Hypersonic Wakes Citation McCarthy, John Francis (1962) Hypersonic Wakes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/63NK-AG38. https://resolver.caltech.edu/CaltechETD:etd-03302009-092458 Abstract An experimental investigation was made of the flow field behind a two-dimensional circular cylinder at a nominal Mach number of 5.7. The free-stream Reynolds number based on the cylinder diameter was varied over a range from 4300 to 66, 500 by changing both the diameter of the cylinder and the stagnation pressure of the wind tunnel. Pitot-pressure, static-pressure, and total-temperature measurements were made at various distances behind the cylindrical rod in order to determine the state properties in the wake. Base-pressure measurements were also taken at various Reynolds numbers. From these measurements, the transition from laminar to turbulent flow in the wake was determined and successfully correlated with other data. A transition Reynolds number based on local conditions and the length of laminar run was determined. Extensive comparison of the experimental data with Kubota's theory for laminar flow was then made. A satisfactory comparison was made between theory and experiment. Because of the nature of the tests conducted, only a qualitative comparison was made with the theory of Lees and Hromas for turbulent flow. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Aeronautics and Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Kubota, Toshi (advisor) Lees, Lester (advisor) Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1962 Record Number: CaltechETD:etd-03302009-092458 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-03302009-092458 DOI: 10.7907/63NK-AG38 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1219 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 31 Mar 2009 Last Modified: 05 Oct 2021 22:09 Thesis Files Preview PDF (McCarthy_jf_1962.pdf) - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

HYPERSONIC WAKES Thesis by John Francis McCarthy, Jr. In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1962 ACKNOW LEDGIIMENTS It is with a deep sense of indebtedness and gratitude that the author presents this expression of appreciation to those individuals without whose assistance this thesis would never be: to Professors Toshi Kubota and Lester Lees for their interest, enthusiasm, guidance, and inspiration during the course of the research; to Dr. H. A. Storms, President of the Space and Information Systems Division of North American Aviation, Inc., for his interest in this project in connection with AICBM research; to Mrs. Truus van Harreveld for her extraordinary competence in performing most of the computations; to Professor C. B. Millikan for his support at GALCIT; to Messrs. T. Dolan, G. Brajnikoff, C. Borteck and R. Sogard of North American Aviation, Inc., for their help in performing the machine calculations and final assembly; to Messrs P. Baloga; S.. Roman, J. van Dijk, S. Hall, and G. van Halewyn of the GALCIT hypersonic wind tunnel for their assistance during the experimental tests; to Messrs. C. Bartsch and R. Bartsch of the GALCIT aeronautics shop for building the experi- mental equipment; to Dr. J. Kendall and Mr. R.Wood of the Jet Propulsion Laboratory for the use of equipment for total-temperature measurements; to Dr. L. Hromas and Mr. W. Murray of Space Technology Laboratories, Inc., for their assistance in the turbulent wake calculations; to Mr. J. Paup, S&ID Vice-president and Apollo Program Manager, and my wife, © Marne, for their perseverance. This study is part of a general investigation of hypersonic wakes being conducted at GALCIT under the U. S. Army Research Office and the Advanced Research Projects Agency, Contract No. DA-04-495-ORD-3231. ABSTRACT An experimental investigation was made of the flow field behind a two-dimensional circular cylinder at a nominal Mach number of 5.7. The free-stream Reynolds number based on the cylinder diameter was varied over a range from 4300 to 66,500 by changing both the diameter of the cylinder and the stagnation pressure of the wind tunnel. Pitot-pressure, static-pressure, and total-temperature measurements were made at various distances behind the cylindrical rod in order to determine the state properties in the wake. Base-pressure measurements were also taken at various Reynolds numbers. From these measurements, thetransition from laminar to turbulent flow in the wake was determined and successfully correlated with other data. A transition Reynolds number based on local conditions and the length of laminar run was determined. Extensive comparison of the experimental data with Kubota's theory for laminar flow was then made. A satisfactory comparison was made between theory and experiment. Because of the nature of the tests conducted, only a qualitative comparison was made with the theory of Lees and Hromas for turbulent flow. Section Tl III IV VI vu CONTENTS INTRODUCTION General Discussion Technical Approach EXPERIMENTAL TECHNIQUES GALCIT Hypersonic Wind Tunnel Pitot-Pressure Measurements Static-Pressure Measurements Total-Temperature Measurements Base-Pressure Measurements DATA REDUCTION Pitot Pressure Static Pressure Total Temperature IBM 7090 Computations Discussion of Accuracy CONDITIONS NEAR BODY Separation on Cylinder Conditions at Neck Neck Width and Neck Drag THEORETICAL CONSIDERATIONS AND COMPARISON WITH EXPERIMENT Definition of Wake Edge Laminar Wake Tubulent Wake TRANSITION . . . . "Universal'' Turbulent Reynolds Number Experimental Determination of Transition CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK REFERENCES | Page 10 10 13 24 51 55 60 60 61 62 63 64 68 69 75 80 83 83 85 100 105 105 108 112 APPENDICES Appendix Page A Flow Calibration and Visualization . . . . . 119 B Heated Thermocouple . . . . . . . 125 Cc Transport Properties . . . . . . . 128 D Laminar-Wake Theory . . . . : . . 134 Figure Ow WwW Hh IO 9b 10 11 12 13. 13, 13, 13. 13. 13. 13. 13, 13. 13. 13. 13. 13. 13. 13. 13, 13, 13. 14 15 16 Schlieren Photograph of the Flow Around Cylinder The Flow Field to be Analyzed ILLUSTRATIONS Model Installation Leg, I, GALCIT Hypersonic Wind Tunnel Installation of Variable-Reluctance Pressure Transducer Pitot-Pressure Probe . Pitot-Pressure Recording System . Electronic Circuitry of Recording Systems Typical Pitot-Pressure Traces: Typical Pitot-Pressure Traces: Typical Calibration of Pitot Pressure Recording System Mechanism for Obtaining Angle- -~of- Attack Sensitivity of Probes Angle-of-Attack Sensitivity of Probe Isoaxiometric Isoaxiometric TIsoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Isoaxiometric Tsoaxiometric Isoaxiometric Isoaxiometric Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Pitot-Pressure Static-Pressure Probe Static-Pressure Recording System Typical Vertical Static-Pressure Survey Pitot-Pressure Traces Traces Traces Traces Traces Traces Traces Traces Traces Traces Traces Traces Traces Traces . . Traces Traces Traces Traces Laminar Flow Turbulent Flow Page eS) OT eB 16 17 18 20 20 21 23 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Figure 17 18a 18b 18c 18d 19 20 Zla 21b 2lc 22 23 24a 24b 24c 24d 25 26 27 28 29 30 31 32 33 34 35a Horizontal Traversing Mechanism for Static-Pressure Probe . . : . : Typical Horizontal Static-Pressure Survey: x = 2.00 Inch . . . Typical Horizontal Static- Pressure Survey: x = 3.50 Inch : . . . Typical Horizontal Static- Pressure Survey: x = 4.97 Inch . . Typical Horizontal Static- Pressure Survey: x = 7.00 Inch . . . Two-Dimensional Region of Tests : . Angle-of-Attack Sensitivity of Static- Pressure Probe Axial Static-Pressure Traces: d= 0. 300 Inch Axial Static-Pressure Traces: d= 0.200 Inch Axial Static-Pressure Traces: d= 0.100 Inch Typical Temperature Distribution for Wire with Unheated Supports Heated Thermocouple . . . . Total-Temperature Surveys: Pog = 85.00 psig Total-Temperature Surveys: Pog = 60. 00 psig Total-Temperature Surveys: Pog = 35.00 psig Total-Temperature Surveys: po = 10.00 psig Apparatus for Measuring Base Pressure Experimental Pressure Distribution on Cylinder Transducer Circuitry for Base-Pressure Measurements . : . : Variation of Measured Impact Pressure With Reynolds Number (Figure from Reference 28) . Pressure Distribution on Static-Pressure Probes (Figure from Reference 28) . . Recovery Temperature Versus Knudsen Number (Figure from Reference 14) Experimental Static Pressure . Pitot-Pressure Survey Close to Model Step Model : . . Boundary Layer Separation and Plateau Pressure Rise Coefficients (Figure from Reference 33) : . . Base Pressure Versus Reynolds Number: Relative to Total Pressure Page 46 47 47 48 48 49 49 50 50 50 52 53 54 54 54 54 56 57 58 60 61 63 68 71 71 72 76 Figure 35b 36 37 38 39 40a 40b 40c 4la 4lb 4le 42a A2b 43a 43b A3c 43d 44 45 46 A7 48 49 50 5la 51b 52 Base Pressure Versus Reynolds Number: Relative to Static Pressure . . Separation Angle and Turning Angle Versus Reynolds Number Angle of Shear Layer Versus Reynolds Number Mach Number Versus Reynolds Number Wake Width at Neck . . . Experimental Wake Widths: d= 0.300 Inch Experimental Wake Widths: d= 0.200 Inch Experimental Wake Widths: d= 0.100 Inch Comparison of Theory and Experiment—Simplified Laminar Theory: x/d = 6.00 Comparison of Theory and Experiment— _Simplifiea Laminar Theory: x/d = 9.00 . Comparison of Theory and Experiment— Simplified Laminar Theory: x/d = 15.00 Typical Results: Normalized Static Enthalpy Typical Results: Normalized Velocity Comparison of Theory and Experiment ~ Laminar Theory with Effective Origin: Normalized Static Enthalpy, d = 0.300 Inch . . . Comparison of Theory and Experiment - Laminar Theory with Effective Origin: Normalized Velocity d = 0.300 Inch . . . . . Comparison of Theory and Experiment - Laminar Theory with Effective Origin: Normalized Static Enthalpy, d= 0.100 Inch . . . Comparison of Theory and Experiment - Laminar Theory with Effective Origin: Normalized Velocity, d = 0.100 Inch . Axial Variation of Static- Enthalpy Excess Axial Variation of Velocity Defect Static-Enthalpy Excess Along Wake Centerline Velocity Defect Along Wake Centerline Momentum Thickness : . : Theoretical and Experimental Wake Thickness Theory of Lees and Hromas Experimental Shock Shape With Derived Static-Enthalpy Profiles: Shock Shape Experimental ‘Shock Shape With Derived Static-Enthalpy Profiles: Inviscid Static Enthalpy Profiles Theoretical Turbulent Wake Growth | Page 76 77 77 79 81 84 84 84 88 89 90 92 93 94 94 95 95 96 96 97 98 99 100 101 103 103 104 Figure 53 54a 54b 5Ac 55 56 57 58 59 60 61 62 63 64 Table Il III IV Diffusivity Parameter Variation Transformed Diffusivity Number versus Transformed Axial Distance: Turbulent Case Transformed Diffusivity Number versus Transformed Axial Distance: Transition Case Transformed Diffusivity Number versus Transformed Axial Distance: Laminar Case Transition . . System for Calibrating Wind Tunnel. Reservoir Temperature Rake Installation . Pressure Contours for Empty Wind Tunnel Schematic of Heated Thermocouple Probe Coefficient of Viscosity for Air Versus Temperature Values Used for Coefficient of Viscosity ‘for Air at Low Temperatures Thermal Conductivity of Dry Real Air Specific Heat at Constant Pressure for Dry Real Prandtl Number for Dry Real Air TABLES Test Summary . Conditions for Separation on Cylinder Conditions Around Neck . . Coefficient of Viscosity for Air Based on Lennard-Jones Potential 109 109 109 111 120 121 123 126 128 131 132 133 133 130 NOMENCLATURE Constant determined from initial conditions Constant determined from initial conditions Chapman-Rubesin constant in linear viscosity-temperature relation, coe eT eoeT r Pressure coefficient, ae Skin~-friction coefficient, q Specific heat at constant pressure 2 Diffusivity number, iim *— yo In —-= w . 2 . — lim y Transformed diffusivity number, — W y*o 1, —& w Cylinder diameter Total-enthalpy excess, = - 1 e, Total-enthalpy ratio, = e Total enthalpy Static enthalpy Thermal conductivity Length of two-dimensional flow Mach number Mach number which would exist at 0. without separation Wake-width constant defined by, wee = Static pressure Atmospheric pressure Base pressure Free-stream stagnation pressure, absolute Free-stream stagnation pressure, gage Pressure increment Prandtl number Heat transfer to the body Dynamic pressure Gas constant Free-stream Reynolds number based on cylinder diameter, Poe Yoo mi 00 Reynolds number based on distance along body and conditions at edge of boundary layer Au Ye Townsend Reynolds number, - T Absolute temperature Adiabatic wire temperature Stagnation temperature Component of velocity parallel to wake centerline u Transformed u, — Ve a) Angle of cylinder measured from stagnation point 0. Turning angle 7] Wave angle w BL Dynamic viscosity coefficient E Variable in x-direction p Density a o Prandtl number, > r Shear stress T Time constant p Angle of trailing shock 3 . : : : M,, Co Xx Hypersonic viscous interaction parameter, Re x oe Subscripts: & Centerline e Conditions at outer ''edge'' of wake m Measured r Reference Ss Conditions at separation point 0 Initial conditions (usually taken at neck) fos) Free-stream conditions < =| | <| Component of velocity normal to wake centerline Transformed v Dependent variable oo Laplace transform of W, f We “pé ag fo) Velocity defect, 1 - — e Length along wake centerline measured from center of cylinder - PoHe Me dx Transformed xf —— —, measured from neck Po HU 0 d Effective origin for x upstream of neck Distance measured from beginning of boundary-layer growth Length normal to wake centerline measured from center of cylinder u y Transformed y, 1 [Re, f > x ioe] Oo 0 Angle of shear layer VM* - 1 Specific-heat ratio Wake thickness, 2y, Wedge angle Angle of wake growth Turbulent diffusivity Variable in y-direction [oa] u pel -—) ay Momentum thickness, 2 f Pele Ue -~1l- I. INTRODUCTION GENERAL DISCUSSION Wakes present one of the oldest basic problems in the field of clas- sical fluid mechanics. Although the low-speed regime of wakes has been discussed in several treatises, (e.g., see References ] to 6), it was only until recently that research was directed toward the phenomenon of high- speed flow. The development of intercontinental ballistic missiles and hypersonic reentry vehicles has rekindled interest in high-speed wakes. One of the more interesting aspects of hypersonic wakes is the extreme length of observable phenomena. For example, it is reported by Jacchia (Reference 7) that during the descent of the 1957 Soviet satellite, Beta One, the burning object trailed a luminous tail approximately 100 km long A representative comment of Jacchia's relevant to the visual observation of Beta One's descent cites the spectacular character of the object's reentry: "Tail was 25-30 nautical miles long; from white it degraded into dark red and then into a black smoke trail without seeing the end of it." Contemporary techniques employed in predicting observables are as yet inadequate because the theoretical models devised for this purpose have not received experimental confirmation. While some information about gross quantities (i.e., wake width) has been obtained from downrange -2- measurements and ballistic ranges (e.g., References 8 and 9), itis difficult to obtain detailed state properties which would permit the basic structure of the wake to be determined experimentally. For example, Lees and Hromas (Reference 10) were able to confirm their theory only by shadowgraph and schlieren measurements taken in ballistic ranges; conse- quently, no confirmation of state properties has thus far been possible. For the laminar flow regime, the severity of the situation is even more critical because of the dearth of existent experimental data. Perhaps the major obstacle in correlating theory and experiment in hypersonic wakes resides in the difficulty of obtaining reliable and detailed experimental data. Atmospheric reentry involves a temperature/Mach regime which cannot be effectively simulated by known devices for extended periods. Therefore, only limited data have been obtained, either from short-duration experimental facilities (shock tunnels and ballistic ranges) or from actual flight -- an expensive, unreliable technique. Unfortunately, these data have been inadequate in proving even the most unsophisticated theories of hot, viscous hypersonic wakes. One tool generally overlooked in the study of hypersonic wakes is the conventional wind tunnel. Although the wind tunnel does not simulate the high Mach number and the high temperature attained during reentry, it does make possible the check-out of theoretical models by permitting the measurement of state properties throughout a hypersonic flow field. One major advantage inherent in wind tunnel testing is the ability to vary ~3. Reynolds number -- an extremely important parameter in viscous and wake phenomena -- independently of other quantities, enabling the Reynolds num- ber effect to be determined explicitly. A systematic hypersonic wake study program employing the hypersonic wind tunnel as the experimental tool has been in progress at GALCIT for some time. Figure 1, which is a schlieren photograph of a circular cylinder at Mach number 5.7, enables the flow field to be segregated into various classical regimes. These regimes are presented in Figure 2. Each of these regimes can be treated analytically by suitable techniques. After individual analyses, these regimes can be figuratively assembled ina manner analogous to piecing together a jigsaw puzzle -- matching boundary conditions and, if necessary, using iterative procedures. The present investigation is focused upon the wake region shown in Figure 2. At GALCIT the problem has been grossly divided into three regimes: (1) neck, (2) near wake, and (3) far wake. This thesis is one of several wherein the near wake is investigated. In the near wake regime the static pressure has not yet reached the ambient, an effect which may contribute to the wake structure. Previous investigations have been only concerned with specialized measurements. For example, Demetriades (Reference 11) has investigated transition by hot-wire anemometry; Mohlenhoff (Reference 12) and Kingsland (Reference 13) have studied mixing by means of helium and argon diffusion; Dewey (Reference 14) is extending the work of Demetriades and will study the neck region; Behrens (Reference 15), gepuljAD punorzy MozTq ey} jo ydeasojoyg usreTTyIg “T Ansty Ol X $799 = Poy 69°G =O W Jo£9Z 2|Sd 00°S8 “NI 008°0 SER SARI OO pia SEPARATION POINT BOUNDARY LAYER SONIC LINE SHEAR LAYER NECK FLOW fo a que = SHOCK WAKE INTERACTION REGION SUBSONIC /TRANSONIC REGION ee INVISCID SUPER/HYPERSONIC REGION DEAD AIR AND REVERSE FLOW REGION BOW SHOCK WAVE Figure 2. The Flow Field to be Analyzed investigating the far wake by utilizing very thin heated wires as models, is engaged in determining flow properties in the far wake by means of hot-wire anemometry. Lees and Hromas (Reference 10) have indicated several important differences between the low-speed and high-speed wake structure behind blunt bodies, viz: the existence of a stable shear layer (see Figures 1] and 2); the very small initial momentum thickness or drag at the neck compared to the total drag of the body; and the absence of characteristic shedding frequencies, at least in unionized air at supersonic and hypersonic speeds. The theoretical problem of hypersonic wakes was first treated by Feldman (Reference 16) who devised a simple model of the wake flow. -6- Feldman's basic approach has been extended by Lykoudis (Reference 17) who, however, added no analytical innovations. More recently, Lees and Hromas (Reference 10) have attacked the problem of turbulent diffusion in the wake by using integral methods to solve the boundary-layer equations. The two-dimensional laminar hypersonic wake with streamwise pressure gradient has been solved by Kubota (Reference 18) using linearized equations. The purpose of the present investigation is to obtain reliable experi- mental data of the wake flow behind a two-dimensional circular cylinder -- the simplest model that could be devised. With these data supplemented by other experiments, transition from laminar to turbulent flow in the wake can be defined as a function of Reynolds number. Comparison between theory and experiment for the laminar region using Kubota's theory is then possible. Comparison between theory and experiment in the turbulent region will then be made to the extent possible as limited by the accuracy of the data. Whenever possible, an attempt will be made to define "univer- sal'' distributions of state properties based on experimental data and to determine the variation of centerline values with Reynolds number. TECHNICAL APPROACH The configuration for the experimental investigation is shown in Figure 3 where the basic model, a cylindrical rod, is mounted in the side plates of the hypersonic wind tunnel. The number and spacing of the vertical supports used for holding measuring probes were such that the TYPICAL TWO - DIMENSIONAL \ CYLINDER MODEL SUPPORTS / ae INCHES FROM 0 THROAT 10 Figure 3. Model Installation wake could be investigated at any desired downstream axial station. To obtain a systematic variation of Reynolds number, the 11 combinations of cylinder diameter and wind tunnel stagnation pressure shown in Table I were tested. The 0.300-inch-diameter cylinder was closeto the model which could be conveniently tested in the wind tunnel without starting difficulties. The 0. 100-inch-diameter model represented the smallest model that could be tested with enough resolution in measurements to give an adequate definition of state properties. The wind tunnel stagnation pressure, as limited by the compressor plant, could be varied between the 0.00 and 100.00 psig limits. Table I Test Summary MODEL Pog TS Reg INCH PSIG OF x1073 0.300 85.00 263 66.5 0.300 60.00 263 49.4 0.300 35.00 262 32.7 0.300 10.00 262 16.7 0.200 96.68 264 49.3 0.200 59.48 263 32.7 0.200 22.48 262 16.6 0.200 3.80 260 8.58 0.100 59,42 263 16.3 0.100 22.54 262 8.28 0.100 3.84 260 4.29 Table I indicates that an attempt was made to duplicate Reynolds number by varying both the density and the model size. An exact duplica- tion was not possible because of the day-to-day variations in atmospheric pressure and the variations in free-stream Mach number caused by variable boundary-layer thickness on the nozzle walls with stagnation pressure. The 3-to-1 ratio in model size and the 6-to-1 ratio in absolute stagnation pressure, promised the possibility of obtaining a sufficient variation in Reynolds number necessary to the determination of its effect on wake properties. After selecting the combinations shown in Table I, a complete defini - tion of the flow field was attempted through experiment. Three state -9- properties were selected for experimental investigation: (1) pitot pressure, (2) static pressure, and (3) total temperature. Pitot-pressure measurements drew initial attention because these were obtained accurately and quickly; they could also be used to define the geometry of the flow field as well as determine one state property. Regarding static pressure, an experimental verification was attempted to demonstrate that the static pressure was constant with verti- cal distance in the wake (as expected from theory). Static pressures were then employed to determine the two-dimensionality of the flow field. Finally, axial traverses were made along the centerline to obtain a second state property. Minimum effort was expended in determining total-temperature because the model used was almost completely insulated -- a factor which evoked only slight variations in total temperature. However, sufficient total-temperature data were gathered to determine the dependence of this state property on free-stream Reynolds number. In the process of investi- gating the 'near'' wake characteristics, it was necessary to make base- pressure measurements in order to adequately extrapolate static-pressure data upstream. Once the flow field was established experimentally, the data were used to define transition from laminar to turbulent flow in order to separate these two regimes. ''Universal'' curves were defined whenever possible and comparisons with theory were made as limited by the nature of the data. - 10 - ll. EXPERIMENTAL TECHNIQUES GALCIT HYPERSONIC WIND TUNNEL All tests were conducted in the GALCIT Hypersonic Wind Tunnel, Leg 1 (Figure 4). The test section of this tunnel is 5 inches in width and 5-1/4 inches in height. The wind tunnel is a continuous-flow, closed- return device with a nominal fixed Mach number of 5.7 in the region where the model was ultimately placed (17. 34 inches from the throat). A complete description of the compressor and the associated instrumenta- tion is given by Baloga and Nagamatsu (Reference 19). The reservoir pressure ranged from 0.00 to 100.00 psig, with an accuracy of +0. 02 psig with corresponding Reynolds numbers between 33,000 and 260,000 per inch based on free-stream conditions. The automatically controlled reservoir temperature can be varied between 225 and 325 F. A reservoir temperature of 262 F was selected for all tests. This temperature closely approached the minimum for good flow without condensation effects for the nominal operating dew points (based on the test section survey discussed in Appendix A). The maximum temperature was limited to approximately 275 F by the saran tubing used in the pressure-recording systems. The reservoir temperature could be maintained constant to within +l] F. The nominal operating dew point was less than -40 F; however, after approximately eight hours of steady il Leg I, GALCIT Hypersonic Wind Tunnel Figure 4, -12- testing the dew point began to deteriorate. No data were taken at dew points higher than -25 F. To obtain maximum distance for the wake studies, two methods were employed to determine the most forward location of the test section rhombus. First, the entire test section was surveyed by a conventional rake, and pressure contours were established by feeding data into a conventional IBM 7090 program. (Details of this survey are discussed in Appendix A, where contours for a low stagnation pressure are presented. } The flow deteriorates with decreasing stagnation pressure. The second method for determining the most forward position suit- able for wake measurements was to change the axial location of the cylindrical model. Five different axial locations were selected. The most forward location was 13.12 inches from the throat; the most aft, 24. 82 inches from the throat. Pitot-pressure surveys were made at each of these locations at various axial distances behind the model, and duplicabil- ity was determined. This procedure was precipitated by the unfortunate experiences of Mohlenhoff (Reference 12), Kingsland (Reference 13) and the author (Reference 20) each of whom tested too far upstream. In these latter tests unusual pitot-pressure surveys were obtained, which could not be repeated. On the basis of these two methods, it was determined to locate the model position as far upstream as possible~—17. 34 inches from the throat (Figure 3). The downstream limitation of flow uniformity was determined -~13- by disturbances orginating slightly upstream of the junction of the model with the wind tunnel's sidewalls (discussed later in this section) and not by the test-section rhombus. PITOT-PRESSURE MEASUREMENTS Obtaining reliable experimentally verified total-pressure surveys at various axial distances in the wake and the adjacent inviscid flow region constituted one of the experimental program's major efforts. Fortunately, total pressure is one of the quantities which can be easily and accurately measured by experiment. This quantity not only provides one flow parameter, but also represents an accurate measurement of wake geometry. Perhaps the most difficult problem encountered in obtaining satis - factory pressure surveys was the attainment of a low time constant. As shown in Reference 21, assuming Hagen-Poiseuille flow in the tubing, the time constant of a pressure recording system is 7 (Vol)Zz 4 pd where 7 is the time constant fis the coefficient of viscosity of the measured gas Vol is the internal volume of the system £is the length of the tubing dis the diameter of the pressure probe p is the measured pressure Therefore, the pressure recording system was designed so as to minimize f. The small variable-reluctance pressure transducer shown in Figure 5 was selected as the basis for a very-low-time-constant pressure-recording system. Consequently, the volume of the transducer with its glass measuring tip was very small; also, the length of the tubing from pitot tip to diaphragm was minimal so that the time constant as defined by Equation 1 was minimized. The time constant of this pressure- recording system was about 50 milliseconds, as determined by driving the probe through the bow shock generated by the cylindrical model. The magnetic circuit of this transducer consists of a shell, a magnetic coil, and an air gap—the air gap is varied by the deflection of WATER IN/OUT (TYPICAL) VACUUM REFERENCE TUBE TRANSDUCER WIRES MEASURING PROBE “\ CALIBRATION PROBE SHELL DIAPHRAGM 0.001" TO 0,005" MAGNETIC COILS Figure 5. Installation of Variable-Reluctance Pressure Transducer - 15 - the diaphragm due to differential pressure. The range of the transducer can be varied from about +0.1 to +0.5 psig for the diaphragm thicknesses (0.001 to 0.005 inch) shown in Figure 5. Obviously, the range of the transducer decreases as the sensitivity is increased. Details of the theory and construction of this transducer are given by Smotherman (Reference 22). The pressure transducer was excited by a 5-volt 20, 000-cycle carrier signal which was modulated by the magnetic circuitry, producing as final output a dce-signal up to 75 millivolts. Unfortunately, the final dc output of the transducer system was very sensitive to temperature. Since thermal environment to which the transducer was subjected varied over a large range, it was housed in a head and water-cooled so as to provide a uniform temperature during a vertical transverse (Figure 5). The temperature of the transducer was monitored by cementing a standard thermocouple on the transducer case (see Figures 5, 6 and 7). Because of the characteristics inherent in the variable reluctance transducers used for the total-pressure measurements, it was necessary to use extreme care in the design and employment of the associated electronics. All wires had to be carefully shielded and excessive lengths avoided. An isolation transformer was used between the power source and the equipment (Figure 7). The transducer, installed as a conventional two-arm Wheatstone bridge (Figure 8), incorporates provisions for balancing the circuitry. AXIAL DRIVER. WATER IN ia 4CUUM REFERENCE VERTICAL. out SUPPORTS 5 ries WIRES FROM TRANSDUCER THERMOCOUPLE WIRE CA LIBRA TION TUBE TUNNEL PORT 1D. o, 013% OLD, 0, G4H! Figure 6. Pitot-Pressure Probe Recording System Before the pressure signal was introduced into the autograph recorder, it was cleared of high-frequency noise by the low-pass filter shown in Figure 8. The filter was designed by conventional techniques (Reference 23). In designing the electronic circuitry, a floating shield was necessary to prevent ground loops. The entire electronic circuit was grounded at one location only (Figures 7 and 8). The pitot-pressure probe was moved vertically from outside the tunnel by a rack which was activated by a motor-driven pinion (Figure 7). The probe-positioning gearing turned the helipot potentiometer which converted the probe position to a linear electrical signal. The electrical Signal, in turn, was fed into the autograph recorder. l? MALAIT HIN Uy Gasol) t YoLLAWONV AW NAdO F NadoO € Gaso1D @ C(HSOTD T OUaZ UNV NMOCULOHS AFONUNOAS GAIVA SONSAYAAAYH WOANDVA YAONAGSNVUL OL urajshkg BuIpsooay sanss9Ig-1Old YANYOASNVUL NOILWIOSI ~\ Ss$vd MOT @ qAGHOoONY SUN LVYFdIWaAL LNIOd MHC TANNOL NMOUeE ANIM ATA NOCOWNYAHL YAMOd DILVINAHDS ‘), oanst gy YWAGHOOSY AIvVudDoL nay Tt SOLLVWaHDS LINDUID DNIGUOORY GONVILSIC _ WAGNITAD NdOud NOILVUATTYD GONG YAM WANoOVA HAONUSNVEL _+— adoud Aunssadd-IVLOL NOINId ONINOILISOd AdOud MALAWOLLNGLOd LOdITSH dvul TLLLLLLL ELL © — TANNOL MoV 18 - 1.34 Vv SW ono. 150n N 4| tf {BATTERY PROTECTION) 20 Ka(ZERO ADJUSTMENT) A AWN v BANANA PLUG TO X-INPUT CONNECTOR 200A SLIDER Cw CCW DISTANCE RECORDING CIRCUIT SCHEMATIC i HELIPOT BOX (FLOATING SHIELD) | R= 478 | am Ce L “30.22 MFD (420%) IN L=305h | + ° + Rp = 82000 —_ ouT + CI Lc dc R - 2 2 1 OUT-FEMALE l | | Lo aM IN-MALE om Loo LOW PASS FILTER SCHEMATIC 2 FLOATING SHIELD —. _ _ TRANSDUCER a -----==- | | | a t | OSCILLATOR 5 VOLTS R, Ry 20 KC | | | DC OUTPUT | DEMODULATOR AMPLIFIER J CARRIER AMPLIFIER SCHEMATIC 3 Figure 8, Ri7 RESISTIVE BALANCE Ry> CAPACITY BALANCE Electronic Circuitry of Recording Systems - 19 - The autograph recorder is a null-balancing device which plots the data automatically. The electronic circuit for recording distances is shown in Figure 8. A 200-ohm helipot with linearity of 0.1 percent was used. The 200-ohm resistance of the helipot was much less than the resistance of the autograph recorder (2000 ohms); consequently, the linearity of the recorder was unaffected by the input signal. All distances could be recorded to £0,001 inch. The final data were obtained as plots of pitot-pressure versus distance. In the final test setup shown in Figures 6, 7, and 8, the reference side of the transducer was connected to a vacuum reference with a large effective capacitance, consequently, fluctuations in any component of the vacuum reference pressure system had little effect on the reference vacuum pressure. The measuring side of the transducer was connected to the glass tip. The tip was about ]l-inch long and had the dimensions shown in Figure 6. The outside diameter of the apex of the glass tip was selected at 0.039 inch—small enough to obtain satisfactory resolution (Figure 9) and large enough to minimize the effects of Reynolds number (Section III). To calibrate the transducer, another glass tip of exactly the same geometry as the original measuring glass tip was located a known distance, 0.149-inch, (Figure 6) below the measuring tip. This calibration tip was connected to the system shown in Figure 7 and was calibrated bya mercury micromanometer with which pressures could be measured to 0.001 cm mercury. - 20 - d =0.30 IN. x/d = 9.00 Po = 35.00 PSIG g P= 74.49 CM Hg atm T =262F fo] _ 3 Re = 31.7X 10 0 | N l | i ~3 -2 -] 0 1 2 y/d (a) Laminar Flow le d =0.300 IN. x/d = 20 .00 PS = 85.00 PSIG 9 P am = 74.75 CM Hg T= 263° F Reg =65.1 X 10° i 1 it n =3 0 1 y/d (b) Turbulent Flow -2 -1 Figure 9, Typical Pitot-Pressure Traces _21- The calibration procedure was to record the final voltage output of the pressure-recording system, move the probe vertically 0. 149,inch, and measure the pressure with the mercury micromanometer. A typical calibration of the pressure-recording system is shown in Figure 10. All calibrations were obtained with the pressure-recording system installed ready for use. Pressure was varied by changing the stagnation pressure of the wind tunnel. The calibration resistances A - Gof the carrier ampli- fier allowed for easy adjustment of the sensitivity of the autograph recorder. It is interesting to note that the linearity of the system was limited by the accuracy in reading (p/P, = 41077) rather than by any electronic component. 2.07 © PRESSURE INCREASE APRESSURE DECREASE OUTPUT (volts ) 0.8+ 0.45 0 i 1 1 L 1 f j 0 4 8 12 16 20 24 26 DIFFERENTIAL PRESSURE (cm Hg) Figure 10. Typical Calibration of Pitot Pressure Recording System - 22 - Axial positioning was provided by the driving mechanism shown in Figure 6. Because of the nature of the installation, the probe was pivoted about a point downstream of the diffuser. An error induced by the angular motion about this pivot point would be introduced during a vertical survey if no provisions were made. Angular errors in axial positioning were eliminated by the locking nut shown in Figure 6. To preclude angular errors in axial positioning, the probe was driven to the desired axial position in the forward direction before tightening the locking nut from outside the wind tunnel. The axial position of the probe was determined by a counter attached to the axial driver. The counter was calibrated by inserting machine shim stock between the measuring probe and the cylin- drical model after temperature equilibrium was established through several hours of wind tunnel operation. Finally, because flow inclinations up to 15 degrees were encountered in the flow field, the device shown in Figure 11 was designed to determine the angle-of-attack sensitivity of the pitot probe. The long lever arm afforded sood accuracy for small angles of attack, especially since the calibration counter had a turn ratio of 100 to 1. The results of this calibration are given in Figure 12, showing that the pitot probe is relatively insensitive to angle of attack in the range of interest (+15 degrees). Typical pitot-pressure traces are shown in’Figure 9. Isoaxiometric visualization of the entire flow fields for the combinations given in Table I ~ 23 - CURVED RACK [7 SETAINER BAR STATIC PRESSURE PROBE INTERCHANGEABLE WITH ATION COUNTER GLASS PITOT TIP tl PIVOT Sse E-OF-ATTACK INPUT . SET SCREW TO ANGLE-OF-ATTACK RECORDING SYSTEM TO PRESSURE RECORDING SYSTEM TUNNEL PORT & Figure 11. Mechanism for Obtaining Angle-of-~Attack Sensitivity of Probes 0.04 Py = 35.00 PSIG Potm = 74,34CM Hg g T = 263 F ° actual or a _— = AP, — — -S _ — 7 Pp = “~ Po, 85.00 PSIG 0.04 F- l L j | ~0..08 ! l i L i -25 -20 715 -10 “5 0 5 10 15 20 Figure 12. Angle-of-Attack Sensitivity of Pitot-Pressure Probe - 24 - are shown in Figure 13. Figures 13.la, 13. 2a, 13. 3a, and 13.4a represent enlarged versions of the flow field around the neck. Figures 13.9a, 13.10a, and 13.1lla are the same as Figures 13.9, 13.10 and 13.11, respectively, but they are presented in reduced scale for vertical distance. Tunnel-empty surveys were made for each station of Figure 13 as a check on the quality of flow. STATIC-PRESSURE MEASUREMENTS Because of the extremely low static pressures measured, it was not practical to use the small variable-reluctance transducers shown in Figure 5. Attempts were made to use adaptations of these transducers with the static-pressure probe (shown in Figure 14) inside the wind tunnel; however, the inherent inaccuracies of the carrier-amplifier system prevented this scheme. Therefore, a very sensitive pressure transducer was employed. Transducer size, however, made the assembly ofa pressure probe around the transducer impractical for measurements inside the wind tunnel. The time constant of the static-pressure recording system was approximately two seconds. Despite extensive precautions to prevent leakage (i. e., use of vacuum grease and glyptol at joints), it was necessary to encase the transducer in a vacuum vessel to obtain the required accuracies. In this manner, the pres- sure differential across any face was made very small. The overall schematic of the static pressure recording system is shown in Figure 15. With the silicone manometer used, pressures could be read to an accuracy of 1/10 micron of mercury. The valve sequence -~ 25 =- Model: d = 0.300 inch Pog = 85. 00 psig Datm = 74.69 cm Hg To = 2639F M.. = 5. 69 Reg = 66.5 x 10 FLOW Py x 1? Po yid Figure 13.1, Isoaxiometric Pitot-Pressure Traces - 26 - Model: d = 0.300 inch Pog = 85. 00 psig patm = 74.69 cm Hg to = 263°F M = 5. 69 Req = 66.5 x 10° 58; y Figure 13.la. Isoaxiometric Pitot-Pressure Traces - 27 = Model: d = 0.300 inch Pog = 60. 00 psig Ty = 263°F M,25.71 Reg = 49.4 x 10? FLOW Py x 102 Po Pe) 4 es Figure 13.2, Isoaxiometric Pitot-Pressure Traces _ 28 - Model: d = 0.300 inch Pdg = 60. 00 psig patm = 74.53cm Hg tg = 263°F Mo = 5.71 Req = 49.4 x 10? Bp x 102 Po FLOW xid= 683 yld Figure 13.2a, Isoaxiometric Pitot-Pressure Traces - 29 - ‘Model: d = 0.300 inch Pog * 35. 00 psig Patm = (4.33 cm Hg To = 2620F M75. f1 Reg = 32. 7 x 107 a, yi Figure 13,3, Isoaxiometric Pitot-Pressure Traces -~ 30 - Model: d = 0.300 inch POg = 35. 00 psig patm = 74.33 cm Hg ip = 2620F M2 5.7] Reg = 32.7 x 10° Pp x 102 FLOW 5 683 yd Figure 13.3a,. Isoaxiometric Pitot-Pressure Traces ~ 3) - Model: d = 0.300 inch Datm = 74.36 cm Hg To = 262°F M wo = 5. 64 Req = 16.7 x 10° FLOW Pox 102 yld Figure 13.4. Isoaxiometric Pitot-Pressure Traces - 32 - Model: d = 0.300 inch POg = 10. 00 psig patm = 74.36 cm Hg tg = 2620F M. = 5.64 Reg = 16.7x 10° Bp x 102 FLOW Ln yid id= | 683 yd Figure 13.4a. Isoaxiometric Pitot-Pressure Traces ~ 33 = Model: d = 0. 200 inch Pog * 96. 68 psig Datm = /4.44cm Hg To = 2640F M.. = 5.69 Reg = 49.3 x 10 yld Figure 13.5. Isoaxiometric Pitot-Pressure Traces _ 34. Model: d = 0. 200 inch Pog * 59. 48 psig Tp = 263°F M.=5.71 Rey = 327x 10° o3 yid Figure 13.6, Isoaxiometric Pitot-Pressure Traces -~ 35 - Model: d = 0. 200 inch Pog * 22. 48 psig Ty = 262°F Me = 5, 68 Req = 16.6 x 1? FLOW Bp x 102 Bo » yd 4 Figure 13.7. Isoaxiometric Pitot-Pressure Traces ~ 36 - Model: d = 0. 200 inch Pog * 3. 80 psig Datm = £4.69 cm Hg Tp = 2600F M,, = 5.58 Req = 8.58 x 10 2 P p3 yld Figure 13.8, Isoaxiometric Pitot-Pressure Traces - 37 - Model: d = 0.100 inch Dog = 59. 42 psig Datm = 74,54 cm Hg To = 263°F M25. 71 Reg = 16.3 x 107 Figure 13.9. Isoaxiometric Pitot-Pressure Traces - 38 - Model: d = 0. 100 inch Pog * 59. 42 psig Datm = /4.54 cm Hg To = 263°F M.=5.71 Reg = 16.3 x 10° Figure 13.9a. Isoaxiometric Pitot-Pressure Traces - 39 - Model: d = 0. 100 inch Don = 22. 54 psig Patm = 74.56. cm Hg To = 2620F M..= 5.68 Reg = 8. 28 x 10? Figure 13. 10. Isoaxiometric Pitot-Pressure l'races - 40 - Model: d = 0. 100 inch Pog * 22. 54 psig Daim = /4.56 cm ‘Hg T- = 2620F M. = 5. 68 Reg = 8. 28 x 10? Figure 13.10a. Isoaxiometric Pitot-Pressure Traces ~ 4) . Model: d = 0.100 inch Do, = 3. 84 psig Daim = /4. 71 cm Hg To = 260°F M,, = 5.58 Reg = 4. 29 x 10? Figure 13.11. Isoaxisometric Pitot-Pressure Traces -~ 42 - Model: d = 0.100 inch Patm = 74. 71 cm Hg To = 260°F M.. = 5.58 Rey = 4. 29 x 102 Figure 13.lla. Isoaxiometric Pitot-Pressure Traces ~ 43 - AXIAL DRIVER VERTICAL SUPPORT KEYWAY TO SILICONE MANOMETER SET SCREW i, 264" |_ —— - LOCKING NUT STATIC PRESSURE ORIFICES 0,042" 3 HOLES #79 DRILI. 0.0145" DIA 30° ANGLE TUNNEL PORT Figure 14. Static-Pressure Probe was designed to prevent two occurrences: the loss of silicone into the wind tunnel and the entry of atmospheric air into the pressure-recording system. To evacuate the entire system, two ]1-hp vacuum pumps were operated constantly for several weeks before recording final data. The reference pressure on the transducer was maintained at about 4.0 microns of mercury; it was measured by a conventional Stokes gauge equipped with a liquid-nitrogen cold trap. As with the pitot-pressure measurements, the static-pressure recording system was calibrated by varying the stagnation pressure of the wind tunnel over a wide range. The electronic circuitry shown in Figures 7 and 8 was also used for the static-pressure recording system. wtejshg SuIprooey oanss eTq-oeig WA LSAS DONIGUODTY DINOULOATA OL ~ 44. WANnoVWA UIONASNVAL Ya LAWONVA ANOOITIS dasold/Nado 9 NadoO 1 Nado ¢ aasoto + Nado € [sic tela semr4 "GT eans1g NMOCLOHS WONOVA/AVaE dgaso lod 9 NadoO Zz dasolo € Nado tT NadoO ¢ __a4s0T9 + UaALAWONVA Nado 9 Nado € GaAsOTO @ Nado + dgsoto i agsoto ¢ OuazZ ANV NMOG.LOHS SONGNOTS FATWA dwnad OVA OL gonvo SaMoLs oL } Vb LL YALINOLLNALOd \ Ass LOdI1aH nomnta {9} ~———___// ; a | = MOVa LINOYID ONIGUOODS GONVILSIG OL =|e -~- 45 . To establish the fact that the pressure was constant across the wake, as expected from theory (discussed in Section VI of this thesis), several vertical static-pressure surveys were made for various-sized cylinders at different axial locations. A typical vertical static-pressure survey is shown in Figure 16. It should be noted that the static pressure is constant across the wake and almost constant up to the trailing shock. To test the two-dimensionality of the flow field behind the cylindrical rod, the apparatus shown in Figure 17 was designed. Here, a conventional lathe bed was used to drive a static-pressure probe horizontally. Again, distances were converted to voltages by the electronic circuitry shown in Figures 7 and 8. Typical horizontal static-pressure traces are shown in Figure 18. Several trace surveys were made at various axial positions 1.6 ) x 10° 0.8 — d =0.300 IN. x/d = 15.00 Po = 85.00 PSIG g P am 7 74:13 CM Hg 0.4 T= 26S F = 3 9 Re, =65.0 X 10 down——+—~ up 0 | 5 -4 =3 -2 -1 0 1 y/d Figure 16. Typical Vertical Static-Pressure Survey - 46 - TUNNEL PORT TO PRESSURE RECORDING SYSTEM Y CALIBRATION POINTER TO DISTANCE RECORDING CIRCUIT HORIZONTAL POSITIONER CONVENTIONAL LATHE BED Figure 17. Horizontal Traversing Mechanism for Static-Pressure Probe downstream of the 0. 300 and 0.100-inch models. These surveys revealed that a disturbance originated ahead of the intersection of the model with the side walls of the wind tunnel. As a result of these tests, the limits of two-dimensional flow (Figure 19) were identified. As noted, the distance Lop? was a function of model size and wind tunnel stagnation pressure (Figure 21). As with the pitot-pressure probe, the angle-of-attack sensitivity of the static-pressure probe was obtained through use of the mechanism shown in Figure 1l. The results of this experiment are shown in Figure 20. Unlike the pitot-pressure measurements, however, the flow inclination around the static-pressure probe is very nearly zero; therefore,. — ulo ° ) x08 e/p,) x 10° _ 47 - 1.8 or MODEL: d=0.100 IN, x =2.00 IN. 14h 047 39 PSIG P itm = 73-84 CM Hg 12p 1, = 263°F | NY 0.6F 0.4- 0 RIGHT —«-——}+——_ LEFT (LOOKING UPSTREAM) 0.2b (CENTERLINE) z-INCHES z-INCHES 0 4. 1 ye rm i -3 -2 a) 0 1 2 (a) 2.00 Inch Downstream Station 1.8 1.6 MODEL: d = 0.100" x = 3,50" 1.45- Pog = 59.42 PSIG Patm = 73.82 cm Hg 2b Ty = 263° F 1.0 0.6-- 0 0.4 RIGHT} LEFT (LOOKING UPSTREAM) 0.2 z-INCHES (CENTERLINE) z-INCHES | i | | | -3 -2 -] 0 ] 2 (b) 3.50 Inch Downstream Station Figure 18. Typical Horizontal Static-Pressure Survey (p/p) x 10° (p/po) x 108 ] ] 1. I _ 0. 0. 0. 0. 8 ~ 48 . - OF- MODEL: d = 0.100" x = 4,97" 4 Pog = 59.42 PSIG Patm = 73.83 em Hg -2/- To = 263° F 0) 8 ee aa 6 4b 0 RIGHT —_——!_|—> LEFT (LOOKING UPSTREAM) 2r- (CENTERLINE) | z~INCHES , | \ z~INCHES | 0 3 -2 ~| 0 ] 2 (c) 4.97 Inch Downstream Station 1.8 MODEL: d=0.100" 1.6 x =7.00" Pog = 59.42 PSIG 1.4 Patm = 73.84 cm Hg Ty = 263°F 1.2 1.0 0.6 0 0.4 RIGHT + LEFT (LOOKING UPSTREAM) 0 | | | l l ~3 -2 cd | 0 1 2 (d) 7.00 Inch Downstream Station Figure 18, Typical Horizontal Static-Pressure Survey - 49 - FLOW INTERACTION REGION CAUSED BY SIDEWALL BOUNDARY LAYER, MODEL, MODEL AND MODEL'S BOW SHOCK BOW SHOCK BOUNDARY vver\\ TUNNEL SIDEWALL —\ A\AAL |) \ ™ a ) — | _ —, ™—~ Mo | —™ ——_ — NL | REGION OF MEASUREMENTS -—>— — | __— | — — — | — — | — — || en —~| dL TUNNEL SIDEWALL — bg (d,P.) Figure 19. Two-Dimensional Region of Tests .02 re x Pog = 85.0 PSIG 01 Ay 0 ~ Pog = 35.0 sig -.01 F -.02 F Patm =74,12 To = 263° F -.03 : 4 1 -2.0 -1.0 0 1.0 2.0 @ (DEGREES) Figure 20. Angle-of-Attack Sensitivity of Static-Pressure Probe - 50 - S90eA], OINSSOTG-Olelg [eIxy ‘[?7 ernst TapurtAg Toeyeuretgq yout OOT‘O (2) zSputpA9 tejyeuretq yout 9az‘o (q) ort OL 08 o or oz 0 oS or oe 02 OL Q — _- — i pe oat 10 «pI ° on oooh cone * : - wow eo =» 0 T00W Y ' R00" _ nl bo —t| a0 P. 1 - - wn gO OU ogo Keone ~ HH OL ov x ozo = Py i 5 . a fot xe 91 = Pm Xen as yw Te —- cee x : op 2228) 1. Ol X Fal = Pe ‘ og JP, - i X eS Ge ~ yoo poe pee ie " — — ~~ — a a . P, 01 X Cor = Poy | | ce — —F —. 4. oe ee ce OLX Lee w Pay - i i ° ou Pay 01x 62 a Pay “gt WO wad Oh X 9H wy UX eee Poy epee ow Ob inawalixda—— nn - i— — 7 AOL X 85° 8 m= Pay ctxt ne? eee aw o ! i | 0X ere Poy age OW a oNniva == : i P : ee rr — + LW pl X 1 "oes we LNW 183d Xj me | |. nr hn ! XL ee ey ge OW a — sy = — TP rt rt ban Oi x ear = Py “sree ow o + [ a i _ _ i oe oF ! ONIIVS ===! L | | ore TepurfAg reyoureiq yout go¢e ‘0 (e) P/x oe St 0% $1 ot s (0) T HONI O0€°O =P JaGOW > | oa [nd WN ofa i238 om tO K $99 = “~N an IN Ot X 964 = Pay WN a OLX 2'ze = Poy —- vt OLX 2°91 « Poy NY. | | ‘ "1 01x £91 = Poa y9*g = @Wo avs 9-H]. 01 X 969 = Poy NN, . e e t—-pOLX LE = MH izeg= ewofl. — € P, INIWR3EXS EE ae Poy t t / t o_o eda Ao 01 X ¥'ér = Wiese ews p | ,01 X $799 = Pay gore = wo pel Xela ONISIV4 ----- - 5] - the only error in angle of attack is due to installation which is estimated to be accurate to less than 1 degree (the order of magnitude of the flow uniformity). The static pressure is not a minimum at zero angle of attack (Figure 20) because the three static-pressure holes (Figure 14) were not symmetrical about the horizontal axis of symmetry. After establishing that the pressure was constant across the wake — and after identifying the boundaries of the two-dimensional region behind the model —static-pressure traces were taken along the centerline of the wind tunnel and behind the models for the 1] conditions cited in Table I. The results of these tests are shown in Figure 21. As with the pitot traces, the static-pressure traces were again checked by measuring the pressure at several different stations on various days. Estimated accuracy of the static-pressure measurements is ae = +0.005. As with oo the pitot-pressure measurements, all axial static pressure traces were supplemented by corresponding tunnel-empty surveys. TOTAL-TEMPERATURE MEASUREMENTS At first, attempts were made to measure total temperature with an unheated thermocouple. However, this naive experiment using a bare wire was unsatisfactory because the flow Reynolds number was very low and varied over a wide range. Also, as shown in Figure 22, the adiabatic wire temperature (i.e., the temperature associated with infinite aspect ratio) is higher than the temperature measured by the thermocouple; this -~ 52 - ADIABATIC WIRE TEMPERATURE TEMPERATURE SUPPORT TEMPERATURE -1 0 1 FRACTION OF SEMI-SPAN Figure 22. Typical Temperature Distribution for Wire with Unheated Supports is attributable to the fact that the support temperature is lower than the temperature at the center of the wire. A detailed discussion of this problem is presented by Dewey (Reference 14) wherein the end losses for hot wires are investigated. The correction associated with the tempera- ture difference between the adiabatic wire temperature and the thermocouple-measured temperature can be of the same order of magnitude as the total-temperature differential being measured, especially for turbulent flow. ~53- 3 THERMOCOUPLE WIRES —— 3 HEATING WIRES POSITIONING ME CHANISM SET SCREW 5 MIL. WIRE ADAPTER CALIBRATION NEEDLE Figure 23. Heated Thermocouple Total temperature was ultimately measured by the heated thermocouple shown in Figure 23, the details of which are given in Appendix B. Although the autograph was again used to record data, it was employed as a potentiometer rather than a null recorder. Because each total temperature trace required several hours, only the traces shown in Figure 24 were taken with the corresponding tunnel - empty data. These traces were considered adequate for defining total - temperature variation in the wake as a function of Reynolds number — especially since the percentage variation of absolute temperature is small (except in the vicinity of the neck) even for laminar flow, - 54 - P, = 85.00 PSIG a Re, =66.5X 10° ‘d 475 cal a 4.50 — fe x/4= 15.00 MILLIVOLTS 4.25 L 4.50 be Tw x/q = 9.00 4,35 4.50 SS 4.25 ! 1 i -l 0 1 yd (a) Pog = 85.00 PSIG P, 735.00 PSIG 8 3 Re, = 32.7 X10 4.75 4.50 — x/4 = 25.00 = 4.75 L 4.50 x/g= 15.00 MILLIVOLTS > 4,75 4.50 x/4= 9.00 Sad 4.50 x/g = 6.00 4,25 1 Nl L -1 0) 1 y/¢ (c) Po, = 35.00 PSIG g Figure 24, P, = 60.00 PSIG § 3 Re, = 49.4 X 10 2 4.75 = 4.75 4.50 b i an MILLIVOLTS Sy 475 b 4.50 b / =“ a x/g = 9.00 = 4,50 a 7 \ / aa x/q> 6.00 4.25 ‘ L L <1 0 J y/s . (b) Po, = 60.00 PSIG ml Me Lin > 5.0 x/,= 15.00 4.75 be x/g = 9-00 ~ J 4 = 6.00 wh ae 5.00 + x/y= 1.00 MILLIVOLTS 4.50 F- NOTE: ALL FIGURES PLOTTED TO SAME SCALE 4.00 Pp, 710.00 PSIG 9 3 Re * 16.7 X 10 d 3.50 ; py t 1 WV i y/a (a) Po, = 10.00 PSIG o Total—-Temperature Surveys ~ 55 - When the tunnel-empty data were reduced, they confirmed the calibration of reservoir temperature (discussed in Appendix A) to within 3 F. This difference is well within the value of T ow! TS = 0.95 +0. 01 for continuum flow given by Laufer and McClellan (Reference 24). Although the absolute level of free-stream total temperature was accurate to within 3 R, the total-temperature distributions are even more satisfactory because they represent modulated values about an absolute level. The accuracy of the thermocouple measurements is estimated at +1/2 R. BASE-PRESSURE MEASUREMENTS To extrapolate the measured static-pressure traces in the wake, upstream, it was necessary to measure the base pressure of the model accurately (Section IV). The apparatus shown in Figure 25 was designed for this experiment. To obtain base-pressure measurements, the model was placed in the region of best flow—22 inches downstream of the throat. The best flow region was determined by the wind tunnel calibration discussed in Appendix A. Only the 0. 300-inch diameter cylinder was used because of the finite size of the static-pressure orifices. The 0. 009-inch diameter (Figure 25) was as small as practicality permitted, and it corresponded to about 3 degrees for a 0. 300-inch diameter. This experiment utilized three holes to check data duplicability; this was accomplished by blocking one or two holes. After duplicability was established, the time constant of the total system was considerably reduced by simultaneous use of the three holes. Pressure system volume was kept - 56 - a TO PRESSURE RECORDING SYSTEM / CALIBRATION HOLLOW CYLINDER 0,306" DIA A : COUNTER TUNNEL PORT TO ANGLE-OF-ATTACK RECORDING SYSTEM Figure 25. Apparatus for Measuring Base Pressure to a minimum by filling the hollow cylinder with solder, leaving only a minimum diameter outlet open. The pressure-recording system for this experiment is identical to the system discussed earlier for static- pressure measurements. The calibration counter (Figure 25) hada 100-to-1 turn ratio, permitting the angles to be measured to +0. 01 degree. The base-pressure measurements determined by this experiment are shown in Figure 26b. This figure also plots the author's best estimate of the data of Tewfik and Giedt (Reference 25) whose experiments were made at a much lower Reynolds number than the present tests. In Figure 26b the region of reverse flow is evident, especially at high stagnation pressure. Accuracy of the base-pressure measurements is e(6) p(0) 0.8 0.6 0.4 0.2 0) p(d) ~57- o TEWFIK & GIEDT DATA (Re, =4.1X 10°, M=5.73) M =5.75 ‘ = - 4X10 a NOMINAL KP Rey = (15.9% 10 - @1 " a ij ND, d =0.300IN. 1 i ! it l il ee eee ae Daren [] aamene ee 0 20 40 60 80 100 120 140 160 180 @ (DEG) (a) Pressure Distribution on Cylinder 0.08 0.07 F 3 Ch TEWFIK & GIEDT DATA (Re, = 4.1 X 10°, M= 5.73) 0.06 F 3 Res =9.6X 10 O Re, = 15.9 10° Pale 0.05 + 3 Re, = 31.6 X10 Rey 46.9 109 M =5.75 L NOMINAL 0.04 3 ; . Re g=61.4X 10 i d=0.300 IN 0.03 F a mn, OO a (> : L) CT} 0.01 4 { 4 { 1 { { I ” 100 110 120 130 140 150 160 170 180 190 @ (DEG) (b) Base Pressure Figure 26, Experimental Pressure Distribution on Cylinder ~ 58 - estimated at Ap(€)/p(0) = +0. 02 for the lowest stagnation pressures as determined by the symmetry of the experimental data about the rear stagnation point. For the highest stagnation pressures, the accuracy is estimated at Ap(@)/p(0) = +0. 01. In addition to the base-pressure measurements, the pressure distribution was measured over the entire surface of the cylinder with the setup shown in Figure 25. However, in this case, a conventional unbonded- strain-gage transducer was used because the sensitivity required was not large. The arrangement is shown in Figure 27. With this arrangement either atmospheric or vacuum reference could be utilized depending on the range of pressures being measured. The manometers A and B shown in Figure 27 refer to the outlets shown in Figure 7. TO VALVE SEQUENCE CYLINDER TO OR PROBE MANOMETER (A) ATM ZERO REAR MANOMETER t 1 CLOSED 1 OPEN 1 OPEN 2 OPEN 2 CLOSED 2 OPEN 3. CLOSED 3. CLOSED 3 OPEN 3 4 OPEN 4 CLOSED 4 CLOSED TO 5 OPEN 5 CLOSED 5 CLOSED ATM 6 CLOSED 6 OPEN 6 OPEN 4 5 6 TO VACUUM PUMP Q—=- & STOKES GAUGE | & MANOMETER (B) TRANSDUCER CALIBRATING CIRCUIT R, = CALIBRATING RESISTOR TRANSDUCER Ne ZERO \ ADJUST CABLE i CIRCUIT TO AUTOGRAPH Figure 27. Transducer Circuitry for Base-Pressure Measurements - 59 - Demetriades (Reference 26) integrated the pressures obtained from this experiment and calculated a drag coefficient of 1.27 for the cylindrical rod as compared to a value of 1.2 to 1.3 as predicted by Ferri (Reference 27). The pressure distributions obtained are shown in Figure 26a. This figure also plots the values of Tewfik and Giedt (Reference 25). The accuracy of the data of Figure 26a was determined only by graphical reading error (p(6)/p(0) = #0. 002). As seen in Figure 26a, the dimensionless pressure distribution around the cylinder up to 6 = 90° is practically independent of Reynolds number in the range considered. This is attributable to the fact that the effective shape of the cylinder does not change appreciably. It is obvious that the base pressure is a function of Reynolds number which will be discussed in Section IV. -~- 60 - QI. DATA REDUCTION PITOT PRESSURE Reynolds number corrections to measured pitot pressures are shown in Figure 28, which is taken from Matthews (Reference 28). For the present tests, the outside diameter of the glass tip used for pitot measure- ments (Figure 6) was chosen to preclude corrections when reducing the experimental data. The outside diameter allowed sufficient resolution in pitot pressure (Figure 9); the inside diameter was large enough to prevent the time constant of the pressure-recording system from being noticeably penalized. No account was taken of the effective displacement of 1.12 qd © MATTHEWS! RESULTS FOR FLAT NOSE ~ = 5.4<M<5.7 d A SHERMAN, NACA TN 2995, FOR 10° INTERNAL CHAMFER {= 1.7<M<2.0 1.08 F- V SHERMAN, NACA TN 2995, FOR 10° INTERNAL CHAMFER | 28<M<3.4 —_ . . Pm 10st \ 4 = REYNOLDS NUMBER BEHIND NORMAL SHOCK | ! I 1] 20 40 60 80 2 200 400 10° Figure 28. Variation of Measured Impact Pressure With Reynolds Number (Figure from Reference 28) - 61 - 0.30 © 5° CONE NOSE Meo -— _ —_—> <n [?) A 10° CONE NOSE P. I 0.20 F 1 20° CONE NOSE d Re/in = 194,000 PRESENT _ TESTS Pore O10 F ; eo 0.10 | i | ! ! | N i 0 5 10 15 20 25 0 35 40 45 50 x /d s Figure 29, Pressure Distribution on Static-Pressure Probes (Figure From Reference 28) streamlines caused by velocity gradients present in the wake. Hill, Baron, and Schindel, (Reference 29) and Marson and Lilley (Reference 30). This correction was estimated to be less than 10 percent of the wake width for the worst case, i.e., in the vicinity of the neck. This subject is presented in greater detail in a subsequent section dealing with a discussion of accuracy. STATIC PRESSURE To correct the values of static pressure for boundary-layer effects as measured by the probe shown in Figure 14, Matthews' data (Reference 28) were used as shown in Figure 29. These data were taken for static-pressure probes of the same geometry as that used for the present tests. The calibration point for the present tests is shown in Figure 29. Based on this point, a value of the viscous interaction parameter, X, was calculated. A linear variation was then assumed for the value of measured pressure to ideal pressure as a function of the viscous interaction parameter, p_/p=1+40.235 xX. (2) m as expected, this variation is less than that for an insulated flat plate as discussed by Hayes and Probstein (Reference 31, p. 349). TOTAL TEMPERATURE The temperature as measured by the heated thermocouple (discussed previously in Section II) is the adiabatic wire temperature associated with infinite aspect ratio. However, the adiabatic wire temperature must be corrected for Knudsen number to obtain the true total temperature. This correction was accomplished with the data shown in Figure 30 which is based on Dewey's work (Reference 14). The ordinate is normalized so that the asymptotic value for free-molecule flow is equal to one and the asymptotic value for continuum flow is equal to zero. The values for free-molecule flow and continuum flow correspond to ratios of adiabatic wire temperature to total temperature of £ and 0. 95, 6 respectively. - 63 - Kn, >10 | THEORETICAL LIMIT OF FREE-MOLECULE FLOW ° ~ = —_— 0 t, wo 7/6 1 0.9 F O DEWEY'S RESULTS , MACH 5.80 0.8 + WX LAUFER & MC CLELLAN, MACH 1.33 - 4.54 (WITHOUT END CORRECTIONS) x o7 @ LAUFER & MC CLELLAN, MACH 3.05 (WITH END CORRECTIONS) 06 - + SHERMAN, MACH 2.0 & 4.0 x STALDER, GOODWIN & CREAGER i 05 KNUDSEN NO. = \/— ee . 2 Rewire 0.4 - Poo co SWIRE 0.3 - Rewige = —2— WIRE Ho O.2- Ss EE EXPERIMENTAL LIMIT OF CONTINUUM FLOW ToT. 70-9 —=—0 TSS ~} () i LOG}9 KNUDSEN NUMBER Figure 30. Recovery Temperature Versus Knudsen Number (Figure from Reference 14) IBM 7090 COMPUTATION As explained previously, two iterations were necessary to reduce the experimental data. The static pressure as measured by the probe required a correction for the viscous interaction parameter, and the total temperature as measured by the thermocouple required a correction for Knudsen number. Neither the viscous interaction parameter nor the Knudsen number were known beforehand. Also, inasmuch as the laminar theory (Section V) is developed in terms of transformed coordinates which involve running integrals, it was decided to reduce the experimental data by programming them for 7090 computation. - 64 - To provide total-temperature distributions for all the cases of interest (Figure 24), all the available total-temperature measurements were first reduced. Utilizing Reynolds number correspondence, the distributions were then used as input data to the IBM program for all the cases. This procedure was considered satisfactory for two reasons: (1) data existed for both laminar and turbulent flow, and (2) the defect in total temperature was less than 3 percent in terms of absolute temperature— except for the cases upstream of the neck. In the computations, the Mach number was then determined by standard compressible-flow techniques (Reference 32) using the corrected values of static pressure and pitot pressure. With the Mach number and the assumed value of total temperature, all other pertinent quantities could be computed. The IBM 7090 program was written so that all quantities of interest were printed out and automatically plotted by a cathode-ray tube. DISCUSSION OF ACCURACY As noted in the preceding portions of this section, the accuracy of each measurement was carefully identified. However, because of the many involved calculations necessary in obtaining final data, the accuracy of the final product can only be roughly estimated. To estimate the cumulative error, the following factors were considered. All location and dimensional measurements were made to an accuracy of +0. 001 inch and all distance-measuring devices were - 65 - carefully calibrated at thermal equilibrium. A minimum of several hours was allowed for reference-pressure stabilization, and a minimum of several minutes was allowed for model surface and base static-pressure stabilization. Flow uniformity was perhaps the most critical parameter in the experimental data, This fact became evident when models were tested at various axial locations in the wind tunnel (Section II). Closely associated with flow uniformity is the region of two-dimensionality cited in the static-pressure measurements discussion in Section II The effect of the variation of static pressure due to disturbances from the model supports is somewhat alleviated, however, because the theory (discussed later in Section V) is based on actual edge conditions. The manner in which the wake edge was defined (discussed later in Section VI) presents a potential source of final data error; however, even at best the wake edge is an arbitrary quantity. The extrapolation of static pressure upstream in the region of the neck (a detailed discussion is presented in Section IV) is estimated to be within 5 percent. The use of Reynolds number correspondence to determine total-temperature profiles in data reduction (already cited in this section's discussion of IBM 7090 computation) is not considered a large source of error. Although repeatability of experimental data was relatively simple to establish quantitatively, all errors associated with probes could not be determined. For example, the pitot-probe error associated with velocity gradient was not determined. As discussed by Hill, Baron and Schindel - 66 - (Reference 29) and Marson and Lilley (Reference 30), impact measure- ments do not reflect the pitot pressure on a streamline aligned with the probe axis; rather, these measurements reflect pitot pressure on the Streamline displaced towards the regionsof higher velocities. This phenomenon was not considered because reliable correction data were not available. This effect was minimized, however, by keeping the diameter of the impact-pressure probe as small as possible (Figure 6), and it is estimated that the maximum displacement due to this effect is 0.007 inch. Probe errors caused by misalignment with the flow direction were not considered significant (Figures 12 and 20) because the error in the probe angle of attack was less than 1. 0 degree. Fortunately, the machine computations do not introduce additional errors due to arithmetic. However, when the experimental data was digitized some error was accepted because only a finite number of points could be taken. As discussed previously, the methods of reducing experimental data to ideal quantities are somewhat inaccurate. In all cases however, the experimental apparatus was designed to ensure that these corrections were small; consequently, little accuracy loss was experienced. Assuming that the edge of the wake was properly defined, it is estimated that the overall accuracy of the final experimental data is from 2 to 4 percent for the distributions of normalized velocity, static enthalpy, and total enthalpy except in the immediate region of the neck. ~ 67 - It is estimated that the computed absolute values of the centerline quantities is accurate to within 5 percent. The major source of error is considered to be the extrapolation of static pressure upstream which is worse in the neck region. - 68 - IV. CONDITIONS NEAR BODY The orifices of the static-pressure probe (Figure 14) were located a fixed distance behind the apex of the probe cone; therefore, it was not pos- sible to obtain experimental values of static pressure close to the neck. The difficulty this presents is illustrated in Figure 3l—where data between Stations 1 and 2 are undetermined. Consequently, it was necessary to estimate the static pressure in the vicinity of the neck from available data. Unfortunately, because the axial gradient in static pressure is high the extrapolation upstream of the measured data is inaccurate unless a value of the static pressure at the neck is available. FLOW FIELD BOW SHOCK TRAILING SHOCK JL. —_—" € MEASURED STATIC PRESSURE 4 nf 0 ® Py BL ° 2 ee Py a Poo = _ —_ Figure 31. Experimental Static Pressure - 69 - SEPARATION ON CYLINDER The first step in estimating the static pressure at the neck was to define the conditions at separation on the cylinder. From the base-pressure measurements (Figure 26), a qualitative definition of the separation angle with Reynolds number is possible; however, the 0.009-inch diameter of the static-pressure orifices of the hollow cylinder (Figure 25) prevented the separation angle from being defined closer than 3 degrees in angle which corresponds to the orifice diameter for the 0. 300-inch-diameter cylinder. The problem is further aggravated by the fact that at separation the flow turns through an angle before sliding down the shear layer. To attack this problem, the simple schematic shown in Table II was developed to define the quantities of interest. The base-pressure measure- ments discussed in Section I] were then exploited. From pitot-pressure surveys immediately behind the body (Figure 32), it was demonstrated experimentally that the separation shock is negligibly weak—i.e., there is no total pressure loss through it locally. Therefore, the separation shock was neglected in the simple model of Table I]. The turning angle at separation was approximated by the step model shown in Figure 33. As shown by LeBlanc and Webb (Reference 33) in Figure 34, for laminar flow the pressure at separation, Ps: is halfway between the initial and final pressures. This conclusion, which is supported by experimental evidence taken from Chapman's work (Reference 34), is applicable even for finite pressure gradients. - 70 - Table If Conditions for Separation on Cylinder Po Moo Pog Moo Reg 65 M, Mg Mo Be a Pb/Po psig x 1073 | degrees degrees | degrees | x 103 85.00 | 5.87 | 61.4 | 123.3 | 3.72 | 3.24 | 3.03 | 18.3 | 15.0 | 0.849 60.00 | 5.83 | 46.9 | 123.0 | 3.74 | 3.26 | 3.01 | 19.6 | 13.4 | 0.892 35.00 | 5.79 | 31.6 | 123.8 | 3.79 | 3.30 | 3.00 | 21.8 | 12.0 | 0.927 10.00 | 5.74 | 15.9 | 127.0 | 3.88 | 3.38 | 2.99 | 26.2 | 10.8 | 0.990 0.00 | 5.71 | 9.60 | 131.5 | 4.01 | 3.51 | 3.02 | 30.3 | 11.2 | 0.960 Tewfik | 5.73 | 4.10 | 147.0 | 4.29 | 3.77 | 3.10 | 38.8 | 18.2 | 0.839 & Giedt -~71- SEPARATION POINT Po 1 SEPARATION °h Figure 33. Step Model 4 -——] | l DATA DATA 17 OFF = OFF SCALE SCALE 3r BOW SHOCK p P 2 — 2b p, x10 d =0.300 IN. 7d =0.50 "on = 35.00 PSIG \ 1) = 263 °F —— 1p Pam == 74.72 CM Hg | Reg = 32.7 X 103 LINE OF PRESSURE SURVEY 0 1 r L L L a -3 -2 -] 0 1 2 3 y/d Figure 32. Pitot-Pressure Survey Close to Model INHERENT uy SHOCK lay Wy HT [atl LAMINAR BOUNDARY - 72 - BOUNDRY LAYER SEPARATION AND PLATEAU PRESSURE RISE COEFFICIENTS 10 v 1/2 _ 1.77 TURBULENT | "= am oe A | B, R10 — ] FLOW ~ ~s eS. L x) Oe ee me me oe ee oe ee ‘aP 12, 1.% Pe eg 1 q 1 g 0 Po ~P, , 1/2 ~ B 2 1m, | x 2!) gB 107! Fg q 1 ny S 1 aq! —_ a Ane PL Ri/4 D top oO ~SnL bs) Pom, wd ig: in Oe, Pom LAMINAR Pod FLOW de] p 1/2, 0.889 “=a, q 1 Ri/4 S a , ALL DATA FROM CHAPMAN, NACA REPT 1356 107 i l ' 104 10° 10° 10" 10° Py yx ial xy MY Figure 34. Boundary Layer Separation and Plateau Pressure Rise Coefficients (Figure from Reference 33) Hakkinen, Greber, Trilling and Abarbanel (Reference 35) have shown that for the step model of Figure 33 va c -c \Ye (3) p 1" B, where C is the pressure coefficient, Ap Pp q C) is a constant . . i 2 q is the dynamic pressure, x PU Ce is the skin friction coefficient, — B is equal to V MM“ -1 ~ 73 - M is the Mach number. Utilizing first-order perturbation theory (e.g. Reference 36, p. 92) Cc p B, where y is the specific-heat ratio 6. is the turning angle Since the skin-friction coefficient varies as (Reference 37) cof : (5) £ \/ Relocal is the Reynolds number based on local conditions and the where Clocal length of laminar run, and assuming that where Rg is the Reynolds number based on free-stream conditions and d the model diameter, then JB, ~ 74 - where for our case (Table II) 6.=0.-F-a (9) The assumption of Equation 6 is reasonable based on Chapman's (Reference 38) order-cf-magnitude analysis of base-pressure coefficients. It should be noted that the step model used above assumes zero pres- sure gradient. With finite pressure gradient, the-constant, C, in Equations 7 and 8 merely changes value as shown by Greber (Reference 39). To determine the constant, C, for this case, base-pressure data were corre- lated with schlieren photographs from which the angle of shear layer was obtained. (It should be noted that schlieren photographs in cases where the shear layer was well defined were available only for the highest pressures. ) For the circular cylinder, the value of the constant, C, in Equations 7 and 8 was found to be equal to 2.66 for the circular cylinder based on these experimental data. It is interesting to note that, as pointed out by Guman, (Reference 40) C = 1.1 for the following cases: (1) a compression corner, laminar separation upstream of the corner but downstream of the leading edge, (2) a step corner the height of which is on the order of several boundary-layer thicknesses, laminar separation upstream of the corner but downstream of the leading edge, and (3) pure laminar separation induced by - an incident shock. For a compression corner with laminar leading~edge separation, a value of 2.1 checks with available experimental data. An attempt was made to predict the angle of the shear layer by expanding the flow from 90 degrees on the cylinder to the base pressure by -75- simple wave mechanics. However, it was found that this procedure produced shear-layer angles approximately 4 degrees too high, based on experimental data. From these calculations, it was concluded that the expansion from parallel flow at @ = 90 degrees to the base pressure cannot be approximated by a simple wave. Once the constant, C, was established in Equations 7 and 8, the data given in Figure 26b was then utilized to give the quantities tabulated in Table Il. Final results of these calculations are shown in Figures 35, 36, and 37. In Figure 35, two methods of non-dimensionalization are used; the different curves represent the effect of the change of free-stream Mach number with wind tunnel stagnation pressure. To extend the present tests into the regime of lower Reynolds number, the data of Tewfik and Giedt (Reference 25) are also presented. The points represent the author's best interpretation of the data of Reference 25. CONDITIONS AT NECK Having obtained plots of separation angle, @,, and effective initial Mach number, M); as functions of Reynolds number (Figures 36 and 38) from the base-pressure measurements by means of the simple model dis - cussed in the preceding portion of this section, these data were then used to obtain values of static pressure in the neck region. The Mach number M, is a fictitious quantity and can be interpreted as the Mach number that would be obtained at 6, if separation did not occur. The quantities of interest in this analysis are shown in Table II]. The shear layer was assumed to bea 0.9 -76- Mae (, —Pe Po Figure 35, Rey X 1079 (b) Relative To Static Pressure Base Pressure Versus Reynolds Number 1.0 0.9 0.8 (J TEWFIK & GIEDT © PRESENT TESTS 0.7 | | i | ! i 10 20 30 40 50 60 70 Re | x 107? (a) Relative To Total Pressure Moo = P= O— O — (] TEWFIK & GIEDT © PRESENT TESTS | | | | | 20 30 40 50 60 70 - 77 - 160 Mo 150 O = ANGLE OF SHEAR LAYER 140 -- CO TEWFIK & GEIDT © PRESENT TESTS @ (DEG) 130 120 F 110 100 ! \ | at ! ! 0 10 20 Ky) 40 50 60 70 -3 Re, X 10 Figure 36. Separation Angle and Turning Angle Versus Reynolds Number a 30 2 “ / a T\ > " a at (DEG) 10 0 | | | | | 0 10 20 30 40 50 60 Re, X 107° Figure 37. 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The separation angle @., and the fictitious Mach number, 5? M,, were used as basic quantities to define the flow. They were thus employed because their variation with Reynolds number was smooth (Figures 36 and 38). Using Equations 7 and 8 as well as the experimental value of the trailing shock angle, ¢, which was obtained from pitot-pressure measurements, the quantities given in Table III can be easily calculated through use of standard two-dimensional compressible-flow techniques. It should be indicated that at least one redundancy is present for these calculations (the experimental value of the pitot pressure at the neck) which serves as a check. 6 ; , ’ ry ee V ae Pn i ee ee Om Va QO TEWFIK & GIEDT DATA © PRESENT TESTS Re, X 107? Figure 38. Mach Number Versus Reynolds Number - 80 - Both the pitot- and static-pressure gradients are high in the neck region; however, the Mach number at the edge of the wake is relatively constant. Also, the precise axial location of the neck is unknown. There- fore, the static pressure, P3, was obtained using the calculated value of M, and the experimental value of the pitot pressure at the axial stations of interest. These data were then used to obtain the extrapolated curves shown in Figure 21. In Table III, the effective wake growth at the neck, €, is negative in most cases because the flow is accelerating in this region. NECK WIDTH AND NECK DRAG As will be discussed in Section V, the value of the momentum thickness and wake width at the neck must be known as initial conditions for theoretical analyses. Lees and Hromas (Reference 10) have estimated “1/2 based on ) these quantities and shown that they should vary as (Re, estimates of skin friction around the body and the pressure rise at the neck. Experimental values are shown in Figure 58 based on the present tests. In Figure 39, 11 points are shown corresponding to the combinations of cylinder diameter and Reynolds number given in Table I. The wake width was obtained from pitot-pressure measurements and the momentum thickness was calculated from experimental data. The axial location of the neck was taken at x/d= 2.50 for the 0. 300-inch diameter cylinder and x/d = 3.00 for the other models (Table III). Figure 39 shows that these quantities do yd /2 vary as (R The scatter of Figure 39 is probably due to the effect €d of the velocity gradient on pitot measurements discussed in Section II. - 81 - 1.0 Y/D neck ~ (Rey)“1/2 Y/d neck 0.10 -— (6/D neck (8/4) neck ~ (Rey) 72 Od = 0.300" Z\d = 0.200" (Jd =0.100 Moo = 527 nominal -0] | 102 104 105 Reg Figure 39. Wake Width and Momentum Thickness at Neck ~ 82 - Marson and Lilley (Reference 30) have shown this to be on the order of 0.18 times the outside diameter of the pitot probe —approximately 0.007 inch in the case under discussion (Figure 6). Also, the exact axial location of the neck, which varies with Reynolds number, was not known (Table III). - 83 - V. THEORETICAL CONSIDERATIONS AND COMPARISON WITH EXPERIMENT DEFINITION OF WAKE EDGE Before the laminar wake can be treated theoretically, the edge conditions of the wake must be known--either from inviscid flow field calculations or from experiments (Appendix D), In the case under discussion, several exploratory computations were made to determine the best definition of the wake edge. It was finally concluded that the wake edge as seen from the pitot-pressure profiles was the best definition of this nebulous quantity. For its determination, the intersection of the Gaussian-type, pitot-pressure profile with the inviscid outer wake was selected from the experimental data, A typical pitot-pressure profile, where this intersection is fairly well defined, is shown in Figure 9a, It was extremely difficult to define the wake edge for the turbulent wake (Figure 9b) from this program's experimental data, Therefore, the hot-wire- anemometry data of Demetriades (Reference 11) was used to define the wake edge in the turbulent region, Wake widths are shown in Figure 40, For x/d< 6, the wake is still undergoing internal adjustment processes; therefore, no simple similarity has yet been established, This adjustment is evident in the wake widths shown in Figure 40, x/d ~ 84 — % ub Wd 32h apr W/o af L—__ WAKE €-= = 20h Po/Po 16 wb TRACED FROM PITOT DATA i OF RUN: | Po, = 85.00 PSIG n! x/q = 7.00 h MODEL: d =0,300" oe ab © —— Po, = 85.00 PSIG DO ==== Fy, = 60.00 PSIG @ = Pog = 35.00 PSIG B —— Pog = 10,00 PSIG ak 0 i ae L. _L LL wk. 0 2 4 <6 8 1.0 1.2 14 W/de (a) 0. 30 O Inch Diameter Cylinder % ] % BE 4b 1 Wd | 32 ' 32 F J | y/d 1 ty! b ! ' J _) Hott fa 26+ a ty 2 y/de j t t i ' (y/d) . . ia] al ey 1 24 if aah I ee , | WAKE G —> po ! “d —tL.__- wakeg —--— | I { | 20 ° 4 at oe; oo ! t ! ' 1 16 Pp/Po L lok Pp/P, eo i I TRACED FROM PITOT DATA i 2 OF RUN: 1 | fi Pog = %.68 PSIG f 2 x/d = 9.00 1 . TRACED FROM PITOT DATA | MODEL: d = 0.200 ' . OF RUN: Pog = 59.42 PSIG MODEL: d=0.100 ce) Pog = 96.68 PSIG. x/d = 15,0 O-—= = Pog 57,42 PSIG 8 ak a Pog 7 22.54 PSIG Q----- Pag = 59.48 PSIG © ~~ py, = 3.84 P16 © ——-— Pog = 22.48 PSIG 4 © n= Poy = 3.80 PSIG ak i) ve A i 1 4 4 0 | 1 1 1 i 2 4 é 8 1.0 1.2 14 0 2 4 8 1,0 1.2 Wde 0,200 Inch Diameter Cylinder Figure 40, (c) Experimental Wake Widths &/d), , 0.100 Inch Diameter Cylinder ~ 85 - LAMINAR WAKE As shown in Appendix C, the governing linearized equations for the static-enthalpy excess and velocity defect in a laminar wake with stream- wise pressure gradient are dM 2 (mM “w) +6 — = 2 (uM * w] (11) e dx ~2 e dy lr ) xl where the transformed coordinates are defined as x - _ Po He dx 12) ~*~ "“[pku od ( ao 0 0 (o} y u - = — /R, oy (13) y Weg d P,, d [e) and the subscript ( )e refers to the edge conditions, The static enthalpy excess can also be written in terms of the total enthalpy excess, G, h 2 Y- 1, 2 h -1 = (y-1)M, wt (vs tm.) (14) where H > = —- i! C (15) (0°) - 86- Equation 14 allows for the solution of G after the static enthalpy excess and velocity defect-are known, It should be noted that the only assumption of the above analysis are the usual boundary-layer approximations and that w<<1l, The Prandtl number, 0, is assumed to be constant, but it can be any value--not necessarily equal to one, Since for our case the total enthalpy excess in the wake, G<<w (16) Equation 11 can be simplified to read The assumption that G<< w is true only if there is little heat transfer, For actual reentry, this assumption may not be valid, and the exact in- tegral solution given by Kubota (Reference 18) would have to be used, Equations 10 and 17 can be recognized as the simple heat diffusion equation; therefore, because the boundary conditions are homogeneous, simple closed-form solutions can be obtained if the initial conditions at the neck are assumed to be delta functions, These solutions are 2 _ oF - 2 -leee * (18) e VE _2 _¥ r Miwsete ™ (19) - 87 - It should be noted that the static-enthalpy excess is similar regardless of the value of the heat transfer. The constants A and B can be determined from the initial drag in the wake (not necessarily at the neck) and the heat transfer from the body to the fluid. Re p46 dj eve 2 A = l/2 7 (; afd m,?) (20) oo 0 0 2 Tv B Yeg Q= 29? d ——h,.- A 21 00 60 VReg fo eo Me2 (21) 0 where 0 is the momentum thickness and the subscript ( do refers to the initial conditions (usually taken at the neck), Experimental comparison of the simplified laminar-wake theory for the normalized distributions of velocity, static enthalpy, and total enthalpy are shown for three stations in Figure 41, It can be seen from Figure 41 that the assumption of a delta function as the initial condition affects the theoretical solutions; therefore, a satisfactory experimental check is not obtained until about 15 diameters downstream of the center of the cylinder. The experimental check of total enthalpy is exceptionally good at x/d = 15 because the tail portions of the velocity and static enthalpy curves (which do not compare favorably with experiment), tend to cancel out. Gold (Reference 41) has shown that for initial Gaussian velocity and enthalpy distributions of the form - 88 - 0.8 THEORY EXPERIMENT MODEL: d=0.300 IN. P, = 10.00 PSIG 9 x/d = 6.00 3 Re, = 16.7 X 10 Me =5.64 0.2 0.4 0.6 0.8 0.6 veu ‘3 447% 0.4 0.2 1.0 0 0.8 \ \ THEORY 0.6 - 0.4 \ EXPERIMENT \ MODEL: d=0.300 IN. P, = 10.00 PSIG 3 0.2 x/d = 6.00 3 \ Rey * 16.7 X10 \ Ma = 5.64 0 J S_eete + LY 0.2 0.4 0.6 0.8 1,0 Wy, 1.2 H-H g H, “He Figure 4la, MODEL: d =0.300 IN. *o, = 10.00 PSIG 0.2 x/d = 6,00 3 Re, =16.7X 10 Mo™ 5.64 0 i ! Ll 0 0.4 0.6 0.8 1.0 We Comparison of Theory and Experiment---Simplified Laminar Theory: x/d = 6,00 0.2 - 89 - Figure 4l1b, Laminar Theory: x/d 9. 00 1.0 7 — —_ — 0.8 / THEORY 0.6 | EXPERIMENT uu € % = od | MODEL: ¢=0.300 IN. 0.4 p 5" 10.00 PSIG x/d = 9,00 / Mo=5.64 4 Re = 16.7 X 10 0.2 | 0 i i ! i \ 0 0.2 0.4 0.6 0.8 1.0 \ ¥/Ye \ THEORY \ EXPERIMENT \ MODEL: d =0.300 IN. P, = 10.00 PSIG g \ x/d = 9.00 ~ Ma = 5.64 3 \ Re, = 16.7 X10 d “es ene 0.2 0.4 0.6 0.8 1.0 Wy, 1,2 1.0 / ~ WS 0.8 | 4 | THEORY -H g H -H 2G 06 | EXPERIMENT | MODEL: d =0.300 IN. 0.4 P, = 10.00 PSIG 9 | x/d = 9.00 Mo=5.64 4 Re, * 16.7 X 10 0 | i | | 0.2 0.4 0.6 0.8 1.0 WY, Comparison of Theory and Experiment--- Simplified 1.0 0.8 06 ueu ¢ Ye G 0.4 0.2 1.0 0 9.8 MODEL: d=0.300 IN. P, = 10.00 PSIG 9 x/4 = 15,00 3 Re, =16.7X1 0.6 - g 16.7 X10 Me = 5.64 0.4 \ THEORY \ EXPERIMENT o2- \ 0 i i “ae 0 0.2 0.4 0.6 0.8 1.0 Wye 1.2 1.0 0.8 H =H t 0.6 A -H : e 0.4 0.2 0 Figure 4lc, Comparison of Theory and Experiment---Simplified Laminar Theory: x/d = 15,00 Yo / THEORY / EXPERIMENT MODEL: d=0,300 IN. P_ = 10.00 PSIG — %5 x/d=15.00 4 Re, = 16.7 X 10 Mo =5.64 \ i | N 0 0.2 0.4 0.6 0.8 1.0 ye U U L. U U Ul [— i MODEL d= 3.00 IN. p_ = 10.00 PSIG o 9g ‘d= 15,00 1 Re 5 = 16.7 X 10 THEORY Ma = 5.64 EXPERIMENT l ! ! i 0 0.2 0.4 0.6 0.8 1.0 Wy 2 -ay wilh) = Ale” (22) h _2 = -1= Be AY (23) e the problem of the laminar wake with streamwise pressure gradient can be dealt with theoretically by putting an initial delta function velocity distri- 1 bution at a distance Xo = ya upstream of the origin and an initial delta * 7 . * . - o . . function enthalpy distribution at a distance Xo = 4B upstream of the origin, These results are equivalent to matching the initial distributions by suitable choice of XQ Typical distributions of normalized static enthalpy and normalized velocity are shown in isoaxiometric form in Figure 42, where a non-zero value of Xo was used, In Figure 42 the change in shape of the normalized quantities from a Gaussian to a parabolic distribution at transition is evident, In Figure 43, a more practical scheme of non-dimensionalization, which does not depend on the wake "edge" is used for the abscissa, The results of Figure 43 are encouraging, considering the many varied inputs needed to obtain the experimental results, ‘'Universal'' curves of normal- ized static enthalpy and normalized velocity are indeed possible, Values of the state properties along the wake centerline as predicted by the theory are also very good, The constants A and B were calculated according to Equations 20 and 21 for each case. Comparison of theory and experiment was within 10 percent for all cases, A typical calculation of the constants A and B is shown in Figures 44 and 45, To obtain the heat - 92 - h-he h -he MODEL: d = 0.100" FLOW, | Pog = 59,42 PSIG y al .0 Tg = 263 F /- Req = 16.3 X 10 TRANSITION —-49 K > Figure 42. Typical Results (a) Normalized Static Enthalpy - 93 - MODEL: d= 0.100" ¢ Pog = 59.42 PSIG neg To = 263° F me Req = 16.3X 103 pi Mgo> 5.71 ‘a0 25 “a > ea 40 > TRANSITION ea 50 > ea a " Ww 80 » 5 x5 9° x x/ = 90 -1.0 Figure 42. Typical Results (b) Normalized Velocity 94 - 1,0 oo MODEL: d = 0,300" : Pog = 10.00 PSIG Pore = 74.36 cm Hg x/d = 10,00 To = 262°F - Moy = 5.64 og b x/d = 5,00 Rey = 16.7 X 103 x/d = 20.00 x/d =25,00 O7- x/d = 15.00 0.6 \ ‘I note ost Ee 0.4 | THEORY q 03k x/d = 25.00 x/d = 20,00 x/d = 3.50 o2b \ x/d = 15.00 . \ x/d = 10,00 \ x/d = 5,00 ort NOTE: K, = 3.00 SS “ 0 aL i ! L ~~F 1 $ L 0 i 2 3 4 5 6 7 9 10 u 12 13 14 15 (a) 8 Normalized Static Enthalpy, 0.300 Inch Diameter Cylinder Lob 0.9 0.8, 0.7/ 6b i MODEL: d= 0,300" i Pog = 10.00 PSIG ve O05 H Patm = 74.36 cm Hg were i To = 262°F a UE H Mo= 5.64 ' Reg = 16.7 X 10 h THEORY a ~x/d = 25.00 oat f Sk x/d = 20.00 Ser 15.00 9.27— x/d = 10.00 x/d = 5.00 WIE =3 ° x/d = 3.50 NOTE: %, = 3.00 0 ! 1 ! 1 1 il t 1 0 2 4 6 8 10 12 “4 16 f/f A + Xo (b) Figure 43, With Normalized Velocity, 0.300 Inch Diameter Cylinder Comparison of Theory and Experiment- Laminar Theory Effective Origin _ 95 - MODEL: d= 0.100" Pa. = 3.84 PSIG Pare = 74.71 cm Hg To = 260°F Me = 5.58 Red = 4.29 X 103 NOTE: X, = 7.26 (c) Normalized Static Enthalpy, 0.100 Inch Diameter Cylinder “ 30 = x/d 0 | — 25 0 20 J 1.0 g ” 5.00 80 ao - 7 9.00 MODEL: d= 0,100" : Pog = 3.84 PSIG Pate, = 74.71 om Hg THEORY tor 5 Mec= 5.58 os Reg = 4.29 X 10° O7 - 0.6 ue 4g Ye %@ O5 b a3 THEORY x/d = 90; 80; 70; 60; 50; 40; 30; 25; 20 — x/4=9; 0.2 x/d = 9; 5 ot NOTE: X, = 9.05 0 1. 1 i 1 H i i i 0 2 4 6 8 10 12 4 16 18 20 (d) Normalized Velocity, 0.100 Inch Diameter Cylinder ‘Figure 43, Comparison of Theory and Experiment-Laminar Theory With Effective Origin -96- 32 28 L 24 1 42 2F h li] he 16 Pe 8 MODEL: d=0.300 IN. Pp. = 10.00 PSIG oJ 9 To = 262 °F 3 4 Rey =16.7X 10 Me =5.64 0 1 i 1 L J l 1 i I i] i 0 2 4 6 8 10 12 14 16 18 20 22 24 x Figure 44, Axial Variation of Static-Enthalpy Excess i.) Lak 1.25 1.0 - 2 . 2 0.8 M, “G 0.6 F- 0.4 - Re, =16.7X 10 Mo = 5.64 0.2 0 i L i L 1 ~4 | ! | 0 2 4 é 8 10 12 14 16 18 20 22 24 Figure 45, Axial Variation of Velocity Defect -97- transfer, the total enthalpy distribution was integrated across the wake, If the centerline distributions of static-enthalpy excess and velocity defect are plotted for all the runs of Table I, the data shown in Figures 46 and 47 are obtained, Figures 46 and 47 show that there is definitely a Reynolds number correspondence for the laminar theory. The two runs which do not corre- spond with the base line are fully turbulent. It should be noted that the static-enthalpy excess and the velocity defect for the turbulent case also varies as (Req)? when turbulence is fully established. Figures 46 and 47 can also be used as a measure of transition, At transition, the values of velocity defect and static-enthalpy excess decrease much more rapidly than for the laminar case because of the more efficient mixing processes, 19.0 KEY d (INCH) Pog (PSIG) O 0,300 85.00 O 0.300 60.00 a 0,300 35.00 rs 0,300 10,00 @ 0.200 96.68 a 0,200 59.48 wa? 4 0.20 0 tab 1,.0-- w 0.200 3.80 x 9,100 59.42 a 0,100 22,54 3.84 e ia} = ° ° oe oan - 0,10-- Vv fa] s ay bx Veh _ Q+ (pyug 6), Po, 4 *y “ Virco" od hog ° ° . é ° ee 01 1 i av 1.0 0.0 100.0 Figure 46, Static-Enthalpy Excess Along Wake Centerline - 98 - KEY AUINCH) Pa, (PSIG) © 0.300 85.00 @ 0.300 60.00 ® 0.300 35.00 € = 0.300 10.00 @ 0.200 96.68 ® 0.200 59.48 @ 0.200 22.48 & 0.200 3.80 V 0,100 59.42 lol B 0.100 22.54 B 0.100 3.84 C) (Mw) © ° wn 6 > Reg [lve OMe? e eo ve A="Vo 7 (Se2e* ) ° . 2th fo) . o ° ve aed s [3 L ° 0.10 o ® ao ah o° » Vv © 40 ee o « e 0.03 i L 1.0 10 100 Figure 47, Velocity Defect Along Wake Centerline Appendix D shows the momentum thickness as predicted by the simplified theory is constant if modified by the edge conditions (2% Me) © 0 OL = var 9% Re “e e€ To compare this aspect of the theory with experiment, Figure 48 was drawn where a typical laminar case is presented. The experimental point on the extreme right is in a region of turbulent flow, As discussed in the previous portion of this section, the wake is "adjusting" until x/d = 6, There was no similarly satisfactory correlation of momentum-thickness data possible for the turbulent cases, - 99 - 1.0 oO Ome O omm 6 am O me Ooms O O° MODEL: = d = 0.300" O° Po, = 10.00 PSIG pe Ue Mez 76 aa NECK _ 3 (Pe ue M2). Rey = 16.7 x 10 Mo = 5.64 0 i l i L | l i I j j 0 2 4 6 8 10 12 14 16 18 20 22 Figure 48, Momentum Thickness Also, as discussed in Appendix D, the laminar wake width can be predicted theoretically, p pu ee 6 _ . 2"e e 0 re a Reg = At+(y¥Y-1) (, Pa, bi) (25) c ao Us where 2\ _ 2c cf fom -k1nM V& (26) 0 Comparison of experimental and theoretical laminar wake widths is shown in Figure 49, Again, the experimental point on the extreme right is ina turbulent region. N= 100 means that, for the theory, the edge of the wake - 100 - 68 d =0.300 IN. Py = 10.00 PSIG ” | °%s 4 a2 F ° 3 Ray 15.7 X 10 we 52 pyr, Vu, Peo Yoo Figure 49, Theoretical and Experimental Wake Thickness was assumed to be at the point where the wake velocity = 0.99 ua; N = 1000 means that the wake velocity = 0.999 ues Comparison between theory and experiment is surprisingly good, considering the nebulous nature of the wake width, TURBULENT WAKE The turbulent wake has been adequately treated theoretically by Lees and Hromas (Reference 10); therefore, only a few general remarks will be made here, The theory presented in Reference 1 0is depicted graphically in Figure 50, Physically, it can be thought of as the penetration of an inviscid static-enthalpy profile by a turbulent boundary layer with initial width and momentum thickness, - 101 - aa INVISCID y TURBULENT BOUNDARY WAKE Figure 50, Theory of Lees and Hromas Lees and Hromas (Reference 10), in order to obtain a definite inter- section of the wake static-enthalpy profile with the static inviscid-enthalpy profile, assume that the static enthalpy distribution in the wake is parabolic. Integral methods are then used to solve the turbulent boundary-layer equations; thus, any definition of the distributions--except those originally assumed--are not products of theory. However, some assumption had to be made concerning the turbulent diffusivity at the centerline of the wake. They assumed that Townsend's (Reference 4) "universal'’ Reynolds number for incompressible flow behind circular cylinders is also valid for the hypersonic case, In particular, for the two-dimensional case, they asserted a constant value across the turbulent wake for - 102 - where Au = u -u e © Ye T ~ Ep is the equivalent turbulent diffusivity Y_, is the Howarth-Dorodnitsyn variable for the wake cage [ Pay re) ioe] The total temperature in the wake was assumed to be constant in the theoretical treatment, Comparison of the Lees and Hromas turbulent theory with the present experiments was not rewarding because experimental data were not obtained far enough downstream. The wake grew very slowly and little change in state properties was obtained. However, theoretical calculations were made for this case utilizing a program written at the Space Technology Laboratories by Hromas, Inputs to this program involve the inviscid static- enthalpy profile and initial conditions. To obtain the static-enthalpy profile, the shock shape as obtained from schlieren photographs was approximated by the equations shown in Figure 51, As suggested by Lykoudis (Reference 17), the body was then omitted and the flow expanded from the pressure at the bow shock to the ambient pressure in the free stream, The results of this calculation are shown in Figure 51, It should be noted that the slope of the inviscid static- enthalpy profile is very steep near the axis, Also of interest is the fact that, for x/d>9, the static-enthalpy profile obtained in this manner checked satisfactorily with experiment and with inviscid-flow calculations. YL - 103 - 3.5 500 me ewpinican | 1/4 = 158 AO? 5 w/a <1 3.0 F . 0.564 ; x/d 21 y/d = 1.58 (x/d) z a EXPERIMENT 25h CYLINDER ¢ AT x/d = 0.720 x/d y/d 2.0 0 fe) 0.0241 0.2191 y/s 0.0723 0.4130 0.1205 0.5354 MODEL: 4 =0.300 IN, 0.2169 0.7301 rs p, = 85.00 PSIG 0.3133 0.8824 9 0.4337 1.040 Po, = 74.69 CM Hg 0.5542 1a? AM avr 0.6747 1,295 ° 0.8193 1.430 rok Me = 5.69 0.9639 1,555 1.205 1.748 1.44 1.925 1.928 2.263 2.410 2.566 ost 3.133 2.986 3,855 3.37% 6 i ! ! Ll ! \ 1 | 9 0.5 O12 1.0 15 2.0 2.5 3.0 3.5 4.0 45 5.0 x/d (a) Shock Shape g 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.0 T T T T T T Uy T q hy g=h é6F oh ¢ -1 40.6 Fo x Y, ={d o dfx L i 5 (t) 5st oo 40.5 4r 40.4 ant 3b 40.3 2 + 0.2 IF 70.1 0 L L 1 L L 1 \ ! 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Q Figure 5l. (ob) Inviscid Static Enthalpy Profiles Derived Static-Enthalpy Profiles Experimental Shock Shape With Y/de 3.2 3.0 2.8 2.6 bg ho 24 2.2 2.0 104 - h¢/h wo O/ae Figure 52, x/d 100 200 300 500 1000 Theoretical Turbulent Wake Growth The results of the Hromas program are shown in Figure 52 which plots the static-enthalpy ratio and wake width. Figur e 52 shows that the swallowing process does not become effective until about 300 diameters downstream of the body because of the nature of the inviscid static- enthalpy profile. This result was predicted by Lees and Hromas (Reference 10) based on an approximate form of their final results. - 105 - VI. TRANSITION "UNIVERSAL" TURBULENT REYNOLDS NUMBER As discussed in the previous section, Lees and Hromas (Reference 10) assumed the Townsend value of "universal" turbulent Reynolds number of 12.5 for the centerline conditions in the turbulent wake. To investigate this assumption experimentally, consider the momentum equation du qu ap C] gu pu +pyet = - 24 fo &) (28) ax ay ax ay PT ay at the centerline a a ¢ Pe Ye (24) + 22s vo (c (?~) (29) axie" ax ‘“gi\T/_\, 2 @ ¢y ¢ With w=l-— (30) Uu e then au ap aw Pag a(S), tHe7 > Aer) Ye(—F) en e “* Gay” '@ 2 To obtain ad experimentally without the employment of doubtful oy means would be very inaccurate, because this quantity involves two 1.3 ldo F 0.9 + 0.8 + 0.7 + 0.6 | 0.5 0.4 |} 0.3 0,2 - 106 - 4,50 4 1 Hi 9,00 10,00 15.00 20.00 85.00 PSIG 0.300 IN. Figure 53, 0.1 0.2 y/d 0.3 0.4 Diffusivity Parameter Variation - 107 - differentiations of experimentaldata. Therefore, the method first proposed by Kingsland (Reference 13)to reduce diffusion measurements was used. Assuming — =l-ay +--- (32) we then 9° w — 7 7 awe t- > (33) ay In — ve 5 =-a (34) y giving aw Awe _——s =- (35) a2 D y “¢€ N where the diffusivity number is defined by 2 D. = —~— (36) N w In © lim yo w “¢ The experimental plots of the diffusivity parameter V In a versus vertical distance are remarkably good (Figure 53). From these data, it was then possible to obtain the slope of the diffusivity parameter versus distance to calculate the diffusivity number discussed above. The diffusivity parameter was plotted in terms of transformed coordinates, but, - 108 - as expected, the results for the turbulent case became worse and those for the laminar case better. (The transformed diffusivity number was also used to give another indication of transition. ) To obtain experimental values of the Townsend Reynolds number via Equation 31, it is necessary to differentiate two experimental quantities which subtract from each other as well as the diffusivity number discussed above. Therefore, the determination of this quantity is crude at best. Other means of obtaining quantitive values of "universal! Townsend turbulent Reynolds number would be more desirable, e.g,, diffusion measurements. In any case, the tentative value, oe = 10 +5, was obtained. This value agrees very well with that used by Lees and Hromas in their theoretical treatment (Re, = 12.5). The results also compare favorably with those of Kingsland (Reference 13) where he estimated the value, Re, 212, EXPERIMENTAL DETERMINATION OF TRANSITION As indicated in the foregoing discussion, a qualitative definition of transition could be obtained from plots of centerline state properties. Another definition is possible by calculating the Townsend Reynolds number discussed in this section. Perhaps the best definition, however, is obtained from plots of transformed diffusivity number versus axial distances (Figure 54). It should be noted in Figure 54 that the value of transformed diffusivity - 109 - 1000 : ae a a a 1 MODEL: — g = 9,300" Pog = 85-00 PSIG 100[- To = 263°F [ Reg « 66.5 X 103 [ _ Moo 5.69 r =2 n Ww. | ma WwW 10 J J IL L i ! i ! i 1 0 2 4 6 8 10 12 14 16 18 20 22 (a) Turbulent Case xt 10,000 = 1000 . - d= 0.300" D a Po, = 35.00 PSIG " TRANSITION 19 = 262°F i Reg = 32.7 X 103 M, = 5.71 100 — ’ « 5 - = &) r Dn i r (4) 10 1 1 1 1 l 1 j 1 i 1 i 1 1 1 0 2 4 6 8 10 12 4 16 18 20 22 24 2% 28 30 (b) Transition Case ® 1000 MODEL: d= 0.200" _ Pog = 3.80 PSIG _ BD To = 260°F DB, = Req = 8,58 X 10 inf “& Mo 5.57 Ww 10 | ! ! H | lL | ! 1 J 1 ! xi (c) Laminar Case Figure 54, Transformed Diffusivity Number Versus Transformed Axial Distance - 110 - number in the turbulent region is much higher than for the laminar case. {In Figure 54a, Demetriades, Reference 11, found transition for these conditions near the neck, but no data point was claimed here. ) Using all available inputs, the results for these tests are superimposed on those of Demetriades (Reference 11) in Figure 55. For the 1l runs shown in Table I, seven transition points could be obtained~two cases being fully laminar and two cases fully turbulent. The comparison with Demetriades' data is surprisingly good. Since the edge properties could be determined with the present measurements, it was possible to define a transition Reynolds number based on local conditions and distance from an effective Origin. Its value was calculated as 85,000. Some question may arise as to the mechanism of transition, because some difficulty has been experienced in obtaining laminar flow by other investigators. In anticipated rebuttal, it should be pointed out that Demetriades' data (shown in Figure 55) is a combination of several different-sized cylinders at several different axial locations in the wind tunnel. In addition, the regimes were confirmed by Kingsland's data (Reference 13) wherein diffusion measurements were employed to investigate a fully turbulent and a fully laminar wake. Demetriades (Reference 26) has also successfully correlated the data of Slattery and Clay (Reference 42) on the present plot. Conclusive proof of the self-induced nature of the transition in the wake would be presented by the experimental investigation of the - lll - DEMETRIADES DATA X PRESENT TESTS 4 a ray TURBULENT \ P Xr 5 e Re — (INCH)7! wo (INCH) of 4 oO 76 2/- , a & LAMINAR ao VW Ao Oe 4 4 J 0.4 , 0.8 1.2 — 1.6 i Re X 109 (INCH) Figure 55. Transition. character of the boundary layer onthe wind tunnel's side walls. This was not done for the nozzle used in the testing reported here; however, Since the present program was initiated, a new nozzle has been installed in Leg 1 of the GALCIT hypersonic wind tunnel; also, the boundary layer on the wind tunnel's side walls has been surveyed for Pog = 35 psig. The results of this survey show that transition of the boundary layer on the side walls does not take place until 25 inches downstream of the throat (Figure 3). In this respect, the previous nozzle should be better than the new nozzle, because it was considerably shorter. In the present tests, for the wind tunnel stagnation pressure of 35 psig, transition took place 23.1 inches downstream of the nozzle throat. - 112 - VII CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK An experimental investigation has been made of the flow behind a circular cylinder at a nominal free-stream Mach number of 5.7. The Reynolds number was varied by changing both the diameter of the cylinder and the stagnation pressure of the wind tunnel -- the former over a three- fold range and the latter over a six-fold range. Enough measurements were made to completely define the flow field as far downstream as 90 diameters for the smallest cylinder and 30 diameters for the largest cylinder. From these measurements, transition from laminar to turbulent flow was defined in the wake and successfully correlated with other data. A "universal'' Reynolds number for transition based on edge conditions and the laminar length of the run was determined as 85, 000. Correlation of the experimental data with Kubota's theory (Reference 18) for laminar flow was then made. The theoretical prediction of center- line values of static enthalpy and local velocity based on this theory were in excellent agreement with the experimental results. Distributions of static enthalpy, and local velocity in the wake agree with the theory if provision is made for an "effective'' origin based on the wake width at the neck. The theoretical predictions of wake width and momentum thickness are also in close agreement with the theory. - 113 - Because of the nature of the tests conducted, it was not possible to correlate the experimental data with the turbulent theory of Lees and Hromas (Reference 10) since only very small differences in state properties were available in the turbulent flow regime. However, a qualitative experimental value of the Townsend Reynolds number for turbulent flow on which the theory of Lees and Hromas is based was obtained. Its value was determined as Re, = 10+ 5. In order to study the turbulent regime in more detail, it is necessary to either heat or cool the body. This project has already been started at GALCIT. It is suggested that the theory be extended to account for real-gas and relaxation effects. Experimental verification could then be accomplished in facilities such as the hotshot and shock tunnel for the two-dimensional case. The laminar theory should then be extended to three-dimensions so that the wake properties behind an arbitrary body of revolution can be predicted with confidence for actual reentry conditions. 10. - 114 - REFERENCES Goldstein, S. (editor), Modern Developments in Fluid Dynamics, Vol. Il, Chapter XIII, Oxford University Press, London, 1938. Schlichting, H., Boundary Layer Theory, 4th Ed., McGraw-Hill Book Company, Inc., New York, 1960. Birkhoff, G. and Zarantonello, E.H., Jets, Wakes, and Cavities, Academic Press, Inc., New York, 1957. Townsend, A.A., The Structure of Turbulent Shear Flow, Cambridge University Press, London, 1956. Roshko, A., On the Drag and Shedding Frequency of Bluff Cylinders, NACA TN 3169, July 1954. Cooper, R.D. and Lutzky, M., Exploratory Investigation of the Turbulent Wakes Behind Bluff Bodies, The David W. Taylor Model Basin, Research and Development Report 963, October 1955. Jacchia, L.G., The Descent of Satellite 1957 Beta One, Special Report No. 15, Optical Satellite Tracking Program, Smithsonian Institution Astrophysical Observatory, July 20, 1958. Slattery, R.E. and Clay, W.G., Width of the Turbulent Trail Behind a Hypervelocity Sphere, Report 35 G-0004, M.1I.T. Lincoln Laboratory, 6 June 1961. Dana, T.A. and Short, W.W., Experimental Study of Hypersonic Turbulent Wakes, Report ZPh-103, Convair, San Diego, May 29, 1961. Lees, L. and Hromas, L., Turbulent Diffusion in the Wake of a Blunt-Nosed Body at Hypersonic Speeds, Aerodynamics Department Report 50, Space Technology Laboratories, Inc., July 1961. Also, Institute of Aerospace Sciences paper No. 62-71, presented at the IAS 30th annual meeting, New York, New York, January 22-24, 1962, ll. 12, 13. 14, 15. 16. 17. 18. 19. 20, 21. - 115 - Demetriades, A., Some Hot-Wire Anemometer Measurements in a Hypersonic Wake, Proceedings of the 1961 Heat Transfer and Fluid Mechanics Institute, Stanford University Press, 1961. Mohlenhoff, W., Experimental Study of Helium Diffusion in the Wake of a Circular Cylinder at M = 5.8, GALCIT Hypersonic Research Project, Memorandum No. 54, May 20, 1960. Kingsland, L., Jr., Experimental Study of Helium and Argon Diffusion in the Wake of a Circular Cylinder at M = 5.8, GALCIT Hypersonic Research Project, Memorandum No. 60, June 1, 1961. Dewey, C.F., Jr., Hot-Wire Measurements in Low Reynolds Number Hypersonic Flows, GALCIT Hypersonic Research Project, Memorandum No. 63, September 15, 1961. Also, American Rocket Society Journal, pp 1709-1718, December, 1961. GALCIT Hypersonic Research Project, Semi-Annual Status Report, April 1962. Feldman, S., On Trails of Axi-Symmetric Hypersonic Blunt Bodies Flying Through the Atmosphere, AVCO-Everett Research Laboratory, Research Report No. 82, December 1959. Also, Journal of the Aerospace Sciences, Vol. 28, No. 6, pp 433-448, 470, June, 1961. Lykoudis, P.S., Theory of Ionized Trails for Bodies at Hypersonic Speeds, The RAND Corp., Report No. RM-2682-1-PR, May 29, 1961, revised October 5, 1961. Kubota, T., Laminar Wake With Streamwise Pressure Gradient, GALCIT Internal Memo, April 1962. Baloga, P.E. and Nagamatsu, H.T., Instrumentation of GALCIT Hypersonic Wind Tunnels, GALCIT Memorandum No. 29, July 31, 1955. GALCIT Hypersonic Research Project, Quarterly Status Report No. 19, April 1 to July 1, 1960. Lilley, G.M. and Morton, D., The Response Time of Wind Tunnel Pressure Measuring Systems, College of Aeronautics, Cranfield, Report No. 141, November, 1960. 22. 23. 24. 25, 26. 27. 28. 29. 30. 31. 32. 33. - 116 - Smotherman, W.E., A Miniature Wafer-Style Pressure Transducer, Arnold Engineering Development Center, Report No. AEDC-TR-60- ll, October 1960. Nodelman, H.M. and Smith, F.W., Mathematics for Electronics, McGraw-Hill Book Company, Inc., New York, 1956. Laufer, J. and McClellan, R., Measurements of Heat Transfer from Fine Wires in Supersonic Flows, J. Fluid Mech., Vol. 1, No. 3, pp 276-289, September 1956. Tewfik, O.K. and Giedt, W.H., Heat Transfer, Recovery Factor and Pressure Distributions Around a Cylinder Normal to a Super- sonic Rarefied Air Stream, PartI, Experimental Data, University of California, Berkeley, Technical Report HE-150-162, 30 January 1959, Also, Journal of the Aerospace Sciences, Vol. 27, No. 10, pp 721-729, October, 1960. Demetriades, A., Private Communications 1961-2. Howarth, L. (editor), Modern Developments in Fluid Dynamics, High Speed Flow, Vol. Il, p. 684; Clarendon Press, Oxford, 1953. Matthews, M.L., An Experimental Investigation of Viscous Effects on Static and Impact Pressure Probes in Hypersonic Flow, GALCIT Hypersonic Research Project, Memorandum No. 44, June 2, 1958. Hill, J. A. F., Baron, J.R. and Schindel, L.H., Mach-Number Measurements in High-Speed Wind Tunnels, MIT Naval Supersonic Wind Tunnel, Technical Report 145, January 1956. Marson, G.B. and Lilley, G.M., The Displacement Effect of Pitot Tubes in Narrow Wakes, College of Aeronautics, Cranfield, CoA Report No. 107, October 1956. Hayes, W.D. and Probstein, R. F., Hypersonic Flow Theory, Academic Press, New York, 1959. Ames Research Staff, Equations, Tables, and Charts for Compressible Flow, NACA TR 1135, 1953 LeBlanc, L.P. and Webb, H.G., Jr., Boundary Layer Separation in a Supersonic Stream, North American Aviation, Inc., Report No. SID 61-72, 29 March 1961. 34. 35, 36. 37. 38. 39. 40. 41. 42. 43. 44, - 117 - Chapman, D.R., Kuehn, D.M. and Larson, H.K., Investigation of Separated Flows in Supersonic and Subsonic Streams with Emphasis on the Effect of Transition, NACA Rep. 1356, 1958. Hakkinen, R.J., Greber, I., Trilling, L. and Abarbanel, S.S., The Interaction of an Oblique Shock Wave With a Laminar Boundary Layer, NASA Memo 2-18-59W, March 1959. Liepmann, H.W. and Roshko, A., Elements of Gas Dynamics, John Wiley & Sons, Inc., New York, 1957. Eshbach, O.W., Handbook of Engineering Fundamentals, 2nd Ed., p. 7-121; John Wiley & Sons, Inc., New York, 1952. Chapman, D.R., An Analysis of Base Pressure at Supersonic Velocities and Comparison With Experiment, NACA TN 2137, July 1950. Greber, I., Shock-Wave Laminar-Boundary-Layer Interaction on a Convex Wall, NASA TN D-512, October 1960. Guman, W.J., On The Plateau and Peak Pressure of Regions of Pure Laminar and Fully Turbulent Separation in Two-Dimensional Supersonic Flow, Journal of the Aerospace Sciences, Readers' Forum, Vol. 26, No. 1, p. 56, January 1959. Gold, H., Laminar Wake with Streamwise Pressure Gradient for Arbitrary Initial Velocity and Enthalpy Distributions, GALCIT Internal Memo, May, 1962. Slattery, R.E. and Clay, W.G., Experimental Measurement of Turbulent Transition, Motion, Statistics and Gross Radial Growth Behind Hypervelocity Objects, M.I.T. Lincoln Laboratory Preprint JA-1891, April 1962. Wood, R.D., An Experimental Investigation of Hypersonic Stagnation Temperature Probes, GALCIT Hypersonic Research Project, Memorandum No. 50, July 15, 1959. Gorecki, J., An Investigation of Temperature Fluctuations on Bluff Bodies, PhD Thesis Department of Aeronautical Engineering, California Institute of Technology, 1960. 45, 46, 47. 48, - 118 - Hirschfelder, Bird, Curtiss, and Spotz, The Transport Properties of Gases and Geseous Mixtures from the Princeton Series on High Speed Aerodynamics and Jet Propulsion, Vol. I, Thermodynamics and Physics of Matter, Rossini, F.D. (editor), Princeton University Press, 1954. Keyes, F.G., The Heat Conductivity, Viscosity, Specific Heat and Prandtl Numbers for Thirteen Gases, Project Squid, TR 37, MIT, 1 April 1952. Graves, J.C., Quiel, N.R. and Nagamatsu, H.T., Impact Pressure and Total Temperature Interpretation at Hypersonic Mach Number, GALCIT Memorandum No. 20, July 26, 1954, NBS-NACA Tables of Thermal Properties of Gases, National Bureau of Standards, 1949 with later additions. - 119 - APPENDIX A FLOW CALIBRATION AND VISUALIZATION The calibration of the Tate-Emery gauge used to control the reservoir pressure was checked against a mercury manometer and found to be within the accuracy of automatic control (+0.02 psig). Atmospheric pressure was measured by a conventional precision mercury barometer to an accuracy of 0.01 cm mercury. The calibration of the reservoir temper- ature was checked using the iron-constantan thermocouple shown in Figure 56, employing conventional apparatus. The thermocouple was located just upstream of the throat at the centerline of the wind tunnel. It was found that the dial setting was high because the thermocouple used for automatic temperature control was located about 12 inches upstream of the throat, consequently, losses through a section of the wind tunnel walls were present before expansion through the nozzle. This fact accounts for the odd nominal temperature of 262 F selected for the tests and the slight variation with pressure. The dial setting for reservoir temperature was maintained constant throughout the tests, and it was calibrated for the full range of reservoir pressures available (Table I). In the calibration of wind tunnel reservoir temperature, the recovery factor of the iron-constantan thermocouple was assumed equal to one HOT JUNCTION D-C GALVANOMETER MELTING ICE COLD JUNCTION = STORAGE CELI. TYPE K-2 POTENTIOMETER CELL Figure 56. System for Calibrating Wind Tunnel Reservoir Temperature because the hot junction was located in a subsonic region of high pressure. The time constant of the temperature measuring system was very short because of the relatively high-pressure air engulfing the thermocouple. The accuracy of the temperature calibration was well within the capability of the automatic control system (+1.0F). The results of these measure- ments agree with those conducted by Wood (Reference 43) and Dewey (Reference 14) as well as with the wind tunnel empty measurements made by the author in conjunction with the wake surveys using a chromel-alumel heated thermocouple (Section II). To calibrate the wind tunnel, a conventional rake shown in Figure 57 was used. The rake was connected to a bank of silicone manometers which Figure 57. Rake Installation - 122 - were read manually. These data were then fed into an IBM 7090 program to obtain pressure contours, a few typical contours are shown in Figure 58. From these contours, it was evident that the flow was very uniform in the region immediately downstream of the cylindrical model (19 to 24 inches from the throat). It should be noted that the flow deteriorates with decreas- ing wind tunnel stagnation pressure so that the contours of Figure 58 represent close to the worst conditions. A schlieren photograph of the flow around the model is shown in Figure 1. To obtain this schlieren, the model was embedded in the glass ports of the wind tunnel in which were drilled 1/4-inch holes approximately 1/4-inch deep for support. These holes were very carefully and slowly drilled into the ports in order to avoid stresses in the glass. This procedure, which allowed for easy model installation and minimum leakage at the model supports, was possible because the dynamic pressure on the model was very low; consequently, high forces due to air loads were not present. It might be mentioned that the schlieren shown in Figure ] was taken with the knife edge horizontal. Photographs were also taken with the knife edge vertical; however, the density gradients in the axial direction were too low for satisfactory visible display. Some effort was expended in trying to take spark pictures in a time interval short enough to see the microscopic wake structure. At a nominal velocity of 2740 ft/sec (which results from isentropic expansion to Mach 5.7 with the wind tunnel stagnation temperature of 262 F) the flow advances about 0.033 inch during 1.0 microsecond. - 123 - Pry a | VT | U 14 |i — 700 > oN AT ee0°0 L 8 g 8 a 8 8 ~ ° ° = z 7 Zz %, \ To N PyYs/ = s s 8a. Ne 9.040 2.035 C2 ve - os 0.037 rto"o il \ : L. 200, r~ pennn— Tog" 0 ———f | a Va SePa_| Olle mags z _————+-—— > <5 8 i es P~— ¥ i LO ccoo <= —— — i Stoo VEO] OS 8 5 S & 8 2 & Rr 8 ~ s s 3 9 3 3 = & CeeNe Tf | Ee ea . e Se SHS \$ Pressure Contours for Empty Wind Tunnel Figure 58, - 124 - One microsecond, then, appears to be the largest order of magnitude of the time interval which must be achieved for visible microscopic wake investigation. In this time interval, it is very difficult to get light intensi- ties high enough for schlieren photographs. An open air spark gap developed by Gorecki (Reference 44) was tried. This source provided time intervals of several microseconds with enough intensity to give photographs using sensitive film. However, due to the flow motion and the resultant smearing, no additional phenomena were visible over the longer - exposure-time schlierens. The project was therefore abandoned. - 125 - APPENDIX B HEATED THERMOCOUPLE The heated thermocouple used for total-temperature measurements is shown in Figure 23. The supports were heated to the same value as the measured wire temperature by the design shown in Figure 59. In Figure 59, it will be noted that three thermocouples were used, one for measuring the total temperature of the airstream, and one for measuring the support temperature at each end of the wire. The output from the thermocouple at the alumel support was fed into the servo along with the output from the thermocouple at the center of the wire. The servo compared these two outputs and heated the alumel support, making the difference in outputs equal to zero. The temperature of the chromel support was then adjusted by comparing the output of its thermo- couple with that of the alumel support with a sensitive galvanometer. The chromel-heated or the alumel-heated (or both) thermocouples were adjusted by varying the resistances shown in Figure 59; therefore, at equilibrium, both the servo and the galvanometer were nulled. ‘The servomechanism used was developed by Wood (Reference 43) for a heated shielded thermocouple. The system consists of a de null detector, a CAT (current adjusting type) control unit, and a power manipulator. - 126 - ELECTRIC HEATER CIRCUIT ------- — CHROMEL Figure 59. Schematic of Heated Thermocouple Probe These elements are all commercially available. The de null detector measures the difference in output between two thermocouples. Its signal is fed into the CAT control unit which, in turn, adjusts the power manipu- lator. The CAT incorporates provisions for obtaining stability and minimum response to demand changes. The calibration needle shown in Figure 23 was used to determine axial distances accurately. First, the distance between its tip and the thermocouple junction was measured with an optical comparator. Then the tip was allowed to barely touch the model during tunnel operation after thermal equilibrium had been established. After axial distance was fixed and the axial counter set, the calibration needle was then removed. - 127 - The chromel-alumel cold junction was coated with wax to eliminate spurious voltages induced by the melting ice. A chromel-alumel thermo- couple was selected for this experiment for the following reasons: neither metal is corrosive, the combination gives out a reasonably large output compared to other metals, and both metals are readily available. As indicated in Figure 23, the heated thermocouple allows for the use of 5-mil wire rather than the very thin wire necessary with conventional means. For this reason, the chances of breaking the wire due to wind tunnel starting loads are minimized. - 128 - APPENDIX C TRANSPORT PROPERTIES Because of the very low static temperatures in the test section, the conventional Sutherland formula for coefficient of viscosity was suspected -- especially since no experimental data for coefficient of viscosity for dry real air exists below 180 F. Therefore, the coefficient of viscosity was computed using the Lennard-Jones potential as outlined by Hirschfelder, Bird, Curtiss, and Spotz (Reference 45). The results of these computations are shown in Figure 60 and Table IV. Along with the 1,4 —— FROM NBS-NACA TABLES OF THERMAL PROPERTIES OF GASES a “TABLE 2.39/1 3/2 x SUTHERLAND'S FORMULA 107 = 145.8 T T+110.4 WHERE 4= ABSOLUTE VISCOSITY, POISE 1.2 T= TEMPERATURE, -KELVIN —7 =~ BASED ON LENNARD-JONES POTENTIAL USING Eq. (6-1), p.371, SECTION D: THE TRANSPORT PROPERTIES OF GASES AND GASEQUS MIXTURES BY HIRSCHFELDER, BIRD, CURTISS, AND SPOTZ, FROM THE PRINCETON SERIES HIGH SPEED AERODYNAMICS AND JET PROPULSION, VOL. I (ROSSINI, F.D, EDITOR) —-~ BASED ON KEYES EQ.: 1.0 LE. = 0.06462 —gpy-4— r 1+— Tx 1077 (Tin °R) REF: KEYES, F.G., HEAT CONDUCTIVITY, VISCOSITY, SPECIFIC HEAT AND PRANDTL O.8b NUMBERS FOR THIRTEEN GASES, PROJECT SQUID TR 37, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, 1 APRIL 1952 0.6- i 180 280 380 480 580 TEMPERATURE (R) Figure 60. Coefficient of Viscosity for Air Versus Temperature - 129 - values based on the Sutherland formula and the Lennard-Jones potential, a curve based on Keyes! equation * (Reference 46) is also shown in Figure 60. Reynolds numbers based on the coefficients of viscosity given in Figure 61 were used for all computations. In a manner similar to that discussed above for viscosity, the thermal conductivity of air was calculated for low static temperatures, based on the Eucken assumption of fast energy transfer as presented by Hirschfelder, Bird, Curtiss, and Spotz (Reference 44). The results of these computations are shown in Figure 62. To compute the Prandtl number, the experimental values of specific heat for dry real air presented in Figure 63 were taken from Reference 48. The Prandtl number based on these properties is shown in Figure 64. *It should be noted that Keyes' equation as given by Graves, Quiel, and Nagamatsu (Reference 47) and Matthews (Reference 28) contains errors. 130 - Table IV Coefficient of Viscosity for Air (M/M,) = (T/T) Choose: T r Hr 1/2 g (T) Based on Lennard-Jones Potential ¥(2,2; T ,) i tt g(r.) Y(2,2;7T) 273. 16°K (= 491. 69°R) 1716 x 107? poise (g/sec-cm) T = (kT/€) k,€ = force constants forair €/k = 97.0°K Viscosity Functions Comparison With Experiment Hexptl. Healc. rT Y(2, 2) g T Y(2, 2) & T(°K) -7 ; (10° poise) 0. 30 2.785 1.0014 2.7 1,069 100 713 702 0.35 2. 628 2.8 1.058 120 846 840 0.40 2.492 2.9 1, 048 140 975 972 0.45 2. 368 3.0 1.039 1. 0034 160 1101 1099 0.50 2.257 1, 0002 3.1 1.030 180 1221 1221 0.55 2.156 3,2 1,022 200 1336 1337 0. 60 2.065 3.3 1,014 220 1448 1447 0.65 1.982 3.4 1,007 240 1556 1554 0.70 1.908 3.5 0.9999 260 1659 1657 0.75 1.841 1. 0000 3.6 0.9932 280 1756 1756 0.80 1.780 3.7 0.9870 293. 16 1819 1819 0.85 1.725 3.8] 0.9811 300 1851 1851 0.90 1.675 3.9 0.9755 400 2294 2290 0.95 1. 629 4.0} 0.9700 | 1.0049 500 2680 2678 1.00 1.587 | 1.0000 4.11 0.9649 800 3613 3680 1.05 1.549 4.2] 0.9600 1000 4165 4257 1.10 1.514 4.3) 0.9553 1200 463} 4761 1.15 1.482 4.4] 0.9507 1500 5262 5494 1.20 1.452 4.5] 0.9464 5000 12080 1.25 1.424 | 1.0001 4.6] 0.9422 10000 18870 1.30 1.399 4.7} 0.9382 1,35 1,375 4.8 0.9343 1.40 1. 353 4.9] 0.9305 1.45 1. 333 5 0.9269 | 1.0058 1.50 1.314 1. 0004 6 0.8963 1.55 1.296 7 0.8727 1. 60 1.279 8 0.8538 1.65 1. 264 9 0. 8379 1.70 1. 248 10 0.8242 | 1.0075 1.75 1,234 20 0.7432 1. 80 1.221 30 0.7005 1.85 1, 209 40 0. 6718 1.90 1 197 50 0.6504 | 1.0079 1.95 1.186 60 0.6335 2.00 1 175 1.0014 70 0.6194 2.1 1.156 80 0. 6076 2.2 1.138 90 0.5973 2,3 1.122 100 0.5882 71 1.0080 2.4 1. 107 200 0, 5320 2.5 1,093 1.0025 300 0.5016 2.6 1.08] 400 0.4811 1. 0080 131 soanjyeroduroy, MoT ye Aty I0F APISOOSTA JO JUOSTOTJZIOD TOT pesy sontear Qo) UNLVYIdWIL ‘Tg eansty Ovz 00z O9L Ozt 08 14 T T T T T ole arr’ 2n0l * £609°6 j- NI - 34S (W) @1 262 cyl g-O1 * LeSt"t j- 4d p- 33S (W) 29 LL zor’ | z-OL * OES" E (D3S 14/OM1S) z- 14 DS (4) 81 96/ 6Le"t 6-Ol * 688H°Z z- NI 23S (4) #1 sez 9SE" g-Ol * 86rZ"L z- W Das (4) OX 024 zee" 7-0 * 86rZ" z- W245 (4) 9 Z0L Boe" | g-Ol * 9IZI 3SIOd|LN3> 3 OL * 9IZL ~WI- 789 €8Z"t Z D - D3S (W) 9 YO ASIOd n n/n 999 662" 4 rot 7 8¥9 vez" /. 0c9 602'1 4O GaNIWA . z19 esi" Ag Alaitinw SNOISNOWIO SEL DLN o1 | aaivinevi / 6S “Sit 1u3ANOD OL / 925 OL L . 8GS EOL" L 7 ors 9L0°1 7 zs 8¥0" | yA 0S 0z0°1 J 9RP 6066°0 ore" z9L 7 . GEZE rl . 897 £196°0 . 92 4 osy | 6tesro | 0882 l zer $106°0 ya pip 9028°0 tn Yo a 96€ 16£8°0 / 1 go 8Ze |_0408"0 ONIUIVS f 09€ Z¥ZZ°0 a Zve L0%2°0 Aaa pze | $90Z°0 7 90E 91L9°0 88Z 65€9°0 9%¥z" 80l 0£z 766S°0 v8Iz" z°L6 . tage : ‘ 72 12960 riz: +°S6 (WOLIGA “d *4 ‘INISSOY) 1 “IOA “NOISINdOUd vez 6Ez5°0 epiz" 9°86 13f GNV SDIWVNAGOU3V G33dS HOIH SI1W3S NOLIDNIYd 3HL 91z ar8r"0 +202" 06 WOdd “ZLOdS GNV ‘SSILUND “GYId “YIDTSAHDSYIH A SINLXIW I erryo | SE91° zt SNOISYD ANV SISWD JO SHLYIdOYd LYOUSNVUL FHL ?G NOILDIS OBI ecorvo | ZSz1°0 ys "UZe “d “(L-9) "OF ONISN IWILNaLOd SJNOP-GYVNN3T NO G3SV@ => —— ; ; , YIV AYD JO ALISODSIA JO INFID1SIIOD /1/6e°% FAVI - Yo, n/y n/n 7) S3SWD 4O SIILYIdONd TWWYIHL JO IIGVL VOVN-SEN WOU sa1avl TWIINI10d ONNIVI see WOVN-SEN S3NOf-G¥¥NN31 L°O €°0 Jn/n S°0 9°0 iz - 132 - — — — — FAIRING THERMAL —————_ DATA FROM NBS-NACA TABLES OF THERMAL PROPERTIES (evra OF GASES = TABLE 2.42, THERMAL CONDUCTIVITY OF DRY AIR TEMP A oO —-—=+ BASED ON EUCKEN ASSUMPTION OF FAST ENERGY TRANSFER R USING EQ. (10-2), P. 405, SECTION D: THE TRANSPORT PROPERTIES OF GASES AND GASEOUS MIXTURES BY 144 0.3092 HIRSCHFELDER, BIRD, CURTISS, AND SPOTZ, FROM THE . 162 0.3450 PRINCETION SERIES HIGH SPEED AERODYNAMICS AND JET 180 0.3831 PROPULSION, VOL. | (ROSSINI, F. D. EDITOR). 198 0.4203 216 0.4576 0.44 234 0.4948 COEFFICIENT FOR THERMAL CONDUCTIVITY OF DRY REAL AIR 252 0.5318 BASED ON EUCKEN ASSUMPTION 270 0.5687 288 0.6052 306 0.6418 p- —4“_ [oc tse GC _(T) 324 0.6780 TG Pot 4) (T) 342 0.7138 0.40+ 360 0.7494 HM = COEFFIECIENT OF VISCOSITY 378 0.7846 = MOLECULAR WEIGHT (28.967) 26 Dees Cp = SPECIFIC HEAT AT CONSTANT PRESSURE 432 0.8885 R= GAS CONSTANT FOR AIR 450 0.9225 “C= REDUCED TEMPERATURE, KT 468 0.9561 0.36 J = TEMPERATURE, °K € py seed k,€ = FORCE CONSTANTS pet 1055 FOR AIR (€/;) = 97.0°K 340 1087 J (T) = VISCOSITY CORRECTION FACTOR 558 hug CIT) = THERMAL CONDUCTIVITY CORRECTION 576 1.151 FACTOR 594 1,182 612 1.213 0.32-- o 630 1,244 T 6 648 1.275 666 1.305 0.30 1.0014 | 1.0022 684 1.335 0.50 1.0002 }| 1.0003 Y/, 702 1.365 ° 0.75 1.0000 | 1.0000 Py) 720 1.394 1.00 1.0000 | 1.0001 ” 738 1.423 0.28;- 1.25 1.0001 1,0002 / 756 1,452 1.5 1.0004 | 1.0006 4 774 1,481 2.0 1.0014 | 1.0021 2.5 1,0025 | 1.0038 3.0 1.0034 | 1.0052 , 4.0 1.0049 | 1.0076 5.0 1.0058 | 1.0090 0.24- 10.0 1.0075 | 1.0116 50.0 1.0079 | 1.0124 / 100.0 1.0080 | 1.0125 / 400.0 1.0080 | 1.0125 1 / 0.20} / o / / TO CONVERT ° TABULATED {TO INDICATED REP ENSIONS MULTIPLY BY / VALUE OF 0.16 . - -5 / cal cm7! sec7! oK-! 5.770 x 10 ° k/ko k Bru ft7! hem! ogé} 1.395 x 10-2 / watts em=} OKT! 2.414 x 10 e 0.12 l e I I l l 0 40 80 120 160 200 TEMPERATURE (R) Figure 62. Thermal Conductivity of Dry Real Air - 133 - 3,88 SOURCE: TABLES OF THERMAL PROPERTIES OF GASES, U.S. DEPARTMENT OF COMMERCE, 3.84 NATIONAL BUREAU OF STANDARDS, CIRCULAR 564, NOVEMBER, 1955 3.80 }- CONVERSION TABLE TO CONVERT | TO HAVING THE DIMENSIONS MULTIPLY BY 3.76 L TABULATED INDICATED BELOW , VALUE OF Cy CAL MOLET! °K"! (oR ecm!) 1.98719 3.72 + CAL G7l eK"ToReCHl) 00686042 JOULES G7? OK! (OR PC™") 0.287041 BTU (LB MOLE)" 1oR™! (oRer!) | 1.98588 BTU LB™!R™! (OR OFA1) 0.0685590 Cp 3.68 TAKEN AS 7/2 FOR T <90°R 3.64 | “ft \ 0.1 AT 1 ATM 0.01 ATM 3.60 -- 3.48 ! ! ! 40 80 120 160 200 3.48 “0,01 ATM ! 1 1 . 180 280 380 480 580 680 780 TEMPERATURE (°R) Figure 63. Specific Heat at Constant Pressure for Dry Real Air 0.88 Coa 0.86 Pr= Pr = PRANDTL NUMBER Cp= SPECIFIC HEAT AT CONSTANT PRESSURE 4 = COEFFICIENT OF VISCOSITY 0.84 [- k = THERMAL CONDUCTIVITY 0.82 + 0.80 + 100 ATM 0.78 0.76 9 0.74 0.72 0.70 0.68 64. 480 580 TEMPERATURE (R) 680 780 Prandtl Number for Dry Real Air - 134 - APPENDIX D LAMINAR-WAKE THEORY Using Kubota's approach (Reference 18), it is assumed that the flow in the viscous wake can be described by the boundary-layer type equations: a(pu) , a(Pv) = 0 1 ax ay (1) du du du. a du ee eee “ax pv dy Pete dx * ay bay (2) @P _ 9 (3) dy @H aH F) ( a) a (2 -¢o au o —— - 2/(f no). (st a= 4 pug PNG ay \oey dy \ o B dy (4) where the total enthalpy, 1 2 H =ht+>u, (5) 2 and assuming a perfect gas, y- 1 p = y Ph , (6) The boundary conditions of symmetry and free-stream flow at infinity apply, viz: ou (x,0) = 0 ; u(x, £0) = u (x) dy e (7) oH (x, 0) = 0 ; H(x, +0) = H dy e - 135 - The following transformation is used x x (x) = f eles ax (3) Po» Boot 00 d le) y u - e p dy youy) = = [re f 2S (9) Uy d 5 P,, a -du -98 _ 2, 1 e au last ag * (8-8) NO ae to (10) e ay 2 (y-1) M7 -ag _-ag _lag l-o ‘Me 9g faa a26 4598 . 28 _ a oS (11) ax ay ay 7 1+%- Mo % ay \ ay where, from continuity (12) and, by definition It is assumed that the Prandtl number is constant (not necessarily = 1), and, analogous to the Chapman-Rubesin relation for linear temperature-viscosity relationship, PH = pK, (14) In order to linearize Equations 10 and 11 first, let u= l-w (15) G= ¢g-l (16) - 136 - and assume the velocity defect, w<< 1. Substituting into Equations 10 and 1l and retaining lowest-order terms, we obtain 2 rs) h ) l 9a (= ) —~(—-1) = =-—>(—- 1). (17) -2 ax ho ° a5 he dM 2 ) 2 2 im “w)+G—~ = “4 (m “w) (18) ox e - - e ax oy where 2 - 2 poli = (y-1)M w+(1 +*2im “\o (19) h e 2 e e with the boundary conditions G = 0] X = 0 bo at (20) w = 0 y = £0 ay , at y = 0 (21) aG “= o ay J w(-y) = w(ty) (22) G(-y) = G(+ty) if G<<w (23) Equations 17 and 18 reduce to the diffusion equation 2 oW _ a W (24) ef an? - 137 - with homogeneous boundary conditions and symmetry € = w W = of}at 7 = +o aw oat = oO e) an 7° W(-1) = W(+n) Using Laplace transform techniques, the general solution of Equations 24 and 25 is <= 1 fy i b= pin -n) W=35 J womre « dy n Pp (26) n ee f wom Ec VPM = ay “ VP. where w =x {wh - Jf we PS ag (27) and W, (7) represents the initial conditions. If Wo (7) is a delta function of strength Q, Equation 26 reduces to WF - 2 e VP (28) VP or 3 NR > AN wy - 138 - Thus, the solutions of Equations(17) and (18) for G <<w and initial delta functions at the origin of x are -2 M Oy = A exp [- y (30) e Vx Ax -2 = - 1- exp (- gy ) (31) e [z Ax where A and B are determined from the initial condition (usually taken at the neck) R, pu @g As a( C (32) ore 3 B = E + (Pia, 0),| (33) 2VT (pu dajh © © “oO where @ is the momentum thickness 6 = | Ba (tae (34) pu u ee e fo) and Q is the heat transfer to the body. From Equation 32 bom 9 eee 0 > (35) % puM - 139 - To determine the wake semi-width, Yo one notes that y pu piu ~ -1 a20- - 2 S IR - 2% /5 +4 ay +4 (m 222 R (36) d e piu e 2 e e 2 e pu_d e d ee 0 d /o where Ye is the wake semi-width in the transformed plane. Assuming that at the wake edge (37) then e N > (38) It is interesting to note that Ye -1 26 (=), ~ Xm, 2), (69)