A modified law of the wake for turbulent shear flows

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1 AUG. 1979

ARCHIEF

Del

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

Bethesda, Md. 20084

A MODIFIED LAW OF THE WAKE FOR TURBULENT

SHEAR FLOWS

4I,4.

by

Paul S. Granville

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SHIP PERFORMANCE DEPARTMENT

RESEARCH AND DEVE LOPM ENT R EPORT

Lab.

v. Scheepsbauwwn1e

Technlcrhe Hngesrhool

May 1975 - Report 4639

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4. TITLE and Subtitle)

A MODIFIED LAW OF THE WAKE FOR TURBULENT

SHEAR FLOWS

S. TYPE OF REPORT a PERIOD COVERED

-6. PERFORMING ORG. REPORT NUMBER 7. AUTHOR(e)

Paul S. Granville

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-Naval Ship Research and Development Center Bethesda. Maryland 20084

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IS. SUPPLEMENTARY NOTES

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-Turbulent boundary layer Law of the wake

20. ABSTRACT (Continue en rvrae wide ii n.c.eemy end identity by block mmib.r)

The analytical description of similarity-law velocity profiles for turbulent shear flow has

been improved by a proposed polynomial which modifies the Coles law of the wake to satisfy the outer slope condition.

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: TABLE OF CONTENTS

ABSTRACT'

ADMINISTRATIVE INFORMATION

INTRODUCTION . . . .. .

SIMILARITY LAWS AND THE LAW OF ThE WAKE

MODIFIED LAW OF THE WAKE ...

A NEW MODIFIED LAW OF THE WAKE

SUMMARY CONCLUSION

REFERENCES

LIST OF FIGURES

1 Ccmparison Of Wake Functions . 5.

2 - Comparison of Adjustment Profiles .. 8

3 - Outer-Law Deviation for Pipe Flow 11

4 - OuterLaw Deviation for Flat-Plate Boundary Layer 11

Page 2 6 9 10 10 12

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NOTATION

A Slope of logarithmic velocity law

B1 Law-of-the-wall factor

B, Velocity-defect factor

q Wake-modification function for outer slope condition

U Velocity at pipe center or outer edge of boundary layer u Streamwise velocity component

Shear velocity Wake fünctioñ

Normal distance from wall

a

Factor in Equation (16)

BOundary-layer thickness p Density of fluid

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-ABSTRACT

The analytical description of similarity-law velocity profiles for turbulent shear flow has been improved by a proposed polynomial which modifies the Coles law of the wake to satisfy the outersiope condition.

-ADMNISTRA1lVE INFORMATION

The work described in this report was authorized and funded by the Independent Research Pgrarn of the Naval Ship Research and Development Center under Project ZR 023-OIOl,WorkUñit 1-1541-002.

INTRODUCTION

The analysis of turbulent shear flowspipe and boundary-layerrequires a valid representation of the mean velocity profile. The two velocity similarity laws based on the

wall sheanng stress provide a general statement of the velocity profile which correlates well with experimental data. The classicalsimilarity laws originally developed fOr pipe flow by Prandtl and von Ka'rmn have in time been extended to boundary-layer flows, including

flows in pressure gradients approaching separation. The first analytical representations have been logarithmic expressions for the overlapping region of the inner and outer similarity

laws. In the nonlogarithmic outer part of the shear flow, Coles observed similarity in

deviations from the logarithmic law and promulgated the well-known law of the wake. Values of the wake function as tabulated by Coles have been fitted by analytical expressions by other investigators.

An inherent discrepancy in the law of the wake as expressed by Coles1 is that the zero slope condition of the velocity profile at the outer edge of the shear layer is not satisfied. To remedy this, Cornish2 proposed a modification profile given by tabulated .average values

'Coles, D., "The Law of the Wake in the Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 1, Part 2, pp. 191-226 (Jul 1956) A complete listing of references is given on page 12.

2Comith, J.J. 111, "A Universal Description of Turbulent Boundary Layer Profiles with or without Transpiration," Mississippi State University Aerophysics Dept., Research Report 29 (Jun 1960).

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of experimental data. Analytical procedures modifying the law of the Wake to satisfy the OUter lOpe condition have been proposed by othet investigators, particularly by Allan3 and by Rotta4, Which are shown to be Unsuitable.

In this report a simple polynomial expreSsiOn is developed for the modifiëatiOn profile that is based on end conditiOns, including the proper outer slope condition. Aeement with

existing experimental data is shown to be excellent.

SIMILARITY LAWS AND THE LAW OF THE WAKE

The velocity similarity laws, the law of the wall and the velocity-defect law, provide a two-parameter velocity profile for the analysis of pipe flow and boundary-layer flow. For smoOth surfaces the law of the wall ot the inner law which applies from the wall outward may be stated in nondimensional form as

=

[U.Y]

(I)

and the velocity-dçfect law or the outer law which applies from the pipe centerline or the boundary-layer edge inward as

Uu

Here u :Streamwise velocity component normal distance from wall

p = kinematic viscOsity of fluid .uT. thear velocity u,.

wall shearing stress (3)

density of fluid

U velocity at outer edge of boundary layer or centerline velocity

for pipe floW

8 boundary-layer thickness or pipe radius

B2 velocity-defect factor which varies with type of shear flow and

pressure gradients

2

34jian, W.K., "Velocity DistribUtion in Turbulent Flow," Journal of Mechanical Engineering Science, VoL 12, No. 6,

pp. 391-399 (Dec 1970).

4Rotta, J.C, "Control of Turbulent Boundry Layers by Uniform Injection and Suctkn of Fluid," Jah±buch 1970 der DGLR, pp. 91-104 (1970).

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Not too far from the wall, the two similarity laws are considered to hold in a common!. overlapping region of the shear layer, a generic term for the region offully developed pipe flow ãnd.fOr boundary layrs. As shown in Reference 5, based on tile classical aralysis Of Milhikan, overlapping of the similarity laws lea4s to logarithmic relations. The law ofthe wall is given by

U

=AQn

-.uT V

and the velocity-defect law is given by.

Uu

_=AQn

_+B

UT 6

where B1, the law-of-the-wall factor, is a constant for smooth surfaces but a variable for rough surfaces and/or drag-reducing'polymer solutions, and B2 is a constant in the streamwise direction but with different values for pipe flow, flat-plate boundary layers, and boundary layers in various equilibrium pressure gradients. In nonequiUhrium pressure gradients, B2 is, however, a variable in the streamwse direction for boundary layers.

The anaytical description of the velocity profile outside. of the logarithmic

region

which encompasses in fact a major part of the shear layer-85 percent for flat-plate boundary layersrelies.mostly on studieS of experimental data. This region may be termed the non-logarthimic velocity-defect layer As shown by an early study performed, by Millikan,6 the most fruitful approach is to consider deviations from the logarithmic relation in this region.

Coles, while examining experimental velocity profiles, observed a similarity in deviations from' the logarithmic, relation for the nonlogarithmic velocity-d'efect layer 'which may be ex pressed for the entire velocity-defect region as

Uu

_=AQn

UT 6

5Granville, P.S., "Integral Methods for Turbulent Boundary Layers in Pressure Gradients," Journal of Ship Research, VoL 16, No. 3, PP. 191-204 (Sep 1972).

C B A Cntical Discussion of Turbulent Flows in Channels and Circular Tubes Proceedings of Ftfth International Congress for Applied Mechanics, 1938, John Wiley and Sons, New YOrk, pp. 386-392 (1939).

3

(* [1]

_-

_.f

rJ)

(6)

(4)

(10)

w 'is called the wake function. This similarity in deviation is termed the law of the wake since the similanty is independent of the wall like that of a wake The limiting value of w at - 1, wEll =2. Coles presents tentativevaluesofw ,based on an averaging of experimental data. For convenience, w is considered to apply right to the wall, where w[0] 0. In lav-Of-thewall Variables for the entire shear layer

ruT.y1

=f

I 1+

uT

LJ

or for the log and nonlog velocitydefect region together

u U.3' w

- An - +B1 +B2

I-I

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ur v 2

_L8J

An analytiëal fit to the. wake function by Hinze7 in trigonometric function is given as

r

1

Li

i-j

=1cos

L

]

A simpler polynomial fit by Moses8 is

()2

(-i-)

(10)

A comparison Of the Moses fit with the Coles tabulated values is shown in Figure 1, Where the aeemeht is seen to be satisfactory.

The Moses polynomial fit may be derived from end conditions

dw

y0,w0, -

_dy

7Hinze, J.O., "Turbuience," McGraw-Hill, New York (1959).

H.L., "The Behavior ofTurbuient Boundary Layers in Adverse Pressure Gradients," Gas Turbine Laboratory, Massachusetts InstitUte of Technology, Report 73 (Jan 1964).

4

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1.0 0.8 0.6 0 z U. w 0.2 MOSES

EO(1O) COLES (TABULATED)

/

V

6

Figure 1 - Comparison of Wake Functions

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y = 6, w = di' d(y/6) B2 d(w12)

=Oaty=6

dy

Differentiating the formulation for the law of the wake, Equation (6), with respect to y/6 and incorporating the outer slope condition, Equation (11), results in

d(w/2) A y

t

'6

6

'These end conditions are empiriôally compatible with the Coles wake function.

MODIFIED LAW OF THE WAKE

Physical considerations and experimental observations indicate that the slope of the velocity profile should be zero at the outer edge of a boundary layer or at the center of a pipe; thus

An examination of the plot of the Coles values of the wake function in Figure 1 and differentiation of the fits to the wake function in Equations (9) and (10) show that

0'at - =1

d(y/5) 6

Hence, there is oily agreement between Equations (12) and (13) at. B2. '°°. which occurs at

separation. . .

ryi

.

'.

.

If a modification function, say Aq

LT]

is applied to the law of the wake, or B2

--+ Aq, then the outer law becomes. .

____ =- A £n

+ B2 (1

--

E-F1)

Aq (14)

Then it is evident the boundary conditions on q to conform W'th those on w should be

dw

=0

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7

9Nelson, D.M.., '-A Turbulent Boundary Layer Calculation Method Based on theLãW Of the Wall and the Law of the Wake, U S Naval Ordnance Test Station (China Lake CaM) NavWeps Report 8510 (NOTS TP 3493) (Nov 1964)

d(y/6) = I and q = Oat y/6 1 dq

and also

q= 0 aty/6 =0 and

d(y/b) Oat '/& = 0

The discrepancy in the outer slope condition has interested previous investigators who have sought the necessary modifications to the law of the wake. ComiSh,2 after examining ex

-

ry

penmental velocity profiles, proposed tabulated values for Aq

Allan3 proposed art analytical modification Of the law of the wake to Satisfy the outer slope condition. The adjustthnt is based On a manipulation of the siflusoidal flt,

Equation (9).. -

-r

y

1 cos I a

-w :

L..

(15) 2

lcoscz

where a is implicitly determined from the outer slope condition

asina

-,

co&a-This is a very awkward equation to solve Furthermore, no unique function q

reults

as shown in Figure 2.

Rotta4 presented an analytical expression for q[y/] without an explanation for its derivatiOn save that it satisfied boundary conditionS

q

(+). ('-+)

(+

-3)

(17)

Still other investigators have recognized the defect in the Coles law of the wake in not satisfying the outer slope cOndition. Their solutions, however, have been both cumbersome and complicated Nelson9 approximated the wake function by a combination of two power

A.

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0.24 0.20, 0.16 qO.12 00 _0.08 ao4 0.04 0

/

p._.0'

-RNgs

Figure 2 - Comparison of Adjustment Profiles

(TA8UL

Cu,

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laws wherein the coefflcients are related to conform to the outer slope con4ition. The boundary-layer thickness is also adjusted to agree with velocity-profile integrations performed with the Coles wake function. Bull10 kept the Coles formulation but defined a flew

boundary-layer thickness to satisfy the outer slope condition. This led to further compli-cations. Sun and Childs,11 instead of modifying the wake function, derived a complicated logarithmic expression for the inner law to satisfy the outer slope condition

A NEW MODIFIED LAW OF THE WAKE

A polynomial expression is now proposed for the wake-modification function q. which correctly incorporates the outer slope condition

di

d(y/ 0)

dq

d(y/ô)

The unique solution is a cubic polynomial

Iyy I

q=

_i)

'-i

q[l] = 0

- [01=0

[II =- 1 (outer slope condition)

'0BuIl, M.K., "Velocity Profiles of Turbulent Boundary Layers," AeronAutical Journal, Vol. 73, No. 698, pp. 143-147

(Feb 1969).

"Sun C.-C. and M.E. Childs, "A. Modified Wall Wake Velocity Profile for Turbulent Cémpressible Boundary Layers,'

Jàurnal of Aircraft, Vol. 10, No. 6, pp. 381 383 (Jun 1973).

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q=

Jr

a

( (18)

As stated before, the end conditions determining the coefficients are

(16)

A comparison of the various formulations for q is presented in Figure 2. It is seen that the proposed formulation agrees well with the Comish results obtained from experimental data. The equation due to Allan does not give a unique result for q. The Rotta formula Of Equation (17) agrees poorly with the ethpirical results ofCornish.

A comparison with experimental data is shown in Figure 3 where deviation froth the log law is plotted for pipe flow. Agreement with experimental data is excellent. A further cm-parison for flat-plate boundary layers is shown in Figure 4. Agreement with experimental data from Reference 12 is also excellent.

SUMMARY

The modified law of the wake may now be stated as

_--

+ Aq or the outer law as

Uu

_{--AQn -}

y UT 5 + B (1

wy

2

U

ry

U

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'2lboff P.S. and Z.W. Diehi, "Some Features of Artificially Thickened Fully DevelopedTurbulent Boun4ary Layers

with Zero Pressure Gradient," National Advisory Committee for Aeronautics Technical Report 1110 (1952). 'with the wake function as

=3

(y

2

( y

\3

(10)

2

and the wake modification function as

/y\2 I

T)

l-

-'CONCLUSION

The new wake modification function, Equation (19), improves the agreement of the similarity laws with experimental data.

)

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0.6 LU i- 0.4 0 0.2 0 02 I

0*.

0

':, 4 - -a' _b

,,

0.4 0.6 0.8 1.0 FLAT PLATE EXPERIMENTAL DATA 0 SCHULTZ-GRUNOW - =2.7 x iO4

(FROM FIG 20 OF REF. 12) = 4.8 x iO4

= 1.52 x i05

0 0.2 0.4 0.6 0.8 1.0

Yb

Figure 4 - Outer-Law Deviation for Flat-Plate Boundary Layer yTh

FigUre 3 - Outer-Law Deviation for Pipe Flow

PIPE FLOW

z

0 1.2

--I

EXPERIMENTAL (FROM U DATA OF HINZE FIG. 7-30. -LAUFER REF. 7) 1.0 O.8 KLEBANOFF& DIEHL O KLEBANOFF& DIEHL

0

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REFERENCES

Coles, D., "The Law of the Wake in the Turbulent Boundary Layer," JOurnal of Fluid Mechanics, Vol. 1, Part 2, pp. 191-226 (Jul 1956).

Cornish. J.J III, "A Universal Description of Turbulent Boundary Layer Profiles with or without Transpiration," Mississippi State University Aerophysics Dept., _Research Report 29 (Jun 1960).

AUan, "Velocity Distribution in Turbulent Flow," Journal of Mechanical Engineering Science, Vol. 12, No. 6, pp.391-399 (Dec 1970).

4: Rotta. J.C.. "Control of Turbulent Boundary Layers by Uniform Injection and

Süction..of Fluid," Jahrbuch 1970 der DGLR. pp. 91 104 (1970).

Granville, P.S., "Integral Methods for Turbulent Boundary Layers in Pressure Gradients," Journal of Ship Research. Vol. 16, No. 3, pp. 191-204 (Sep 1972).

Millikan, C.B., "A Critical Discussion of Turbulent Flows in Channels and Circular Tubes," Proceedings of Fifth International Congress fOr Applied Mechanics, 1938, John Wiley and Sons. New York, pp. 386-392 (l939)

Hinze, J.O., "Turbulence," McGraw-Hill, New York (1959).

Moses, H.L., "The Behavior of Turbulent Boundary Layers in Adverse Pressure Grädieñts," Gas Turbine Laboratory, Massachusetts Institute of Technology, Report 73 (Jan

1964).

Nelson, D.M., "A Turbulent Boundary Layer Calculation Method Based on the Law of the Wall and the Law of the. Wake," U.S. Naval Ordnance Test Station (China Lake, Calif.) NavWeps Report 8510 (NQTS TP 3493) (Nov 1964).

Bull. M.K., "Velocity Profiles of Turbulent Boundary Layers," Aeronautical

-Journal, Vol. 73, No. 698, pp. 143-147 (Feb 1969).

Sun, C..C. and M.E. Childs, "A Modified Wall Wake Velocity Profile for Turbulent Compressible Boundary Layers," Journal of Aircraft, V.ol. 10, No. 6, pp. 381-383 (Jun 1973).

Klebanoff, P.S. and Z.W. Diehl, "Some Features of Artificially Thickened Fully Developed Turbulent Boundary Layers with Zero Pressure Gradient," National Advisory Committee for Aeronautics Technical Report 1110 (1952).

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