Fluid Dynamics Research 12 (1993) 163-171

North-Holland

FLUID DYNAMICS RESEARCH

Asymptotic analysis of turbulent Couette flow

Noor Afzal Depurlment qf Mechanical Engineering. Aligarh Muslim Unioersir~, Aligarh - 202002, India

Received 5 November 1992

1st Revised version received 7 January 1993

2nd Revised version received 30 March 1993

Abstract. This paper deals with asymptotic analysis of turbulent Couette flow at large Reynolds number using

open equations of mean motion. The flow is divided into three layers (central core region, inner region I near

stationary wall and inner region 11 near moving wall) and asymptotic expansions are matched in two overlapping

domains. It is shown that the velocity at the center line is one half of the velocity of the moving wall. In the regions

of stationary and moving walls the relative velocity obeys a logarithmic law with universal constants. Asymptotic

analysis of turbulent kinetic energy shows algebraic rather than logarithmic behaviour in the overlap region. It is

shown that the predictions of the asymptotic theory compare well with the measurements.

1. Introduction

It is well known that in the asymptotic theory of turbulent flow based on open equations, the matching requirement leads to a certain functional relationship (Narasimha, 1977), the solution of which involves some unknown constants that can be determined either from measurements or by some kind of closure model. These functional relationships, which are independent of any closure hypothesis, are of great practical significance and must be satisfied by a closure model of turbulence. In modeling of turbulent flows, however, these relationships are generally not very well appreciated and as a result many existing models for turbulent flow do not satisfy the constraints of the asymptotic theory.

Fully developed turbulent Couette flow (upper plate moving with velocity U) was studied by Gersten (1985a,b, 1987) using the method of matched asymptotic expansions at large Reynolds number. The following points that arise out of these works need further consideration.

(i) Earlier works (Gersten, 1985a,b, 1987) presume that the center line velocity U, = U/2, which should rather be a deduction from the asymptotic theory.

(ii) In matching the velocity profile the classical Millikan’s argument is not invoked. Matching in the overlap region is rather based on certain adhoc assumptions (see relation 22 and 26 (Gersten, 1985a)) to obtain logarithmic laws.

(iii) The asymptotic analysis of turbulent kinetic energy A, and the matching of inner (viscous wall layer) and outer (fully developed core) layer results in a logarithmic behavior of A, similar to the well-known logarithmic laws for mean velocity profile. Here also the matching is based on adhoc assumptions (relations 22 and 26 of (Gersten, 1985a)) and Millikan’s argument is not invoked. Further, Schneider (Schneider, 1989; Schneider et al., 1990) has shown that the data do not support the theory of Gersten. He further speculates that how logarithmic behavior of A, in

Correspondence to: Dr. N. Afzal, Department of Mechanical Engineering, Aligarh Muslim University, Aligarh - 202002, India.

0169-5983/93/$03.25 a 1993 - The Japan Society of Fluid Mechanics, All rights reserved

the overlap region, could be in accordance with the matching conditions and suspected the coefficient of the logarithm in the asymptotic expansion of A to be zero.

The present paper is an attempt to provide a further insight into the asymptotic structure of mean velocity and turbulent kinetic energy and to clarify the matching process. In particular, it addresses the three points mentioned earlier. Asymptotic analysis of turbulent flows has been studied by many workers, and Gersten (1987) has listed many references. Apart from this asymptotic analysis of turbulent CouetteePoiseuille flow based on eddy viscosity closure has been carried out by Lund and Bush (1980) while Afzal (1976, 1982) has studied plane Poiseuille flow

using open equations.

2. Velocity profile

Fully developed mean turbulent flow of a viscous incompressible fluid between two parallel plates (distance 2h) where the lower plate is stationary and the upper plate moves with velocity U. as shown in fig. 1, is governed by

y = 0, u=t=o. (2)

y = 2k, u = u, 5 = 0. (3)

Here y is the normal coordinate measured from the lower stationary plate, u is the velocity in the direction of the plate, r is the appropriate Reynolds stress, 11 is the density and \I is the molecular kinematic viscosity of the fluid. Let U, be the center line velocity at J = k. The appropriate Reynolds number R, is

R, = u,k/v, u, = (~,/p)“~,

where u, is the friction velocity.

(4)

Asymptotic analysis of eq. (I) in the limit FZ = l/R, + 0 is carried out. The flow field is divided into three regions where appropriate variables and corresponding limits are given below:

(i) A fully developed central core (outer) region, defined by limit J_ = J’/k fixed for 1: -+ 0. (ii) Inner region I in the neighborhood of the stationary plate, defined by limit I’+ = JU,/V fixed

for E -+ 0. (iii) Inner region II in the neighborhood of the moving plate, defined by limit

Y+ = (2k - y)u,/v fixed for E -+ 0.

Fig. I. Turbulent Couette flow

N. A&al 1 Turbulent Couette pow 165

In the three regions (outer, inner I and inner II) appropriate asymptotic expansions for velocity and Reynolds stress along with limiting equations are summarized in table 1.

The matching requirement of the asymptotic expansions in the outer and inner regions I, described in table 1, leads to the following functional equation

IJCIU, -.f1 (Y- -0) - F1 (Y+ + a). (5)

As E + 0, UC/u, + cc the left-hand side approaching to infinity and thus the right-hand side must diverge at large y+. Using Millikan’s argument (Narasimha, 1977; Afzal, 1976) we differentiate with respect to Y to get a functional equation of variable separable type

Y+ aF,laY+ - - Y- afilay-, (6)

as y+ + w and y- + 0 and its solution is

F1 = k-‘lnY+ + B, y+ + a,

f,=-kk’lny_+D, y-+0,

(7a)

(7b)

where k is the Von Karman constant. The matching of velocity (5) using relations (7) we get

UC/u, = kpl In R, + B + D. (8)

Further, the matching requirement of the asymptotic expansions in the outer and inner regions II, also described in table 1, leads to the following functional equation

UC-u& (y--+2)-U--u,Fi(Y++co). (9)

Using the procedure analogous to (5), the solution to eq. (9) is

Fi =k-‘lnY+ +B,, y+ +c=, (10)

fi=k-‘ln(2-Y-)-D,, y-+2 (11)

Table 1

Summary of relations from asymptotic analysis of velocity and Reynolds stress distributions in the three layers for plane

turbulent Couette flow

Outer region

(core region)

Inner region I

(stationary wall)

Inner region II

(moving wall)

(1) Independent variable

.vm = y/h

(2) Appropriate limits

y_ fixed&-O

(3) Velocity and Reynolds stress

(U, - m/n, =f(Y-)

zlr, = S(Y-)

(4) Equation

g=l--ES

(5) Asymptotic expansions

f=fi(Y-) + sf2W) + “.

9 = g*(Y-) + Q*(Y-) + “’

(6) Lowest order equation

Y1 = 1

y+ = yu,,‘v

y+ fixeds+0

U/UT = F(Y+) T/L = G(Y+)

Fy,, + G = 1

F = F,(y+) + EF*(Y+) + “’

G = G,(Y+) + ~Gz(y+) +

F,,., + G, = 1

Y, = (2h - y)uJv

Y, fixed c+O

(cl - u)/u, = F( Y,)

A = qy+)

9,, +9= 1

3 = P-,(Y+) + EFz(Y+) + “’

Q = 9,(Y+) + E&(Y+) + .”

Ply. + 91 = 1

and (U - lJ,)/u, = K’lnR, + B, + II,,,.

Eliminating R, between relations (8) and (12) we get a relation

2U,/U = 1 + (B + D ~ B, - D,,)u,;‘C’

(12)

(131

3. Turbulent kinetic energy

The equation governing the transfer of turbulent kinetic energy of fluctuations in non- dimensional form is given by

5 d u dB d’(il + .I) o

T, dYlc, 11 ~ TV + i: ~--d = . (14)

productIon dl,,lp‘ltlon turhulenl .~nd VI\COLI?. d~ltu\wn

where

42 = (u” + 1’2 + W’Z), d = hc:,,!u; = DC.

A = +q’/u;. J = (~~“)/~$ (15)

B = (($4’ + p’/p)r’)/u;, Y = y/h.

Here, A is the non-dimensional turbulent kinetic energy. C, is the turbulent dissipation and ( J denotes the ensemble average.

Analysis of eq. (14), using the results of the mean velocity profile described in table I, is carried out in three layers (outer, inner I and inner II) for t: 4 0. Analysis of eq. (14) and the asymptotic

expansions in the three layers are given below:

(i) Outer luyrr. Equation:

Expansions:

d = Ir,(.L) + EC/,(?._) + ... .

B = h,(~j_) + r-:h,(~‘_) + . . . .

A = O”()._) + CU,(&) + “..

J =.jo(ym 1 + r:j,(ym) + ..‘.

(ii) lnnev layer 1. Equation:

(17a)

(1%)

(17C)

(17d)

+D-$+&+J)=~. (18) .+ .+ Expansions:

D = D,,(y+) + ED, + . . . .

B = B,(y+) + EB~(_v+) + ‘.. ,

A = A,(y+ 1 + Cal + “. ,

J = J,,(y+) + cJ1(_i*+) + ....

( 19a)

(1%)

(19c)

(19d)

N. A&al / Turbulent Couetteflow 167

(iii) Inner layer II. Equation:

ydP-

dY+ D + $ + ddy2L -(A + J) = 0.

+ + (20)

Expansions:

D = 9,,(Y+) + EQ~(Y+) + ..., (214

B = &(Y+) + &aq(Y+) + “‘) (21’4

A = do(Y+) + Edl(Y+) + . . . . (21c)

J = $o(Y+) + Ef1(Y+) + “‘. (214

(iv) Matching. The matching of outer and inner I region to the lowest order requires

&, (Y- -0) _ E-‘DC, (Y+ + ~01, (224

bo b-+0)-&1 (Y+ -+ ~), GW

a0 (Y - + 0) - A0 (Y + + a), (2W

.io (Y- -0) - Jo (Y+ -+ =Jo). (224

As E -+ 0, the right side of (22a) approaches infinity and the matching requirement demands that the left-hand side must be unbounded. Using Millikan’s argument we get

d, - ~o/Y~ 3 as y_ +O, (23a)

D - aoly+, as y++co. (23b)

Regarding B, A and J, both sides of (22b), (22~) and (22d) are of order unity. Thus to the lowest order B and A must approach to constants as

bo (Y- -0) = BO (Y+ + ~0) = eo, (244

a0 (Y- -0) = A0 (Y+ + ~0) = PO, WW

j, (y- -0) = Jo (y+ + co) = co. (24~)

Gersten (1985a, 1987) in his work has proposed logarithmic behavior of A in the overlap region. Moreover, from analysis of the data Schneider (1989) suspected that the coefficient of the logarithmic term in the asymptotic expansion for A was zero. To gain a further insight into the structure of A, in the overlap region, it is instructive to consider Millikan’s argument and formally differentiate relation (22~) with respect to y to get

Y+ aAolay+ - y_ aao/ay_. (25)

Integration of the relation can be expressed as

A0 = al1 lny+ + fi12. y+ -, co, (26a)

a0 = cr,,lny- + PII, y- -+O (26b)

and matching of A (22~) requires

~lIln~,+BI~-B~I=O.

Using skin friction law (8), In R, can be eliminated to get

a~~kUc/n, - XII@ + D) + BIZ - PII = 0.

(27)

(28)

As E + 0 and U,/U, + ~8, the first term on the left-hand side becomes unbounded, whereas the second and third terms are of order unity, and matching therefore demands that

@ll = 0, PI2 = PII. (29)

The same conclusion could have been derived directly from relations (27) if we note that R, 4 x,

In R, -+ ~ ‘cc and therefore X~, = 0. Thus to the lowest order matching shows that A approaches a constant value rather than logarithmic behavior (Gersten, 1985a. 1987) in the overlap region. It is instructive to consider higher-order matching requirement of A as

q, (_L -0) + wI (y_ -0) - A. (y+ + x) + LA, (y+ + x’). (30)

The lowest-order relation (24b) can be written as

uO(y_) = [&, as J_ +O, (3la)

A,(y+) = [&,, as .r+ + X. (31b)

where J_ = I:Y+. In order to match the higher-order terms it is essential to consider what Kaplun (1967) calls “switch back”, that is, to say that in trying to find terms of certain order, one is forced to reconsider lower-order terms. Extension of Millikan’s argument to match higher-order terms was considered by Afzal (1976) and the only assumption needed was the order of higher-order terms. As the next order terms in (3 1) are of order c, following (Afzal, 1976) it necessarily follows that higher-order terms in (3 1) as J -+ 0 and _V + + X, as a consequence of switch-back, should be

in integral powers of y_ and l/y+ and relation (3 1) becomes

ao(y_)-/j” +cc,_L_ + “‘, as !‘_ -0. (32a)

&(y+ 1 - PO + T,/Y+ + t’.. &S y + + ‘X (32b)

Based on (32) the matching relation (30) reduces to

(I,(?._) - ‘r’,/J’~_ 4 A,(.v+) - %I?‘+ (33)

and the matching leads to

u1(.1‘- 1 - ~,/JIL + /j, + a,~‘- + ..‘. as x_ -0. (34a)

A,(p+) - cc,y+ + /I, + ]‘z:~+ + ..., as y+ + X. (34b)

The matching of D, Band J may be carried out and the results are summarized in the next section.

4. Results and discussion

Analysis of velocity distribution in fully developed turbulent Couette flow at large Reynolds

number leads to the following results:

(a) Core region

(U, ~ u)/u, = - li~‘ln~.. + D, J’L -+ 0. (35a)

=k-‘ln(2-y-)-D,,. ~7. 42. (35b)

(b) Stationnr~~ wall region

u/u, = k-‘my+ + B, J’+ + y_. (36)

(c) Moving wall region

(U-u)/U,=k-‘lnY++B,, Y++-*_. (37)

N. AJzal / Turbulent Couette jlow

(d) Skin friction law

U/u, = 2kK’lnR, + B + D + B, + D,,

2UJU = 1 + (B + D - B,,, - D&,/U.

169

(384

Wb)

As R, + c(i, q/U + 0 and above relation reduces to

u&J = l/2.

Furthermore relation (39) becomes exact if

(39)

B = B,, D = D,, (40)

which implies that the law of the wall for the stationary wall (36) and moving wall (37) have the same constants. A comparison of the above relations with the measurements (Leutheusser and Chu, 1971; El Telbany and Reynolds, 1982) near the stationary and moving wall show k = 0.41, B 21 B, N 5.1. In contrast to the wall law, not much attention has been paid to describing the flow in the core region. From the limited results (El Telbany and Reynolds, 1982) it may be inferred that D 2 D, ‘v 1.15. Thus the skin friction law (38a) becomes

U/u, = 2k- ’ In R, + 12.4. (41)

Analysis of turbulent energy equations at large Reynolds number leads to the following results in the overlap domain of core and stationary wall region 1 as:

(1) Dissipation

d + + ~1 + bly_ + ...+ECg+02+... +... ( Y- )

1 D-p+&

CO

b+ a,+-+ . . . , y++rj.

Y+

(2) Kinetic energy

A N po + a1y_ + ... + E(yl/y- + p1 + 22y- + . ..) +

‘4 - /30 + h/Y+ + ... + E(~IY+ + h + h/y+ + . ..) +

(3) Turbulent difusion

as y_ -+O,

, as y_-

(424

(42b)

0, (434

. .

, as y, +a. (43b)

B - e. + soy- + ... + E(C~/JJ_ + b,, + ally_ + ...), as JJ_ +O, (44a)

B h e. + cl/y+ + ... + &(u,y+ + b,, + Q/Y+ + . ..). as y+ -, co. (44b)

The results for overlap region between core region and moving wall region are similar to above except that y_ is to be replaced by 2 - y-

These results may be compared with Gersten (1985a,b, 1987) where logarithmic behavior of A in the overlap domain was proposed. The data of El Telbany and Reynolds, as shown by Schneider (Schneider, 1989; Schneider et al., 1990), do not support a logarithmic region.

It is well-known (Schneider, 1989; Schneider et al., 1990) that the classical k - E model in case of Couette flow fails to predict distribution of kinetic energy of fluctuation. Schneider (1989) modified the turbulent diffusion term by incorporating a diffusion term that takes into account the combined variation of mean velocity and integral length scales, and a reasonable agreement between predictions and experimental data was reported in the bulk of outer layer. An asymptotic

analysis of the model shows (Schneider. 1989; Schneider et al., 1990)

A = C,“(I + K:y+‘+ ‘..). as j‘+ + L.

A - C,‘“(l + K,y+x + ..‘). as j‘_ -0.

where

(45a)

(45b)

(46)

The constants CD = 0. I I. C,, = 3 were determined (Schneider. 1989; Schneider et al., 1990) by comparing the predictions with experimental data. Based on these constants along with I; = 0.4. the value of 3 calculated from (46) gives r = 0.847, whereas present results (43) predict r = I. Furthermore, relation (43) shows that there exists a maximum value of if at

The data (El Telbany and Reynolds, 1982) for A displayed in terms of outer variables ~1 in fig. 9 show that substantial linear regions do in fact exist for J_ + 0. Fitting of data by linear relation as

4.0

3.0

A

0.0 0.2 0.4 0.6 0.8 1.0

Y/h Fig. 2. A comparison of present relation (48) with experimental data (El Telbany and Reynolds. 1982) and predictlons

(Schneider. 1989: Schneider ct al.. 1990) of non-dimensional turbulent kinetic energy in plane Couette Aow.

N. A&l / Turbulent Courtrejo~ 171

proposed by lowest-order outer solution (43a)

A = 5 - 7y_, (48)

agrees well with the data. In the same figure Schneider’s (Schneider, 1989; Schneider et al., 1990) predictions are also displayed which deviate considerably from the data for small y_ . It may be mentioned that one of the reviewers has pointed out to the author a recent work of Gersten (1989) where a constant value of A is proposed in the overlap region using dimensional and similarity

arguments.

5. Conclusions

(a) In turbulent Couette flow the velocity at the center line is one half of the velocity of the

moving plate. (b) The velocity distribution near the stationary and moving sheets satisfies the logarithmic

law of the wall with universal constants. Furthermore, the velocity defect law in appropriate coordinates also exists in two overlapping domains.

(c) The kinetic energy of fluctuations, in the overlap region, shows algebraic behaviour rather

than logarithmic.

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361-372.

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