the local Richardson number of the spreading layer in terms of the frontal velocity. Fay (1971) (see Liang, 1971) used von Karman's solution, H(O) = g~h(O,t) R2(t) 1 =2 (C.O.S) 403 as the boundary conditions. Hoult (1972) and Fannelop and Waldman (1972) used the downstream spreading layer thickness as the boundary condition of their similarity solutions. More generally, a variable A can be used so that (C.0.6) H(O) In this appendix two possible sets of boundary conditions at the front will be derived based essentially on Whitham's rules (1974) for the jumps in conserved properties across the front. C.l Surface Density Spread The spreading flow motion involves discontinuities in three variables: density, velocity and thickness. While we have two equations, continuity (A.l.lS) and momentum (A.l.16), which are n a(ph) + ~ a(pr uh) at n ar =0 (C.l.l) r and a(pu) + u a(pu) + g~ a(ph) at ar ar o (C.l.2) Equation (C.l.2) can also be rewitten as a(puh) + ~~(~pr~n~u~2~h~) + 1 g6 a(ph2) = 0 at n ar 2 ar r (C.l.3) 404 Now suppose that there is a single discontinuity located at r=rd(t) between two positions r=r (t) and r=rl(t), Le., 2 r (t) <rd(t) <rl(t). 2 Integrating the above three equations from r= r (t) to r= r (t) gives 2 l d(ph) n r dr = 0 at a(pu) (C.l.4) o , (C.l.S) at and a(puh) at (C.l.6) Using Leibniz rule, performing the integrations and then allowing r (t) + rd+ (t) and r (t) + rd - (t) gives the conditions 2 l 0 - rd[ph] + [puh] = 0 0 rd[pu] + 0 or [1:2 pu 2 + pgt.h] rd[puh] + [pu 2h + i (C.l.7) 0 pgt.h2] = 0 (C.l.B) (C.l.9) 405 where [ ] denotes the jump value inside brackets between righthand side and lefthand side of discontinuity at rd(t) and dt (C.1.l0) dt Explicitly, the three boundary conditions at the front for surface density flow are 0 R(t) [puh] [ph] (C. loll) [~ pu 2 + pgilh] 0 R(t) = (C.1.l2) Cpu] and 0 R(t) [pu 2h + 1. pgilh2 ] 2 [puh] (C.1.13) It is interesting to note that the radial flow jump conditions are essentially the same as the plane jump conditions. Both can be easily derived from Whitham's (1974, pp. 138) rules, by three partial differential Eqs. (C.l.l), (C.l.2), and (C.l.3). C.2 Submerged Density Spread The two equations of continuity and of momentumn (A.3.l7) and (A.3.l8), are 406 a(ph) at a(pu) at + ~ a(pr n uh) rn o ar (C.2.I) + u a(pu) + gEh a(ph~ = 0 ar and the second equation (C.2.2) dr (C.2.2) can be rewritten as o (C.2.3) Suppose that there is a single discontinuity located at r=rd(t) between two positions, r=r (t) and r=rl(t), Le. 2 r (t)::;r (t)::;r (t). 2 d l If we integrate the above three equations from r = r 2 (t) to r = r I (t) following the same limiting procedures as previously, we find o R(t) = ~[pc....::u~h,.....] [ph] 0 R(t) = (C.2.4) [12 pu 2 + 12 pgEh2] [pu] (C.2.5) or 0 R(t) = [pu 2 h + 13 pgEh3] [puh] (C.2.6) 407 where [ ] denotes the jump value inside brackets between the righthand side and the 1efthand side at a frontal discontinuity. These three jump discontinuity relationships can also be derived easily from Whitham's (1974, pp. 138) rules applied to Eqs. (C.2.1), (C.2.2), and (C.2.3). 408 APPENDIX D SEDOV'S (1957) LINEAR SIMILARITY VELOCITY FUNCTION A linear similarity function for the mean particle velocity was used by Sedov (1959) to solve the problem of spherical expansion of a gas cloud into vacuum. (For details of this gas expansion flow, see Zel'dovich and Raizer (1967), pp. 276-281.) The reason for selection of this linear function was clearly explained independently by Stanyukovich (1960). The concept of his derivation of this linear function is used here to show that the linear function is a good approximate solution to the initial phase of surface and subsurface density spread flows. The two sets of shallow-water wave equations, which were derived in Appendix A, and used in Chapter 3 to solve the initial stage of density spreading flows, are ah + u ah + h ( au + nu) = 0 at ar ar r au + at u au + ah o ar gL\ ar A (D. 1) (D.2) for the free surface bottom surface spreading and ah + u ah + h (au + nu ) = 0 at ar ar r (D .3) au + au + h ah 0 at u ar ge: ar = (D.4) 409 for sumberged interf1ows. Consider the case when the gradient of the spreading ar ' · k ness, d h .1S rat h er sma11 . t h 1C Then, two pressure gradient terms in both momentum equations (D.2) and (D.4), g~ ~~ and geh dh ar ' are small in comparison with the inertial term. Therefore, the two momentum equations can be approximated by the equation dU + dU dt u ar o (D.5) Equation (D.5) implies that the mean particle velocity must have the form u(r,t) r (D.6) = t which means that the mean fluid particle velocity is proportional to the location r. Substituting Eq. (D.6) in Eqs. (D.1) and (D.3), gives (D.7) or D Q,n h + n+ 1 Dt t o (D.8) 410 which implies that h(r,t) ' \ 11- - (D.9) t n +l For a fixed location r = r , spreading layer thickness is inversely 1 proportional to tn+l. These two forms were found in Section 3.2.7 for the surface spread layer as R h(r,t) = hl (R(~) ) (l+n) (1 - ~2) (D.lO) and for the submerged spread layer as (D.ll) in which two thickness ~cales hI are listed respectively in Table 3.2.1 (for surface spreading layer) and Table 3.2.2 (for submerged spreading layer) for each initial shape of released volume. The frontal distance R = R(t) is proportional to time t when t is very large. 411 APPENDIX E DERIVATION OF VISCOUS LONG WAVE EQUATION FOR DENSITY SPREADING FLOWS Two viscous long wave equations describing the viscous-buoyancy region of both surface and submerged density spreading flows will be derived here. E.O Basic Equations and Assumptions The Navier-Stokes equations describing viscous surface and submerged density spreading flows are (E.O.I) dU + u ~ + w dU = _ llP. + v \/2 u dt dr dZ P dr s dW dt + u dW + W dW = _ llP. + g + v \/ 2 W dZ dr P s dZ- (E.O.2) and the continuity equation is l r n d (r u) n dr + dW = 0 dZ (E. 0.3) in which u and ware respectively the unsteady horizontal and vertical velocity of the spreading fluid, and P is its mass density and v is s its kinematic viscosity. \/2 is the Laplacian operator. The coordinate system for free surface spreading flow is shown in Figure 3.2.l(a) and that for submerged spreading flow in Figure 3.2.I(c). The positive sign before the gravitational term g is for the surface spreading case and the 412 negative sign for the submerged spreading case. In order to simplify the above three equations, the following assumptions are made: 1. A long wave approximation is applicable. The horizontal length scale is much larger than the vertical length scale so that the value of the vertical component of fluid particle velocity is negligible in comparison with that of the horizontal component, i.e. w « 2. The inertial force is negligible in comparison to the viscous force, i.e. Du/Dt « 3. u, v V2 u and Dw/Dt « v V2 w, A boundary-layer type assumption for the viscous force terms is applicable so that the viscous force terms are dominated by the vertical gradient terms rather than horizontal gradient . a2 u terms, 1.e. W » a2 u ~ and a2 u W » 1 au r ar • With these assumptions, the Navier-Stokes equations are simplified to a relationship giving a balance of horizontal pressure force with viscous force, (E.O.4) and a hydrostatic pressure distribution given by o (E.O.5) 413 E.l Viscous Surface Density Spread Since the horizontal gradient of the spreading layer thickness is rather small, the super~elevation of a spreading layer above the free surface will be given by p o(r,t) = a ( - p ps s ) h(r,t) (E.1.l) which can be derived from the equilibrium of hydrostatic pressure along the interface. The pressure force inside the spreading layer is then p(r,z,t) = fZ -o(r,t) P g dz = p s g(z+toh) (E.1. 2) s where the dimensionless density difference to is defined by to = (p - p ) / p. ass Thus, the horizontal pressure gradient becomes dp(r,z,t) dr = (E.1.3) We assume that the vertical gradient of horizontal fluid velocity at the level of free surface will be vanishing, i.e. du(r,O,t)=0 dZ (E.1.4) 414 and that the value of the horizontal fluid velocity at the interface is negligible, i.e. u(r,h,t) o (E.1.5) Integrating the approximated Navier-Stokes equation, a 2u _ _ 1_ 2.E. _ ~ ah az 2 - v Ps ar - v ar (E.1.6) twice, gives the horizontal component of fluid velocity, u(r,z,t) = - gbh 2 ah 2v ar Integrating the continuity equation from z (E.1. 7) -o(r,t) to z = h(r,t), gives I h (r,t) ..1.- a (rnu) -8(r,t) r n ar dz + w(r,h,t) -w(r,-8,t) = 0 (E.1.8) so that with the kinematic boundary conditions w(r,h,t) ah ah = at + u(r,h,t) dr w(r,-o,t) = a8 at a8 u(r,-8,t) ar (E.1.9) (E.1.10) 415 one has (E.1-11) From Eqs. (E.1-11) and (E.1- 7), the integral term in Eq. (E. 1.11) becomes , Jh u dz - -gt. ah (h+o) (h 2 - 1. ho _ 1. 0 2 ) -0 - 2v ar 3 3 :t - gt.(h+o)h 2 ah 3v dr (E.1.12) Thus, the continuity relationship (E. 1.11) is reduced to o (E.1-13) or (E.l.l4) For the case of a plane viscous spreading layer, Eq. (E.l.l4) becomes (E.l.l5) 416 which is in the same form derived by S. H. Smith (1969), P. Smith (1969) and Buckmaster (1977) for a plane viscous sheet horizontal dry bed (see Figure 3.2.l(b» advancing over a except for the reduced gravita- tional term gIl. We assume that the similarity solution for the thickness of a viscous spreading layer will be 9, h(r,t) ::: R (t) H(~;) (E.!.l6) Substituting this assumption into the viscous long wave Eq. (E.l.14) gives (E.!.l7) In order to homogenize the dimension in both sides of Eq. (E.l.17), one must set (E.1.l8) so that the similarity assumption (E.l.16) becomes h{r,t):::(~~) 1/3 H(t:) (E.1.l9) where the dimensionless variable t: is defined as r/R(t), R(t) being the frontal displacement at time t. 417 E.2 Viscous Submerged Density Spread The coordinate system for the present case is shown in Figures 3.2.l(c) and A.3.l. Assume that the density of interflowing fluid is uniform and the flow is radially symmetric. The Navier-Stokes equations for this case become (E.2.l) for the horizontal component and o = - J:... P ~ dZ - g (E • 2 • 2) s for the vertical component. The unsteady pressure distribution at each point (r,z) becomes p(r,z,t) ~ dz = _rh dZ 1a g dz - p a rz p s g dz ~ s (E.2.3) and the horizontal pressure gradient becomes (E.2.4) 418 where (E.2.5) e: = is considered constant. The interflowing fluid particle velocity can be found by integrating Eq. (E.2.l) or Eq. (E.2.6): o2u __1_lE. _ ge:h oh oz2 - vp or - v or (E.2.6) s from z = 0 to z for the first integration ou -= oz z o2u _ .8.fh 2Q. ° oz2 dz v or z I (E. 2.7) and from z = h to z for the second one (E. 2.8) u(r,z,t) and using the boundary conditions ° (E.2.9) u(r,±h,t) = 0 (E.2.l0) ou(r,O,t) = oz and 419 Integration of continuity Eq. (E.O.3) from z -h(r,t) to z = h(r,t), with the boundary conditions ah + u(r,±h,t) ar ah w(r,±h,t) = at (E.2.1l) gives 2 ah 1 a (h n at + -n a; Lh (r u) dz = 0 r (E.2.l2) - Substituting Eq. (E.2.8) in Eq. (E.2.l2), gives the viscous long wave equation for the interflowing spreading layer as o (E.2.13) or more simply s 2 S ah _ ~ (a h + .!!. ah ) at - l5v ar2 r dr (E.2.l4) Assuming a similarity solution of the form (E.l.16) gives the reduced ordinary differential equation as o l5v R 4£-1 (~H - F,;H') gER 420 so that for dimensional homogeneity (E.2.l6) and the similarity solution must be in the following form o h(r,t) = (~~) 1/4 H(~). 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