ELSEVIER Fluid Dynamics Research 17 (1996) 181-194

FLUID DYNAMICS RESEARCH

Turbulent boundary layer on a moving continuous plate

N o o r Afzal

Faculty o)C Engineering, Aligarh Muslim University. Aligarh 202 002, India

Received 29 June 1995; revised 15 November 1995; accepted 28 November 1995

Abstract

The turbulent boundary layer on a moving continuous plate is studied at large Reynolds number. The flow field in the boundary layer is divided into two layers: outer wake layer and inner wall layer. The matching of the velocity profile in the overlap region by the Millikan-Kolmogorov hypothesis leads to logarithmic laws. The relative wall velocity on the moving continuous plate satisfies the law of the wall with universal constants. To the lowest order, the outer wake solution reduces to the Sakiadis equation (analogous to laminar flow) with finite velocity slip and wall shear stress. The wall shear stress Cf and drag coefficient Cv are compared with the data and earlier results.

1. Introduction

In certain industrial processes, the turbulent flow on a moving continuous surface through a quiescent fluid environment has been of interest in design, optimization and control of the thermal processing of moving materials. The heat treatment of material travelling between a feed roll and a wind-up roll on conveyor belts, lamination and melt spinning processes in the extrusion of polymers, all possess the characteristics of a moving continuous surface. The turbulent boundary layer on a moving continuous plate was first considered by Sakiadis (1961), where an approximate solution using a one-seventh power law velocity profile was studied. The experiments of Tsou et al. (1967) for a moving continuous plate displayed the law of the wall, where the wall shear stress was determined by Clauser's plot. Further, the eddy viscosity closure with the Deissler model was employed to integrate the turbulent boundary layer equations to obtain the law of the wall and the wall shear stress. When compared with the values obtained from data by Clausers' plots, the skin friction predictions by Tsou et al. were lower by 6% and that of Sakiadis by 15%. The mixing length model due to Cebeci and Smith has been studied by Moutsoglou and Bhattacharya (1982) but predictions of skin friction were 8% below the data.

The present work deals with the turbulent boundary layer on a moving continuous sheet based on under-determined equations by the method of matched asymptotic expansions at large Reynolds number. The related problem of the turbulent boundary layer on a stationary plate, subjected to a free stream, has been studied by several workers by using the asymptotic methods

0169-5983/96/$15.00 ~) 1996 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved SSD1 0 1 6 9 - 5 9 8 3 ( 9 5 ) 0 0 0 3 2 - 1

182 N. Afzal / Fluid Dynamics Research 17 (1996) 181 194

(Yajnik 1970, Mellor 1972, Afzal 1973) where the velocity defect U~ - u in the outer layer is small. The velocity defect layer and the wall layer were analysed and the aysmptotic expansions were matched to obtain classical log laws. For the flow on a moving continuous plate there is no free stream velocity (U~ = 0) thereby the classical velocity defect limit and Karman defect law do not exist. In the present work, the analysis of the outer wake layer in the wake limit is governed by equations of a nonlinear wake (where inertia terms, Reynolds stress and pressure gradient, if present, are of the same order).

On a stationary plate subjected to a free stream, the wake layer occurred to Afzal (1980, 1983) as an outer layer of the turbulent boundary layer subjected to strong adverse pressure gradient. In the wake layer, the nonlinear inertia terms, the pressure gradient and the Reynolds stress terms are of the same order. In the turbulent boundary layer with moderate pressure gradient, Ponomarev (1978), Sychev and Sychev (1987) and Silva-Freire (1989) regarded the wake layer as an intermediate layer between the Prandtl wall layer and Karman velocity defect layer. Ponomarev assumed that the solution of the intermediate (wake) layer equation can be obtained by direct matching of zero order terms for the wall and the defect layer in the intermediate matching zone (Silva-Freire 1989). Sychev and Sychev matched the intermediate layer with the defect layer to predict singular algebraic behaviour (like y-~ as y --, 0 for ~ ---, 0) and with inner layer to get log laws. Afzal (1995) has shown that the wake layer equation is "rich enough" to contain the equation of the Karman defect layer and therefore Karman's defect layer may be dispensed with. For a shallow wake, the linearization of the outer wake layer equation reduces to the classical Karman velocity defect layer and the works of Coles (1956) and Clauser (1956) are mutually consistent. For a constant eddy viscosity closure (Clauser 1956), the lowest order outer wake layer equation reduces to the well-known Falkner-Skan equation with finite velocity slip and wall shear stress (see also the Appendix in Afzal (1983)).

In the present work on a moving continuous plate, the asymptotic expansions of the outer wake layer and the inner wall layer are analysed and matched by the Millikan-Kolmogorov hypothesis to obtain logarithmic laws. The lowest order outer wake layer equation under constant eddy viscosity closure reduces to the Sakiadis (1961) equations (analogous to laminar flow) on the moving continuous plate. The results for skin friction and drag coefficient are compared with the earlier results.

2. Equations

The turbulent boundary layer equations for the two-dimensional mean motion of an incom- pressible fluid with constant pressure are

~u ~v + - - = 0 , ( 1 ) tgy

U ~ x + V ~ y = V g T + - - ' p 0 y (2)

For a moving continuous plate, the boundary conditions are

y = 0 , u = U, v = z = 0 , (3a)

y / 6 ~ oc u, r ~ O. (3b)

N. Afza l / F lu id Dynamics Research 17 (1996) 1 8 1 - 1 9 4 183

Here x is the coordinate measured from the slit in the direction of plate mot ion and y is the coordinate normal to the plate, u and v are the velocity components in the x and y directions and r is the appropria te Reynolds shear stress. U is the velocity of plate motion, Zw is the shear stress on the plate and c5 is the boundary layer thickness.

The K a r m a n m o m e n t u m integral is

d 0 - Zw 0 = dy. (4) dx p U 2 '

The drag D per unit width of the plate obtained from integration of the wall shear stress is

D = f ] Zw dx = p g 2 0 . (5)

The coefficient of skin friction Cf and drag coefficient Co are given by

dRo Cf = 2 ~ = 2e 2, (6)

2D 2Ro pU2x - CD - Rx ' (7)

where the various Reynolds numbers are defined as

Rx = Ux/v, Ro = UO/v, R~ = u~cS/v, (8a)

2 u~/U, ~r g/u~ (8b) u~ = "Cw/p, ~ = = .

3. Analysis

The turbulent boundary layer on a moving cont inuous plate has been studied for R~ ~ ~ , Cf --, 0. The analysis shows existence of an outer wake limit (y/6 and x /L fixed, g/L = (u~/U) 2, R~ ~ oo) and the wake layer, where non-linear inertia terms and Reynolds stress are of the same order (see also Afzal (1995)). In the wall layer, the relative velocity with respect to the plate U - u is significant where viscous terms and Reynolds stress are of the same order. The asymptot ic expansion for the wake layer and the wall layer are matched in overlap domain.

3.1. Outer wake layer

In terms of outer variables

= U3F(X, Y), r = pu}G(X, r ) , (9)

Y = y/6, X = fo~ L - 1 dx, (10a)

184 N. Afzal / Fluid Dynamics Research 17 (1996) 181 - 194

the wake limit is defined as

6/L = (u~/U) 2 for R~-, o% (10b)

where 6 is the order of the boundary layer thickness and L is the order of streamwise variations of flow in the x direction. Based on the Karman momentum integral (4), L may be taken as L = 0/c a. The turbulent boundary layer equations (1) and (2) in the wake layer reduce to the equation of a non-linear wake

~3G 6,~ 02F OF ~32F ~32F OF 1 ~33F

0----Y + ~ F~-Y 5 + •Y 0Y ~3X (~y2 0X - eRe 0Y 3 ' (1 1)

where Reynolds stress and non-linear inertia terms are of the same order and the effects of viscocity are of higher order. The outer expansions for velocity and Reynolds stress are

u = U [ F ; ( X , Y ) + ~F;(X, Y) + ..-], (12)

= pup[Go(X, Y) + eGI(X, Y) + . . . ] . (13)

The first and second order equations are

G'o + aFoF~ = F~F'ox - F~Fx, (14)

G] + a(FoF~' + F ~ F 1 ) = F'IF6x + F6F~x - Fi'Fox - F~Flx , (15)

Y --* ~ , F; , F~, Go, G1 - , 0, (16)

where

a = bx/b. (17)

Here the prime denotes differentiation with respect to the normal coordinate Y.

3.2. Wall layer

The inner variable for the velocity (the relative velocity with respect to the plate) and Reynolds stress are

( u - u)/u = f ' ( x , y+), = rw0(X, y+), y+ = yu /v, (18)

and the inner limit may be defined as y+ fixed for Re - , oe. Based on inner variables (18), the boundary layer equations (1)-(3a) reduce to

_ f , , , + g , = e e 2 • ~ [--fix + m l ( f ' - - y+f")] + ~ [ f x f " - - f ' J J + m l ( 2 f f ' - - f " ) ] ,

y + - , 0 , f ' - - ,0 , g - , 0 , (19)

where rnl = ~/8. Here the prime denotes differentiation with respect to the normal coordinate y+.

N. .{[~al / Fluid Dynamics Research 17 (1996) 181-194

The inner expansions are

g f = fa (X , y+) 4- ~ r f 2 ( X , y+) + . . . ,

g g = - g l ( X , y + ) - F ~ r g 2 ( X , y + ) + "".

The lowest order equat ion is given by

- f i " + 9] = 0, y+ = 0,

An integral of relations (22) is

+ g l = 1

185

(20)

(21)

where

Us

The

The

y ~ 2 F ° - ~ O as Y ~ 0 , ~y2

02fl 02F1 Y+ @2 - - Y ay---5-.

integration of the first order terms (26b) leads to

f ~ = k - l l n y + + B , y + - ~ m ,

F~ = - k - l ln Y - D, Y ~ O .

matching of velocity profile from relation (24) yields

(U - U~)/u~ = k - 1 In R~ + B - D,

= U F ; ( X , o).

(26a)

(26b)

(27a)

(27b)

(28)

which shows that the viscous stress, the Reynolds stress and the wall shear are of the same order.

3.3. Matching

The matching of the inner wall and outer wake layers of the velocity profile demands

U - u~f{ (X, y+ --, oo) = UF6(X , Y --. O) + u~Fi (X, Y --. 0). (24)

This is a functional equat ion and its solution can be obtained by the Mi l l i kan -Kolmogorov hypothesis, to differentiate it with respect to y to get (Afzal 1976)

02fl U y 632Fo ~2F1 -Y+ @z+ - u ~ ~ + YOY--~ (25)

as y+ --, oc and Y--, 0. As R~---, 0% U/u~--. oo the matching of the lowest and first order terms require

(23)

A = f l = gl = 0. (22)

186 N. A/'zal / Fluid Dynamics Research 17 (1996) 181-194

The matching of Reynolds shear stress demands

r w g a ( x , y + ~vo) = rw[Oo(X, Y --*0) + 8GI(X, Y ---~ 0 ) ] .

The solution of this equation leads to

g l = 1 + ( k y + ) -1 , y + -'* (it3,

Go = 1, G1 ~ 0 as Y ~ 0 .

(29)

(30)

(31)

4. Other results

4. l. Outer layer

The lowest order outer wake layer equation (14) subjected to matching and boundary conditions (26) and (16) may be expressed as

G'o + aFoF~ = F~F6x - F~Fox , (32)

Y ~ 0, Fo ~ O, YF[~ ~ 0, Go ~ 1, (33a)

Y ~ ~,, F6 -~ 0, Go ~ 0. (33b)

In view of Clauser's (1956) constant eddy viscosity closure, it is assumed that

r = v, Su/Sy, v~ = a U 8 I ,

(~ ~- a a l ( x - - Xo) , a l = ~1/(~

for 82/0( fixed, and the relations for Fo and Go reduce to

t / t !

Go = aFo, al Fo = - Go. o(

Based on relations (35), the wake equations (32) and (33) become

t t t t ! " ! t t t

Fo + FoFo = 6(FoFo6 - FoF6) ,

;o Fo(~, O) = O, f"((3, O) f'o d Y = _~2/~ ,

If the outer layer is in equilibrium

Fo((~, Y ) = fih(~l), Y = rl/fl,

then Eqs. (36) and (37) reduce to

h'" + h h " = O ,

h(O) = O, h'(O) = 1, h'(oo) = O,

F ; ( 6 , = o.

(34a)

(34b)

(35)

(36)

(37a, b,c)

(38)

(39)

(40 a, b, c)

N. Afzal / Fluid Dynamics Research 17 (1996) 181-194 187

and Eq. (37b) becomes

~2 = e(~m)- z/z, m = -h"(O)h(~) . (41)

Eqs. (39) and (40) are analogus to equat ions of Sakiadis (1961) for laminar flow on the moving cont inuous plate, where

h"(O) = -0.62755, h(oc) = 1.14272, m = 0.7172. (42)

In conclusion, the solution to lowest order wake layer equat ions is given by

Fo(& Y) = ~h(rl), Y = rl/3, (43a)

f i e = e(~m) -1/2, m = 0.7172. (43b)

The slip velocity Us on the wall, to the lowest order is

Us = UF6(X, 0), (44a)

F~(X, 0) = e(c~m)- 1/2 (44b)

The skin friction relation (28) in view of relation (44) becomes

U/u~ = k - 1 In Re + B + C, (45)

C = 27r/k = (am) a/2 _ D. (46)

4.2. Uniformly valid solutions

For the velocity profile, the uniformly valid solution is

u 1 + e [ - f ~ ( y + ) + (mc~)-X/Zh'(flY) + F~(X, Y)] Uc, (47) U

where Uo is the c o m m o n part

U~ = 1 - e ( k - 1 In y + + B ) , (48a)

U o=~[ (~ m) 1 / 2 _ k - l l n Y - D ] . (48b)

For Reynolds stress, the uniformly valid solution is

z = Zw[y(X,y+) + Go(X, Y) + eGI(X, Y) - 1]. (49)

The velocity profile (47) using (48b) may be expressed as

(U - u)/u~ = f i ( Y + ) + ½(me0-1/2wl(X, Y) - ½ Dw2(X, Y),

wl(X, Y) = 2 [ - h ' ( f l Y ) + 1], w2(X, Y) = 2(F~ + k -11n Y + D)/D.

(50)

(51a)

The two wake functions wa(X, Y) and w2(X, Y) would be needed, which may be zero at Y = 0 and two at Y = 1. These functions may have similar qualitative behaviour and to the lowest order

188 N. Afzal / Fluid Dynamics Research 17 (1996) 181-194

the two wake functions may be presumed equal

Wl(X, Y) = wz(X, Y) = W(X, Y). (51b)

The composite velocity profile may be expressed as

(U - u)/u~ = f i ( y + ) + ½CW(X, Y). (52)

For y+ > y* the expression for the velocity profile may be written as

( g - u)/ur = k -1 In y+ -~- B -~- 1CW (X, Y ) , (53)

u/u~ = - k - ' In Y + C[1 - 1 W ( X , Y)], (54)

where

W(X, O) = O, W(X, 1) = 2. (55)

Further, the additional boundary conditions at Y = 0 and Y = 1 can be obtained. The conditions for Y = 1 are

1 1 W'(X, 1) = - - , W"(X, 1) = - . [56)

717

The wake function may be expressed as

1 W(X, Y) = Wo(X, Y) + - Wl(X, Y) + ... (57)

In the present work, the wake function which satisfies the boundary conditions (55) and (56a) is adopted (Granville 1975; Afzal 1995)

_l(y2 W(X, Y) = 6Y 2 - 4Y 3 + _ y3). r~

4.3. Skin friction and drag

An integral of Karman momentum integral (4) leads to

R x = f f Z R o - - 2 f ] aRo da.

The momentum Reynolds number Ro may be expressed as

Ro = UO/v = R~I2/a,

where

12 --- (U/Ur) 2 d Y .

(58)

(59)

(60a)

(60b)

N. Afzal / Fluid Dynamics Research 17 (1996) 181-194 189

Using the composi te velocity profile

u , v y* (61a) --=0---fl(Y+),u~ 0<~; <~ r--~

y~ y = - k - l l n Y + C [ 1 - ½ W ( X , Y ) ] , ~ < ~ < 1, (61b)

the integral 12 may be est imated as

';? f/ = --f l (Y+)) dy+ + ,:/R~ 12 - ~ (0- ' 2 (u/uO 2 d Y

1 = ~ (221o- - 22) + b, (62)

where

213 13 b - 4 8 1 9 A 2 + A C + C 2, A = Ilk, (63) 2520

21 = (A l n y * + B - A ) y * - f ~ ( y + ) d y + , (64a)

2 * I~* 22 = [(A In y* + B) 2 - 2 A ( A In y* + B) + 2A ]y+ - f~2(y+)dy+. (64b)

The expression (62) for R0 becomes

Ro = (bR~ + 2)ola - 22)/0-, (65)

R~ = exp[k(a - B - C)] . (66)

Based on the integral relation (59) Rx can be estimated

Rx = bR~[a - 2A(1 -- e-a/A)] + 220". (67)

The relation (7) for drag coefficient Co based on (65) and (67) becomes

2e(ebR~ + 221 - 22e) Co = bRo i l - 2Ae(1 - e-k/~)] + 22 ' (68)

Cf = 2g 2. (69)

For a given o-, the functions R~ and Rx may be obtained from relations (66) and (67) to obtain Cf and CD.

5. Results and discussion

The main results for velocity profile in the turbulent layer on a moving cont inuous plate are given below:

190

Wall layer:

( u - u)

Ur

N. AJ}al / Fluid Dynamics Research 17 (1996) 181-194

- - = , f ' ~ ( y + ) + O(e/RO

= k - l l n y + + B , y+-- ,oc.

Wake layer

u = UF;(X, Y) + u~Fi(X, Y) + .. . ,

(Us - u)/u~ =- k - l ln Y + D, Y ~ O.

Skin friction law

(70a)

(70b)

(71a)

(71b)

(U - Us)/u~ = k- 1 In R~ + B - D. (72a)

Here Us = UF;(X, 0) is the slip velocity at the wall due to the lowest order outer wake layer. Using the solution (43) and (44) of the lowest order outer wake layer based on constant eddy

viscosity closure, the wake law (71b) for Y ~ 0 and skin friction relation (72a) become

u / u ~ = ( a m ) - l / 2 - k - l l n Y - D , Y - , 0 , (71c)

U/u r ~--- k - 1 In Re + B + (am)- 1/2 _ D, (72b)

C = 2n/k = (c~m)- t/2 _ D. (73)

The relation (72b) may also be expressed as

Cf = 2 I I n ~ G(A, OI , (74)

A = 2 In R~, E = 2 In (2k) + 2(B + (c~m)- 1/2 _ D), (75a)

where the function G(A, E) is given by (Gersten and Herwig 1992)

A A + 2 In ~ - E = A. (75b)

For A ~ c~ the function G may be expanded as

G - I ( A , E ) = I - 2 1 n A - E ( I _ 2 ) + O ( 1 A A ) A - - . (75c)

The velocity profile above the viscous sublayer is

(U - u)/u~ = k -1 l ny+ + B + ½ C W ( X , Y), (76)

u/u~ = - k -x In Y + C[1 - ½ W ( X , Y)], (77)

N. Afzal / Fluid Dynamics Research 17 (1996) 181-194 191

where

1 W ( X , Y) = Wo(X, Y) + - W x ( X , Y) + .-..

7~ (78)

The measurements of the turbulent boundary layer on a moving continuous plate were reported by Tsou et al. (1967). The measurements were made on a rotating drum (12.6 in. diameter and 24 in. long) fitted with a skimmer plate (12 in. diameter and ~ in. thick) to fix the origin of the boundary layer. The effects of curvature were negligible. The boundary layer was tripped between the second and third station of measurements by a wire (0.006 in. diameter) parallel to the cylinder surface (at 0.003 in. from it). The turbulent velocity profile along with surface speeds at various circumferential stations were measured for Reynolds number Rx from l0 s to 6 x 105. The wall shear was deduced from the Clauser plot. The measurements in wall layer coordinates are displayed in Fig. 1, and the data supports the law of the wall

U - - b / - - k - 1 In y + + B (79)

Ur

with universal constants k = 0.41 and B = 5. The data of Fig. 1 also show that at large y + departure of measurements from wall law (79) is very weak and consequently the constant C in the outer layer is very small. Further, as the universal function f [ (y+) describes the law of the wall, the values of 21 and 22 defined by (64) were estimated by Landweber (Granville 1975):

21 = 32.72, 2z = 434.9. (80)

The skin friction Cf is displayed against Rx in Figs. 2 and 3 for C = 0. Fig. 2 shows that Cf values for data of Tsou et al. obtained from Clauser's plot, are 5% higher than the computed values. The

U - u

u t :

2 0 - -

1 5 - -

1 0 - -

5 - -

0

:•/I STATION U ( m / s ) v No. 10.6 17.4 2.4

7 o • Z B t, -~,

II n ¥

|3 ×

5 10 50 I00 500

yu1:;

Fig. l. Data of Tsou et al. (1967) for a moving continuous plate in wall layer coordinates which supports the law of the wall. The solid line represents the universal relation (U - u)/u~ = k 1 In y+ + B, k = 0.41, B = 5.

192 N. Afzal / Fluid Dynamics Research 17 (1996) 181-194

5 0 x

,.T /4

Present Work

O O

U ( m l s ) ~ ~ l~

10.6 o

17.4 A

24.0 []

[ I ( , , , / , I ,

1 1 5 2 3 4 5 6.10 5 Rx

Fig. 2. Comparison of predictions for wall shear stress between the present work and Clauser plot of the data on a moving continuous plate.

k \ \ \ Present "I i %. " \ . . . I Continuous 5iN- % Isou e m l t ~ , k ' ' ,N "" Sakiodis J ~urTace F \ . ' ~ \ \ Flat Plate

1

0 I I I L I __L I 10 5 10 6 10 7 10 8

Rx= Ux/~'

Fig. 3. Comparison of theoretical predictions for wall shear stress Cf on a moving continuous plate.

estimate of Cf from Clauser's plot was based on wall law intercept B = 4.75 (Tsou et al. 1967, p. 231). If the universal wall law intercept B = 5 is adopted then the revised Cf values would be more appropriate, which also improve the comparison with theory. For a wide range of Reynolds number R~, the wall shear stress Cf, displayed in Fig. 3, is compared with earlier computations.

N. A[zal / Fluid Dynamics Research 17 (1996) 181-194 193

4

1 10 5

\

~'\ \ . . . . . / Surface .,~.k~X ~ ~ ~. Flat plate

"~'~..~ ~

10 6 10 7 10 8 Rx: Ux/ZJ

Fig. 4. Comparison of theoretical predictions for drag coefficient CD for a moving continuous plate.

The predictions of Tsou et al. are about 1%, and Sakiadis power law analysis are 8% below the present findings. Further, Cf for a stationary plate is also displayed in Fig. 3. It may be seen that the result for a stationary plate is higher than for a moving continuous plate. This is in contrast to the laminar flow in the boundary layer where the friction coefficient Cf for a moving continuous plate is higher than for the stationary flat plate.

The drag coefficient CD for the moving continuous plate is displayed in Fig. 4. The comparison with the prediction of Tsou et al. and Sakiadis for a moving continuous plate and CD for a stationary flat plate are also displayed. The trends are similar to those already discussed in connection with the results for the local friction coefficient.

Acknowledgement

It is a pleasure to thank Dr. S. Rani for constant encouragement and support.

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194 N. Afzal / Fluid Dynamics Research 17 (1996) 181-194

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