Analysis of a turbulent boundary layer subjected to a strong adverse pressure gradient

0020-7225/83106056)_l4$0J.lW0 @ 1983 Pergamon Press Ltd.

hf. J. Engng Sri. Vol 21. No. 6. PP. 563-576, 1983 Printed in Great Britain

ANALYSIS OF A TURBULENT BOUNDARY LAYER SUBJECTED TO A STRONG ADVERSE PRESSURE GRADIENT

Department

NOOR AFZAL of Mechanical Engineering, Aligarh Muslim University, Aligarh-202001,India

Abstract-We define two non-dimensional parameters A = T,JP~Sand R, = &S/v where T+ is the wall stress, p,(>O)is the pressure gradient to which the turbulent boundary layer (of thickness 6) is subjected, Y is the kinematic viscosity, U, = (v~dp)“~ is a characteristic velocity and p is the density. The limit corresponding to the strong adverse pressure gradient is formulated as A+O, Rp +m, AR, finite. Using appropriate inner and outer asymptotic expansions, both above a wall layer possibly scaling with 7wand v, it is found by an application of Millikan’sargument that there is an inertial sublayer where the streamwise velocity distribution obeys a half-power law, whose slope depends on A, and intercept on ARp Indeed comparison with available experimental data shows the inner law to be well represented by u/U, = (3.5t 19A)(yU,/v)“*t 2.5AR,. The outer flow obeys a generalized defect law; use of constant eddy viscosity closure yields results in good agreement with experiment. I. INTRODUCTION

STRATFORD[~] was first to investigate the turbulent boundary layer under the influence of a strong adverse pressure gradient. He through an ad hoc attempt has shown that the velocity distribution, away from sublayer, is governed by half power law. Townsend[21 has studied the self-preservation of a zero stress layer (see also Townsend[3]). Following the work of Stratford and Townsend, various proposals (Perry[4], Perry et aL[5], McDonald [6], Mellor [7], Perry and Schofield [81and Chawla and Tennekes [9]), have been made for the description of the velocity profile in a turblulent boundary layer approaching separation under the influence of strong adverse pressure gradients. Particular attention has been given by all workers to the wall law in the overlap region. None of the proposals made so far is in satisfactory agreement with experiments (Samuel and Joubert [lo]). Very little attention has been paid to describing the flow in the outer region. Chawla and Tennekes[9] have studied the problem of Stratford, using the method of matched asymptotic expansions. In their analysis these authors have assumed the outer layer to depend on viscosity. The measurements (Simpson et al. [ill) in the turbulent boundary layer under influence of strong adverse pressure gradient, show that the viscosity is not important in the outer layer. The present work is an attack on the problem using asymptotic arguments of a kind that the author has employed in other situations (for example Afzal and Narasimha[l2]. When this work was in an advanced stage, the paper of Kader and Yaglom[l3], using similarity and dimensional arguments, appeared in print. A detailed comparison of the present work with that of Kader and Yaglom shows that while there are some similarities, the present results are different on a variety of points; the author believes that he has achieved a neater and more rational solution to the problem; comparison with experiment shows the present description also to be in better agreement. 2. GOVERNING

EQUATIONS

The governing equations of mean motion for a twb dimensional turbulent boundary layer of an incompressible fluid are

(1) (2) PX = - PUUX, 563 ES Vol. 21. No. &A

(3)

N.AFZAL

564

The boundary conditions are y=o

u=u=T=r=()

y+@J u+U(x),

(4) r,r+o.

(5)

Here x is the streamwise coordinate, y the normal coordinate measured from surface. u and o are the mean velocity components in x and y directions respectively. T and r are the Reynolds shear and normal stresses divided by density of the fluid and v the kinematic viscosity. Further U(x) is the free stream velocity, V,(x) the streamwise velocity gradient and p* the pressure gradient. 3.ANALYSIS

In the present asymptotic analysis the chief non-dimensional parameters are

where U, = (~p,/p)“~ is a characteristic velocity; 8 is the boundary layer thickness and r, is the wall shear. We propose that the limit that is appropriate to the boundary layer subjected to strong adverse pressure gradient [21]

h-+0, &+m,

Oa)

AR, = O(1)

and the stress gradient of the order of pressure gradient

(7b) In what follows we consider the flow above a wall layer scaling possibly with 7wand v. (a) Outer layer An estimate of flow variations in the streamwise direction is both necessary and useful. If I_. is the scale of these variations, we may infer from momentum integral or otherwise that L= -

u/u,.

(8)

The outer length scale in the transverse direction is 8, the boundary layer thickness, and from (7b) it follows that the scale of stress is U,‘, where V, is defined by

u, = (~p~/p)l’* = u(~/~)“*.

(9)

We now consider the following outer variables U,lU dx,

u=Ug$X,Y),

Y = y/8 T =

U,*T(X, Y),

l- =

U,*i"(X, Y)

(10)

and study the limit X, Y fixed for R, + m. In terms of outer variables (10) the boundary layer eqns (l)-(3) reduce to

u, ‘ai;

v

a3F

( 1ax+.2-&P

= u

(11)

Analysis of a turbulent boundary layers

565

The outer boundary conditions are

In the outer layer the effects of normal Reynolds stresses is of the order of (UJU)’ when compared with the inertial and Reynolds shear stress terms, where as effects of viscosity, of order v/S’&, are quite insignificant at large Reynolds numbers. The outer asymptotic expansions are F = FO(X,Y) + A,F,(X, Y) + O(A,) T = To(X, Y) + AzT,(X, Y) + O(AJ

(13)

i; = r,(x, Y) + O(1) where Ai and AZare unknown gauge functions to be determined from matching with inner layer. The equation satisfying the leading terms in expansions (13) is

aFo a2Fo aFo a2Fo _

---+-7-o

aY axaY ax aY

(14)

and the outer boundary conditions are (15) The equation governing the next approximation with A, = A2are

aF, a*F, +ZP-S

aF1 a2Fo aFoa2F1_ -+-7-0[(+~/A,]. axaY ax aY

(16)

(b) Inner layer We consider the following inner variables (17)

and study the limit 6, 5 fixed as R, +w, Here Z(x) is the scale of streamwise flow variations in the inner region, smaller than that in the outer layer L. The boundary layer eqns (l)-(3), after integrating once with respect to 5, may be written as

(18) The inner expansions are f =

f1& 0 + 9f*(5,5) + O(d

f

=

d&5)

f

=

At,

+

E272(5,0

5) + O(1)

+ O(4

(19)

N. AFZAL

566

where ll and E?are the gauge functions. In view of the fact that R, + m, A --) 0, AR, = O(1) the leading terms of inner expansions satisfy the following equation

q+&+hR al

P+o

(v 1 up6p .

(20)

In the inner layer the viscous, Reynolds shear stress and pressure gradient terms are of equal order, where as the normal Reynolds stresses are of order v/U,~?. If the scale of streamwise flow variations L!?is chosen to be of order v/U, then the normal stress also becomes of same order as shear stress. (c) Matching

We now match the inner and outer layers following the arguments of Millikan[ 141and Afzal and Narasimha[l2]. Using the matching principle (Afzal[21]) the outer limit of inner solution equal to the inner limit of outer solution, the matching requirement for streamwise velocity in the overlap region is lim U %-

lim U

(21)

,The above relation can also be written as functional equation

(22) the first term on right hand side approaches infinity and matching As Rp-, UlU, +q requires the function f, on 1.h.s. be unbounded at large 5, say like S(b). The functional eqn (22) can be differentiated with respect to y (note that the procedure adopted would be invalid if 9([ + m) did not diverge; see Afzal and Narasimha[l2]) to get (23) At large Reynolds numbers U/U, is large and the matching to the lowest order requires

d(Y@+(0) as

Y-+0.

(24)

If we choose UAJU, to be of order unity, say A, = UJU

(25)

then the matching condition (23) reduces to lim d(l) aZfl = lim d( Y) 3. 852 yA f-

(26)

The matching condition (26) implies that each side of it approaches a constant, say A/2, independent of Y and 4’(but could depend on a variety of parameters in the problem-this does not affect the argument) so that

Wa) Wb)

Analysis of a turbulent boundary layers

567

where C and D are functions of integrations independent of S and Y. respectively, but could depend on the parameter of the problem. On matching the streamwise velocity we get a relation

(27~) The matching of normal velocity component leads to

Fo(X, 0)= 0, F,(X, 0) = 0.

(28)

The matching of Reynolds stress in inner and outer layers leads to functional equation

As R, + ~0,here again the matching requires r1 be unbounded at large 4. Differentiating it once with respect to Y leads to

(2%) The solution to above functional equation is

where a20 and a30 are functions of integration. The matching of the shear stress requires

Similarly the normal Reynolds stress r satisfies the functional relation

whose solution is

ro=a3,+a,,y

Y-*0

WI

with Q31 =

wpDJ*h.

(32~)

Substituting the matching relations (30) and (32) in the inner equation (18) we get ax)=

AR,, aa= A

a,o=l+O

(& 1. P

(33)

In the matching procedure described above the dependence of the constants A, C and D in the relations (27) remains hidden on the parameters of the problem. The parameter A appears both in the inner and outer equations, so that A = A(A).

(34a)

N. AFZAL

568

The inner equation involves a parameter A$, implying

c = C(AR&J.

Wb)

The outer layer equations involves a variety of parameters i.e. A, US,/U,S, x&/U and history of the motion, therefore D could, in general depend on these parameters i.e.

4. COMPARISON

WITH EXPERIMENTS

The main results for the streamwise velocity profile are

u/U~= Am

-I-CfA&Jt 5-+”

(35)

u/U = F&(x, 0) + (U$ U)[A(A)V’( Y) + D], Y + 0

(36)

along with the relation F&(X,0)=

2

C(AR,)

-$

D

(37)

Equations (35) and (36) show that there is an overlap region governed by a half-power law whose slope A could depend on A. The intercept in the inner law could be a function of the parameter AR,. The outer law has an intercept that includes the parameter Fi,(X, 0), the velocity at the wall of the zeroth order outer wake flow, to be determined (see Appendix). An extensive comparison of the present theory with all availabIe two dimensional measurement of Bell, Clauser, Ludwieg and Tillmann, Newman, Perry, Schubauer and Klebanoff, Schubauer and Spangenberg and Stratford (from Coles and Hirst[ IS]) and Samuel and Joubert [ lo]; have been made for A d 0.2. The data in inner layer coordinates u/U, vs t/(y U,/V) is displayed in Figs. l-g. All these figures show that the substantial half power region do in fact exist, the slope and intercept are not universal numbers. The slope and intercept obtained by fitting (35) to these measurements are shown in Figs. 9 and 10 as functions of A and A% (note that a value of skin friction is necessary only for estimating A, and not for deriving A and C in

Fig. 1. Velocity distribution in inner layer coordinates. Data of Bell (Series E) from Coles 3201;@, 3202; x, 3203; t, 3204;8.3205; V, 3206;A, 3207;8,3208.

andI&St.0,

569

Analysis of a turbulent boundary layers

Fig. 2. Velocity distribution in inner layer coordinates. Data of Ludwieg and Tillmann for mild adverse pressure gradient from Coles and Hirst. l , 1102;0, 1103;8, 1104: 0, 1105;+, 1106,A, i 107;V, 1108;a, t1O9;o,lllo;m; 1111;0,1112.

I Ludwieg

8

Tillman

II

30

40

Fig. 3. Velocity distribution in inner layer coordinates. Data of Ludwieg and Tillmann for strong adverse pressure gradient flow Coles and Hirst 8 1202;; $.!..3: 0, 1204;0, 1205; 0, 1206;0, 1207;0, 1208;

570

N. AFZAL

Fig. 4. Velocity distribution in inner layer coordinates. Data of Newman from Coies and Hi&. .,3502; 3503;9, 3504;0, 3505;0,3506; 0,3507.

b

I

IO

I 20

1

.Jyw-

I

30

I

I

I

40

Fig. 5. Velocity dist~bution in inner layer coordinates. Data of Perry from Coles and Hirst. A, 2901; X, 2902; V,2903; 0,2904; A, 2905;6,2906; 0,2907; G, 2908; 0,2909; %2910.

I 50

571

Analysis of a turbulent boundary layers

I Schubauer

300

I 8 Klebanoff

f 250 t

I

IO

0

I

8

I

I

J-T&x7

I

I

I

40

30

2o

Fig. 6. Velocity dist&ution in innerlayer coordinates. Data of Schubauer and Klebanoff from Coles and Hirst 0, 2130;0.2131; 0, 2132; V,2133;A,2134;0.2135; 8,2136; t ,2137; X, 2138.

c

Schubouer

150

I

I

I

8

Sponge&erg

+++t

1

xx

U/l

Fig. 7. Velocity distribution in inner layer coordinates. Data of Schubauer and Spangenbere from Coles and

HistA,4405;+,4406;x,4407;.,4804;0,4805;~,4806,V,4807;~,4;0,4809;*,45~;~~4505~0)~4 qL4507.

512

N.AFZAL

150.

I

I

I

Stratford

t xxx

0 ox

125-

o*

+tt+ AAdA

A

too-

A+ +

0

0

X

x

x

x

0

A+

X

X

U/Up 75

::

~~~~~~*~ane@@.

1

0 2

“.” I

1 5

Y 0

‘0

mG7

I

I

15

20

25

Fig. 8. Velocity distribution in inner layer coordinates. Data of Stratford from Coles and Hirst. Cl,5302;0, 5303; A, 5304; t, 5305;0, 5306; x, 5307.

(35); we have taken

the skin friction from estimates of Coles provided in the Stanford conference). Fig. 9 shows that for A < 1 the data of Schubauer and Klebanoff yields A = 4.16 (a value recommended by Townsend [3] and Perry [4]) where as data of Perry, Newman and Stratford yields A = 3. Figure 11 shows that a reasonable fit to the data is the relation. A=3.5+19&

OsAsO.2.

(38)

This may be compred with the relation proposed by Kader and Yaglom A = (20 + 2OOA)“’ 8

Kader UYaglom A=(20+2OOA)

‘/2

6

A 5 4 Present A=3&+19 A + v 3%x += X

2 0

I

I

0.05

0.1

I

I

c-15

02

I\ Fii. 9. The slope of half power law. Bell, A 3200,CJauser,8 2200,Cl2300,Ludwieg and Tillmann, I 100.Q l2~;Newman.~ 35OO;Perry.+ 2990;Scbubauer and Klebano~.O21~; Schubauer and Spangen~rg.044~. @4500,O 4800,Stratford, x 5300.Samuel and Joubert V. -- Kader and Yaglom relation A = (20+ 201lA)“~. -Proposed relation A = 3.5+ 19‘4.

573

Analysis of a turbulent boundary layers

150

80

e

0

C

20

40

60

80

l

60 40

I 30

I

10

0

20

I

40

;;,

ARP Fig. 10. Intercept of half power law in inner region. Legend same as in Fig. 9.

also displayed in Fig. 9. The differences seen here are due to the manner in which data have been fitted by Kader and Yaglom on a computer. It may also be pointed out that the channel flow measurements of El Telbany and Reynolds [16, p. 61 also suggests a rather low value A = 3 for A Q 1 compared to Kader and Yoglom A = 4.5. Figure 10 shows that the intercept in the inner law is well described by the linear relation (39)

C = 2.5AR,, 0 I ARP 5 25 and quadratic relation C = 2SAR, - 0.012 (AR,)‘,

w

0 5 ARP 5 100.

1-o -

l

COLES

WAKE

FUNCTION

F,’ 0.5 -

0

0.2

a4

06

ylb

0.8

1.0

I.2

Fig. 11 Solution of Falkner-Skan equation for wakes at separation. 0 Coles Wake function.

N. AFZAL

I

-04

I

I

- 0.35

I

-0.3

-0.25

I

-0.2

P Fig. 12. Intercept of half power law in the outer layer. Legend same as in Fig. 9.

This relation (40) is much simpler than the one proposed by Kader and Yaglom C = (AR,)“‘[2.44 In r - 15/I”” - 6/r] r = (AR,)3’2/(5+ SOA) which implies that as A + 0 the intercept C + - m. However, in our analysis of the experimental data or that of others, the plots u/U, vs (~U,/V)“~have never indicated a negative intercepts (as shown in Fig. 4 of their paper). Furthermore, the scatter in Fig. 10 is appreciably less than in Kader and Yaglom’s correlation (see Fig. 4 of their paper). In the outer layer the governing equations show that the flow behaves like a non-linear wake with prescribed stress. An equilibrium analysis with constant eddy viscosity closure for UC& leads to the Falker-Skan equation for wakes with p = a/(1 + a) as parameter (see Appendix). The solution to this equation at separation (A = 0) for F’(0) = 0 leads to p = PO= - 0.1988 (a = - 0.166) and the normalised velocity profile displayed in Fig. 11 compares very well with the wake function of Coles [ 171.The solution to the Falkner-Skan equation (Berger [18]) for a given p gives the value of F;(O) the velocity of the outer flow at the wall. An approximate closed form solution (see Appendix) is F’(O) =

1-

Z(1 + 2@/(1+ @), Z = 1.363.

(41)

This enables us to obtain the parameter D in eqn (36). In view of the relation (A.14) the parameter D is displayed against p in Fig. 12. A line shown in the figure corresponds to the relation D = 6.5(p - PO)+ O(A).

(42)

The two expressions (41) and (42) predict the intercept of outer law (36) quite accurately. Further, it may be pointed out that the present outer law (36) is not the usual defect law, as the reference velocity is not independent of Y. The Kader and Yaglom for the outer layer have proposed the usual defect law (U - u)/l& with intercept depending on A. REFERENCES [l] B. S. STRATFORD, J. Fluid Mech. 5, l-35 (1959). [2] A. A. TOWNSEND, J. Fluid Mech. 11,97-120 (l%l). [3] A. A. TOWNSEND, The Structure of Turbulent Shear Flows. Cambridge University Press, (1976).

515

Analysis of a turbulent boundary layers [4] A. E. PERRY, J. Ffuid Me& 26,481-506 (1%6). [5j A. E. PERRY, J. B. BELL and P. N. JOUBERT, J. Fluid Me&. 25,299-320 (1%6). [6] H. MCDONALD,J. Fluid Mech. 35,311-336 (1%9). I?1 G. L. MELLOR, J. Fluid Me& 24,255-274 (1964). [S] A. E. PERRY and W. H. SCHOFIELD, Phys. Fluids 16,2068-2074(1973). [9] T. C. CHAWLA and H. TENNEKES, Int. J. Engng. Sci. 11,45-64 (1973). flO] A. E. SAMUEL and P. N. JOUBERT, J. Fluid Me&. 66,481-505 (1974). [ll] R. L. SIMPSON, J. H. STRICKLAND and P. W. BARR, f. Fluid Mech. 79,553-594 (1977). [12] N. AFZAL and R. NARASIMHA, J. Fluid Mech. 74,113-129 (1976). 119 B. A. KADER and A. M. YAGLOM, .l. Fluid Mech. 89,305-342 (1978). 1141C. B. MILLIKAN, Proc. 5th Intern. Congr. Appf. Mech. Cambridge, Massachusetts (1939). 1151D. COLES and E. HIRST, Proc. 1968AFOSR--IFP-Stangord Con!., Vol, 2 Da!o Compilation (1959). 1161N. M. M. El TELBANY and A. I. REYNOLDS, J. &id Mech. 100,l-29 (1980). [17] D. COLES, J. Fluid Mech. 1, 191(1956). [18] S. A. BERGER, Laminar Wakes, Elsevier, New York pp. 69-76 (1971). 1191F. H. CLAUSER, Adu. in Appl. Mech. 4,2-51 (1956). [20] M. H. STEIGER and K. CHEN, A.J.A.A. f. 3,528-530 (1965). [Zl] N. AFZAL, J. Heat Transf. 184, 397-402(1982). (Received 28 September 1982)

Ac~no~fedgement-It is a pleasure to thank Prof. Roddam Narasimha for many stimulating discussions during the course of this work.

APPENDIX Prediction of outer velocity profile

The flow in the outer layer, to the lowest order is governed by the equation

The boundary conditions are

and the matching conditions are 643, a, b, c)

The eqn (Al) along with the boundary conditions (AZ) and (A3) are that of a non-linear wake with a prescribed stress. If the outer layer is in equilibrium in the sense of Clauser[l9], then the equation (Al) reduces to TQ-(lta-‘)F~FZ~F~*-l=o

(A4)

where a = U,slU& is a constant independent of X. The dash denotes the total differentiation with respect to the lament Y. The momentum integral of (A4) for A 6 l shows that a=-1/(2+W)

(As)

where H is the shape factor. In order to make some predictions, let us employ the constant eddy viscosity closure

where Y, the eddy viscosity, is given by (Townsend[31) v,= Uw?s

W7)

where S* is the displacement thickness and Rs is the eddy Reynolds number. In~~ucing the following variables FO= ag($,

Y = at)

(A8f

a2 = - /3US*/(S*U,Rs), fi = a/(1 C a) along with the relations (A6) and (A7) in the eqn (A4) and the boundary con~tio~s (A2) and (A3) we,get a F~er-Sky equation g”+ gg”+ 8(1- 8’2)= 0

(A91

N. AFZAL

576 with the boundary conditions

g(0) = 0, g”(0)= a,oARs( f_JdU)‘/l- 1 - g’ dn, g’(m)= 1.

(AlO, a, b, c)

Here the second boundary condition (AlO, b) prescribes the wall stress and arises from eqns (A3, c) and (A6). When A = 0, the solution of the above problem for g’(0)= 0 leads to 6 = - 0.1988(or a = - 0.166)and the normalised velocity profile, shown in Fig 11, compares very well with the Coles[l7] wake function. Further as A 4 1, (UdU)2 d 1and R, - 50 (Townsend(31)the boundary condition (46b) may be approximated by g”(0)= 0.

(All)

The eqn (A9) along with the boundary conditions (AlOa), (AlOc) and (Al 1) is the Falkner-Skan equation for the wakes whose solution is well known (Berger[l8]). An approximate solution to the problem gives (Steiger and Chen[20]) g’(0)= 1 - Z(1 t 2f3)/(lt p)

(A121

where Z = d(2). The accuracy of the solution (A12) in the domain of data, is largely improved if Z = 1.363is adopted. Further, in view of the outer expansions (13) and (27) and the fact that A( I!&/IQ2 is regarded as a higher order terms, it follows that the intercept D in the law (27) may be written as D = D(/3,A).

(A13)

D = Do(P)+ AD@) f O(A).

(A14)

For A + 0, the above relation may also be written as

Note added in proof. The eqn (2) shows that the maxima in Reynolds stress, around which inertia, pressure gradient and viscous terms are of same order, is associated with an intermediate layer whose length scale A = (uL/U)“* = (- v/U,)“*.The intermediate layer theory for a pipe and boundary layer is described in Afzal (Ingenieur-Archiu, Vol. 52, No. 6,1982,andAfzal (J. de Mecanique Theo. ef Appl. Vol. I, No. 6, pp. 103-I 13,1982).In contrast to Long and Chen (JFM, Vol. 105,p. 19,198l) the present theory re-establishes the credibility of the classical theory to explain the intermediate layer and shows the existence of two log. regions. The predictions based on three layer theory are better than the classical two layer theory.