Structure of turbulence in the boundary layer near the wall

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THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

DEPARTMENT OF AEROSPACE ENGINEERING

AERODYNAMICS LABORATORY

Technical Report

Structure of Turbulence in the Boundary

Layer Near the Wall

W. W. WILL MARTH

BO-JANG TU

Under contract with:

Department of the Navy

Office of Naval Research

Cont.ct No. Nonr-1224(30), NR-062-234

Washington, D. C.

t:

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ANN ARBOR

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ARC:

02920-4-T

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Reproduction in whole or in part is permitted for any purpose of the United States Government.

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THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

Department of Aerospace Engineering Aerodynamics Laboratory

Technical Report

STRUCTURE OF TURBULENCE IN THE BOUNDARY LAYER NEAR THE WALL

W. W. Willmarth Bo-Jang Tu

ORA Project 02920

under contract with:

DEPARTNT OF THE NAVY OFFICE OF NAVAL RESEARCH

CONTRACT NO. Nonr-l22LO), NRO6223L

WASHINGTON, D.C.

administered through:

OFFICE OF RESEARCH ADMINISTRATION ÂNE ARBOR

February 1968

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Reprinted from

BOUNDARY LAYERS AND TURBULENCE THE PHYSICS OF FLUIDS SUPPLEMENT, 1967

Structure of Turbulence in the Boundary Layer near the Wall

I. TRODUCTION

BACKGROUND

turbulence in the boundary layer has been pro-

knowledge of the structure of

vided by the extensive experimental investigations of Townsend,' Schubauer and Klebanoff,2 Laufer3

and Klebanoff4 in the period 1950-1954. In this

period the structure of turbulence was inferred

from measurements of spatial correlation and power spectra of turbulent velocities. More recently Grant5 has studied the structure of large eddies as inferred

from spatial correlation of velocity. The work of

Favre, Gaviglio, and Dumas6 introduced

measure-ments of spacetime correlations of streamwise

velocity components using a multiehannel tape re-corder for time delay. Using their ideas Willmarth and Wooldridge have measured spacetime correla-tions of the wall pressure7 and of the spacetime cor-relation between wall pressure and two velocity

com-ponents u and y.8 The introduction of spacetime

correlation makes it

possible to investigate the

evolution of turbulent eddies.

The new measurements reported here extend the

spacetime correlation measurements to the

cor-relation between pressure and the third velocity

component w. The rather extensive set of

measure-A.A. Townsend, 'I'he ,Struclure of Turbulent Shear Flow (Cambridge University Press, London, 1956).

2 G. B. Schubauer and P. S. Kiebanoff, NACA Report

1030 (1951).

J. Laufer, NACA Technical Note 2954 (1953). P. . KlebanofT, NACA Technical Note 3178 (1954).

H. L. Grant, J. Fluid Mech. 4, 149 (1958).

6 A. .1. Favre, J. J. Gaviglio, and R. Dumas, J. Fluid Mach. 2, 313 (1957); 3, 344 (1958).

w. w. Willmarth and C. E. Wooldridge, J. Fluid Medi.

14, 187 (1962).

8 w. W. Willmarth and C. E. Wooldridge, NATO AGARD Report 456 (1963).

W. W. WILLMARTU AND Bo JANO Tu A eros pace Engineering Department

The University of Michigan, Ann Arbor, Michigan

Previous measurements of spacetime correlations of turbulent wall pressure and velocity in the boundary layer were extended to include spacetime correlations between the wall pressure and all

three velocity components in the vicinity of the wall. Additional measurements of spacetime

correla-tions between various velocity components have also been made. The velocity correlacorrela-tions include measurements of the spatial correlation of the streamwise component of the fluctuating wall shear stress. A qualitative model is proposed for the structure of that portion of the turbulent field near the wall that is correlated with the wall pressure. The model outlines the sequence of events that result in the production of intense pressure and velocity fluctuations by stretching of the vorticity

produced by viscous stresses in the sublayer. All the measurements show that the shape and size of the contours of constant correlation and the sign of the measured correlations are in agreement with the proposed model for turbulent structure.

Sl 34

Printed in USA.

ments9 has encouraged us to try to formulate a

qualitative model for turbulent structure near the

wall. Before discussing the model we review some of

the previous results and present some new results

of our recent experiments.

IL DISCUSSION OF EARLIER SPACETIME CORRELATION MEASUREMENTS OF WALL

PRESSURE AND VELOCITY.

The wall pressure fluctuations are produced by

turbulence in the boundary layer covering the wall pressure transducer. It has been shown'° that when

the boundary layer is laminar in the region

very

near the pressure transducer the measured wall

pressure fluctuations are very small. When this layer is tripped and a turbulent boundary layer covers the

transducer the fluctuating signal increases by an

order of magnitude. The wall pressure signal is not increased

until the tripped turbulent boundary

layer (which spreads with the contamination

half-angle _{8.6°) covers the transducer. This meaiìs that}

at the low Mach numbers, M < 0.4, of these tests

appreciable radiation of sound from turbulence in the boundary layer does not occur.

Furthermore, the spacetime correlations of wall

pressure show that the pressure is produced by

disturbances in the turbulent boundary layer that

travel at a convection speed U. less than the free

stream speed7 actually 0.56U,. < U. < 0.83U, for

various spatial separations and, or frequency bands. The pressure fluctuations on the wall are produced

by the pressure field of moving turbulent eddies

within the boundary layer.

More extensive results are given in B. J. Tu and W. W. Willmarth, University of Michigan Technical Report ORA

02920-3-T (1966).

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r

+0080 .0.020

-O0?0 -0040 -0,60 +0 "+004.

In later work8 the spacetime correlation between

pressure and the velocity component y normal to

the wall has been measured. One result of these

measurements is the discovery that the spacetime

correlation

p(x,, 0, x3, I)v(x1 + x',, x, x, + .r, I + r)

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appears to be produced by a convected disturbance

moving at approximately the local velocity in the

boundary layer at a distance x (the position where

y is measured) from the wall. The correlation, pv,

is an odd function of x wheìi r = O and if the time

delay necessary to allow the disturbance to move

from the point x, to the point x, + x

is taken into

account the correlation is also an odd function of

z' measured with respect to a new origin at xÇ

Ur.

The correlation pv is positive for xÇ > O negative

for zÇ < 0, passes through zero at x = 0, and

vanishes for xÇ large.

From the measurements, contours of constant

correlation pv with zero-time delay have been

plotted in planes parallel to the wall. For planes

x > 0.5* the correlation pv is positive downstream of the pressure transducer and negative upstream of the transducer. In planes closer to the wall x = 0.2 ö' and x = 0.1 o" the symmetry of the contours in the

stream direction is destroyed and the correlation

becomes positive upstream and to either side of the

pressure transducer but remains negative directly

upstream of the pressure transducer. This swept

back structure is shown in Fig. 1.

The correlation of wall pressure and velocity

normal to the wall is undoubtedly produced by the convected vorticityin the turbulent boundary layer.

The correlation pv that would be measured by

50 0020 -0010 -0020 5 6 7 8 9 0 II 0 0010 -0014

-.010

0 20

passing a pressure transducer and hot wire through

a Rankine vortex has been computed" for various

spatial separations x. The computations show that

the shape of the correlation,

,

as a function of

spatial separation xÇ is remarkably similar to the

measured correlation, .

Experimentally we are

measuring jii produced by a random distribution

of eddies whose axes at the point where y js measured ire oriented obliquely to the wall and stream.

III. DISCUSSION OF CORRELATION MEASUREMENTS OF PRESSURE AND VELOCITY

COMPONENT NORMAL TO STREAM AND PARALLEL TO THE WALL

Recently we have reported9 measurements of the correlation

The spacetime correlation

also shows the effects of convection in a manner similar to the convection effects found earlier8 for the cor-relation i. Contours of const.ant correlation in

planes normal to the wall and stream direction are shown iR

Fig. 2. The contours are symmetric, but

with opposite sign, about the x,, x2 plane and con-tours for x3 < O are not shown. The pressure trans-ducer is located at the origin and the correlation in

planes upstream and at the transducer is positive.

Downstream the correlation becomes negative near the wall and further downstream the region of nega-tive correlation near the wall grows larger.

IV. MODEL FOR TURBULENT STRUCTURE NEAR THE WALL.

The results of the above measurements have led us to propose a model for the average turbulent eddy

11_{F. W. Itoos (unpublished).}

STRUCTURE OF TURBULENCE NEAR WALL

S 135

-0080 -0120 +0100 .0040 +0010 - .0.51

0020 0040 00O/ 00I X,

o

FIG. 1. Correlation contours of R', correlation coefficient nori,ialized on the value of the velocity fluctuation at x2/ô8 _{= 0.51.} Origin of coordinate system at pressure transducer.

+0

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Ii

FIG. 2. Three-dimensional diagram of contours of correlation

coefficient = constant.

structure near the wall. We have discussed how

eddies of any inclination or obliquity with respect

to the wall or free stream direction can produce the observed correlation far from the wall. However,

near the wall the contours of constant correlation

i show a swept back structure so that the eddies

must he primarily oblique. The contours of constant correlation in planes normal to the wall show ari

oblique disturbance moving away from the wall.

In the problem of the transition from laminar to

turbulent boundary layer theoretical and

experi-mental studies127 show that in the course of devel-opment from two-dimensional Tolimien-Schlichting

waves to the final stage when the turbulent spots

are formed, a necessary intermediate step is the

appearance of streamwise vortex components.

Ac-cording to Stuart,'7 it

is this streamwise vortex

component which produces the vertical convection of the spanwise vortieity component which is at the saine time stretched along the streamwise direction so as to be intensified and eventually generate

turbu-lence. We believe that such a streamwise vortex

component, together with the other two components of a three-dimensional vortex line of a hair-pin shape, also exists near the wall in the turbulent boundary

layer and is an important part of the physical

mechanism which maintains the turbulence.

To explain physically how a three-dimensional

vortex line is formed we first refer to the work by

12 D. J. Benney and C. C. Lin, Phys. Fluids 3, 656 (1960).

' _{F. R. llama, Phys. Fluids 6, 526 (1963).}

14 F. R. llama and J. Nutant, in Proceedings of the 1963

Heal Transfer and Fivid Mechanics Insiliate (Stanford Univer-sity Press, Stanford, California, 1963), p. 77.

15 P. S. Kiebanoff, K. D. T,dstrom, and L. M. Sargent, J. Fluid Mech. 12, 1 (1962).

16 L. S. G. Kovasznay, H. Koinada, and B. R. Vasudeva, in Proceedings of the 1962 Heat Transfer and Fluid Mechanics

Insiliate (Stanford University Press, Stanford, California,

1962), p. 1.

J. T. Stuart, National Physical Laboratory Report

NPL Aero Report 1147 (1965).

Browand'8 who studied the instability of a shear

flow. Browand gave a qualitative explanation for

his observation of subharmonic

waves in a shear

flow. His explanation used the idea that a small

vortex of opposite circulation from the mean circu-lation should experience a restoring force, when it is displaced either upward or downward; while a vortex with circulation in the same direction

as the mean

circulation should experience a destabilizing force

when displaced. Using this idea we consider the

region near the wall in a turbulent boundary layer. This region including the viscous sublayer is a region in which the mean vorticity parallel to the wall and

normal to the stream is large and disturbances

are also large. Suppose that random disturbances which are initially two-dimensional (i.e., correlated over some distance in a spanwise direction) cause motion of vortex lines near the edge of the sublayer either toward the wall or away from the wall. If the

vortic-ity in motion normal to the wall has the

same sign as the mean vorticity the motion will tend to continue and a rolling up and stretching process in the shear flow starts simultaneously. Finally the vorticity with

the same sign as the mean vorticity near the wall

will take the form shown in Fig. 3(a). Note that_one apex of the deformed votex line is "anchored" at the wall while the other is carried off down stream. With

the above qualitative physical model in mind

it

seems that one can understand qualitatively how

an originally two-dimensional motion becomes un-stable and develops into three-dimensional mótion in a shear flow.

V. COMPARISON OF THE MODEL WITH EXPERIMENTAL MEASUREMENTS.

The correlation jÏi2 measured in planes normal to

the wall and stream, Fig. 2, can be qualitatively

explained using the above model for average eddy structure near the wall. Figure 3(b) shows the sign_of the spanwise velocity and lig. 3(c) showsthe sign of the correlation

_{in various regions of the flow}

field of the vortex line. When correlation is measured we observe (Eulerian correlation) a random

collec-tion of disturbances passing over our instruments

and this results in a high noise level. Therefore, the observed correlation coefficients

are rather small

(of the order of 0.1).

We have also made measurements of the correla-tion between velocity components.° We have found that when the measuring points are separated in the

spanwise direction in a plane parallel to the wall,

' _{F. K. Browand, Massachusetts Institute of}

Technology Report ASRL TR 92-4 (1965).

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nd the velocity correlation ii is measured, there

is good agreement with Grant's5 results far from the wall. When the measuring points are near the wall,

x < 0.2ô*, the correlation ¿i5 becomes negative.

This indicates that stream-wise vorticity components

of small scale transverse to the stream are present

near the wall. In this connection we should mention that BakeweW°2° has recently studied the sublayer

structure in a flow of glycerine and has found

evi-dence that streamwise vorticity ispresent in or near the sublayer.

We have also measured the velocity correlation

ii at two points in a plane near (x2 = 0.1ô*) and

parallel to the wall. When the measuring points are slightly separated in the spanwise direction and the point at which r is measured is well upstream of the point where w is measured the correlation is of one sign but passes through zero and changes sign when the velocity r is measured at a point downstream of the point where w is measured. This result shows that at this distance from the wall oblique eddies inclined

at a small angle to the wall are indeed present.

One additional experimental result can be

men-tioned. There is not enough space available to

dis-cuss this result in detail. A number of hot wires

were glued on the wall x2u,/v = 5, and are used

to measure the spatial correlation of streamwise

component of wall shear stress. The correlation con-tours are elongated in the stream direction and show symmetry about a line parallel to the stream. When

the correlation between the wall shear stress and

the streamwise velocity fluctuation in a plane further

from the wall, x2u/v = 200, is measured; two

maxima of the correlation are observed on either

side and downstream of the point where the wall

shear stress

is measured. The location of these

maxima can be approximately predicted if we

as-sume that a vortical disturbance is produced near

the wall and is convected downstream with the

local flow velocity as it diffuses in the plane normal to the wall and stream. The diffusion distance used is the local root-mean-square disturbance velocity times the time interval required for the local velocity to carry a disturbance downstream.

We have proposed a qualitative model for turbu-H. P. Bakewell, Jr., Ph.D. thesis, The Pennsylvania

State University (1966).

20 F. R. Payne and J. L. Lumley, Phys. Fluids Suppi. lo,

S194 (1967).

Pressure

measured at O

pressure, p

Fxu. 3. Structure of an average model of vortex line near the

wall and the explanation of measurements of constant

cor-relation coefficient Rpv.

lent structure near the wall. All the measurements

we have made are in qualitative agreement with the model. Significant spatial ordering of the fluctuating

velocity field appears to exist near the wall. We

plan to use multiple arrays of hot wires to determine more details of the turbulent structure, and perhaps

gain some understanding of the nonlinear process

which maintains the turbulence in the boundary

layer.

ACK(OWLEDGEMENTS

We wish to thank Professor A. M. 1uethe and

A. F. Messiter, and F. W. Roos for many helpful

discussions.

This work was supported by the Fluid Dynamics

Branch of the Office of Naval Research under

Contract Nonr-1 224(30).

STRUCTURE OF TURBULENCE NEAR WALL

_S137

Vortex line,

-

average model

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Structure of Turbulence in the Boundary Layer Near the Wall

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Technical Report (Reprint from Physics of Fluids Supplement,

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Willmarth, W. W., and Th, Bo Jang

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13. ABSTRACT

Previous measurements of space-time correlations of turbulent wall pressure and velocity in the boundary layer were extended to include space-time correla-tions between the wall pressure and all three velocity components in the vicinity of the wall. Additional measurements of space-time correlations between various velocity components have also been made. The velocity correlations include meas-urements of the spatial correlation of the streamwise component of the fluctuating wall shear stress. A qualitative model is proposed for the structure of that portion of the turbulent field near the wall that is correlated with wall pressure

The model outlines the sequence of events that result in the production of in-tense pressure and velocity fluctuations by stretching of the vorticity produced

by viscous stresses in the sublayer. All the measurements show that the shape and

size of the contours of constant correlation and the sign of the measured correla-tions are in agreement with the proposed model for turbulent structure.

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ROLE CT ROLE w'r ROLE CT

Turbulence structure Boundary layer structure Turbulence

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