A law of the wall for turbulent boundary layers with suction or injection

CoA Report A e r o No. 166

Bibliotheek TU Delft

Facuttait der Luchtvaart- en Ruimtevaarttechniek | Ktuyverweg 1

2629 HS Doift

T H E COLLEGE OF AERONAUTICS

C R A N F I E L D

A LAW O F THE WALL FOR TURBULENT BOUND AY LAYERS WITH

SUCTION OR INJECTION

by

CoA, REPORT AERO NO. 166 July. 1963.

THE C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

A Law of the Wall for Turbulent Boundary Layers with Suction or Injection

by

T . N. Stevenson, B . S c , D . C . A e .

SUMMARY

It is found from experiment that the law of the wall with suction and injection is

Jul r f i + M ^ . 1 ] 1 log ïHr+A

vw L \ " r V J K ^e V

where A and K are independent of v^ and hence have values obtained for the case of no blowing.

The equation is compared with many experimental results obtained by the author and others. It is shown how the skin friction may be obtained from the velocity distribution for a given injection or suction velocity.

B i b i 1 ot|-.eel. TU De 1 f t. L & P

CONTENTS

Page Summary

List of Symbols

1. Introduction 1. 2. A review of turbulent boundary layer 1

theories with suction and injection.

2.1 The momentum transfer theory 1

2.2 Turcotte and Leadon 4 3 . A law of the wall 5 3.1 A comparison between equation (18) and that

suggested by Black and Sarneckl. 5 3.2 To obtain the skin friction from a velocity

t r a v e r s e . 6 4. Discussion 6 5. Conclusions 7 6. Acknowledgements 7 7. References 8 Tables Figures

LIST OF SYMBOLS

constants of integration with respect to y local skin friction coefficient

mixing length constant

mean velocity in the x - direction friction velocity , u^

friction velocity obtained by experiment

friction velocity estimated using a velocity profile together with the law of the wall equation (18) velocity in the y - direction

co-ordinate along the wall co-ordinate normal to the wall boundary layer thickness viscosity

profile paramieter defined by equation (20) density

sum of the viscous shear s t r e s s and the Reynolds s t r e s s kinematic viscosity

wall condition

free s t r e a m condition

n-condition at y = y^. ya is the position at which the sublayer equation intersects the law of the wall equation

1. Introduction

There is no adequate theory for shear flow turbulence. There have been a considerable number of experiments in turbulent boundary layers over solid surfaces but there are relatively few which includetranspiration, and these show considerable scatter in the r e s u l t s .

The semi-empirical theories available for the turbulent boundary layer with transpiration a r e reviewed in Section 2. An equation which was suggested by the experimental results of the author is presented for the inner turbulent region. The skin friction is given by this equation if the velocity profile and the transpiration velocity are known. The equation is analogous to the law of the wall for flow over solid surfaces.

It is shown how this equation correlates the suction and injection experimental results obtained by the author, and others.

2. A review of turbulent boundary layer theories with suction or injection 2.1 The momentum transfer theory

Most of the theories for suction or injection a r e based on the momentum transfer theory of Prandtl, together with the assumption that the mixing length is proportional to the distance from the wall. This yields

where K is independent of y.

This equation is further substantiated by Townsend (ref. 9) who considers regions of turbulent shear flow in which there is equilibrium between the local r a t e s of energy production and dissipation.

The usual turbulent boundary layer approximation (which is exact for the asymptotic layer with suction) results in the equation

- f" -'Ï f

<2)

w d y P d y

au —

T is the sum of the viscous shear s t r e s s ^-^ and the Reynolds s t r e s s -pu'v'.

Equation (2) is assumed to hold for both the suction and blowing c a s e s , but there would seem to be less justification for its use with blowing than with suction.

In the sublayer where the Reynolds s t r e s s e s are assumed negligible, equation (2) may be integrated twice with respect to y. After substituting the wall conditions, the solution reduces to

2

-The velocity distribution for the inner turbulent region will now be obtained quite generally without using the boundary conditions. It will then be shown how this equation reduces to those obtained by other workers, by sub-stituting the appropriate boundary conditions.

If equation (1) is substituted into equation (2), then

This equation is integrated twice with respect to y, to give 1 log i - ^ = 2uz ( l w _ V , y , , .

K

V

Vuf-

1 1 » * '

(5)

c and d are the constants of integration. They a r e , in general, functions of V , u , u^, and 6.

w 1 T' 2 . 1 . 1 Kay ( r e f . l )

Kay considers the asymptotic suction case and uses the boundary conditions -r- = 0 and u = u, at y = 6

dy ' ^

If these conditions are used together with equation (5) then,

vw"i ^ 1 , 6ur , „ , and

u * ' 4K'u ""^e

"

^oë;(jlb) (7)

This is the equation obtained by Kay, but his experimental results for suction did not agree with the equation. This is because the mixing length equation (4) is not valid at the outer edge of the layer where the boundary condition was applied.

2 . 1 . 2 Clarke, Menkes and Libby(ref.2) write equation (5) in the following form

H =A + Biog ( y r x ) + 1 , . I Ï : l o g ' ^ f H r i ) (8)

When V = o this equation reduces to the accepted logarithmic form

w

3

-Clarke et al overlook the implicit relation between A, B, and v (see equations 19

and 20). ^

2 . 1 . 3 Rubesin (ref. 3), Dorrance and Dore (ref.4) and Mickley and Davis, (ref. 5) Rubesin and Dorrance and Dore consider the compressible boundary layer and obtain integral equations for the sublayer region and for the inner turbulent region. Mickley and Davis write the equations in incompressible form., assuming that they hold on either side of a transition point y = y^. At this point the velocities and shear s t r e s s e s are matched

A . , , , . , u = „ . . „ a ( „ | a ) ^ . ( . K y ( | ^ ) - ) ^ ,10)

The velocity-shear relationship which is valid in the sublayer, and which is assumed to hold in the inner turbulent region, is obtained by integrating equation (2) and substituting wall conditions. Hence

puv = r - r (11) "^ w w

Equations (10) and (11) are combined at y = y^ to give

aVw + 4 =

\^^)

UaVw + ul = K y ^ ~ - (12)

If this condition together with u=u^ at y=ya is used to determine c and d in equation (5) then

1

Therefore equation (5) may be written

This is the equation presented by Mickley and Davis. Experimental curves of

VwU \ 2

1 + —5" ) are plotted by them and straight lines of the predicted 2Ku

slope y ^ are obtained over the inner turbulent region. Mickley and Davis obtained the skin friction by momentum t r a v e r s e s . They are not able to correlate the results to obtain satisfactory variations in the position of y .

2 . 1 . 4 Black and Sarnecki (ref. 6)

If it is assumed that the shear relation of equation (11) is valid at the same time as the linear mixing length equation (1), then c = -1 and equation (5) may be written

- 4

2u

(As V •• o then d •• B - — ^ where B is the constant in the no-blowing law of the

wall.) w v^

1 ku If d is made equal to — log —''then _{K e y}

< ^-w" = ( 1 l°ge k) <^«)

k is the constant of integration.

This is the equation which Black and Sarnecki call the bilogarithmic law. They plot curves of

- + y^« ^ log i i ^ against ^ ^ |+ ^ . log ^ and obtain the u, 4K u, ^e v ^ 2K ^|- u ^e v

value of u from the straight line part of the curve.

Karman's constant ,K, in the mixing length should be independent of the transpiration velocity providing I v^i is small compared with the velocity u. The experiments of Black and Sarnecki and those of the author confirm that K is independent of v .

Black and Sarnecki try to find the variation in the position y = ya with v^ and

\Xy . This will be discussed in section 3 . 1 .

2. 2 Turcotte (ref. 7) and Leadon (ref. 8).

Turcotte assumes that the shear s t r e s s in the fully turbulent portion of the boundary layer is unaffected by injection and suggests a similarity parameter,

JiL . (The subscript o refers to zero blowing conditions.)

In a reply to Turcotte's paper, Leadon shows that the shear s t r e s s assumption is incorrect. Leadon suggests that a proper similarity param.eter should include the free s t r e a m velocity.

A further review, including foreign gas injection, and compressible effects, is given in reference 11.

5

-A law of the wall

Equation (15) may be written

This is similar in form to that given by Townsend (ref. 9). Equation (17) reduces to the familiar law of the wall equation when v = o.

The experimental curves for flow over a permeable or impermeable wall may now be compared on one figure if log ZÜT is plotted against

— I ( — ^ + 1 ) - 1 1 • The inner turbulent region should plot as a s e r i e s of

VwL\

parallel lines if the mixing length coefficient K is independent of v .

The author used this method in plotting his own experimental results (ref. 12) and found that they plotted close to the accepted impermeable wall curve (fig. 1). The experimental results show that the t e r m ( d + ^ÜM varies very little with

suction or injection, and that the variations obtained by other authors are possibly due largely to experimental e r r o r s .

There still remains some doubt as to the appropriate values for the constants in the case of no blowing in the 'law of the wall'. The values which were found by Dutton (ref. 10) will be used in this paper.

The law of the wall with suction or injection is therefore

3.1 A comparison between equation (18) and that suggested by Black and Sarnecki (ref. 6)

Black anr! Sarnecki wrote their bilogarithmic law in the form

\XT e V 4K u T \ ^e V /

with a = 7 ^ U^ - y . b = - and

1 '^w "rk , >

Several non-dimensional parameters which might provide a possible criterion for specifying conditions at y = y were considered by Black and

Sarnecki. Eight possible equations for X were obtained by considering the sublayer equation (3) and the bilogarithmic law, equation (16). X was then plotted agadnst ^^ and compared with experimental r e s u l t s . There is a considerable scatter in the experimental r es ul ts , but Black and Sarnecki chose the equation for X which predicted most accurately the actual variation for layers on smooth and nearly homogeneous walls. The equation is

X = V T T ^ - m '°^e [ £ '°ê, (1 + 2m)] (21)

"a"*^w " a

where m = „ - & N = — ; m > - i . 2 u f Uj.

This equation is presented in Fig. 2, together with experimental r e s u l t s , and is reproduced from Black and Sarnecki's report.

The equivalent variation which is implied by the law of the wall with suction and injection is

This equation is also shown in Fig. 2.

3. 2 To obtain the skin friction from a velocity traverse Equation (18) can be written in the form

2

5.5

^[(-^-J*-0-e-%.(^)--^)

(23)

Vw y " ,

A set of curves of — against log — has been evaluated for particular values of CJ. at fixed values of — (Fig. 3a, b, & c.)

f u, Vw

If the value of — is known for a particular profile, then Fig. 3 may be used to plot the curves of — against log,,, ^ ' for particular values of c,. If the experimental profile is plotted on these curves, the skin friction

values of Ur which are obtained in this way will be < values obtained from momentum t r a v e r s e s by Ur^,.

profile is plotted on these curves, the skin friction may be estimated as in F i g . 4 . The values of Ur which are obtained in this way will be denoted by u^ , and the experimental

Discussion

Some of the author's experimental results are plotted in Figs. 1 and 5. The values for the skin friction evaluated from momentum t r a v e r s e s and from the law of

t h e wall equation a r e c o m p a r e d in t a b l e 1. T h e d i f f e r e n c e s a r e within the e x p e r i -m,ental a c c u r a c y .

T h e e x p e r i m e n t a l r e s u l t s p r e s e n t e d in Black and S a r n e c k i ' s r e p o r t a r e plotted in F i g . 6. Only the s t r a i g h t l o g a r i t h m i c p o r t i o n s a r e plotted for c l a r i t y . T h e p o s i t i o n s of the e x p e r i m e n t a l s t r a i g h t l i n e s with r e g a r d t o that of equation (18) do not show any t r e n d with c h a n g e s in v ^ .

T h e p o s i t i o n of the s t r a i g h t line i s v e r y dependent on the a c c u r a c y of u r g ( F o r v ^ ~ o , a ± 10% e r r o r in u ^ „ r e s u l t s in equation (18) plotting a s the chain dotted l i n e s in F i g . 6 . )

M i c k l e y and D a v i s do not s e e m t o include the p r e s s u r e g r a d i e n t t e r m in t h e i r s k i n f r i c t i o n c a l c u l a t i o n s . Some of t h e i r v a l u e s for c . a r e p r e s e n t e d in t a b l e 2, t o g e t h e r with modified v a l u e s which include the p r e s s u r e g r a d i e n t t e r m . T h e s e a r e c o m p a r e d with v a l u e s of c which w e r e e s t i m a t e d f r o m the law of the w a l l e q u a t i o n . S o m e of the v e l o c i t y p r o f i l e s a r e plotted in F i g . 7.

5 . C o n c l u s i o n s

It i s found from e x p e r i m e n t that the g e n e r a l i s e d law of the wall for t u r b u l e n t b o u n d a r y l a y e r s with s u c t i o n and injection and z e r o p r e s s u r e g r a d i e n t i s

u^ f / VwU \ 2 "1 1 UT-y

w h e r e A and K a r e independent of v and h e n c e have v a l u e s obtained for the c a s e of no blowing.

It i s shown how the skin f r i c t i o n m a y be obtained f r o m the v e l o c i t y d i s t r i b u t i o n for a given injection o r s u c t i o n v e l o c i t y .

6. A c k n o w l e d g e m e n t s

T h e a u t h o r w i s h e s to thank h i s s u p e r v i s o r s P r o f e s s o r G. M. L i l l e y and P r o f e s s o r A . D . Y o u n g for the helpful d i s c u s s i o n s d u r i n g the p r e p a r a t i o n of t h i s p a p e r ,

A m o r e r e c e n t p a p e r , "A modified v e l o c i t y defect law for t u r b u l e n t b o u n d a r y l a y e r s with i n j e c t i o n " i s to be published a s College of A e r o n a u t i c s R e p o r t A e r o No. 170.

References

Kay, J . M . Boundary Layer flow along a flat plate with uniform suction.

A . R . C . R & M. 2628. 1948.

Clarke, J , H . , A provisional analy.iis of turbulent boundary layers Menkes, H . R . , and with injection.

Libby, P . A . J . A e r o . Sci. , Vol. 2 2 . , 1955, pp. 255 - 260

Rubesin, M.W. An analytical estimation of the effects of transpiration cooling on the heat-transfer and skin-friction

characteristics of a compressible turbulent boundary layer.

N . A . C . A . , T . N . 3341, 1954.

Dorrance, W.H. , and

Dore, F . J .

The effect of m a s s transfer on the compressible turbulent boundary layer skin friction and heat transfer.

J . A e r o . S c i . , Vol. 21, 1954, pp. 404 - 4 1 0 .

Mickley, H.S, and Davis, R.S.

Momentum transfer for flow over a flat plate with blowing.

N . A . C . A . , T.N. 4017, 1957.

Black, T . J . , and Sarnecki, A . J .

The turbulent boundary layer with suction or injection. A . R . C , Report 20,501, 1958.

Turcotte, D . L . A sublayer theory for fluid injection into the incompressible turbulent boundary layer.

J . A e r o S.Sci.Vol.27, No.9, pp. 675 - 6 7 8 . Sept. 1960.

Leadon, B.M. Comments on "A sublayer theory for fluid injection". J . A e r o S.Sci. , Vol.28, 1961, pp. 8 2 6 - 8 2 7 .

Townsend, A.A. Equilibrium layers and wall turbulence. J . F l u i d Mech. , Vol.11, 1961, pp. 9 7 - 1 2 0 .

Dutton, R.A.

Craven, A.H.

Stevenson, T.N.

The velocity distribution in a turbulent boundary layer on a flat plate.

A . R . C . , C . P . 453, 1959.

Boundary layers with suction and injection. College of Aeronautics Report 136, 1960.

Experiments on injection into an incompressible turbulent boundary layer. (To be published)

9

-TABLE 1. EXPERIMENTAL RESULTS

u, = 50 f t / s e c .

Ui ^E \ u, / f \ u, /

From Momentum From the law of T r a v e r s e s the Wall Equa

-tion 0 II ir .0013 It .002 II .003 .004 .0057 .0037 .0036 .0035 ,0028 .0026 .0021 .0020 ,0019 .0014 .0013 .0010 .0006 .0003 .0040 .0038 ,0037 .0026 .0025 .0023 ,0020 ,0016 .0015 .0011 .0009 .00046 .00032

10

-T A B L E 2 . MICKLEY & DAVIS' RESUL-TS ( R E F . 5.)

Vw u« .003 0 Station E G H I J K L M N J K L M Cf As Evaluated by Mickley & Davis

.00136 .00097 .00087 .00068 .00066 .00054 .00058 .0005 .0005 .00349 .00329 .00332 .00307 Including the Pressure Gradient T e r m f A fB' .00175 .00165 .0008 .0016 .00094 .0014 .00087 .0013 .00098 .0014 .0005 .0013 .0005 Only a very small correction required '^f F r o m the Law of the Wall Equation .0019 .0017 .0014 .0013 .0011 .00105 .0009 .0009 .00085 .00318 .00311 .00292 .00295 F i g u r e 8 shows the d i s t r i b u t i o n of u, along the w o r k i n g s e c t i o n . C u r v e A w a s used to e v a l u a t e "^f. and Curve B to e v a l u a t e

^{D-X + 0 o t t 3L "* ^'- -TE o 0 0 3 5 0 O 6 4 0 I 0 4 0 183 0-44 «1 « T E 2 3 6 28 32 3SS 45 9 7 6 7

FIG. 1. THE AUTHOR'S EXPERIMENTAL RESULTS COMPARED WITH THE LAW OF THE WALL. EQUATION (18).

0 8

0 . 6

0 4

0 3

FIG. 2. VARIATION OF THE PROFILE PARAMETER X WITH TRANSPIRATION

0 0 0 6 0 0 0 5 0 0 0 4 0 0 0 3 0 0 0 2 OOOI O ^ OOOI

FIG. 3a. EQUATION (18) WHEN - =0.15 FIG. 3b. EQUATION (18) WHEN - = 0.3

u. 0 0 0 6 0 0 0 5 0 0 0 4 0 0 0 3 a o o 2 -aooi 0 0 0 6 0 005 0 0 0 4 0 0 0 3 0 0 0 2 OOOI

FIG. 3 c . EQUATION (18) WHEN - = 0 . 5 _{FIG. 3d EQUATION (18) WHEN - = 0 . 8}

0 8

0 3

LAW OF THE WALL EQUATION. EXPERIMENTAL POINTS:

e< FROM THESE Cf FROM MOMENTUvf

CURVES '

OOOOS

FIG.4 THE SKIN FRICTION USING A VELOCITY

TRAVERSE AND EQUATION (18) OR (23)

3 D

FIG. 5 THE AUTHOR'S EXPERIMENTAL RESULTS

USING THE LAW OF THE WALL, EQUATION (18) 20 EQUATION (IB) t K>% CHANGE IN Ur T~5-/ " ^ ( ^ ) ütt. H L

SYMBOL UTE UTÏ

» - O 039 20 SARNECKI « - O O S I IS-8 BLACK « - 0 0 6 14.9 DUTTON » - O 06S 14-7 SARNECKI o - 0 0 6 6 14 2 BLACK 0 - 0 O 6 8 15 4 DUTTON + - 0 08S i\-7 DUTTON

FIG. 8 EXPERIMENTAL RESULTS FROM R E F . 6 COMPARED WITH THE LAW OF THE WALL (EQUATION 18)

O 9 Ui 0 7 O'6 OS 0 4

FIG. 7 VELOCITY PROFILES FROM REFERENCE 5 (MICKLEY & DAVIS)

u, n/sec SI

FIG. 8 THE FREE-STREAM VELOCITY GRADIENT