THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING HYDRAULIC JUMP IN A ROTATING FLUID Chia-Shun Yih Ho Eo Gascoigne Wo Ro Debler April9 1963 IP=613

ACKNOWLEDGMENTS This work has been sponsored jointly by the National Science Foundation and the Army Research Office (Durham)o The authors wish to express their appreciation for this sponsorship, and their thanks to Dr. David Appel for helping in the measurement of the upstream depth. ii

Table 1 Figure 1 2 3 LIST OF TABLES Data for Rotating Hydraulic Jump........................ LIST OF FIGURES Definition Sketch...................................... Schematic Diagram of the Apparatus................... Comparison of Theoretical and Experimental Results,..... LIST OF PLATES A Photograph of the Apparatus Used.o,,,,............... A Photograph Showing Location of the Jump and Streaks in the Flow................................................ Page 12 Page 14 15 16 Plate 1 2 Page 17 18 iii

1. INTRODUCTION It has been generally recognized that the flows of a rotating fluid are, in many respects, similar to the flows of a stratified fluid in the presence of a gravitational field, Since a free surface is a surface of density discontinuity, which is a form of extreme stratification, there is also a similarity of flows of a rotating fluid with a free surface to free-surface flows in the gravitational field. A free surface in the rotating fluid is necessary to ensure similarity of its flow to a free-surface flow in the gravitational field because the quantity corresponding to a discontinuity in specific weight in the latter is a discontinuity in p r2 in the former, p being the density and r the circulation of the flow along any circle in its domain located with axial symmetry. Thus the counterpart of the ordinary hydraulic jump appears to be a hydraulic jump in a layer of liquid flowing down the inner wall of a rotating cylinder, and rotatilg with it. The analysis was carried out in 1961 and construction of the apparatus for experimentation started in October of that year. By the summer of 1962 the phenomenon was clearly observed in a repeatable fashion in the apparatus constructed. When this was mentioned to Mr. A. M. Bonnie of Cambridge University, who was visiting the first author in September of 1962 Mr, Bonnie showed some pictures of a hydraulic jump he observed in a swirling flow down a stationary tube, in an experiment performed to study the effect of a bend on a swirling fluid. His work has since been published [1]. But his tube is stationary and his work is not primarily a study of hydraulic jump in a rotating fluid, -1

-2Due to the difficulty involved in the measurement of the upstream thickness of the water layer, the experiments were finished only in December, 1962o The analytical and experimental results are presented in this paper to provide yet another instance of the similarity of rotating flows and stratified flows.

2. ANALYSIS With reference to Figure 1, b is the inner radius of the tube, dl is the depth of water upstream from the jump, and d2 the downstream depth. The pressure in the fluid upstream from the jump is 2 Pi = r2 a ) (l = () (1) 2 in which I1 is the angular speed of the rotating water film, and is equal to the angular speed C of the rotating cylinder, r is the radial distance from the axis to the point at which the pressure is being considered, and a1 = b - dl. Downstream from the jump, the angular speed of the fluid co2 in general varies from one radial position to anothero Two extreme situations may be considered. If viscous and turbulent mixings are ignored, Kelvin's theroem on the conservation of circulation enable one to compute C2 as a function of r, upon utilization of the equation of continuity and the assumption that the downstream velocity U2 is constanto This would be a very unrealistic situation, because there is violent turbulent mixing at the jump, so that Kelvin's theorem cannot be valido The other extreme condition is the condition of complete mixing, so that after the jump another uniform cu2 exists, which can be computed from cD1 by use of the conservation of the integrated angular momentum. Thus, on the assumption that n2 is uniform., the downstream pressure distribution is given by 2 pO22 2 P P~2 (r - a2 ),(2 ~2in which a2 = b - d2. The total axial force acting at an upstream in which a2 = b - d2. The total axial force acting at an upstream -3

-4 section (Section 1-1) is b P1 = a1 p1 2irdr = Pc 2 (b2 a12) (3) The total axial force acting at downstream section (Section 2-2) is b P2 = /2 a2 P2 2trdr = I- u 2 (b2 - a2) (4) The discharge is given by ib J al b U1 2trdr = / U2 2itrdr, L a2 (5) which is the equation of continuity. The downstream flow is very turbulent, so that U2 can be assumed constant without appreciable error. If the upstream flow is also turbulent, so that U1 can also be assumed constant, the equation of continuity can be written as Ul(b2 - a12) = U2 (b2 - a22). (6) The momentum flux through Section 1-1 is 2 M = pU12 2irdr, a1 and that through Section 2-2 is (7) Mb M2 = 1 a, 2 pU2 2 rdr, (8) If U1 and U2 are assumed constant, 2 2 2 2 2 2 M1 = pItU (b -a1 ), and M2 = pU2 (b -a2 (9) The fluxes of angular momenta are the same before and after the jump, since the torque exerted by the wall of the cylinder -.:i be neglectedo

-5 Thus b b a (pnr2) U1 2jrdr = (p2r2) U2 2rdro (10) Now w is constant, and as explained before U2 and o2 can be assured constant. If the upstream flow is turbulent, U1 can also be assumed constant, and (10) becomes pU (b 4(11) (b4 - a4 ) = p2U2(b - a24) (1) which can be reduced to D(b2 + al2) = a2(b2 + a22) (12) by the use of (6). The momentum equation applied to the fluid between Sections 1-1 and 2-2 is P1 - P2 + = M2- Ml, (13) in which W is the weight of the body of fluid in the region of change of depth. If the inner radius of that body of fluid is assumed to vary linearly (with 3) from. al to a2, and if the length of the jump is assumed to be c(a1 - a2), c being a constant of proportionality, _1~~~ 2 W = gpt(al - a2 (b"2 - 1 2 -2 = gpi(al - a2) [b2 - a1 a2 - (a1 - i)lj 1.4) Equation (13) then becomes

-6 - [w2(b2 - a12)2 22(b2 a22)] + gp (a - a2)[b2 - a a - 1 (al - a2)2] = p[U22b2 - a22) - U12(b2 - al2)1 (15) Now if (6) is used on the right-hand side, (12) is used to eliminate c2 and for simplicity one writes a1 a-. U1 a1 = - C2 = -a- F G - b a I2- w1b' bc =' One obtaines, after simplifications, (1 - ai2 22)(1 - a22) = F12(1 - l2)(l + a22) c G (1 - c22) (1 + a22)2 [3(1 - ala2) - (a1 - a2)2] +,,g. i>(16) 3(a1 + a2) This equaltion enables one to find a2 for given values of al, Fil G, and c. In the experiments performed, d1 and d2 were very small compared with b Hence al and a2 were nearly equal to b and so a, and a2 were nearly equal to 1o Putting a1 and a2 equal to 1 except where differences are involved, one obtains from (16) (1 - ala2)1( - 2) - 2 F12(1 - a) + c G (1 - a) [(1 - ala2) - (c - ( c - 2)2go (17) 3 Now with d2 2 U2 dn.I J - F 2bdl

-7-l one has 1 - a2 = ( 1 + ), (1 - 2) - I F2 (1 - al) b b 2 d1 = Fl - and (aL- _ )2 i- -2 - 2 b Thus (17) can be written as 1dl di (I + 1)(1 - c G) +- 1 c G ( 1)2 2F2o (18) 3 b The depth ratio r had a maximum value of 10o7 in one test, and less than 10 on all the other tests, and dl/b was very smallO Thus, under the experimental conditions, the second term on the right-hand side can be neglectedo The resulting equation can be solved simply. The solution is d2 1i + - [-1 8F ] (19) n-a-e[-+-Vl+1z cG

3. APPARATUS AND METHOD OF MEASUREMENT The apparatus is shown in Plate I and schematically in Figure 2, The working section was a piece of transparent tube of polished cast resin about 50 inches long and 9 inches in outside diameter. The thickness of the wall was 1/4 of an inch. The inner-surface diameter had a variation of at most 0,012 inch. The tube is supported by a rigid hub at the top and a rigid ring at the bottom, A rod running centrally from the top to the bottom carried a movable point gage for measuring depths, A turntable fixed to the bottom ring supporting the transparent tube was driven by a vanable-speed motor of 5 horsepower, Water at 62~ F was introduced into the tube through a rotating union threaded into the top hub from a head tank. The flow was regulated by needle valves through flow meters of the type of the FischerPorter rotometer. The flow meters were calibrated under test conditions and the variation of the discharge was within + 1 per cent in each run. After entering the rotating union, the water was spread onto the inner wall of the best cylinder by a circular plateo Vertical uniform flow was established after approximately one tube diameter and. a half. At the bottom of the tube were efflux ports which could be opened or closed at will to adjust the location of the jumpo The jump could be moved up the tube by reducing the opening at the bottom of the tube, The angular speed of the turntable was measured electromagnetically and was maintained constant. The variation in each run was n- -il re than 1 r.pom., or about + 01. per cent in the tests. This angular speed -8

-9 is the same as a in the analysiso Plate II shows the location of the jump and streaks in the flow both upstream and downstream of the jumpo Since c was known and dl was small, the circumferential velocity upstream from the jump was knowno Assuming the streaks were statistically the same as the streamlines, one could obtain the upstream surface velocity U1 of the fluid in the axial direction from the inclination of the streaks. From U1 one can easily calculate dlo The mean inclination of the streaks was obtained photographically with a variation of + 050o Downstream from the jump the inclination of the streaks were too small to be useful as a reliable means of obtaining d2, which was therefore measured with the point gage. The error was with +- OO007 inch approximatelyo The upstream depth was so small that the waviness of the freesurface would introduce a substantial percentage error in d1 if measured with the point gageo That was why the streaks were utilized upstream from the jumpo The length of the jump was observed to be between 05 inch and 1 inch in the tests.

4. DISCUSSION OF RESULTS The results are shown in Table 1 and Figure 3, As explained in Section 3, U1 was computed from the inclination of the streaks on the upstream free surface, Since the upstream flow was assumed to be turbulent, this U1 was considered to be the axial velocity in the major part of the upstream flowo The Reynolds member Uldl R = U.d based on the surface velocity was recorded in Table 1, with v = 1,2 x 10-5 ft2/sec. The values of R show that the judgment of turbulent upstream flow is not an unrealistic one. It is known that plane Poiseuille flow, which would be the upstream flow if it were laminar and the slight curv ture effect were neglected, is unstable at a value 2000 for the Reynolds member based on the mean velocity, or 3000 for Ro It is also known that a free surface tends to destabilize the flowo But it is important to remember the distinction between stability against surface waves and that against shear waves, For surface waves the flow is unstable at any Reynolds number however small, but at the same time it is shear-wave instability that is responsible for turbulenceo In view of the fact that the Reynolds numbers recorded are from. 930 to 24009 which are of the order of 3000, and considering that the flow was not free from turbulence as it entered the tube, the assumption of turbulent upstream. flow was not unrealistico With Q as the discharge in in3/sec, di in inches was obtained from -10

-11dl = 0.0375 Q/U1, U1 being in in/sec. In Figure 3 the data are plotted in a chart with F as the abscissa and d2/d1 as the ordinate. Equation (19) is plotted with c = 0, and also, for best fit at various values of G, for c = 7o This value for c is the same as for the length of the ordinary hydraulic jump. It can be seen that the agreement between the theoretical prediction and the experimental results is quite satisfactoryo

TABLE 1 DATA FOR ROTATING HYDRAULIC JUMP Q a U dl d2 d b 1 rr i - p11;4 I-cI deg tanac in/sec in in F ER *-"I g 1 462 2 535 3 535 430 5 540 6 (72 7 592 8 465 9 565 10 625 11 455 12 610 13 550 14 450 15 525 16 6oo 17 450 1.8 525 19 6oo 20 440 21 525 22 625 23 575 44,4 29,8 33.0 44 4 31.6 32,8 47,5 47 5 47.5 53.3 53.3 53.3 5970 59.0 59.0 64,2 64,2 6402 71-0 71.0 71.0 76,2 21 19.8 17 21 19.7 16 16.3 21 1878 16.5 23 17.5 20.2 23.7 20 17.8 23.5 19.8 18 24 20 16.5 18.7 0O38 0o36 0o31 0o38 0736 0,29 0,29 0.38 0o34 0,30 0o42 0o32 037 0o44 036 0732 0o43 0.36 0o32 0o44 0736 0,30 0~34 79 0.021 86 oo018 72 07o15 73 o0o017 86 oo019 79 oo015 77 oo016 79 0o022 86 0o021 61 0~029 86 07023 86 0o023 90 0022 88 0o025 85 0~026 86 0o026 87 07028 84 07029 87 0,028 87 0o030 85 0.031 82 0o032 87 0033 0 o18 0.128 O ol6 0 oll O11 0.17 0 o15 o 714 o 21 o, 14 O.16 0,21 0o. 16 0o. 14 0.23 o.18 0o15 023 o,19 o0.16 0o.18 5.5 5~5 5o2 6.1 5.3 4,9 4.7 5.3 4.9 2.7 5.7 4,2 51l 5.7 4~6 4.1 574 4.i4,0 5.3 4.3 3.4 3.8 8,8 879 8.0 10.7 8.1 7.2 6.9 7.8 7,2 4,7 9.2 671 7.4 8.5 6,3 5.5 8,2 6,2 5,4 7.7 6.1 4L8 5.6 i4oo 1300 900 1000 i4oo 1000 1000 1500 1500 1500 1700 1700 1700 1900oo 1900oo 1900oo 2000 2000 2000 2200 2200 2200 2400 25.8 34.6 34.6 22.4 35.2 47.1 42,4 2672 34,7 47.2 25.0 450O 36,5 24,5 3372 4376 24,5 33.2 43.6 23.6 33.2 47.2 40.0o I I

REFERENCE 1, Binnie, A. M,, Experiments on the Swirling Flow of Water in a Vertical Pipe and a Bend, Proc. Royo Soc. A, Vol 270, ppo 452-466, 1962 -13

w I 2 2 Figure 1. Definition sketch.

-15 1. Supporting structure 2. Flowmeters 3. Depth indicator scale 4. Seal 5. Rotating union 6. Flange bearing 7. Circular flow spreader plat 8. Top hub sleeve bearing 9. Cast resin transparent tube 10. Center post assembly 11. Depth gages 12. Control tube 13. Bevel gearing 14. Flow controller 15. Efflux port 16. Collection box 17. Turntable 18. Turntable spindle 19. Drive belt and pulley 20. Input shaft of right angle 21. Magnetic pick-up 22. Electronic counter 23. Shaft extension 24. Threaded nut assembly 25. Structure base 26. Leveling screws and pads 25 Figure 2. Schematic diagram of the apparatus.

I1.0 0 0 - o' c. a C] 10.0 - 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 - 1.0 1.0 bw2 I -b = - 20-25 26-30 31-35 36-40 41-45 46-50 S G. C=7, SYMBOL 0 0 A X 0 0 =50 0 I / C=7,-=20 = c=7, -=30'0 z I o~ l 2.0 3.0 4.0 5.0 6.0 Ui Froude Number, (b2d)1/2 Figure 3. Comparison of theoretical and experimental results. 7.0

Pl:a-te..I.. A pho':to)grapVh o:f' the aptpart-ttu used.

P?..ate 2 A pho-tograth showing location of 1the jumpn and. s tr eaks in the f low.