Flow in a P ipe o f Rectangular Cross-Section.

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691 Flow in a P ipe o f Rectangular Cross-Section. By R. J. Co rnish, M.Sc., Assistant Lecturer in Engineering in the University of Manchester. (Communicated by E. A. Milne, F.R.S. Received June 22, 1928.

1 691 Flow in a P ipe o f Rectangular Cross-Section. By R. J. Co rnish, M.Sc., Assistant Lecturer in Engineering in the University of Manchester. (Communicated by E. A. Milne, F.R.S. Received June 22, 1928.) List of Symbols, a b = half width of pipe in centimetres. = hah depth of pipe in centimetres. m = hydraulic mean depth = - T. a + b p = pressure. Q = quantity of water in cubic centimetres per second. R = resistance per unit area in dynes per square centimetre. S = average velocity in centimetres per second.. t = temperature. w = velocity of a particle parallel to the axis of the pipe. 2 = distance parallel to the axis of the pipe, p = density. H = absolute viscosity. v kinematic viscosity. Note, R X (area of walls) = (difference of pressure) X (area of cross-seotion). R. 4 (a + b) dx = dp. 4ab ; or R = m. dz Hence Object of Research, The object of the research was to investigate the flow of water in a pipe of rectangular cross-section. Much work has been done on similar problems with pipes of circular section,* and pipes of rectangular section have been investigated by Frommf and Davies and White. J Fromm worked with pipes in which the ratio of the sides was never less than 6 to 1 ; his report deals only with turbulent flow. In the case of Davies and White s research, the minimum ratio of the sides was 40 to 1, so that the laminar flow could be calculated from the formula for flow between infinitely wide parallel plates. The present writer used a pipe of section 1*178 cms. by 0*4:04 cms. (ratio of sides = 2*92) ; this presents a fresh problem where stream line flow is concerned, and shows interesting results in the region of the critical velocity. * See especially Stanton and Pannell, Phil. Trans.,5 A, vol. 214, p. 199 (1914). t Z. f. ang. Math. Mech.,5 vol. 3, p. 339 (1923). J Roy. Soc. Proc.,5A, vol. 119, p. 92 (1918). VOL. OXX. A. 3 n

2 692 R. J. Cornish. Apparatus. Fig. 1 shows a cross-section through the pipe. The two main components were two brass castings about 120 cms. long. In the lower casting a channel was cut and finished smooth with emery paper. The upper casting was a plate, planed flat and smoothed with emery paper. Three gauge-holes, a, (3, y, each 116 inch in diameter, were drilled in it. The distance from the entrance to a was 30*2 cms., from a to (3 30*50 cms., from [3 to y 36*43 cms., and F ig. 1. from y to the exit 22*8 cms. The two plates were kept together by 28 -^--incli bolts, the surfaces in contact being covered with a thin film of vaseline to prevent leakage. The width of the channel (2a) was 1*178 cms., and the depth (26) was 0*404 cm. The maximum variation from these average figures was less than 0*5 per cent, in both cases. The pressure differences were measured in three ways, according to magnitude (1) Very small differences, up to about 12 cms. water, were found by observing a differential water gauge with a cathetometer, which could be read to 0*001 cm. by a vernier. (2) Up to about 30 inches water a differential water gauge, read directly, was used. (3) For all higher pressures two mercury gauges were used. A calibrated mercury thermometer was used for temperature, and the quantity of water was found by measuring with a stop watch (calibrated every day) the time to fill vessels whose volume was known within 0*1 per cent. Mathematical Formula for Stream Line Fioiv. The general steps of the theory are given in Appendix II. The formula for the flow is n a Q 3 (x ' i 192 ^5 l ( t an h g + i. t an h 2 +. For the section under consideration, this formula gives Q = dp (x dz -)} (1)

Flow in Pipe o f Rectangular Cross-Section. 693 Now m = 0-150 cm., and A = 0-476 sq. cm. ; using the relations Q = AS and R = m. dpdz, we easily find from (2) that (3) Results.

3 Flow in Pipe o f Rectangular Cross-Section. 693 Now m = cm., and A = sq. cm. ; using the relations Q = AS and R = m. dpdz, we easily find from (2) that (3) Results. The results have been divided into two series, and are detailed in Appendix I. Series 1 includes readings taken at gauge holes a and y and the readings of series 2 w^ere taken at (3 and y. In fig. 2 log (RpS2) has been plotted against log (msv). The observed equation for the stream line portion is The coefficient was found by averaging separately the first 10 readings of series 1 and the first five of series 2. In both cases the average was The difference between this value and the theoretical value 2-12 (see equation (3)) is easily accounted for by slight errors in measuring so small a channel. In calculating the values of RpS2 andmsv, p was assumed to be 1 and p was found from the formula p = ( Z2),* where t is measured in degrees Centigrade. Conclusions. 1. Equations (2) and (3) are confirmed by the experiments. 2. The effect of the distance of the gauge-holes from the entrance is wellmarked. Even with an entrance length of 400 m, as is the case with gaugehole [3, the curve shows a departure from the stream line curve as the critical value of msv is approached. The conclusions of Davies and Whitef from a similar result are referred to below. 3. The critical value of msv for pipes of rectangular section is in the same region as that for pipes of circular section. In fig. 2, additional laminar flow curves have been drawn for a square section and for a section of infinite width. Prom the position of these curves one would expect the critical value of msv to increase as the ratio a to b increases. This is borne out by a comparison of the author s results with those of Davies and White. * Hosking, Phil. Mag., vol. 18, p. 262 (1909). t Roy. Soc. Proc., A, vol. 119, p. 96 (1928). 3 b 2

$R. J. Cornish. s V y y y # ' y * i i i i i i.? 0 T T - 1 J 1 *1---- It 1 y - i. ] 11 v f i y : V y& fr \ y h * - y 7 V y > '> ---- 7*7 > c -? 7> < ' ZS dya?$

4 R. J. Cornish. s V y y y # ' y * i i i i i i.? 0 T T - 1 J 1 *1---- It 1 y - i. ] 11 v f i y : V y& fr \ y h * - y 7 V y > '> *7 > c -? 7> < ' ZS dya?<n r $ C \i O C O * 9 + > \ V) E ICVJ IC \I K \ J ICO,c O, n ^ l c r o * * * Series 1. -f Series 2. Dotted lines show Stanton and Pannell s Curves for Drawn Brass Pipes. Chain-dotted lines l ~ o d0 *-g <D 1 g) PH s 0 0C2 1 TS fl ce eg Q *o 'w

Flow in Pipe o f Rectangular Cross-Section. 695 4.

5 Flow in Pipe o f Rectangular Cross-Section When turbulent flow is fully established, the curve connecting RpS2 and msv approximately coincides with that for drawn brass pipes ;* this is in agreement with the work of many investigators on smooth channels and pipes of various sections. Note. The conclusions of Davies and White referred to above appear to the writer to call for some discussion. They deduce a third critical value of msv, below which eddies cannot be transmitted to, or exist in, the channel. f There is mathematical evidence for believing that there exists a critical velocity below which every disturbance must automatically decrease,and above which it is possible to prescribe a disturbance which will increase for a time.7 J This, of course, is a very different critical velocity from that found by Davies and White, and it could not be detected by their method, the essence of which is that below a certain value of msv a gauge point just inside the entrance wxould register steady flow under all conditions. The experimental evidence which leads them to their conclusion is shown in figs. 3 and 4 of their paper. The present writer has been led from his own work on flow through narrow passages to consider that the dotted curves of their fig. 3 are tangential, or perhaps asymptotic, to the curve for Fl([iv[d), and the points of their fig. 3 would fit in quite well with this view. If this opinion is correct, it would seem to be difficult to find definite values from which to draw their fig. 4. * Stanton and Pannell, loc. cit. f croy. Soc. Proc., A, vol. 119, p. 99 (1928). X Orr, 6Proc. Roy. Irish Acad., A, vol. 27, p. 77 (1907).

696 R. J. Cornish. A p p e n d ix I. Series 1. Temperature F. S. dp. R. ms v. RpS2. 5 7 1 5-28 0-400 cms.

6 696 R. J. Cornish. A p p e n d ix I. Series 1. Temperature F. S. dp. R. ms v. RpS cms. water ,, , * * * ,, * * * * * * * * ,, 17* * * ins. water 31* cms. water * *54,, * ins. water 51* * * *33 ins. Hg * * * ,, * ,, * ,, * * ,, *

Flow in Pipe o f Rectangular Cross-Section. 697 Series 2. Temperature F. S. dp. R. msv. RpS2. 55-7 7-17 0-297 cms.

7 Flow in Pipe o f Rectangular Cross-Section. 697 Series 2. Temperature F. S. dp. R. msv. RpS cms. water »> >> j) * * S * ins. water * cms. water J y ins. water ii cms. water ii * ii ins. water cms. water ins. water yy yy yy * y y yy yy y y ins. Hg y y yy Appen d ix II. Theoretical Formula for Stream Line Flow. Take the origin of co-ordinates at the centre of the cross-section, Ox parallel to the longer side, 0y parallel to the shorter side, and 0 z parallel to the axis of the pipe. (See fig. 3.) The general equations of motion reduce to dp dx = 0, dp _ dy A y B * b0 a * b 0 C Fig. 3. 0, w

8 698 R. J. Cornish. The first two equations show that the pressure is constant over the section. Let t = and w = y + t ( h 2 +?2). Substituting in (6), we have The object of introducing y is to simplify the boundary conditions. Along the boundary <»> Hence, along AB, CD, and along AD, BC x + T X = 0, ( 7 ) x = T (62 (8) From (7) all the terms in y must vanish when y = ± b. satisfied by terms such as (2n + 1) izy 7] COS 2b 9 where 7}is a function of x only and n is an integer. Substitute y tj cos my in (5). Then whence By symmetry about 0 d2, 8a;2 m2yj = 0, 7) = Ancosh mx + B sinh 7) = A cosh mx, y,bk= 0 ; hence This condition is and 7 consists of terms like A cosh mx cos my, where m = (2 + 1)-12b. These terms must now be made to satisfy the other boundary condition (equation (8)). For convenience, let Then (9) ( 10) ( 11) y = 260tc. (12) X 2 Ab cosh ~ cos (2 + 1)6. 2b Substituting from (12) in (8), we have X = t. 4 (62 t-2)tc2. (13) (14)

$Flow in Pipe o f Rectangular Cross-Section. 699 This must agree with (17) when x = + a, i.e., with + 00 a + 1 ) r:0 0 i \ a X 2 cosh ----- - -...cos (2w + 1) 6.$

9 Flow in Pipe o f Rectangular Cross-Section. 699 This must agree with (17) when x = + a, i.e., with + 00 a + 1 ) r:0 0 i \ a X 2 cosh cos (2w + 1) 6. n=q 2 6 (15) Expanding (14) in a Fourier s Series, we have 32 t62 X cos 6. cos 36 + ^. cos 56...j. (16) Comparing coefficients in (15) and (16), we obtain A0, Av etc., whence we find _ 32 t62 (cosh (to26) tuy cosh (3to26) 3ruy \ n - t:3 Icosh ++26) 2b 33 cosh (3tc+26) * 26 Hence... _ 32t62 fcosh (tct26) cq;: izy 1 cosh (3to26) 3m, ) 7^ [cosh (to7:26) ' cosh (3raz26) " 26 *"j The total flow is given by Q = +b rx=+a w dx dy. J y= b J x a + r (6*-<*). (18) The integration of the terms in (18) is a straightforward matter, and gives finally ^ ^ f m 6 ^ 1 t o \l * 5 IX dz { TZ6 a \ This formula is substantially equivalent to that for the torsion of a rod of rectangular section; see, for example, Prescott s Applied Elasticity, p. 148 (1924), where approximate methods of summing the series for particular cases are given. Two Special Cases. (i) a b. It can easily be shown by differentiation of equation (24) that, for a given area, the square is the form of rectangular section giving the maximum flow for a given pressure difference. To find the flow in this case, put a = b in (20). Then (19) Q = (21) [i. dz Since Q = 4a2S, a = m, and R = m. djijdz, we get

700 Flo w in Pipe o f Rectangular Cross-Section. (ii) a infinite ; 6 finite. When a is large, (24) becomes approximately whence Q S il 4 ah ab3 {* 0-630-}, aj - >,» 3 ~ dp. - - as a -* oo.

10 700 Flo w in Pipe o f Rectangular Cross-Section. (ii) a infinite ; 6 finite. When a is large, (24) becomes approximately whence Q S il 4 ah ab3 {* }, aj - >,» 3 ~ dp. - - as a -* oo. dz But m = abj(a + 6) &as a -> o. We therefore arrive at the standard formula for flow between two infinite parallel plates, viz., m2 dp mr (25) or R _ o V ps2~ ms# FZow m Opew Channels. In the case of laminar flow in an open rectangular channel of width 2a and depth 6, the velocity gradient perpendicular to the surface is zero, if the air resistance be neglected. The flow is therefore one-half that in a pipe of rectangular section, 2a by 26, and the formula is (23) (24) (26) Q = 2 of dp U 3 rjt dz l 192 b_ tz5 * a t-anh H + ±tanh 87za (27) The author acknowledges with thanks the valuable assistance he has obtained from Dr. Prescott s Applied Elasticity, and from Prof. E. A. Milne, F.R.S., of Manchester University, in working out the above proofs.

LAMINAR FLOW BEHIND A TWO-DIMENSIONAL GRID

LAMINAR FLOW BEHIND A TWO-DIMENSIONAL GRID BY L. I. G. KOVASZNAY Communicated by Sir GEOFFREY TAYLOR Received 20 May 1947 INTRODUCTION The production of a 'wake' behind solid bodies has been treated by