Repression of High Velocity Flow Phenomena Around Sharp Bends in Supercritically Sloped Open Channels Citation Wagner, Warren Orval (1945) Repression of High Velocity Flow Phenomena Around Sharp Bends in Supercritically Sloped Open Channels. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/e7z9-rm42. https://resolver.caltech.edu/CaltechTHESIS:06302025-210853671 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1945 Record Number: CaltechTHESIS:06302025-210853671 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06302025-210853671 DOI: 10.7907/e7z9-rm42 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17509 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 30 Jun 2025 23:49 Last Modified: 30 Jun 2025 23:57 Thesis Files PDF - Final Version See Usage Policy. 29MB PDF (Layout of Experimental Channel) - Supplemental Material See Usage Policy. 933kB PDF (Closed Circuit Model Basin) - Supplemental Material See Usage Policy. 725kB PDF (Closed Circuit Model Basin II) - Supplemental Material See Usage Policy. 1MB Repository Staff Only: item control page

REE>RESSION OF HIGH VE:WCITY F:WW PHENOMENA .AROUND SH.ARP :SENDS IN SUPER-CRITICALLY S:WPED OPEN ClIDmELS Thesis by Warren o. Wa,gner Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1945 THE PROBL:F.M .Analysis and Repression of These Waves ' TABLE OF OONTfflTS PART I. INTRODUCTION ~ A. S'\llIJIIlary . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 C. Introduction and Statement of Problem.................. 4 PART II. ANALYTICAL STUDY A. Review of Properties of High Velocity Flow............. :S. Review of Previous Studies of Effects of Curves on High Velocity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. D. 6 10 Analytical Approach to Super-Ori ti cal Flow Around a Sha.rp B et1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 l. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2. Analysis of Wave Depths . .. .. .. .. .. .. . . . . .. .. . . .. .. 13 3. Location of Wave Crossings . . . . . . . . . . . . . . . . . . . . . . . . 20 Anticipated Effect of Sills on Wave Repression......... 23 1. Wall Wave Heigh ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. Reduction of Transverse Velocity 24 PART III. EXPERIMENTAL STUDY A. Apparatus and Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ii. B. C. D. l. Preliminary Studies ................................ 29 2. The Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3. Entrance Section of Channel ........................ 33 4. Source of Water and the Circulating System ......... 34 Instruments ............................................ . 37 ......................................... 37 l. Point Gage 2. Pitot Tube 38 3. Velocity Direction G~e ........................... . 39 4. Sills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5. Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 l. Adjustment of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Profile Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3. Velocity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4. Sill Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5. Measurement of Forces Acting on Sills .............. 50 6. Photography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 l. Sill Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ...................................... 55 3. Water Surface Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4. Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5. Momentum Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Wall Profiles iii. PART IV. C0MP.ARIS0N OF .ANALYTICAL AND EXPERIMENTAL RESULTS A. ]. Un treat ed. Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 77 l. Wave Depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2. Location of Wave Crossings . . . . . . . . . . . . . . . . . . . . . . . . 80 3. Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Treated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 1. Wall Wave Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2. Velocity Distribution and Reduction of Transverse Velocity ................................ ~... 87 PART V. C0NCIDSI0NS A. Sl.lmIJlf:JX'Y' of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 B. Application to Field Conditions ........................ 95 BIBLIOGRAPHY .................. 97 .APPENDIX A. List of Tables 98 l3. List of Figures and Drawings .......................... . 99 I. SUMMARY AND INTRODUCTION .A.. SUMM.lRY This investigation was conducted for the purpose of contributing additional information to our present lmowledge of the principles of super-critical flow around bends in open channels. The effects of a mild but abrupt bend in a steeply sloped channel were studied and numerous experiments were made with various types of control devices designed to repress the undesirable waves created by the bend. The channel used for these experiments was 18 inches wide, 12 inches deep, and approximately 40 feet long, with a 10° abrupt bend at its mid-point. Measurements were made of wave heights along the walls and over the water surface, of velocities and their changes (both in magnitude and direction) from section to section, and of forces acting due to the consequent cha.n€es in momentum. The experimental results obtained agree well with the analytical developments propounded. Wall wave heights above those for uniform flow were reduced by the proper control device to as little as onethird the wave heights created by the bend in the untreated channel, with the accompanying establishment of fairly uniform conditions of velocity end cross-sectional area below the control device. With such satisfactory repression of the undesirable waves in a single, small-angle abrupt bend, it is felt that the principle could be extended to a large angle bend by using a series of these small, abrupt bends one below another. B. 0(. = NOTATION angle of sill with upstream channel (3 = wave angle b = width of channel (= 1.50 feet) C = wave celerity C8 = center of bend d = depth of water in channel d, = -wiiform upstream depth dz = depth beyond wave front or downstream depth d, = critical depth D~x = distance x downstream from bend f.. = specific energy F = Froude Number= v,/-1:ji. g = gravity constant h = height of sill \W = inside wall of channel J = force divided by mass of water flowing per second i = length of sill L = r\ = we.ve length coefficient of friction for channel (n = 0.0085) OW= outside wall p = force cf, = effective turning angle exerted by sills Q = discharge in cu. ft. per sec. 3. R = radius of curvature in curved channels. ROUI.1. -- location of first cross-wave on outside wall Ri" = location of first cross-wave on inside wall S = bottom slope of channel e = angle of bend U-x. = distance x upstream from bend V = velocity in feet per second "· = upstream velocity in feet per second w = specific weight of water w = rate of discharge in lbs. per sec. 4. C. INTRODUCTION AND STATD!ENT OF PROl3Ll!M Many sections of the United States are subject to occasional and infrequent rainfalls of in tense magnitudes, while in normal years the rainfall may be so minor as to cause only a small flow in the adjoining streams and their tributaries. In the central, eastern, and northern parts of the United States most rivers flow at a substantial rate throughout the year, being fed principally from areas covered by snow for a portion of the year. In the south-western portion of our country, however, rivers may flow for only a few days annually, at other times completely drying up so far as surface flow is concerned. Such a condition is found in ma.t17 portions of Southern California and parts of Nevada, Arizona, New Mexico, and Texas. With little or no water flowing for most of the d~s of the year, there is a natural tendency on the part of a densely populated area to encroach more and more upon the banks of the streams, confining the river channels to smaller and smaller areas just sufficiently large to handle the normal yearly flood flows. When the sl0pe of the river drainage bed is mild, this encroachment can be greatly counteracted by deeper excavation of the channel and by providing a smoother bottom and side walls. However, when the slope is steep, this corrective method is prohibitive because of the excessive cost of confining the flow within the channel walls. Flow phenomena on steep slopes is entirely different from that on mild slopes . .A.s long as the flow continues without change in direction, no appreciable physical difference in the two ty-pes of 5. flow occurs. It is comparatively inexpensive to confine the flow within the channel walls around a bend where the slope is mild, but it is quite a different problsn where the slope is steep. Waves build up along the walls of the channel at the bend and downstream from it, which mlW become several times the depth of the nonnal flow. These conditions of high velocity flow occur elsewhere as well, so that a study of such flow is not confined to flood channels, although the need for a remedy is most apparent in such channels. Practically no information was available on high velocity flow around bends in open channels until the publication of the results of a test conducted by Knapp and Ippen1 on the flow around curves in rectangular channels. Their results proved so satisfactory and so entirely different from any previously anticipated results that their research was continued in the attempt to find some means for diminishing the dangerous, high wall waves2, This latter study also included flow in a trapezoidal-shaped channel. The problem dealt with in the investigation presented, herewith, had to do with flow around a sharp bend and possible means of repressing undesirable features of the flow created by the bend. l Subscripts refer to material listed in the bibliography. 6. II. ANALYTICAL STUDY A. REVIEW OF PROPERTIES OF HIGH VEU)CITY FI.OW The term "specific energy" was first introduced by l3akbmeteff in 1911. It is merely the total energy per pound of fluid flowing in an open channel using the channel bottom as a datum plane. It can be expressed as: (1) Since V-= i , Equation (1) may be written in the form: E -= d + Q.t. A2. (2) 2.~ The depth for minimum specific energy can be found by differentiating Equation (2): dE d(d) (3) If the area of cross-section A can be expressed as a function ol the above derivation can be completed. A,. lod , and Equation (3) becomes: of de. d(d) or + For a rectangular channel 7. ' from which: (4) since Q-=Vbd . Rearrangement of Equation (4) shows that critical velocity occurs when: (5) If the specific energy equation is divided by clc. or its equivalent values, as above developed, a dimensionless equation results3: +- .L (~.,_ 2 d.-, , (6) a plot of which is shown in Fig. l: .,... tcr -I 3 2 2 3 ◄ .5 ID E/ci&c SPECIFIC ENE.RGY (DIME.NSIONLE.SS) Fig. 1. From this plot it can be seen that two entirely different conditions of flow are possible for a given specific energy, one with a 8. large depth relative to the critical depth and the other small relative to the critical depth. The properties of flow on the upper portion of this curve are well lmown and have been utilized for many years. However, the properties of flow on the lower part of the curve, which are termed 11 su.per-cri tical 11 , are not too well known. With the studies that have been made in recent years on supercritical flow bas come the realization that a definite relationship exists between the velocity of flow in a channel and the velocity of propagation of a gravity wave. The velocity of propagation of an infinitesimal gravity wave, usually referred to as wave celerity, in still water, as first developed by La Grange, is given by the relationship: (7) It can be seen that the wave celerity increases with an increase in the depth of water. If a disturbance causing a wave occurs in a channel where there is already a velocity of flow, the one is superimposed upon the other, and the resulting velocity is the vector sum of the channel velocity and the wave velocity. The similarity of Equations (5) and (7) is at once apparent and examination shows that the two velocities are equal when d = de. . Thus, if a disturbance occurs when the depth is critical; i.e., when specific ener{!;f is a. minimum, the resulting vector velocity is zero and the wave thus created will be a stationary one so far as the upstream face is concerned. 9. When the conditions of flow in a channel are sub-critical; that is, when cl is greater than olc , and V is consequently less than Ve , it can be seen that the effects of a disturbance can be felt upstream from the source of tb.e disturbance, since C will also be greater than V , and the wave created by the disturbance will travel ups'tream at a speed of C-V . tions are super- critical, ( However, when the flow condi- cl <clc and V )Ve), the effect of a disturbance cannot be transmitted upstream, since C than will be less V and the propagating wave consequently cannot travel upstream , but will be limited to a zone downstream from the disturbance. The limits of the zone can be determined readily by reference to the adjoining sketch, showing the vector velocities. C The wave velocity travels in all directions along the water surface at the same time it is being swept along downstream with a velocity V. The effects of any disturbance at 11 .A." can only be felt within the boundaries of this zone. sides of the angle ~ where These boundaries are the 2 (3 , where: : . -·, -C -= SIWl. _, -'gd . -• \ l Cll_C4::: SiVI V V F StY\ F is the Froude number. (8) ~ is known as the wave angle and is analogous to the Mach angle for super-sonic flow in gases4. 10. :B. REVIEW OF PREVIOUS STUDIES OF EEFECTS OF CURVES ON HIGH VEWCI TY FWW The studies made by Knapp and Ippen, as previously cited.1, 2 , yielded results which gave for th/ first time clear concepts of the mechanism of super-critical flow in curves, with an understanding of the underlying reasons for the resulting disturbance patterns, both within the curves and downstream from them. Fig. 3 is a diagrammatic sketch showing the manner in which waves are built up in a curved section: Fig. 3. When the particles of water nearest the outside wall in the upstream straight section of the channel arrive at point A, the beginning of the curve, they are deflected by the curve through the angle (3, whose magnitude is as shown by Equation (8). However, particles of water in the center of the channel continue in their original direction for a further distance downstream and are unaffected by the wall at the same cross-section, because of the inability of the wall to exert any influence on them. Particles near the inside wall are deflected at point :B, the inside beginning 11. of the curve, through the same angle (3 . As the particles a small distance from the outside wall in the upstream section pass through the infinitesimal wave created at point A and meet the curved section of the channel, they too are deflected and create additional waves, building up the wave front shown by points a,b,c. Also, as each particle further and further from the outside wall meets the curved wall, it builds up the depth of water along the wall. Downstream from point B, the wall is convex and the waves thus created are negative, with the consequent reduction in depth within the zone of influence of the disturbance at B. It would, therefore, be expected that the maximum depth along the outer wall would be attained when the effects of point B were first felt across the channel, point C. This is true except for a slight modification. Due to the increase of the wave velocity caused by the increased depth in the area AabC and the change in direction of the velocity in this area, the maximum depth is found downstream a small distance from C, or at point e 1 • Knapp and Ippen have shown that maximum depth occurs where: (9) The depth of flow along the walls, at any angle 8 below the beginning of the curve, which they calculated on the assumption of constant velocity throughout the curve (an assumption justifiable generally for super-critical flow because changes in depth are small compared to velocity-heads encountered) is given by: 12. (10) The plus and minum signs refer to outer and inner walls, respectively. The maximum depth is obtained by substituting Equation (9) in this latter expression and is found to be: (11) It would be expected that these maxima and minima waves would be reflected back and forth across the channel in the downstream section, and so they are, gradually damping out with friction and other disturbances. The distance along either wall between two successive maxima, or between two successive minima, is: L-- (12) The first maximum depth as given by Equation (11) is greater than the other downstream maxima unless unfavorable corrlitions of reflection and interference are encountered further downstream. 13. C. .ANALYTICAL APPROACH TO SUPER-CRITICAL FIOW AROUND A SHABP BEND Even a hasty glance at the problem of attempting to analyze the mechanism of super-critical flow around a sharp bend indicates that it is essentially one involving three dimensions. Analysis of three- dimensional hydraulic phenomena is especially difficult. Since the present problem is concerned with red:u.cing undesirable wall wave heights and the re-establishment of moderately 1miform flow conditions in the downstream section, (both of which are essentially two-dimensional in scope), it is pertinent to make some simplifying assumptions which will allow a rational approach to the problem. 1. Assumptions: The first assumption is that the distribution of pressure is hydrostatic. Actually pressure distribution is hydrostatic only if the curvature is gradual. However, Knapp and Ippens' studies of flow around curves indicated that the assumption of hydrostatic pressure distribution was permissible. Rouse 5 , in an experimental study of high-velocity flow along a vertical plate at various angles to the original direction of flow, also determined that such an assumption was within satisfactory limits. The second assumption is that of a velocity vector of constant magnitude, an assumption in close agreement with measured results. 2. .Analysis of Wave Depths: Fig. 4 indicates the wave which would be created by super- critical flow in a channel where the vertical outside wall changes abruptly in direction: 14. Fig. 4. The velocity relationships are also indicated. At a point sufficiently removed from the wall to insure full development of the shock wave, application of the momentum and continuity principles should yield a relationship between the upstream and downstream depths. Fig. 5 indicates a crosssection through the wave front and normal to it. If the momentum equa- tion is applied between vertical Fig. 5. sections on ea.ch side of the wave front, there results: • (13) Applying the continuity principle yields an additional necessary relationship: (14) 15. use of the first part of which, substituted in Equation (13), results in: . (15) Also, from Equation (14): (16) substitution of which in Equation (15), yields: Equation (17) can be simplified to: (18) which when solved for V.siV\@> gives: (19) This relationship, as pointed out by White and Rouse 6 , is the same as that for a standing surge normal to the wave front. Equation (19) can be rearranged into a more useful form: @ ~ sin-'[-t ft,(<1,+d~ and by recalling that F -= V, (20) /¥ can be simplified to: (o = si..•'[--t,:y;,t (d~+d:)1 (al) 16. It might be pertinent to point out that Equation {21) differs from Equation {8) by the term under the radical, - Equation (21) being that for a wave of finite height, while Equation {8) was that for an infinitesimal wave. One would expect, then, that Equation {21) should become Equation (8) when the conditions for an infinitesimal wave are applied to it. These conditions require d 1 to equal d, , and this substitution is seen to satisfy the requirements. By a few operations on Equation {21), it is possible to obtain a simple relation between d:t and c::1, . Thus: (22) Multiplying both sides by 4 and adding l, gives: which is equivalent to: It- BF .. sir1'{3 - ( d, !~d~y- {24) Further, (25) or (26) Equation {26) then becomes: (27) 17. a relationship giving da. for any value of cl,, provided (?, and F are known. Unfortunately, (3 is not usually lmown but must be deter- mined in some other manner. In this regard, Equation (21) cannot be used because it is merely a rearrangement of Equation (27), and is, therefore, not an independent equation. However, an independent relationship can be obtained with ease. By reference to Fig. 4, it can be seen that the momentum equation applied parallel to the wave front will yield: (28) which, when applied as a divisor on Equation {14), gives: (29) This can then be rearranged to give the second relationship between d2. and ol, : {30) If it is desired to separate {l> from & . Equation (30) can be expanded readily to: (31) Equations (27) and (30) are thus two simultaneous equations, the solution of which gives (3 and the ratio da./d' 'e being usually fixed in value, and F being constant depending on the value of d 1 • 18. (27) (30) Reference to Fig. 4 and the above development, would lead to the belief that the same analysis would apply in the case of a convex wall where a so-called negative shock wave would be encountered. Here, however, the values of 6 would be ng-ative, or Equation (30) could be rewritten as: (32) Little inspection of the two simultaneous equations is necessary to realize the difficulty of their solution. accompanying chart, Fig. 6, has been prepared. For that reason the For any value of 8 and F, the corresponding values of d 2 /d, and (3 can be determined quickly. The area to the left of the line marked "Locus of minimum d1 /d, 11 and above the line d 1 /c:1, : I (or &=o•) is the region to be used for the solution of the equations for positive shock waves, the only stipulation being that the flow at GI~ remain in the super-critical state. The significance of the area above and to the right of the lin'e of minimum d 2 /e&,is believed to account for the flow conditions which would occur because of an obstruction or alteration changing the downstream flow from . super-to sub-critical. 19. I I . " _j_ __ N 11. 'I 0 ,n (\J 20. For example, with an angle ofe,2O0 and F = 4, the depth ratio below the shock wave front would be 2. 7 with a wave angle (3 of 340 if super-critical flow continues downstream. However, if sub-critical flow conditions existed downstream, or if the remedial measures applied within the channel bend caused the flow to change from superto sub-critica.l, it is probable th.at a ju.mp would occur and the downstream depth would then be 5. 2 times the upstream depth and at an angle across the channel of ~ = 950. The region to the left of the line marked "Locus of maximum da./d, II and below the line da./d, =I (or e =o·) is th.at used for determining the downstream depth when so-called negative waves exist; as, for example, on the inside of a channel just below a bend. an example, for a value of As & = -5° and F = 3, the ratio of dz. to el, would be 0. 75 and the wave angle (.?> would be 15½0 . No physical significance of the area below the d~/c:t, = \ line ( & = o• ) and to the right of the maximwn d&jd, locus has been discovered by the writer at the present time. 3. Location of Wave Crossings: While the downstream wave depths just determined apparently exist all along the shock waves, it is at once seen th.at near the boundaries the waves cannot be fully developed. On the outside wall the vertical acceleration produces a value of da./~. considerably in excess of that given by Equations (27) and (30), and on the inside wall the vertical deceleration produces a value of larger than th.at given by the equations. dz/d, also It lll\lst also be remembered that "negative shock waves" a.re not physically possible, the wave 21. actually being broadened out over a considerable distance. The maximum depth along the outside wall could be expected to be located only a short distance below the change in direction of the channel. Unlike flow around a curve, no particular disturbance would be expected where the negative wave from the inside wall crossed to the outside wall. Below the negative wave front the water depth is decreased and this decrease should cause continued decrease in the water height along the outside wall below the point of reflection of the inside wave. In Fig. 7, by the geometry of the figure the inside wall wave should r each the outside wall at: Rout = (33) where ~ is the wave angle for the inside wall and could be determined by (a) Equation (21), (b) solution for (3 from Equation (31), or, preferably, (c) read from the chart of Fig. 6. In a manner similar to the above, the point at which the shock wave created by the outer wall reaches ... the inside wall should be at a distance Rin where Fig. 7. from Fig. 7: 22. In this equation (2> is the wave angle for the outside wall. The disturbance created a.long the wa.11 at this point could be expected to be rather severe in view of the fact that this is the positive wave front. Undoubtedly, the water will attain considerable height here. Immediately upstream from the wave front, because of the negative wave effects, a very small depth of water should be encountered. The wave reaching point Rin could now be expected to be reflected diagonally back to the outside wall of the channel reaching it at a point approximately equ.a.l to 2 Rin· It would thereafter cross back and forth across the channel as it progressed downstream, hitting consecutive points along the channel with the wave length: L= 2. RiW\ (35) It is important to point out that although the two wave angles (?> mentioned above might at first appear to be the same in magnitude, reference to the chart of Fig. 6 will show that (3 for the outside wall will not be the same as (3 for the inside wall. 23. D. 1. ANTICIPATED EFFECT OF SILLS ON WAVE REPRESSION Wall Wave Heights: Fig. 9. Fig. 8. Echelon placEillent of submerged sills, as shown in Fig. 8 a.nd Fig. 9, would be expected to divide the flow of water into two layers, a bottom lEcy"er whose flow direction would be parallel or approximately parallel to the sills, and a top l~er, whose flow direction would depend upon the principles developed in the preceding sections. If the angle of the sills to the upstream channel walls is o<., the bottom layer would be deflected at approximately the same angle throughout the width of the channel. The top lEcy"er would continue on, with a rise due to the bottom pressure influence but with little direction influence from the sills, until affected by the inside and outside changes in wall direction. If the angle a<. were greater than 8 , a piling up of the water on the inside of the channel and a reduction of the qo.antity of water along the outside of the channel could be expected as a result of the turning of the bottom layer. Superimposed on this would be the negative wave on the inside portion of the channel, tending to reduce the depth of flow and a shock wave on the outside tending to increase the depth of flow. Ey proper sill 24. selection, arrangement, and angle of attack, it should, therefore, be possible to reduce drastically or repress the undesirable depths of flow along the outside wall bei.ow the bend and at the same time increase the low depths of water along the inside wall. wall profiles would be those of uniform flow. The ideal While it would be desirable to reduce wave heights within the central portions of the channel, these are not nearly so important as the reduction or suppression of the water heights along the walls. 2. Reduction of Transverse Velocity: In addition to reducing the outside wall water heights and increasing those along the inside wall, any sills to be satisfactory must be capable of sufficient reduction of the overall transverse velocities as to assure positive non-recurrence of objectionable waves anywhere downstream from the bend. There is a certain amount of energy loss within the wave itself. By substitution of the shock wave velocity: (36) (which is the same equation as (19) without the sin@ factor, as previously mentioned), into the specific energy relationship: (37) Rouse7 determined the loss due to tu.rbulence at the wave front to be: (38) 25. Fig. 10 shows a channel with an abrupt bend in which are placed a series of sills, echelon fashion. at an angle c,t... with the upstream walls. The forces which are absorbed by the sills due to the dynamic action of the water impinging upon them, are represented by ~ and f, . being parallel to and norm.al to the original direction of flow, respectively. y, in this fiture represents the effective angle of turn of the water as a whole, due to the effects of the sills. V, ► Fig. 10. Application of the continuity principle to the upstream and downstream sections, assuming that the velocity distribution is uniform throughout each section, gives: (42) Since force can be represented as the time rate of change of momentum, the force exerted by the Fig. ll. sills can be included in the following equation, indicating the transport of momentum in the x-direction: 26. Substitution of Equation (42) yields: (44) or If the total force in the x-direction divided by the mass of water flowing per second is defined as J )(,.=:. J~ ; i.e.: ..., ~w~ '><.. Equation (45) can be rewritten in the form: (46) where Jx bas the dimension of a velocity. Next, consider V. =V-u z.:,.. ' where u is, for the moment, the decrease in velocity in the x-direction due to the effect of the sills. Substituting this in (46), gives: V,+~ I 2..V, (47) or (48) Because the angle of turn is relatively small, no appreciable error will be introduced by dropping the term "1/V,% and expanding the 27. last term by the binomial theorem: 94!l, = J - ll + ~ ( I + 2. ~ -t I·) 2.V, y. 2.V, V. ( 49) This reduces to: (50) from which (51) substitution of which back into the relationship defining u gives: (52) In they-direction of Fig. 10: If now J~ is defined as P~ + 3" , the above becomes: Hence, the effective angle of turn t/, is such that: Jx (55) V, - (1- ~a) or, reconverting J" and Jj : (56) 28. And, finally: ~ - ...,, - + -~wbd:F'2.p~ ~I\' Pl<. (,-~.) The tezm factor. 1 (57) Wbd ,a,F2. is recog;nized as the upstream momentum 1 Thus, Equation (57) tells us that the effective angle through which the water in the channel is turned is that whose tangent is the ratio of the pressure exerted on the sills at right angles with the original direction of flow to the difference between the original momentum and the pressure exerted on the sills in the direction of flow, (corrected by a factor to account for the relative velocities of wave celerity and the channel velocity). 29. I II. A. 1. EXPERIMENTAL STUDY .APPA.BATUS AND EQ,UIPV!ENT Preliminary Studies: Considerable preliminary studies were made on two triaJ. channels before comprehensive detailed experimentation was started. One channel was constructed of galvanized iron and had a bottom width of' 12 inches and a depth of 9 inches. A 15° sharp bend was built into its approximate midpoint, with an approach and downstream section each approximately 25 feet long. Serious difficulties were Fig. 12. 30. encountered in establishing uniform flow in the short distance before the bend, but these were finally overcome by adding a specially designed entrance section. ~ottom stability was difficult to obtain until the channel bottom was heavily braced to prevent deflections in the channel material. Fig. 12 is an over-all view of this channel. Fig. 13 is a view showing the pil. ing up of the water and the crossactions of the waves created by the bend in the channel. The studies made in this channel, the knowledge gained in its construction, and the determination of its unsatisfactory features, led to the belief that a wider channel of heavier material with closer control of the quan~tity of Fig. 13. water flowing would be desirable. A smaller degree of bend was also deemed essential in order to study properly the effects of the sharp bend and the possible corrective treatments to be applied. 2. The Channel: A brass channel, belonging to the Hydraulic Structures Laboratory of the Institute, was made available for the continuance of this work, through the kindness of Drs. Knapp and Vanoni. 31. This new channel was 18 inches wide, 12 inches deep, and 30 feet long. It was equipped with an entrance box and a completely independent water system, including remote control operation of the pumping unit. A 100 bend was constructed in the channel at a. point approximately 20 feet from the entrance section, allowing a 10 foot downstream section. Many runs were made in this channel on a slope of 0.0431. / Fig. 14. Due to the short section below the bend, a full downstream picture of flow conditions could not be obtained, and to this downstream section was then added a 10 foot section of heavy galvanized iron to allow the waves created in the bend to cross and recross the channel before the water spilled out over the lower end. Fig. 14 and Fig. 15 give a general view of the channel and its appurtenant equipment. Drawings A, B, and C indicate the details of construction, wiring, etc. It was originally intended that the slope of the channel be adjustable, but such did not prove practicable, because of the necessity of redesigning the bend for each change of slope and the difficulties encountered in moving the entire channel as a body to prevent wall buckling. Fig. 16 shows the layout for the 10° bend a.s actually constructed and built into the channel. It is pertinent to point out that in order to maintain a level bottom in the channel nonnal to the direction of flow, a modification of this design would have been required for each particular slope. Fig. 15. The slope of the channel was carefully adjusted to obtain a five per cent slope longitudinally along the channel and level conditions across the channel at each cross-section. The entire channel was equipped with a substantial and well graded track on either side which served as a support means for measuring depths, velocity magnitudes, and velocity directions. The channel measurements 33. were made with the inside wall of the bend as a zero point, termed in all the data as CB. 11 U 11 Upstream stations were marked with the prefix while downstream stations were marked with the prefix D 11 • 11 • The outside wall of the channel at the center of the bend was also marked CB, but station U-0 was directly across the channel from the inside center of the bend with relation to the upstream portion of the channel, and D-0 was directly across the channel from the inside center of the bend with respect to the downstream end of the channel. A paper scale, varnished, was placed on the track and served at all times to insure e~ct longitudinal station location with a minimum of error. 3. Entrance Section of Channel: The entrance section provided a ready means of establishing uniform flow conditions in the upstream section of the channel. It consisted .of an adjustable gate with scales and verniers attached for the purpose of obtaining equal water depths throughout the entrance control section. Fig. 17 is a close-up view of the entrance section. A view of the entire entrance box can be seen in Fig. 14. Fig. 17. 34. Fig. 18. 4. Fig. 19. Source of Water and the Circulating System: Water for the operation was stored in a sump below the surround- ing ground level beyond the downstream end of the channel. The circulating pump was driven by an electric motor, the two being interconnected with a variable speed friction pulley, V-belt driven. The impeller of the pump was set below the water level in the sump so as to be primed at all times. Water was pumped from the sump to a 12 inch horizontal pipe which ran the entire length of the channel and discharged into the entr&nce box. The qdantity of flow was deter- mined by a well-calibrated Venturi meter. Fig. 18 is a closeup view 35. of this Venturi. The pressure lines from the Venturi were run directly overhead to a water manometer, which was housed in a weatherproof box and contained, as well as the manometer, a carefully plotted dia~Tam showing the discharge for any differential head reading of the manometer. Provision was made for bleeding all manometer lines at the start of operations. Fi g. 19 is a view of the manometer with its housing and calibration curve. To summarize then, the circulation of the water was as follows: From the sump the water was pumped into the horizontal pipe running the length of the channel. Near its midpoint, a Venturi meter was inserted which allowed accurate determiretion of the quantity of flow at any time. The water from the 12 inch pipe discharged into the entrance box. With proper adjustment of the entrance box, uniform flow could be maintained with only insignificant variations. The water flowing out of the channel at its lower end discharged into a large basin where it was circulated by t he pump back into the pipe to repeat its previous process. The dis- charge could be maintained at Fig. 20. a very constant value by means . 36. of remote control of the pump speed. This remote control permitted experimental runs of fairly long duration to be made at any specific value of discharge, thus maintaining consistent conditions of flow over long periods of time. Fig. 20 is a view of the end basin, the pump house, and the trench which contained the 12 inch circulating pipe. 37. ·B. l. INSTRUMENTS Point Gage: The depth of flow at any point was determined readily by the use of a point gage which was especially built for the channel. As seen in Fig. 21, it consisted of two brass angles spaced 4 inches apart and mounted on a single plate at one end a.nd a 12 inch length of channel iron at the other end. The channel iron contained a grooved roller at each extremity which ran on the track previously described. The plate slid on the track on the opposite side of the channel. Mounted on the supporting plates was a heavy scale which held the point gage. An integral part of the point gage was a vernier which permitted the reading of depths to .001 feet. The point gage could be readily moved up and down, and with a friction spring which was built into the bottom supporting base was held in any vertical position with ease. The upstream cross angle iron was also scaled so that setting of the point gage across the channel could be made readily and accurately. ,Fig. 21, 38. 2. Pitot Tube: A pi tot tube, made of a hypodermic needle, served as the means of determining velocities at any point in the channel. Because of the minute size of the needle, it was possible to determine velocity magnitudes at almost :rmy point desired, including velocities within 0.02 feet of the walls or 0.002 feet of the bottom of the channel. The hypodermic needle was attached to a brass tube which served as a means of support and also as an inter-connecting link between the needle and a glass tube used for measuring the velocity heads. The glass tube was of a sufficient length to permit measurement of the maximum velocities that were encountered. A scale was glued to the vertical support which held the glass tube, and the whole assembly was clamped to the point gage in such a manner as to allow free vertical movement up and down with the point gage and l horizontally across the channel. The needle could be turned in any direction desired so that velocities not only in Fig. 22. the direction of flow were 39. measured, but transverse velocities could be obtained as well. The pi tot tube point, its mounting and inter-connecting linkage with the glass tube, can be seen in the left-hand side of Fig. 22. 3. Velocity Direction Gage: Fig. 22 also shows the velocity direction gage, which was developed and built by the author at the suggestion of Dr. Knapp, to determine the velo·ci ty direction at any point in the channel. The velocity direction gage was also built into the point gage support in somewhat the same manner as the pi tot tube and served as an integral pa.rt of that instrument. Hence, velocity directions could be deter- mined at any point and with but a slight shift of the instrument the corresponding magnitude could be determined at the same point. The velocity direction gage consisted of a 1/16 inch vertical rod with a vane at its lower and upper ends. The rod was inserted within a suitable mounting to allow ease of rotation and to give sufficient support and rigidity against bending which might occur as a result of the high velocities involved. The upper vane was soldered to the rotating shaft after insertion in the supporting member. The posi- tion of the lower vane thus was transmitted directly to the upper vane. Immediately below the upper vane was placed a finely graduated circmla.r scale for determination of the velocity direction. The upper vane was carefully correlated with the lower vane and checked at the beginning and . end of each day's experimentation. Direction · angles could be read to within one-half degree, under nonnal conditions. Movement vertically was provided in the same manner as that for the point gage. 40. This apparatus, with its direction gage, is illustrated in Fig. 23. The friction spring which served to hold the gage at any elevation in the channel can easily be seen. Fig. 24 is a general view of the combined pitot tube and velocity direction gage. Fig. 23, 4. Sills: Preliminary investigations over a period of several weeks disclosed that one or two single long sills stretching diagonally across the bottom of the channel were not a suitable corrective device for reducing the "Wldesirable waves created by the bend. Fig. 24. It may be pointed out that in a 41. Fig. 25, long radius bend in a rectangular channel two or three strategice.lly located sills extending across the channel serve as a fairly suitable remedy. These preliminary investigations, coupled with an attempted mathematical analysis of the problem, led to the belief that a single sill cut up into short lengths and placed in parallel across the chamiel would accomplish the purpose desired. Fig. 25 is a view of a small part of the many different types of sills used in this study. The sill in the center of the picture was made of brass and welded to a heavy steel shank sharpened to a knife edge on both edges to permit studies of the effects of a si ngle sill at any point in the channel. Fig. 26 is a view of a typical set of sills mounted in place in the channel. The sills were fastened to the channel bottom by brass bolts running the full depth of the sill and tapped into 42. Fig. 26 .. the channel. This method of mounting also gave stability to the sills and enabled them to withstand the relatively high dynamic forces exerted against them by the flow at the high velocities used in -this research. 5. Scales: A very ingenious method of measuring the forces acting on the sills was suggested by Dr. Merit P. White and, with minor improvements and variations is shown in Fig. 27. a thin sill plate, as shown in Fig. 28. The sills were mounted on The upstream edge of the sill plate was allowed to move freely longitudinally and laterally, but was prevented from moving vertically any appreciable amount by small clearance washers, as shown between sills 2d and 3d and between sills 4d and 5d in the figure. Two lines were attached to the up- stream edge of the sill plate, one to the downstream edge and one to 43. Fig. 27. either end of the sill plate. Ea.ch line ran horizontally from the sill plate sufficiently far to eliminate any appreciable effects of the minute pulleys and their supports which served to transmit the force from a horizontal direction to a vertical direction, and thus permit measurement of the forces on scales. Regu.lation of the position of the sill plate was made by adjusting nuts on the support rod of the scale. The bolts which held the sills to the sill plate 44. did not extend through the sill plate, but were merely mounted upright upon it with the heads soldered to the plate. Fig. 28. 45. C. l. EXPERIMENTAL PROCEDURE Ad,justment of Flow: The experimental apparatus including the channel. pumping system. and all instruments to be used, were given a quick inspection daily . before each run to insure validity of the basic data. At periodic intervals the channel slope was also checked, but at no time was any variation discovered. While some minor difficulties were encountered during the early stages of the experimentation, they soon led to a standard method of starting the equipment for each day's run. The method adopted was first ·to open slightly the valve on the discharge side of the pump. The pump was then started and by the remote control was slowed down or speeded up to the point where the discharge through the system was approximately 1 cu. ft. per sec. The manometer lines were bled of unwanted air and the entrance box was adjusted. to the proper opening to correspond with the uniform depth of flow for the discharge desired. Provision was made in the entrance box for bleeding all air from the bo~ at its highest point, and once or twice during each experimental run any air collected in the entrance box was let out through the bleed valve. Once all the air was removed from the entrance box and uniform flow conditions obtained, the pump speed was adjusted by remote control at a point near the manometer box to bring the quantity of water flowing up to the desired amount. The remote control permitted adwustment of the flow over an extremely narrow range so that all 46. runs were made at practically a constant discharge. Water passing through the channel and on into the tailrace passed through a fine screen and on into an intermediate sump from which it was again picked up by the pump and circulated back through the system over and over again for as long a period of time as was necessary for the run. The value of 11 n" for use in the Manning formula had been extensively determined from experiments by Knapp and Ippen in a similar channel and also by Rouse in the same channel. It was felt un- necessary to analyze the channel further for its roughness coefficient, in view of the compatible results of these experimenters. 2. Profile Mea.suremen ts: All studies made in the channel, with a few exceptions at the beginning of the experimentation, were those for a constant discharge and steady uniform flow conditions in the upstream section of the channel. Once such flow was established for the desired discharge, profile measurements were made from section to section downstream from the bend by use of the point gage, and also at three or four stations upstream from the bend to assure absolute check on the uniform flow assumption. The point gage described under "Experimental Apparatus" provided a quick means of reading depths of water at any point within the channel. The depth of the water sur- face was first read and recorded in a book suitable for that purpose , and entered immediately below that figure was the depth of the bottom of the channel. Early experimentation was carried on for 47. profile measurements by recording the bottom profiles for the entire channel on a single page of the data book and it was thought that reading and recording of the water surface heights would be the quickest means of determining the complete profile. It took but little experimentation to learn that it was far simpler to set and read the point gage for the water surface and the channel bottom at the same point than to compute the depth of water as a separate step. Complete profile measurements were taken for the untreated channel flow at discharges of 1.0, 2.0, 3.0, and 4.0 cu.ft. per sec. Parti- cular attention was paid to the outside and inside wall profiles in order to have an accurate plot of such profiles throughout the available length of channel. Sill selection was based primarily on the results obtained from wall profile measurements with the many and varied types of sills in place in the channel angle. 3. Velocity Measurements: The velocity direction gage and pitot tube were carefully cali- brated to determine any unanticipated characteristics which they might contain. found. Fortunately, no eccentric characteristics were Readings of velocity magnitude and direction were made by placing the bottom vane of the velocity direction gage at the point where the velocity was to be measured and reading the angle indicated on the circular scale by the upper vane. Careful attention was paid to the location of the point involved and the pitot tube then was moved to the same point. The direction of the velocity having been determined as just described, the pitot tube was pointed in the 48. same direction and the height to which the water rose in the glass section of the pitot tube was read and recorded. The pitot tube was so mounted that it could be turned with ease in a:n:y desired direction. This method of measurement was sufficiently accurate to allow the determination of the velocity at any point in the channel above 0,002 feet from the bottom of the channel, and to within 0.02 feet of either wall. Velocity measurements required considerable time at each station and for that reason comprehensive studies were made only for the maximum discharge with untreated flow and later with the best treatment obtained. 4. Sill Selection: Careful analysis of the many and varied experiments conducted by Knapp and Ippen in long radius curves in a rectanglllar open channel, led to the belief that a single sill extending diagonally a.cross the channel could not be a satisfactory means of repressing the waves created in a bend. As previously mentioned, brief experimentation indicated that great possibilities could be obtained by cutting the _single long diagonal sill into short lengths and placing them echelon fashion approximately at the bend. All ty-pes of sills were tried, ranging in shape from wide, triangular-shaped sections to narrow, vertical walls confining the flow within narrow limits. Many difficulties were encountered and the results obtained by nearly all the types of sills tried were very discouraging. One of the principal requirenents was that the sills should be self- 49. cleaning and under no conditions induce the piling up of any debris that might be present in a full-scale channel. It was soon learned that the sill on the extreme inside wall would have to be removed. It exerted such a drastic diversion effect a s to cause an excessive and prohibitive piling up of water on the inside channel wall. Many sills performed favorably under the largest discharges experimented with, yet under low discharges they were entirely An example of this unsuitable. can best be illustrated by reference to Fig. 29, which shows the disturbance caused by an otherwise satisfactory set of sills. Another serious difficulty encountered with many of the sills used was separation of the water in flowing around the back of the sill. Fig. 29. This separation produced air pockets which entrained air in the water, the entrained air passing on downstream. At the same time the air pockets evidently bordered on the edges of what was believed to be cavitation. The rapid intermittent collapse and reformation of the air bubbles frequently ma.de very loud explosive- 50. type noises, and loosened the bolts holding the sills to the channel bottom. No other effects, however, were apparent on the sills them- selves. In an effort to overcome this defect, airfoil-type sills were installed, but as would be expected, the water in its supercrtical state would not follow the sill contour. Of all the sills experimented with, only two sets or arrangements were deemed sufficiently satisfactory to be worthy of extensive study, and complete velocity measurements and profile measurements were made with these sills in place. In all other cases, the maximum and minimum down- stream distu~bances only were recorded and remarks made as to the separation features and the ability to self-clean. The large sill discussed in the section under "Instruments 11 and pictured in the center of Fig. 25, aided greatly in determining the direction at which the sills should be placed, their location within the channel, and their relative nearness to one another. 5. Measurement of Forces Acting on Sills: The components of the total force acting on a particular set of sills were measured by means of a sill plate, as previously discussed, through which force components were transmitted to scales above the channel. Sufficient disturbance was made on the sill plate before each reading to insure complete freedom of movement of the sill plate. All scales were adjusted to an arbitrary pre-determined zero, with the sills in place but with no water flowing. The water was then allowed to attain equilibrium conditions and the scales read.justed so that a direct reading of the force acting could be made. Little experimentation ~as needed to satisfy the condition of free movement 51. of the sill plate, because readings of the forces for the same flow conditions following disruptions alw~s showed a set consistency. 6. Photograph.y: It was found desirable at the outset, due to the length of each run, to record by pictures as much of the flow conditions as could satisfactorily be so obtained. Pictures of the flow conditions were difficult to get, because of the rapid movement of the water; the poor lighting available because of the channel being located below a large platform; and because of the direction at which pictures bad to be taken. A vertical picture of the flow phenomena was impossible over sufficient range to be of suitable use. Neither could such a picture indicate with any degree of discernment the relative water surface contours and profiles. It was, therefore, necessary to take all pictures either facing the downstream or the upstream sections. Several hundred photographs were taken but only a small number is included in this thesis. D. EXPERIMENTAL EESULTS The experimental results obtained are presented both graphically and by photograph wherever possible. In many instances a photo- graphic view is not at all feasible, as, for example, in the case of velocity distribution within a cross-section. Other results are presented only in photographic form, an actual picture conveying many times more than could be displ~ed in graphic form. 1. Sill Selection: As previously discussed, many and varied were the types of sills used in this research. The series of sills experimented with in the preliminary work were denoted by number, such notation being entirely satisfactory so long as all sills in a set were of the same shape and size. But, with the ad.di tion of the ten foot length to the downstreaJll end of the channel, a new system of designating each sill had to be devised. :Each set of sills was arbitrarily assigned a. letter denoting the serie~ or grouping and each individual sill was assigned a. number to indicate its position with respect to the out' side wall, the sill nearest the wall carrying the number 11 1", etc. The sills shown in Fig. 25 are grouped in such a manner that all sills of the same size and shape are together. In the early experi- mentation each group was used as a whole; i.e., all sills in any group were equivalent in shape and size. Such a set of sills can be seen in Fig. 26, where the 11B11 sills are shown in place in the channel. Each sill is the same as every other sill. 53. The advantages of using various sizes of sills in a series were discovered with the addition of the downstream length of the channel. While it was obviously impracticable to attempt to investigate the characteristics of all the various possible combinations of sills, this was not necessary because certain combinations immediately eliminated themselves and similar ones when tested. The best combin- ation which the writer was able to discover was known as the K series, pictured in Fig. 30. All sills in this series were 3/4 inches wide and 5 inches long, but varied in height. The sill nearest the wall was l inch high, sills Nos. 2, 5, and 6 were 1-1/4 inches high, and Nos. 3 and 4 were l½ inches high. Another series, lmown as the I-series, performed much better at the high discharges and velocities, but were 'Wlsuitable at low values of discharge and velocity. Fig. 30. Table I. contains a sununary of the seven best series of sills tested, along with four series of equal size sills, and conditions of 'Wltreated flow for comparative purposes. The figures comprising the body· of the table indicate inside or outside maximum wall depths for a discharge of 4.0 cu.ft. per sec. 54. TABLE l COMPARl50N OF MAXIMUM WALL OE.PTH~ va. SILL TYPES SU.. L SERIES AT l)•l "·"• 467 56S SH D 418 2.14 415 3"72. 452 .., I 3/1 ;flCNS C 2.,1 387 S ◄ 'l ◄ 20 331 3"'1~ A a,e us oil I i""°' I!!, 342. 304 -405 39'4 call I '4 ifleh~a a 11 I Y'a.. ;.,cM:s MAX.lh/lUM SILi.. l'fE.ICH'TS 1H IHCME~ t WA~L t>E.PTH$ • •s .,._ NEAR D ◄ Mr:AR 1>-IO.S HE,R ll-lfl •z -~ "' "" ,.. a.ills ·~ ·~ K 331 3.52 3'41 34C. W4 I"& I 30(, 3fol 32.0 jV4 IV& F u, 3 ◄!5 3<-8 341 341 ' w.. 1•~ IV.. I 1'4 1'4 I'll. 1'4 H 2.~O 374 E 34~ 34-4 310 355 3SC. ~f.9 J 3z, 3GC. 314 3(.~ L 306 375 312. •Alt dep#!s a,- ifl 1\louseruttl\5 of '" foot, 3-il I I I 114 ·'"' ~ ·~ IV.. 1¥& II'& I I 1"1 1'4 ,.... I 1114 .v.. 1'4 I'-'! I'"& IV& ' "' I t All ~ills .-. 3M" w,de ~~ 5 • •~· Depths under the column headed Station D-2 are depths at that station. Stations D-4, D-10. 5, D-18 merely denote the approximate location of the maxima, the maxima shifting slightly one we;y or the other depending on the sill 'series used. The maxima, of course, occur first on the outside wall, then on the inside wall, and again on the outside wall, literally bouncing from one wall to the other, as indicated by the untreated flow pattern (Q = 4.0 cu.ft. per sec . ) in Fig. 31. It might be pointed out in Table I. that the effects of an increase in height of similar sills (series D, C, ~. A) are to reduce any given maximum, al though the reduction is not a uniform one between different maxima. At Station D-2 a small sill height 55. decreases drastically the original wall wave height, but further increases in sill height act counter to their original reduction effects. 2. Wall Profiles: Table I. summarized the maximum wall wave heights for the seven best series of sills tested. Fig. 32 is a plot showing the inside and outside wall profiles for the untreated channel under various conditions of discharge. Note that the flow Fig. 31. patterns are all similar, the maximum points increasing considerably with discharge, while moving downstream only slightly. Figures 33, 34, 35, and 36 show these same wave patterns (for untreated flow) at discharges of 4.0, 3.0, 2.0, and 1.0 cu.ft. per sec., respectively. The graph of Fig. 37 shows the effects on wall profiles of variations in sill heights where the sills in each series are all the same. with untreated flow. Note the comparison I o.~ a 0 • :; 0.1 I % ... 0 ~ 0.16 ~ '! 0.4 i o.,I o., ' ♦ I "" t : 1 'I ' J I ~ ::,e '' -n ♦ t D ; , I t ic j ♦ -, I • ' I /I 6 ~ Jl W~LL Fig. _32, RECTANGULAR CHANNEL AROUND 10• ABRUPT BEND FOR UNTRE ATEO FLOW 10 -=- - -~ 8 -- ,---,1 'C ti 7 0.) I 0.4 ' O. ♦ 1 0.5 o., 0.2 --10,3 -~ II I♦ " 18 0 o-.J=..:::::::: ~l==.=----10.1 ' r 16 I . \ I o.s 7 -1:· O.•S.O CJl'.S. : - - 14 \ 1 o., ~ ~ 0.t. I ~ '-.I ~ Gl.•LOCl'!'S. t.:1_;1 f'II r r e • Q•"-OC.l':S. 10 12 DISTANCIS aELOW KND IN FEET ' I -- - ~ ______ ~ / ..,........__,, INSIDE 4 I ;,:_r ~ ------ f I / , , - - - . • - . - & . _ ·_fl '-.. ' "'<.,/ / r - • • 1.0 cr.-a. ----...;,,_ i OU.TSIOE WALL WATER SURFACE PROf'IL£5 ~ I I t I n- 1 1 az::::r:::::::) -- ,- - r --- 0 r .-~ I -- ' T T ~l ~l ;~r i! DtSTANUI AaOVE eEND I c::::- + ~ O~I t I- ~ 0,2. ::c !...... 0.1 !: ' "'04 50.!l' [ l- o., r----- I ~ 57. Fig. 34. Fig. 33. Q. = 4.0 cu . ft. per sec. Q. = 2.0 cu.ft. per sec. Fig. 35. Q. = 3.0 cu.ft. per sec. Fig. 36. Q. = 1.0 cu.. ft. p er sec. 58. 0 ·' OS ----0---- ~.:--..LLI~ ~ . --+-- a • SILLS ....::...=- C!.SILL'I - I ! Dr Sll,J.9 o.s 0.4 a., o., ii : ~ 0.2 ~, 0.1 0 '--r------r------,o+-.;_D_1_sT_'IM=z-e_1t_S_M.----,~,---MM_D--.,.IN_•_a_T--,8:----'.':ID----:,2,----7'14e----:15::----:::i1:--""" D ~ ~ ~ &IJ. o., o.s El a., !;;, 04 O, ♦ O.} o., 0.2 .l o., 0.1 10 IN510E WALL o.'5 0.5 0 .4 o. ♦ 0 .3 i 0.2 01 !I o ~ l>I 0.1 .. 2 2. ' 4 & - ·· - 10· - - ,2 ---,.- - -,-, - - -1.- ~ 0 OUTSIDE~ 05 ~ O♦ ij ,;, 3 0.1 . 0.1 0 - - + 2 0 2 ___4_ __ ' e IO ,i INSIDE W.t.LL WATER SURFACE PROFILES FOR TREATED FLOW (VNIFORM-'.:illED 51LLS) .t.11.0UND ~ 1.Q_• AMUPT BEND~ FIECTMGVLAR Cl-iANNcl Fig. 37. 0 59. o.& , 0.5 0,4 - - - - ·~-+-- I - +.~ '----+- - - ,....=..o:.o-F-- -+-- - +---+-- --+-f-- 0.4 1 I 0,3 0,l - 0,1 0 ,1 0'-----,4,----2.---: 0--~~---♦,---...---~8----'~---,~l---K,.____ _~~----'O 0.4 o., ii 0,3 0.2 o., !I o ~-~.--~'Z..-----,,:-,.-----!:-2.---.4 , - - - ~ - - - - - - - - ,10,,....----,1,;,2.----,,.-----,,,1 - - - - ' o '"' ~ M 2) - - - J ! l ~..C. :::_ ~ c;,_~ o,5 l . 0.4 _. ~.NTf/Uflo.TEP F\bllt ,r',, ., 0,4 1 q IN~IDE WALL : I ~ :'. / ,, '~ -~-y~ ' 0 ,.3 ,-- O,l. o., ,, OL - - -...---.,-1.- - --::0 :------,,_:-- -4-:----:-,--~8,-----,,o,:---,:-::'L,---:',14:-----,',:----' 0 4 - - ,-,, .~,- - - ---1 O,?, o. 1 , ,. I , -- ---------~-_,#" I 0.1 0. 'l 0,1 0,1 OL---4----~---0,---,,-----,:r----:----~.,---:-:~----,,;,---~--~,~.--~O WATER !>URFACE PROFILE.~ fQ& TREATED~ (K-~ILLS) MOl.)NP ~ I Q . • ~ ~ L~ f!CCTAMiU\,AI\ OWiNC:L Fig. 38. 60. Fig. 38 is a graph comparing the wall profiles for the best treatment found with those for the 'Wl.treated channel. .A. comparison of three types of treatment versus no trea~ment for 4.0 cu.ft. per sec. are shown on page 61. Fig. 39 is for no treatment, Fig. 40 for treatment with K-sills, Fig. 4l for treatment with I-sills'-, and Fig. 42 for treatment with .A.-sills. For this quantity of discharge it is apparent that the I-sills offer the best corrective prospects. Figures 43, 44, 45, and 46 are the same conditions except for the discharge being 3.0 cu.ft. per sec. The actions of these sills under a discharge of 2. 0 cu.ft. per sec. is shown in Figures 47, 48, and 49, where Fig. 47 is for the untreated channel, Fig. 48 is for K-sills, and Fig. 49 is for A-sills. The effects of the I-sills are almost identical with those of the A-sills. Attention is called to the ve-ry undesirable flow conditions over the A-sills shown in Fig. 49. It was this same feature of the I-sills which prevented their being the best selection. Although this condition appears to exist for the K-sills in Fig. 48, the offense is not serious bec~se it only occurs in an 'Wlfavorable degree on sills Nos. 2 and 3, and has no undesirable effect on the outside wall. The angle at which the picture was ta.ken does not permit a view of the action over sill :fl,l. The splashing actually is more of the magnitude of that over sill f,4. Confirmation of this fact can be made by referring to Fig. 75, which is an upstream view for a similar condition of flow. It can be seen that no objectionable splashing takes place at the wall, due, undoubtedly, 61. Fig. 39. No Treatment Q, = 4.0 cu.ft. per sec. Fig. 40. K-sills Q, = 4.0 cu.ft. per sec. 'Fig. 41. Fig. 42. I-sills Q, = 4.0 cu. ft. per sec. A-sills Q, = 4.0 cu.ft. per sec. 62. Fig. 44. Fig. 43. No Treatment Q, = 3.0 cu.ft. per sec. K-sills = 3.0 cu. ft. per sec. Fig. 46. Fig. 45. A-sills I-sills Q, = 3.0 cu.ft. per sec. Q, = 3. 0 cu. ft. per sec. 63. Q= Fig. 47. No Treatment 2.0 cu.ff. per sec. Fig. 48. K-sills Q = 2. 0 cu. f.t. per sec. Fig. 49. A-sills Q = 2.0 cu.ft. per sec. 64. to the lower #1 sill height and its location near the wall. Veri- fication of this can also be seen by reference to Fig. 29, where sill fl is even there splashing considerably less than the others. Fig. 50. Prohibitive wall water heights due to such splashing over the I-sills led to their rejection as a satisfactory corrective means, regardless of their superior performance over all the other sills st higher discharges, K-sills included. Fig. 50 shows the first maxinrum wave on the inside wall for ~ = 4.0 cu.ft. per sec. (untreated flow) before the addition of the ten foot downstream section. Thorough investigation of wall profiles for various fillets placed at the bend revealed that the fillet bad little effect. Fillets with radii of 18 inches, 36 inches and 54 inches were formed by plasticene so as to obtain a smooth and even transition around the outside wall. . ~o~ ~ ll ~t!-O.R"' ~y:n-' o. t__6 \ 0 , 114 ~ " ~~· 0 () DIST.. NC!t 10 10 10 1?.ELOW BE'.N v .!.J:! F'C:E: ~ Fig_. 51. RECTANGULAR CHANN~L. AROUND IO°ABRUP"T BE.ND FOR UNTREATED FLDW WATER S URFACE CONTOURS .., 1.0 .o < G .2. ' 1.0 I r-- eeM D ~ ~ C,£"LOW 2..0' SCA<. i : 0 1~TANCE5 n -z.?I .'\, ~,§~.- j 15 15 {J) U1 66. 0 --------- ~~~· L_O C.F.S. 14 17 AROUND Fig. 53. IQ" ABRUPT eEND ~ R~CTANGULAR CHANNEL ~-SECTIONS FOR UNTR EATED FLOW ~~1 ~ [ ~ ~ ~ ~ DISCHARGE • DISCHAR<rE ~ 3.0 C.F.S. D15CHARGE s 4.o ~ --.J 0) - ---- . ~·=- \ ---------- - - - - -- ------- ~~~- ......----- - • r 0 ' 2 - · -1-:_- . . ---- 4 ~ ~:~- - - . - 8 ~ - FECT ·- - -- ~ !J .:~ 9 (,_I '.>. fi _ ~-~~ Fig. 54. >,~"'~'- .;LL I I{ N_:., fGR Tk(ATEL> FLOW ·· K- '.:>ILLS II ~:C, C HAF< 0 L DISCHARGE.~ 3.0 C.FS. BE.NO_ ~ 1o ~ g ARC.o i.C. 10" /\~Rt..>-" clENi., 11'1 H!:. C7 A r ~ Cku:, ::, Dl5TANC.£S &LOW --- -~l_-- - ---_,.~:\~]_f l r ,-11 ----- ----lu-J_iLJ JLij -r -~l_-- DISC.HARG£: ~ C.F.S. ()) (X) . 69-. 3. Water Surface Contours: All of the photographic views of the previous section apply also to this section as they indicate not only the wall profiles, but also the variations existent over the water surface within the channel. Contours of equal depths showing the magnitudes of the wave crests and troughs have been plotted for several different ty-pes of treatment. Those for untreated flow conditions may be seen in Fig. 51, while those for the best treatment are found in Fig. 52. Note the drastic reduction of peaks at the three wave crossing points with the K-sills. Changes in water surface from one section to another are best ty-pified by the cross-section comparison of Fig. 53 and Fig. 54 for the same flow conditions just described. 4. Velocity Distribution: Comprehensive velocity studies were made only for the maximum rate of discharge for both the untreated channel and for the treat- o..taW& W.11 - I .f O.IOO 1.1·,- 0·*o~- , -_..... 4 __ 6__.,.....-cl0;::___1t--,4....1oc,~ ;,. t . . t , - - . . . i ~tn'lc;AL. v&LOCITY """011'11.I.S IN • "&CTANGI.ILAR CHAHl«L Q•4.0c.f'.a. <Vol■ • Oh • • o,oao Fig. 55. o.,•1,i J,,.W.W.11- 4 8 It v.i..;ti ,,. ' - ' ,., .. u..4 HO!tl'ZONT-'L VSl.()CITV CONTOURS IN "~CT...H<.ULAA CH,_NHE.1. Q.a4.0_a .i:a. 41.._• V.. $•0 .0SO Fig. 56. 70 . ment with K-sills because of the length of time required for this study at each cross-section. Approximately half a day was devoted to each, several weeks being required to complete the study. Typical vertical and horizontal velocity profiles are shown in Fig. 55 and Fig. 56. Velocity distribution throughout various cross-sections are shown in Fig. 57 and Fig. 58, the former indicating tba.t for the untreated channel, the latter that for the best treatment . Fig. 59 is a very interesting study of velocity distribution immediately downstream from the K-sills (~ = 4.0). Velocity vectors; i.e., magnitude and direction, in a horizontal plane are illustrated in Fig. 60, Fig. 61, and Fig. 62. Fig. 60 portrays conditions at various depths above the channel bottom just downstream from the K-sills, at the same location shown in Fig. 59. O.l 0 .) 11-~,0.1 o.i - JI ~0.1 0.1 o.+ 0.6 VELOCITY DISTRl6UTION ti! CHANNEL f.rn-S&CTION JUST OOWNSTlltE-'M F~Ol'I K-~ll.LS ~- to c.r.s. Fig. 59. ,.w. 71. C,.6 o. , 0 ,5 0 .4 O,♦ o., O.S 0,2 0 ,1 0 0 ST .A.'T I QN O - &1 0 .4 O.J n. , o. f 0 ____--:::_..-,, ~-----------='=c..=·====•~_, S T "T t 0N ~ 0-~ o .4 0.J 0. 2 o., r: t--·-=- 0 VELOCITY DISTRIBUTION FOR UNTREA1ED FLOW AROUND t:_ . !Q. ABRUPT !IENO ~ RECTA.,_UV.R CMANHEI.. 0 q ~ 4,0 c .r.s., '½. ! , s ~ o .o'5o s Fig. 57. 72. r 0 " u .s 0. 4 o .3 _ _ __.r o.'l. 0 .1 __ • - ~ - - ------ ) 0 0 - 0~ 0 -5 0.5 o.+ 0 .'\ o. , o :z-'77:;:,,--:,,,---------:-=~ '--,ci _ __ ~ ____,) 0.'l. "\ 0 .1 "·' .l ' 0 S T AT10 N l.,;- 11 ~ VELOCITY DISTRIBUTION FOR TREATED FLOW (K- 51LL5) AROlM> ~ 1.Q'." ~ ~ 1!:ll RECTAN<i,ULAA CHANNEL Q * ◄ . O C . F. S., ¾ ,• ~, Fig. 58. S : 0-050 73. ,,' ~\l I11!1Jdll1 I11@Ul111ri \\ \ \ Q. 100 'At>O\la. !)O'T "TOIII I t o.=' Ae<NL eo,hc>A 0.0t.!'MICN& eo'TTOM ' ; 11Jl I L..-..__.__,_L......O._._~.....i.... C.MANNl!J.. eaTTOM O.OOl. A00ole.) HORIZONTAL VELOCITY VE..CTORS AT S"TATION 0-0~ roR K·SILL~ Q = 4.0 Cf-!> ~ig. 60. The directions of the lines represent the velocity directions as measured and the length of the lines represent magn.1 tudes as measured. Note how effectively the sills divert the water without 74. I/ I 'lt...l..OC.liY IN ✓ FT . ~R ~E.C.. ·~L 0 o.-z.,04C HANNE,~ VV\OTH IN FIL.ET SC.ALE... ~ORIZ.ON,AL VE-LOCrT"f' VE..c:TORS A, STATION D-5 F-OR "'-SIU-") Q=4 .0cf~ Fig. 61. destroying the nearly uniform velocity magnitude. Fig. 61 p0rtrays conditions at Station D-8 with the same set of sills and the same discharge. Fig. 62 sho-.s the bottom velocity vectors at different stations throughout the channel. 5. Momentum Changes: Table II. is a summary of the average forces exerted on several series of sills for various discharges. As previously mentioned, 75. the sill plate was surprisingly free in its movements, giving consistent readi ngs for any set of conditions. Note the similarity of force components for certain sill types. VE.LDCITY IN FT, PER sec. ✓ CHANNl!:.L 1,0 _ __ iOr}LeNG.TH 5 oS IN F"Lt:T o ~1. o.4C.H,11,MNE.L. WI O'T H IH~T ::x::AL.~ BOTTOM VE..LOCtT'< VEC.TOR S K-51LL..S Q•4 .0c+~ . 'Fig. 62. 76. T.A.BLE II. Summary of Force Measurements (~n lbs.) Sill Series C K I Discharge Py Pxl Px2 Scale No. · 3 Scale No. 4 Scale No. 5 4.0 9.63 3.38 4.63 8.01 3.0 7.63 2.44 3.81 6.25 2.0 5.38 1.50 2.75 4.25 4.0 7.56 2.56 3.94 6.50 3.0 6.00 2.13 3.06 5.19 2.0 4.56 1.13 2.25 3.38 4.0 9.63 3.56 4.69 8.25 3.0 7.50 2.19 3.81 6.00 2.0 5.44 1.50 2.59 4.19 4.0 10.13 2.88 4.88 7.76 3.0 7.63 2.25 3.94 6.19 2.0 5.63 1.19 2.94 4.13 4.0 11.19 3.63 5.81 9.44 3.0 9.19 2. 31 5.00 7.31 2.0 7.19 1.94 3.75 5.59 Px Px1 + Px2 77. IV. COMPAF.ISON OF .ANALYTICAL AND EXPERIMENTAL RESULTS UNTREATED FI.OW A. 1. Wave Depths : In the a.nalytical development of the two simultaneous Equations (27) and (30), it was assumed that the sections of fluid under consideration were sufficiently removed from the wall to insure full development of the shock wave. It would, therefore, be illogical to expect these shock wave equations to indicate the depth of flow along the walls. Because of the nearness of the wave front to the outer wall, the condition of uniform flow on the downstream side (or~~ side) cannot be fulfilled. The influence of the outer wall would, undoubtedly, be anticipated to be an additive one, giving values higher than would be obtained by the shock wave formula. confirms this. Table II I. With lower discharges, and, accordingly, lower depths of uniform flow, conditions become nearer and nearer those assumed, and it is to be expected that the shock wave development could proceed unhampered without too much wall influence. Accord- ingly, the wave formula should give a better approximation to the actual depths. In Table III. the analytical values of dz. do approach more nearly those experimentally obtained. In all runs the water along the outside wall piles up rapidly and give s the appearance of folding _over the water beneath it similar to the folds of a ribbon. Fig. 64 shows details of the flow along the outside wall in the vicinity of the bend(~= 4.0). Although this picture does not show Uniform Depth Shock Wave Depth (Anal.) First Max. Wall Depth (Exp.) %Excess Over Shock Depth Negative Wave I.W. (Anal.) Min. Inside Wall Depth (E:x:p.) Ave. Area Depth Below Wave (Exp.) Location First Max. (I.W.) (Anal.) Location First Max. (I.W.) (Exp.) %Difference (Anal. to Exp.) Location Min. I.W. Depth (Exp.) Wave Length (Anal.) Wave Length (Exp.) %Difference (Anal. to Exp.) Item L L d2 d d2 Rin R1n dl d2 d Symbol 2% 4.0 0.221 0.424 0.487 15% 0.088 0.068 0.085 6.92 8.0 12% 6.4 13.8 13.5 Untr~ted Flow 0.184 0.353 0.388 10% 0.074 0.060 0.075 6.92 7.5 8% 6.3 13.8 13= 6% 6% 0.141 0.271 0.298 6% 0.056 . 0.044 0.060 6.92 7.2 4% 6.2 13.8 13% Discbar~es in cu.ft. ~er sec. 3.0 2. 0 Comparison of Wave Depths and Locations T.A.13LE III. ·0.091 0.175 0.180 3% 0.036 0.025 0.031 6.92 7.0 1% 6.1 13.8 13.5= 2% _l.O (X) -..J . 79. it too well, at large discharges considerable air is entrained at the wave front by the folding effect and is carried on downstream with the flow. Table III. also gives a comparison of the minimum wave depths as determined analytically with the experimentally determined minimum wall depths, which a.gain, as discussed above, could not be expected to be typical of the depths to be encountered just below the wave. For that reason the average depths in the fairly extensive area below the negative wave front have been listed for comparison. compare well with analytical values and they do. These should This area can be seen quite clearly in Fig. 66 . . Note the extensiveness of it. Attention is also called to the rapid drop in water surface along the inside wall as the water passes a.round the bend. To the left can be seen quite clearly the "ribbon-like" folding over of the water as the positive wave reaches the inside wall downstream from the bend and is reflected to the other side again. Further confirmation of the extent of this "hollow area" is also obtained from the water surface contours of Fig. 51. Fig. 63 is a view looking almost directly down at the wave front when the discharge is only 1.0 cu.ft. per sec. It is obvious that the wave front does not lie along a straight line, but curves outwardly as it progresses downstre8Jil in accord with previous analytical developments;(i.e., increases in depth of water au@Ilent the wave celerity, as snomi by Equations (7) and (19), with a resultant effect of enlarging the cross-component of the vector-velocity relationship between C and V). 80. Fig. 63. 2. Location of Wave Crossings: Table III. includes a comparison of the analytical and experi- mental locations of the wave crossings. The agreement is not too good at the higher discharges, 12% difference for a discharge of 4.0 cu.ft. per sec., and 8% for 3.0 cu.ft. per sec. However, as the discharge decreases, which, in turn, indicates an upstream depth decrease, the percentage disparity between analytic and measured values also decreases. Bl. Precise determination of the wave length is difficult experimentally unless the downstream channel is sufficiently long to enable measurement of several .wave lengths. ·The chief difficulty is in selecting the location of the maximum water height along the channel wall because the water surface profile is a broad flat curve in this region. Nevertheless, analytical and experimental agreement was remarkably close, as shown by the figures in Table III. Fig. 32 shows this tendency toward broad, flat peaks, and also emphasizes the fact that maxima and minima locations shift only slightly with changes in discharge. 3. Velocity Distribution: The assumption of uniform velocity throughout the cross-section is an exceedingly good one. In Fig. 55 the velocity profiles taken through vertical sections parallel to the channel walls at the centerline, 0.2 ft. out, 0.4 ft. out, and 0.6 ft. out practically lie on top of each other, am the wall profile, 0.73 ft. out, is made up of velocities sufficiently large in magnitude to fulfill the uniform velocity criter~a within the accuracies of the limitations set. Like- wise, the horizontal velocity profiles, Fig. 56,fall within a narrow range, only the extreme bottom velocity differing from the others. The bottom velocities, as previously mentioned, are within 0.002 ft. of the channel bottom. Fig. 57 confirms the assumption of relatively constant velocity in the downstream section. In this regard, the latter plot is an interesting study of the changes in velocity distribution as the water flows through the channel. sectional shape and area are rapid and violent. Changes in cross- B. 1. TREATED FLOW Wall Wave Heights: The principal requirement of a perfect wave repression device is the ability to reduce the wave peaks immediately below the bend and in the downstream section, and to increase the wave troughs to such a value that 'Wliform downstream wall height will result. One would expect such a corrective device to be extremely complicated becsuse of the multitudinous corrective features it would have to entail. Thinly-constructed, closely-spaced, confining walls or vanes within the bend might serve to re-develop fairly 'Wliform downstream conditions, although the height to which the water would rise along these vanes would approximate that along the outer wall without vanes. Construction difficulties would not be the only disadvantageous factor in their use. They would not have the ability to insure passage of debris down the channel unless the maximum linear dimension of any foreign material in the water were less than the clear passage distance between any two walls, not an assured condition. Althougl:l no corrective devices studied in this research approached closely the ultimate in perfection, many served as a feasible and practicable means of reducing drastically the unwanted and highly undesirable wave heights present in the untreated channel. Early i,n the experimental studies, it was found that echelon-type sills bad great possibilities and rapidly eliminated other means of correction. Experimentation with many types, sizes, and shapes gradually eliminated all but a few, and of these few, one stood out above the others 83. as giving the best corrective measures. .As previously mentioned, these were the K-sills, pictured in Fig. 30. Comprehensive exam- ination of their characteristics was cond:u.cted over a wide range of discharges, and, although far from perfect, they should, nevertheless, serve as entirely satisfactory wave repressing devices. Fig. 38 is a comparative plot of the effect of K-sills on wall water heights downstream from the bend. Note the very favorable reduction in maximum wall heights, the K-sills reducing this height to never more than a value of 0.34, as against "Wltreated flow heights of 0.58 on both outside and inside wall when the discharge is 4.0 cu.ft. per sec. At lower discharges, the K-sills still keep the maximum wall height below the o. 34 value. These sills also have the effect of filling up the wave troughs on both the inside and outside wall. Their effectiveness in repressing the outside wall wave height just below the bend can best be realized by reference to Fig. 64 and Fig. 65, the former figure having been referred to in previous sections. The inside wall effects can·be seen in Fig. 66 and Fig. 67. Note in Fig. 67 the increased, yet not unfavorable, height to which the sills build up the water along the inside wall and the outstanding lack of the high .cross-wave at the left of the picture. As can be seen in the plot of Fig. 38, the inside wall maximums are those most drastically reduced . .A comparative view of the cDoss-sections at various stations along the channel with and without K-sills are seen in Fig. 54 and Fig. 53. Those under the influence of the 84. Fig. 64. Fig. 65. K-sills are much more regular in form. Still another comparison can be seen in Fig. 51 and Fig. 52, where lines of equal water height are plotted to give a contour map of the water surface with no treatment and with the K-sills acting. 85. Fig. 66. F ig. 67 . Uniform Depth, dl First Max. Depth, O.W. Second Max. Depth, I.W. Third Max. Depth, 0. W. Highest Max. less dl Percent Max. K to Untreated Max. First Min. Depth, I. W. Second Min. Depth, 0.W. d1 less Lowest Min. Ratio-K to Untreated Min. Item 0.221 0.487 o. 566 o. 576 0.355 100% 0.069 0.110 0.152 100% 0.221 0.340 0.332 0.340 0.119 33.5% 0.140 0.155 0.081 53% g, = 4.0 cu.ft. per sec. Untreated f_lo'?!_ Wi_t_ll K-sills T.A.'BLE IV. 0.184 0.400 0.473 0.462 0.278 100% 0.056 0.085 0.128 100% 0.184 0.264 0.288 0.261 0.077 27.8% 0.113 0.127 0.057 45% Q = 3.0 cu.ft. per sec. Untreated Flow With K-sills CX> .m 87. Table IV. compares the pertinent changes in wa.11 water heights occasioned by the induced action of the K-sills. The K-sills reduce the maximum wall depths to only 33.5 percent of the worst possible conditions for a discharge of 4.0 cu.ft. per sec. and to only 27.8 percent for a discharge of 3.0 cu.ft. per sec. At the same time they build up the minimum depths to a point such that the drop in depth with K-sills is only 53Cfo that for untreated flow at 4.0 cu.ft. per sec. and 45% that at 3.0 cu.ft. per sec. As previously dis- cussed, additions to low depths are desirable but certainly are of minor significance compared to the lowering of the maxima. For an excellent comparison of the effects of the K-sills throu~ut a wide range of discharges, refer to the photographic study con- tained in Figures 68 through 75, which will be discussed in a later section. 2. Velocity Distribution and Reduction of Transverse Velocity: The validity of the. assumption of unifonn velocity has been previously discussed. It is relevant to point out here, however, the comparative effects of the K-sills upon the velocity distribution at various downstream sections. Proceeding up the left side of Fig. 5?, across the top, and down the right side, one can obtain an excellent idea of the shifting of the velocity distribution contours as the flow undergoes the severe changes ca;u.sed by the bend and progresses on downstream. It is not difficult to visualize the bouncing of the waves back and forth across the 88. channel for a considerable distance downstream. Throughout all this violent agitation, the flow manages to proceed with the same range of velocities at which it originally entered the bend. Nor do the sills have any great retardation effect, either immediately downstream or further downstream, as can be seen in Fig. 58. There are no areas of greatly reduced velocity, and by the time Station D-11.5 is reached, the velocity distribution pattern is rapidly returning to its normal upstream state. the channel walls damp out rapidly. The components normal to Fig. 61 is a schematic summary of a probe of the horizontal velocity vectors at various distances between the channel bottom (0.002 ft. above) and the water surface at a distance of eight feet downstream from the bend and the sills. Throughout the full depth the velocity on almost the entire outside half of the channel is now parallel to the channel walls, and such transverse velocity components as still exist are indeed small in magnitude. That the bottom water area downstream from the sill region has a rapid and effective influence on the upper area and vice versa, can be seen by reference to Fig. 60 and Fig. 61. In the former, the velocity vectors of the lower and upper area are at an angle of approximately 300. In the latter these areas have obviously become so blended as to give a fairly uniform velocity direction. The effective turning angle developed by various types of sills is summarized in Table V. It will be recalled that Equation (57) gave us an analytical approach to the part plqed by the sills in 89. turning the water. The difference between the effective turning angle and the bend angle, () , is supplied by the outside wall. Just what proper proportion of the total bend angle should be developed by the sills is a matter of considerable further research and is outside the scope of the present study. 90. T.A:BLE V. Comparison of Effective Angle of Turning for Various Types of ·Sills Sill Type Discharge Py p X ¢ :s 4.0 3.0 2.0 9.63 7.63 5.38 8.01 6.25 4.25 6.60 7.6° 9. 3o C 4.0 3.0 2.0 7.56 6.00 4.56 6.50 5.19 3.38 4.5° 5.9° 7. 8° K 4.0 3.0 2.0 9.63 7. 50 5.44 8.25 6.00 4.19 6.6° 7.5° 9.4° I 4.0 3.0 2.0 10.13 7.63 5.63 7.76 6.19 4.13 7.0° 7.6° 9. 7o A 4.0 3.0 2.0 11.19 9.19 7.19 9.44 7.31 5.69 7.8° 9.40 13.00 91. V. A. CONCLUSIONS SUMMARY OF RESULTS This investigation has brought to light a means for controlling high-velocity flow around mild but abrupt bends in supercritically sloped- open channels. The methods of control are the application of suitably-shaped and properly-dimensioned sills or low walls designed to repress the undesirable wave heights which exist in channels within and downstream from mild but abrupt bends. The sills should be placed echelon-fashion across the channel at the bend, the inside sill being omitted as it serves a negative purpose. The angle of the sills was fo'Wl.d to be most effective when about three times the angle of the bend. Details of the best sill dimensions,arrangement and placement are given in Section III. 13. These corrective devices have the following advantages: a. They reduce undesirable wave heights along the channel walls to small proportions. b. They are self-cleaning. c. They keep the flow of water in its super-critical state. d. They are inexpensive to construct. Their disadvantages are: a. Their use is limited at present to bends of small angle. b. Applied in flood channels, they would have to be ·replaced occasionally. c. At low flows of water, the channel center might be subject to intermittent spraying (does not affect the walls) 92. The relative merits of these advantages or disadvantages will be discussed in the next section. Fig. 6 is believed to be a chart which mEcy" hold the key to many future problems in super-critical flow. Careful analysis of observ- able and analytically-determined data led to the proposal of the explanation offered for its interpretation. It must be borne in mind that this explanation, before general adoption, should be experimentally investigated, such investigation having been foreign to the principal purpose of the present investigation. Experimental verification and research on this subject would probably also reveal the significance of the area in the lower right-hand corner of the chart. As a final presentation and summary of the effectiveness of the corrective de~ices proposed, Figures 68 through 75 are presented on the following pages, to show side by side the flow phenomena in the untreated channel and in the treated channel. 93. Fig . • 68. Untreated Channel Fig. 70. Fig. 69. . Treated Channel = 4.0 cu..ft._pe_r sec. ' Fig. 71. Treated Channel Untreated Channel Discharge = 3.0 cu.ft. per sec. 94. Fig. ,'Discharge = Fig. 73. 2.0 cu.ft. per sec. Discharge= 1.0 cu.ft. per sec. 95. B. APPLICATION TO FIELD CONDITIONS The present findings can be applied to field conditions wherever the angle of the bend is small and the Froude number is such that the location on the chart of Fig. 6 is well to the left of the line marked "Locus of Minimum c.1 2 /4 1 "· Sizes of sills, location within the channel, and transference of desired or anticipated results should be made in the usual manner, employing the Froude number as the governing criteria. Disadvantages (b) and (c) listed in the preceding section are minor ones and relatively unimportant. Disadvantage (a), limitation to small bend angles, is listed as a disadvantage solely for the purpose of warning against its application in other than small bends. As pointed out in an earlier section, it is believed that this treatment might be applied to large angles by employing a series of small angle bends one downstream from another until the desired angle were attained. However, such application should first be thoroughly investigated, a research project in itself beyond the time and scope of this study. The ad.vantage of reductions in undesirable wave heights applied to flood channels means a much decreased wall height requirement at a correspondingly reduced cost. The self-cleaning characteristic is a requirement in any flood channel which might be required to carry debris. It is important to retain the flow in its critical state on a super-critically sloped channel because reduction to the 96. sub-critical state is accompanied by an hydraulic jump of imdesirable proportions. The treatment proposed, if properly applied, will keep the flow in its super-critical state. 97. BI:BLIOGRAPHY 1. Knapp-, Robert T. and Ippen, Arthur T. - "A Study of High Velocity Flow in Curved Sections of Open Channels" - .An experimental and theoretical study carried out at the Hydraulic Structures Laboratory of the California Institute of Technology for the Los Angeles County Flood Control District, March, 1936. 2. Knapp, Robert T. and Ippen, Arthur T. - "Experimental Investigations of Flow in Curved Channels" - Research project for Los .Angeles County Flood Control District, July, 1938. 3. Rouse, Ho.nter - 11Fluid Mechanics for Hydraulic Engineers" McGraw-Hill Book Co., 1938, pp. 283-4. 4. Karman, Th. von - 5. Rouse, Hunter - 11 A Method of Measuring High-Velocity Flow in Open Channels" - An investigation conducted for the Section of Watershed and Hydrologic Studies, USDA, SCS, Feb., 1939. 6. Rouse, Hunter - op. cit. 3 - pp. 399-401. 7. Rouse, Hunter - op. cit. 3 - pp. 369-370. 11 Eine praktische Anwendung der Analogie zwischen Uberscha.llstr6mung in Gasen 11berkritischer Str6mung in offenen Gerinnen". Zei tschrift fur Angewandte Mathematik 'und Mechanik, Vol. 18, p. 49, 1938. 98 . .APPENDIX A. LIST OF TABLES ~ Table I. Comparison of Maximum Wall Depths vs. Sill Types 54 Table II. Summary of Force Measurements................. 76 Table III. Comparison of Wave Depths and Location Untreated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . Table IV. Reduction of Wave Wall Heights Due to Treatment with K-Sills . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table V. 78 86 Comparison of Effective Angle of Turning for Various Types of Sills.................. 90 99 . .APPENDIX l3. LIST OF FIGURES Ai."ID DRAWINGS ~ Fig. l. Dimensionless Specific Energy Dia.gram.................. 7 2. Wave Propagation in Super-critical Flow................ 9 3. Wave Pattern in Curved Channel......................... 10 4. Velocity Relationship at Shock Wave.................... 14 5. Momentum Near Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ................. 19 7. Wave Crossing Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 B. Divisional Effects on Flow by Sills.................... 23 9. Effective Turning Angle-Velocities . ........ . ... .. ...... 23 10. Forces on Sills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 11. Momentum Transport Past Sills.......................... 25 12. Preliminary Channel View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 13. Wave Pattern l3elow Abrupt Bend......................... 30 14. View of Experimental Channel........................... 31 15. View of_Experimental Channel........................... 32 16. Construction Layout of Bend Section.................... 32 17. Entrance Control Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 18. Venturi Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 19. Manometer l3ox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 20. Pump House and Spilling Basin.......................... 35 21. Point Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 22. Pitot Tube and Vane of Velocity Distribution Gage...... 38 6. Chart for Determination of and 100. ~ Fig. 23. Velocity Distribution Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 24. Pitot Tube and Velocity Distribution Gage............. 40 25. Partial View of Sills Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 26. B-Sills in Place . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 27. Scale Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 28, Detail View of Sill Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 29, Splashing of Unsuitable Sills at Low Discharge........ 49 30. K-Sills in Place . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 31. Wave Pattern, Q = 4.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 32, Water Surface Profiles - Untreated Flow............... 56 Wave Pattern Q = 4, 3, 2, l ...... ....... ....... 57 37. Water Surface Profiles - Uniform Size Sills........... 58 33. - 36. 38. Comparative Water Surface Profiles - No Treatment vs. Best Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 39. - 42. Comparative Photographs - No Sills, K-, I-, A-Sills Q = 4. 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 43. - 46. Comparative Photographs - No Sills, K-, I-, A-Sills Q = 3. 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 47. - 49. Comparative Photographs - No Sills, K-, A-Sills, Q, = 2. 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 50. I. W. First Maximum Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 51. Water Surface Contours - Untreated Flow 65 52. Water Surface Contours - Best Treatment 66 53. Cross-Sections - Untreated Flow....................... 67 101. ~ Fig. 54. Cross-Sections - Best Treatment....................... 68 55. Vertical Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 56. Horizontal Velocity Contours . . . . . . . . . . . . . . . . . . . . . . . . . . 69 57. Velocity Distribution - Untreated Flow................ 71 58. Velocity Distribution - K-Sills ....................... 72 59. Velocity Distribution below K-Sills ............ ....... 70 60. Horizontal Velocity Vectors below K-Sills ...... ....... 73 61. Horizontal Velocity Vectors at D-8, K-Sills ........... 74 62. Bottom Velocity Vectors - K-Sills .... ................. 75 63. Wave Pattern - Small Discharge........................ 80 64. - 65. O.W. - No Treatment vs. K-Sills .......... ...... 84 66. - 67. I.W. - No Treatment vs. K-Sills ............... . 85 68. - 75. Photogr~hic Study of Effectiveness of K-Sills at Various Discharges .......................... . A. La;y-ou t of Experimental Channel · :a. Wiring Diagram - Remote Control Operation C. Construction Details D. Discharge and Velocity Curves for Experimental Channel E. Range of Experimental Runs on Specific Energy Diagram 93-94 . \ ~ 2 L 0 er: :.-: d ' ' ~ ~ T ~ ~ ,t' I 0 0 L.«:::::::::: i I 0 . ~5C ., _,i I~- • •~ ' . '.,_ .,1·..._ .,, n ·,- .,-, r · 0 nJ I,5·r ,~ ~ ;..:_ _l () "";: .. - 0.2,JO •• 7. t· 0.250 ;- ! (!) ! i I.C 5 Di SCHARGE I 0 4 'v =v, , . vt'I_10:a - IN CF.S 2 .0 j I . • v,,ll.9/ !i y . I I() I f\\.!t'RAGE:: Vt~LC1C \ ·rr ! N FT PER <=:,tC, 30 ' I ' j. d. -:. , i8<fl 15 l -- •·- ff I VJC. WAG!✓ f:' R n-.:. .0085 4 .0 ---- -- ·•· •··-• ·-·- . - - -•-- - - - - - - - - - - - ~~~{.\'{ 2° C.1, :q,{ ;_,. 4 o S = 5% ::-. C50 DISCHARGE VS' LOC lTY vs CEPTH RcCTANGU LAR (HM--1~✓ 8- . b = 1. 50 I ~~- ,-- 20 0 0.4 : -- -~·-- : ; i - r .. 1.0 - l' / t I. 2.. . j r-c: • F ;I I I I j l I : -- l ~ - I -j ! , i , ll . l - 1.4 I (. 6 1. 1 t. 8- • k /.,,. J I l.7- 2..4 -- ·- -L -- ~... - "L - ·- 2.o i i j-· · -~ t··· ·- ·- ! ; i 'I ·I : ' ~ EXP;E..~1 ~E..N!;~ RvN~ l AN!.:}._ q . . 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