An unified view of the law of the wall using mixing length theory

A UNIFIED VIEW OF THE LAW OF THE WALL

USING MIXING-LENGTH THEORY

by

V. C. Pate!

Sponsored by

General Hydromechanics Research Program of the Naval Ship Systems Command Naval Ship Research and Development Center

Contract No. N00014-68-A-0198-0002

uHR Report No. 137

Iowa Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa

April 1972

This document has been approved for public release and sale; its distribution is unlimited.

ABSTRACT

It is shown that, if the well-known mixing-length formula is regarded simply as a relationship between the velocity and the stress distributions in the wall region of a turbulent flow, then a truly universal distribution of mixing length is sufficient to describe the experimentally observed departures of the velocity distribution from the usual law of the wall as a result of severe pressure gradients and

transverse surface curvature. Comparisons have been made with a wide

variety of experimental data to demonstráte the general validity of the mixing-length model in describing the flow close to a smooth wall.

An extension of the relaminarization criterion of Patel and Head, and sorne experimental evidence, suggest that the thick axisymmetric boundary layer on a slender cylinder placed axially in a uniform stream

can not be maintained in a fully turbulent state for values of the Reynolds number, based on friction velocity and cylInder radius, below a certain critical value.

ACGOWLEDGEIVENTS

This report is based upon research conducted under the General Hydromechanics Research Program of the Naval Ship Systems Command, technic-ally administered by the Naval Ship Research and Development Center, under Contract NOO011-68-A-Ol96-OOQ2.

The author gratefully acknowledges the assistance in data

reduction and computation he received from Messers K.C. Chang, A. Nakayama

and R. Damian of the Iowa Institute of Hydraulic Research.

LIST OF COETS

Page

I. INTRODUCTION i

II. RELATION BETWEEN THE VELOCITY AND STRESS DISTRIBUTIONS 3

III. MIXING-LENGTH DISTRIBUTION

IV. STRESS VARIATION MID VELOCITY DISTRIBUTION IN THE WALL

REGION 5

Two-Dimensional Boundary Layer 5

Fully-Developed Flow in a Pipe 8

Thiòk Axisymmetric Boundary Layer 8

Fully-Developed Flow Through Concentric Annuli 11

V. EXPERINTAL VERIFICATION OF TI INFLUENCE OF TRANSVERSE

CURVATURE 13

VI. DISCUSSION 18

VII. CONCLUSIONS 20

REFERENCES 21

LIST OF FIGURES

Page

Definition Sketch, Mixing-Length Theory. ₂₃

The Continuous Mixing-Length Model, Equation (8). 21L

The Influence of Pressure Gradient on the

Law-of-the-Wall. A Plot of Equations

₍₁₆₎

and (17). ₂₅ Velocity Measurements of Newman (1951) Compared

with Mixing-Length Theory. ₂₆

Definition Sketch for the Thick Boundary Layer on

a Cylinder. ₂₇

The Influence of Transverse Curvature on the

Law-of-the-Wall; A Plot of Equation (26).

₂₈

Determination of Wall Shear Stress by the Method

of Clauser. ₂₉

Comparison of the Measurements of Richmond with

Equation (26). ₃₀

Comparison of the Measurements of Keshavan with

Equation (26). ₃₁

Comparison of the Measurements of Yu, and

Willmarth and Yang with Equation (26). ₃₂

Comparison of Mixing-Length Theory with the

Measurements of Lawn and Elliott in Concentric Annuli. ₃₃

Figure 1. Figure

2.

Figure

_3.

Figure 1. Figure 5. Figure

6.

Figure

_7.

Figure

8.

Figure

_9.

Figure lo. Figure 11.

iv

LIST OF TABLES

Page Table 1. Summary of Data on Thick Axisymmetric Thrbulent

Boundary Layers 3I

Table ? A Summary of the Concentric Annuli Data of Lawn

A UNIFIED VIEW OF THE LAW OF THE WALL USING MIXING-LENGTH THEORY

I. INTRODUCTION

Eddy-viscosity and mixing-length models daté back to the

earliest days of the study of turbulence. They have received a great

deal of attention over the years both frein research workers who have sought to discredit the concepts involved, or at least to establish their

limitations, and in the opposite camp, from engineers who have _recognized

their usefulness as a basis for performing practical shear-flow calculations. Recent experience in turbulent boundary-layer calculations suggests that

these models work considerably better in practice than one ha any right

to expect on physical grounds.

A number of physical interpretations have been placed on the

well known mixing-length formula,

.jTt

in addition to Prandtl's original _{heuristic one.}

The interpretation that appears to have received the widest acceptance in recent years is the one due to Batchelor (1950). _{According to this, the mixing-length}

expression represents the equilibrium between the production and dissipation of

turbulent knietic-enery which is implied when convection and diffusion are

assued negligibly small. _{Detailed turbulence measurements indicate} that such an equilibrium between production and dissipation is approached most

closely in the wall region of turbulent _{boundary layers. This region is}

commonly referred to as the inner layer _{or the law-of-the-wall region.}

Here, the mixing-length expression is used as a relationship

between the Reynolds stress and the mean velocity to study the velocity distributions in the wall region of boundary layers developing in pressure

2

influence of pressure gradients on the law of the wall in boundary layers developing on nominally plane surfaces has been the subject of a

rnber of previous itvestigations. It is nòw generally recoiized that

the usual law of the. wall,

u/U = f(Uy/v),

T T

wh-ich has been found to apply in fully-developed flows i pipes and

channels, as well as in the flat-plate boundary layer, ceases to be valid in the presence of sbstantial streamwise pressure gradients, and that the

departures from the usual law can be predicted using mixing-length theory.

The results of the earlier investigations are re-examined here, and a

parallel analysis is presented for axially-symmetric flows in order to

show that mixing-length theory once again predicts a significant

influence of transverse curvature on the velocity distribution in the

wall region. As far as the flow in this region is concerned., transverse

curvature effects have been observed in two types of experiments: in

the thick axisyinmetric boundary layer developing on a long slender

óylinder placed axially in a uniform stream (there being no streamwise

pressuré gradient in this. case), and in fully-developed turbulent flow in the concentric annulai' space formed by two parallel pipes. The velocity

laws deduced from mixing-length theory a1,e been compared with experimental

data from these sources. The substantial agreement found between the

measurements and theory once again demonstrates the general validity and the usefulness of the mixing-length model.

The results of the analysis presented here and the experimental

data collected elsewhere show that there is a remarkable. similarity

between the influence of a favorable pressure gradient ön the flow in

the wall region of a two-dimensional boundary 1rer and the ifluence of

transverse surface curvature on the flow in the -wall region of an

axisymmetric turbulent flow. One is therefore tempted to apply the

criterion proposed by Patel and Head

(1968)

for the re-laminarization

of a two-dimensionâ.J. turbulent boundary ayer under the influence of a favorable pressure gradient to the case of turbulent flow on a slender cylinder. An extension of this criterion and. rather limited experimental

way

3

evidence suggest that there may be a lower limit to the cylinder Reynolds number, based on friction velocity and the radius of the cylinder, below which the flow in the neighborhood of the cylinder can not be maintained

in a fully turbulent state.

II. RELATION BETWE THE VELOCITY AND STRESS DISTRIBUTIONS.

We consider a nominally parállel turbulent shear flow such as

occurs in a pipe or the turbulent

boundary layer. Referring to Figure 1

for notation, the total shear stress at a point may be written

T = , (i)

this being the sum of the molecular and Reynolds stresses. If the latter

is related to the mean velocity distribution via the mixing-length theory, equation (i) becomes

au

T =p+p

--- . 2

Introducing the usual non-dimensional quantities defined in the following

= U/U y =

Uy/y

=

UL/v ,

T _{= T/Tw} and UT

where T IS the shear stress at the wall, y = O, we have

(j*. .4.)2

(au:) _{- 1*}

= O

Solving this quadratic, we obtain

au:

21*

ay:

{l

+

and since u = O at y = O, this yields

rw (J

-

* U - j0 _{i +} + (1)

(5)

This is a r-eltion between the velocity distributior ¿nd the stress distribution. In ordeì to proceed further, we must prescribe the

func-tion ve and also the variafunc-tion with y* of the shear stress

T*.

ÏII..

MIXING-LENGTH .ISThIBUTION

For the flow iñ the neighborhood of a smooth wall we identify threé regions in the usual way, namely the sublayer, the blending region

and the fully turbulent region. Through the sublayer (r < 5, say) the

Reynolds stress is assumed to be negligible so that

For the fully turbulènt region the molecular stress is neglected in

comparisiön with the Reynolds stress and dimensional reasoning leads co

the familiar linear väiation of mixing length in. the form

= K,

(7)

where K isthe von Kármn coistant. In subsequent work K will be

assumed equal. to

o.i8

as suggested by Patel

(l965b).

In the blending

region between the sublayer and the fully-turbulent region., both molecular

and Reynolds stresses are present. Composite mixing-length and

eddy-vis-cosity foulae ihich may bé ued for this region. (5 < y < 50, say) have

been suggested by a number of workers on thç basis Qf empirical correlation

of experimental òbsèrvations. Amongst the most well-known fQrm'llae are

those of Reichardt

(1951),

Deissler

(i95),

and varDriest

(1956).

Here

we shall use afi expression proposed recently by Landweber and Poreh (to

be published.)! This is ve = cy* { talill(X2y*2

y

(8)

where X is a constant such that the tanh terni approaches

tu4ty

in the

region of

y

= 50.

It can readily be verified that equation

(8)

applies thí-oughout the wall region since it gives the correct behavior of

for

small and large y*. A plot of equation

(8)

is shom in Figure 2.

In order to obtain formulae for the velocity distribution we need to consider next the variation of the stress through the wall region. _{The stress distribution will in general dépend upon the geometry}

of the flow. _{In what follows, we shall examine a few cases of interest.}

A. Two-Dimensional Boundary Layer. We consider first a

thin two-dimensional boundary layer developing in a pressure gradient.

Using standard notation, the relevant equations may be written

IV. _{STRESS VARIATION AND VELOCITY DISTRIBUTION IN THE WALL REGION}

Du Du

u +v =

_Dx

By

_pdx

pDy

Du + Dv

Dx Dy

O,

where x is measured along the wall and y normal to it. The components

of mean vélocity in these directions are u and y , respectively, and T

is the total shear stress, i.e. the sum of the molecular and Reynolds stresses.

Integration of equation (9) with respect to y gives

= i + py*

+

4-

J(u

+ y

dy

o

- v.

where

=

Ur

_dx is a non-dimensional pressure-gradient parameter.

Equation (u) shows that the stress variation occurs as a result of the

pressure gradient and the flow accelerations, the latter being zero at the wall itself. In the sublayer it is permissible to neglect the flow accelerations so that

= i + (12)

is a good approximation. _{Further away from the wall, in the} fully-turbulent region, say, the acceleration term is not negligible. Experiments indicate that the stress distribution is nearly linear

T* ₌

lTY

₍₁₃₎

6

where A is a non-dimensional stress-gradient parameter,

T pU y

indepefldent of y* in the wall region iediately outside the blending region.

There are two limiting cases of interest. I-nj a flat-plate boundary layer

it can be shown that the accéleration terms are small so that T* = 3.

throughout the wall region. Notice that this, in conjunction with the

-mixing-length relations of equations (6) and

(7)

lead to the familiar

linear relation,

for the sublayer and the logarithmic law,

i

u = - in y* + B

for the fully turbulent region. These laws can be used in equation (n) to check, retrospectively,, the assumptIon that the acceleration terms are indeed negligible. The second limiting Case is the flow in the

neighborhood of separation. Here again the flow acceleration can be

neglected so that A becomes equal to A. The mixing lengthfomula then

leads tO the well known one-half power law for the velocity distribution as suggested by Stratford (1959).

It will be noted that equation (13) is an empirica], fit to experimental data introduced in order to obtain the velocity distribution

in closed form expressions. This relation was first suggested by

Townsend (1961) and has since been used by a number of authors. In

reality, however, the stress gradient falls continuously through the

wall region from its value equal to the pressure gradient at the wall.

An attempt to represent the stress variation through the blending region,

i.e., one in which A = A near the wall and A = constant further away

T

p

T

from it, has been made by McDonald (1969), who suggested a series expansion

for the stress gradient with terms involving a tanh function similar to that in equation (8).

Equations (12) and (13) can now be used in conjunction with the mixing-length model to obtain expressions for the velocity distributions

in the sublayer and the fully-turbulent region. Integration

or

equation

(5)

using equations

(6)

and (12) leads to the velocity distribution

in-in the sublayer

u* =

(15)

In a similar manner, equations (5),

₍₇₎

_{and (13) give the velocity} distribution in the fully turbulent region in the

(1 + *)½

i-u* =

when A , end therefore

p

positive for adverse pressure gradients.

T

form

½

+2{(1+Ay*)½_1]

p+B* ,(17)

T

(1 + Ay) + i

p

A , are both zero. Note that A and A are

T _{p T}

pressure gradients and negative for favorable

In order to obtain the velocity distribution in the blending region and thereby to find B*, it is of course necessary to integrate

equation (5) using the more complicated mixing-length relation _{given in}

equation (8) and a continuous stress variation model buch as that suggested by McDonald. _{The integration can be performed numerically and the results}

should join smoothly into equations (16) and (17). _{Such a calculation is}

not attempted here but it may be of interest to note that B* _{is nearly}

a linear function of A ,viz

p

B* = B + 3.7 A (18)

(Patel and Head 1968). _{Using typical values of A and Aequations}

(i6)

and (17) are plotted in Figure 3.

Equation (17) was first obtained by Townsend (1961) and has been

used, with minor modifications, by Patel (l965a), Mellor (1966) and

McDonald (1969). Experimental verification of this relation is rather scarse.

While measurements of velocity profiles in boundary layers with _{large adverse}

and favorable pressure gradients indicate departures from the usual

loga-rithmic law (equation

₁₅₎

_{in the manner suggested by equation (17), the}

only direct comparisons with experiment made _{so far are those due to}

Patel and. Head (1968.) (reproduced here _{as Figure 1) and to McDònald (1969).} in order to check the validity of this relation it is of course necessary to measure the wall shear-stress and the velocity arid shear-stress profiles.

The wall shear-stress has to be measured quite accurately and by _{a method}

which does not rely upon the usual logarithmic law.

where B * is expected to be some function of the sublayer parameter A

B. Fully-Developed Flow in.a Pipe. Here the stress distribution

is exactly linear and. related to the pressure gradient by

2dx

2 d-x

where. a is pipe radius, r is measured from the pipe axis, while y is

the distance from thê wall. In dimensionless form this may be written

T*=i_

=l+y*=1+½y*

,

a*

t

_p

Ua

where a* . It cafl easily be showi that for moderate to high pipe

Reynolds numbers (say, Re > i0), the pressure-gradient parameter is small

and a* is large (t _2/a* = -20 Re7I'8). It is permissible therefore to assume = i in the neighborhood of the wall. This leads then to the

conclusion that the velocity distributions in the wall region in

fully-developed turbulent pipe flow and. flat-plate turbulént boundary layers are identical. ThIs has indeed been demonstrated to be the case by the

experiments of Head and Rechenbérg (1962).

For RerÍiolds numbers less than about i0 we may epect depä.rtures

from the usual inner-law equations (li-L) and (15). Suc1 departures have

been observed by Patel and Héad (1969) and may be attributêd to the influence

of eithér the radius of curvature (a*) or the increasingly favorable

pressure gradient ( ), the two effects being inseparable in this case.

p

The influence of transverse crvature on the velocity distribution

is considered in the two examplès that follow.

C. Thick Axisetric BOUrIda±Y Layer. Here we shall consider

the turbulent boundary layer develoing on a long cylinder of constant

radius a placed adally in an unifôm stream of velocity U If the

cylinder is long and slender enotigh the b,oündary layer will become thick

in comparison with the rad:ius of the c1inder. We may then expect the flow to be influeuiced by transverse curvature effects. Since the pressure

gradient is absent, the relevant boundary layer equations are

Bu

= )

1 ,tr. ₂₀

3x ay

ray

p

9

and (ur) +

(vr) =0

(21)

the notation being shown in Figure 5. By ana1or with the flat-plate

boundary layer, e again neglect the acceleration terms in equation

(20) to obtain the stress distribution in the neighborhood of the wall. (Note that this assumption can be verified, retrospectively, just as

in the case of the flat-plate boundary layer.) Thus,the stress distribution

is given by a T = T

-vr

or non-dimensionally by = a*+y* a* (22)

This expression is valid throughout the wall region, i.e. in the sub-layer, the blending region, as well as in the fully-turbulent region.

By following a procedure similar to that outlined for the two-dimensional boundary layer one obtains the velocity distribution in the sublayer in the form

u = a* £n (i + y*/a*) and that in the fully-turbulent region in form

1 (

ti +

U

- -

14*'

y,a,,

-- K

(i +

y*/a*)

+ i J

r

where B* is somewhat similar to 3* in that it is expected to be a

function of a* such that it reduces to the constant B in equation (is)

when a* = - , i.e. for a thin boundary layer.

We may note here several points of interest concerning the velocity distribution in the wall region of a thick axisymmetric boundary layer. _{First, we see that for sufficiently small values of a* significant} variations in stress occur through the wall region as a consequence of the flow geometry. The stress continues to fall monotonicälly from its value at the wall. This is to be compared with the situation in a

two-dimensional boundary layer developing in a favorable pressure gradient < O , < o) where the stress falls from its value at the wall as

p

T

lo

It isnot surprising therefore to find that equations (17) and (214) show similar departures, from the usual logarithc law obtaining in

a fiat-plate boundary layer (compäre the velocity distributions in

Figure 3 for t < O with those in Figure

6).

Secondly, vemy compare

the velocity distribution in the fully-turbulent region, equation (214),.

obtained via mixing-length arguments with the one proposed earlier by Rao

₍₁₉₆₇₎

from purely heuristic considerations, namely

= - 2n {a*Zn (i + y*/a*)} + B (25)

where B is the usual additive constant of equation (15). Rao obtained

this expression simply by replacing y* by the quantity a*9..n (i + y*/a*)

in equation (15). This substitution of variables was justified by Rao on the basis of the òbservation that a similar substitution in the usual sublayer law, equation (i14), gives the correct sublayer law, equation (23), for axisymmetric flow. Be: that as it may, it is indeed surprising to

find that, if the expected departure o B from B. is ignored, the velocity

distributions given by equations (214) and (25) are in excellent agreement with each other over a wide range .of variables. This is easily demonstrated by comparing

(i + y*/aI)½

-with { 2n (i _{y*/a*) }}

(i + y*/a*)½ + i

j

to show that the disagreement is of the order of 1 percent for .y* = a'

and less than 8 percent for

y

= 6a*,

the latter value of y* being expected

to be well outside the range of applicabi.ity of the

wall

region analysis.

The,two laws may however disagree or account of the fact that the values.

of Br* in the present formula may be considerably different from B.

In order to obtaIn the velocity distrIbitioi that is continuous

through the sublayer, the biendng region, as well as he fully-turbulent

region it is of course necessary to integrate equation

(5)

using the

continuous mixing-length model of equation

(8)

and the stress di stribut ion

(26)

given by equation (22). Thus, we have y* * = ) a

)]2

;. J

&r

_J { 1 + [1 + 141(21*2 tarxh(X2y*2) _a*+y* 2(

_a**

This expréssion has been integrated numerically for a range of values of a* using K = o.1l8 and X = v'/63, and the resulting velocity distributions are shown in Figure 6. The results of these calculations can be compared

with equation (21f) for alues of y* jying in the fully-turbulent region

to obtain a suitable expression for the variation of B* with a*.

We shall compare the velocity laws obtained in this section with available experimental data from thick axisymmetric boundary layers

in a subsequent seôtion following some discussion of yet another class of flows which has received considerable attention from previous investi-gators.

D. Fully-Devèloped Flow Through Concentric Annuli. Here we

consider fully-developed turbulent flow in the annular space formed by two concentric cylinders of radii a. and a, subscripts i and o denoting

inner and outer cylinders, respectively. This flow has been investigated

experimentally and theoretically by many workers over a number of years. Among the most compréhensive studies are those due to Rothfus, Monrad

and Senecal

_(1950),

Owen

(1951),

Knudsen and Katz

_(1950),

Brighton and

Jones

_(l96I),

Quarmby

(1967),

Levy

(1967),

and Lawn and Elliott

_(1971).

One of the major observations of these studies is that the velocity

distribution near the inner wall does not conform with the usual

logarith-mic law of equation (15). The profiles on the outer wall are, however,

adequately correlated on the basis of this equation. The disagreement

with the usual inner law on the inner surface has generally been attributed

to the influence of transverse curvature. Some authors have indeed used

eddy-viscosity and mixing-length arguments to demonstrate that the observed departures are to be expected. In what follows we shall again apply the mixing-length analysis to analyse the velocity distribution in the region close to the inner and outer walls using the appropriate stréss variation. while this will undoubtedly duplicate some of the previous efforts it

will serve several useful purposes. First of all, it will enable us to

obtain a unified view of the success of mixing-length analyses in predicting the velocity distributions in turbulent flows close to a solId wall regardless of the nature of the flow farther away from the wall. Secondly, it will show that the flow in the wall regions of (i) a

two-dimensional boundary layer developing under a. favorable pressure-gradient, a thick axisyimnetric boundary layer on a long slender cylinder, and

12

closely similar dynaznícs insofar as the velòôity and shear-stress dis-tributiöns are' concerned. ina1ly, unlike the previois studies, we shall

Obtain the velocity distribution not only in the

fully-turbulent

region

but

also

in the sublayer

änd

the blending regions.

If x is measured in the direction of flow

and

r is the radial

distance measured from the common

axis

of the inner and outer cylinders,

it isreadily shown that the stress distribution in the annulus is given by

and

a.

a.

a.

= __:_ + . _2:. (-r- -T r

dxT

'a. r' w.. V. 1

i

or by i = T

r

2dxt

a

r

w w o o

0

where T

and T

are the wall shear stresses at the inner and outer

i o

cylinders, respectively. If we now measure distances y and y0 froni the

inner and outèr walls, and introduce non-dimensional quantities as before,

these expressions beöoine

T*

where the subscripts i and o indicate that.the variables have been made

non-dimensional by using the shear velocity UT ₌ or

U

IT/

. Equations (29) and (30) show immediately. that in this

case the stress variat-iöns occur not

only

as a result of transvere

curvature but also due to the favorable pressure

gradient

driving the

flow.

The relative importance of the two .nfluences will obviously

depend upon the parameters a.*, a0*, 4

and

,

or alternatively,

upon the radius ratio a./a0

and

the flow Reynolds nunber.

(

i

i½

pi

1-

½

a.*

a* (2

y.* y*

(29)

(30)

a*+y.*

a*

°

y*

)

y*

=

a*

y0* _{

13

The velocity distribution in the wall regions of the outer and inner cylinders can now be found by again integrating equation (5)

making use of the stress distributions given above and the mixing-length expressions of Sectionill. The velocity laws 'obtained by performing the integration numerically will be compared with experimental results in the next section. It is, however, interesting to note here that in

situations where the pressure gradient terms can be neglected in equations (29) and (30), the velocity laws can be obtained in closed f rm for the

sublayer and the fully-turbulent regions. The laws for the innêr wall

are then identical with, those for the thick axisymmetric boud' y layer given previously.

V. PERIMENTAL VIFICATION 0F TIE. INFLUENCE OF TRANSVERSE CURVATURE.

We have already mentioned that the influence of pressure

gradi-ents on the law of the wall in the case of thi two-dimensional boundary

layers has been studied by a number of workers. The expressions obtained

from mix±ng-length analysis (equations 16 and 17) have been found to give

substantial agreement vith experimental data. In this' section we shall

therefore restrict our attention to the influence of transverse, curvature. in order to verify the expressions för the velocity distribution obtained in the previous section we shall rely upon data from two types of

experiments: measurements in thick a.xisymmetric boundary layers and

those ïn fully-developed turbulent flow in concentric annuli. Both flows

have been investigated experimentally over a wide range of parameters. Considering the thick axisynimetric boundary layer first, we find that this flow has been examined experimentally by Richmoiid (1957), Yu (1958), Yasu.hara (1959), Keshavan '(1969), and WilJ.marth and Yang(1970),

and theoretically by Landveber (19I9) and Cébeci (1970). The vide range

of variables encountered in these experiments can be' seen from the data summary given in Table 1. It should be emphasized that this table does not contain al]. the measurements that are available; many more measurements vere examined during the course of this investigation and Table 1 was

l4

ICeshavari ánd Yu measured velocity profiles at severa]. streamwise statiQns for each cylinder diameter, we have listed only the measurements at

the more downstream stations so as to emphasize the influence of

trans-verse ciirvature.

It wifl be clear that in order to compare the measured velocity

distributions with the expressions of the previous section it is necesary

to determine, independently, the wall shear stress. In Table 1 we have

listed the values of the wall shear-stress coefficient, Cf quoted by

the originators. The values quoted by Richmond for his measurements

on the smaller cylinder are thought to be in error. In the experiments

of ICeshavan the wall shear stress was deduced from the measured streainwise

momentum-thickness development by applying the momentum-integral equation..

As coimriented upon by a number of previous investigators, this method is

not altogether reliable. In view of this, much of Keshavan's subsequent

analysis of his data must be regarded with caution. In situations where

such doubts existed concerning the correct value of the wall shear stress,

we followed an alternative method to determine Cf. This is the method due

to Clauser

(1956).

In this method, a chart is made by plotting the usual

flat-plate law of the wall in a parametric form, the parameter being Cf.

The measured velocity profile is then superimposed on this chart and the value of Cf read d-irectly by examining the measurements close to the wall where the law of the wall is expected to apply. rpical "Clauser plots"

for two profiles measured by Richmond and one by Keshavan are shown in Figure 7. This method of determining Cf applies strictly to flows in which the usual flat-plate law of the wall is expected to be valid.

Nevertheless, since the departure from this law in the present case is

gradual, being s,n1l in the sublayer and. the blending regions, the method

gives a good approximation to the actual wall shear stress provided due

emphasis is placed only on the experimental points lying in the sublayer and. blending regions. Where such measurements are not available, as for exainple in the profile of Keshavan shown in Figure

7,

a reasonable

extrapolation of the data points can still be made to estimate the value

of C.

The valuès of the skin-friction coefficient determined by the

differences betweén these and the values quoted by Richmond and Keshavan. For the experiments of these two authors we have also tabu-lated the

values obtained theoretically by Cebeci (1970). In this recent paper,

Cebeci obtained numerical solutions to the eq.uatiòns of a thick

axisym-metric turbulent boundary layer on a slender cylinder using an eddy-viscosity model. The skin-friction coefficient was found to -depend primarily upon R, the Rernoìds number based on the cylinder radius, and only marginally

upon R, the Reynolds number based on the streamwise distance. The values

of Cf attributed to Cebeci's theory in Table 1 were obtained from Figure 13

in his paper after estimating the value of R in each case. It will be

seen that the theoretical values of Cebeci are in excellent agreement with those obtained by. the method of Clauser for the experiments of

Keshavan. For the smaller cylinder of Richmond, however, the theory of Cebeci predicts values of Cf which are much higher than those indicated

by the Çlauser plots. A possible reason for this discrepancy will be

discussed later on.

The value of thé parameter a*, which is a measure of the transverse curvature as it áffécts the flow in the wall region, was obtained for each case by using the quoted value of Ra and the value of

Cf indicated in Table 1, since a* = R/Cf/2 . Notice that the available

experiments cover a very wide range of this parameter, the smallest value

of a* being about 6 in the experiment of Richmond. Using the known value

of a*, the integral in equation (26) was evaluated numerically to obtain

the variation of u* with y* as predicted by mixing-length theory. The

results of these computations were compared with the measured velocity distributions. Ty-pical results, covering a representative ránge of values

of a*, are shown in Figures 8, 9, and 10. These, and similar comparisons

not included here, show several features of interest. First of all, it is

seen that the experiments indicate a very significant influence of trans-verse wall curvature ön the velocity distribution in the wall region of the boundary layer. The departures of the velocity distribution from the usual

flat-plate law of the wall are qualitatively similar to those observed in

thth plane-surface boundary layers developing under favorable pressure

gradients. The curvature parameter a* adequately describes the influence

pi

T = w. J-a. 2. i - a.* a J. O

i6

ixing-lént]:i theory appears to predict the velocity distribution. in the

wail region (y* of the order öf' 2a*) quité well for values of a* greater

than about 30. For a* less than this vaitlé (as in the Ra

= 93.8

end. 253

experiments of Richmond), there are signfiòant discrepancies between

the meàsured profiles and mbdng-length predictions While these may, in

part, be due to the difficulty of obtaining thé correct value of the wall

shear stress, rather gross changes in C are required to reconcile

mi,dng-length theory ìith experiment, and such óhanges are not indicated by the C1ausr plots. A alternative explanation must therefore be offered for

the failure of the mixing-length theory, and also for thé observation that

the experiientai values of Cf are considerably lower than the theoretical

values of Cebeci, for a* less than about 30. Such an explanation is

attempted in the next sectiön.

Considering next the measürements in fully-developed turbulent flow in concentric annuli we find that there should be no difficulty in

this case concerning the wail shear stresses on the inner and outer surfaces.

since these are determinéd by thé streamwise pressure gradient (dp/dx) and

the location of the zero stress point (r r) in the annulus. Using

momentulii considerations t is easily verified that

2 - a 2 r 2a. dx 1 T

-a. w _2. + a T o i (31) (33) and.

a 2r

2 T 2a dx z. (3?) o o

Elimination of r between these equations leads to a useful. relationship

between the dimensionlesS essure-gradi-eflt and curvature parameters

17

It will be recalled that the stress distribution in the annulus can be found from equations (29) and (30) once the values of the parameters

a , and are known.

i o

p

As we have already mentioned., the cöncentric annulus flow has

been investigated by a large number of workers. In the earlier

measure-ments the assumption has been made that the zero stress point coincides

with the positioji of maximum velcity. Although all experimenters agree

on the qualitative features of the flow insofar as the behávior of the

-friction factor and veIôci-y distribution is concerned, there is' much cOntroversy concerning the exact location of the zero stress point,' and

its dependence on thé radius ratio and Reynolds number. This controversy

of courSe carries over to the determination of the wall shear stresses

and consequently to the applicability of the law of the wall. In view of

this, ve have chosento analyze the most recent experiments of Lawn an

Elliott

₍₁₉₇₁₎

where, instead of relying upon the location of the maximum

velocity point (Brighton and Jones, ₁₉₆₁₄₎ or upon Preston tube measurements

(Quarmby,

1967),

_{the zero stress point is determined directly from hot-wire} measurements. The authors demonstrated the accuracy of- their stress

measurements by comparing them with equations (27) and (28). Fortunately,

these experiments cover a range of radius ratios and Reynolds numbers that not only duplicates previous experiments but also makes a meaningful

comparison with mixing-length theory possible.

From the values of the wall shear stresses given by Lawn and

Elliott the curvature 'and pressure-gradient parameters were first determined

using the equations given above. These are summarized in Table 2 tO indicate

the range of variables encöuntered in the experiments. It will be seen that

the pressure gradient parameter' for the outer wall is larger than that for

the inner wall but, for the onditions of these experiments, the valués on

both. surfaces are considerably less than those for which significant departures and Po =

Pl

T w. J-3/2 (314) T w o

18

from the usual law of the wall have been observed in thin plane-surface

boundary layers (Figure 3). This suggests that the departures observed

in the annulus experiments are primari1 due to the influence of

trans-verse curvature. The. curvature parameter a* is of course smaller for the inner surface indicating that the curvatuz'e effects v1111 first become

prominant Í'or the flow near the inner walL This is in accordance with

expérirnentál óbservations.

The values of the paraméters a* and shown in Table 2 were

used in equation (29) to find the st±ess variation, and. the velocity

distribution was computed from equation

(5)

using the continuous

mixing-length model of eqjiation (8). Tyieál comparisons with experimental data

are shown in igure 11. Once again it is seen that mixing-length theory

predicts the departures of the velocity distribution from the usual law

of the wll with acceptable accuracy. It is in fact rather surprising to

find that the theory agrees with experimental data right up to the

loca-tion of the zero-stress point in the annulus. This is somewhat similar

to the often Úiade observation that, with minor changes in the constants,

the well-known logarithmic law, equation (15), can be made to agree with

the vèlocity distribution in fully-developed pipe flow right up to the center of the pipe.

VI. DISCUSSION

Attention has already been drawn, to the rather remarkable similarity between thé influence of a favorable pressure gradient and

that of transverse wall curvature insofar as the stress and velocity

distributions in the wall region are concerned. As has been demonstrated

by Patel and Head

(1968),

there is a definite limit to the favorable

pressure gradient that can be imposed on a turbulent boundary layer

with-out destroying the essential equilibrii. that ed.sts between the pròduction

and dissipation of turbulent kinetic enerr in the wall region, and thereby

provoking relaminariZatiOn. For favörable pressure gradients larger than

this limiting value, the flow in the boundary layer can notbe maintained

in a fülly turbulent state. From their experiments, Patel and Head

19

criterion for the onset of this reverse transition process. In view of

the observed similarity between the influence of favorable pressure gradients and transverse wall curvature, we inquire whether there exists a similar

limiting value of the curvature parameter a below which the boundary layer

on a slender cylinder, and also the. flow closéto the inner wall in

concentric annuli, can not be maintained in a fully turbulent state. If

we compute the stress gradient at the edge of the blending region for the former case from equation (22) and set it equal to -0.009, we find that

the corresponding value of a* is about 28. If the similarity between

favorable pressure-gradient and transverse-curvature effects can be extended in this manner, the result shows that the flow in the neighborhood of

slender cylinders with values of a* less than about 28 frust be regarded

as transitional and not fully turbulent. Mixing-length theory will obviously

fail to apply in such cases. This appears to explain the large discrepancy observed earlier between the measurements of Richmond for a* = 6.0 and

13.85

(Figure ₈₎ and the velocity distributions predicted by mixing-length

theory. Furthermore, if such an interpretation is accepted, it is not surprising to find, as commented upon by White (1970), that the wall shear stressés measured at these low ialues of a* are considerably lower than those calculated from theories, such as that of Cebeci (1970), which assume

the existence of fully turbulent flow. The limiting value of a* for fully

turbulent flow on slender cylinders postulated here needs to be confirmed. by direct experimentation.

The present analysis of the influence of transverse curvature on the law of the wall also sheds somelight on the controversy concerning the use of Preston tubes to measure the wall shear stress on the inner wall of

a concentric annulus. Experiments as well as midng-1ength theory have shown that the usual universal law of the wall obtaining in flat-plate

boundary layers and fully-developed pipe flow does not apply in the boundary layer on slender cylinders and also on the inner wall of concentric annülus

flow. This suggests that it is not permissible in these cases to use the

Preston tube method of determining the wall shear stress. Since, however,

the departures from the universal laws are small in the sublayer and the blending regions, it shouldbe possible to obtain the correct value of the wall shear stress by making measurements with several Preston túbes of different diameters and then extrapolating the results to a hypothetical tube of zero diameter.

20

VII. CONCLUSIONS

It has been shown that, if the wéll-known mixing-length formula is viewed simply as a relationship between the velocity and the stress

distributions in the wall region of a turbulent flow, then a truly universal

distribution of mixing-length in the form 9.*(y*), used in conjunc.tiön with

the appropriate stress variation, is sufficient to determine the velocity distribution. The experimentally observed influence of streamwise pressure gradients on plane-surface boundary layers, and. of transverse wall curvature on thiòk axisymmetric boundary layers and fully developed flow in concentric

annuli, is then predicted with acceptable accuracy. It is' not necessary, as

has been attempted by a number of previous workers, particularly with regard to curvature effects (see, for example, Ktiudsen and Katz or Keshavan), to

correlate the experimental data on the basis of variations of the. constants

K and B appearing in the usüal logarithmic law, equation (15).

The general success of the mixing-length relation shown here may

be regarded as a demonstration of the universal nature of wall turhulence.

as postulated in recent turbulent kinetic-enerr theories. Furthermore, the

present work lends support to the often made claim, particularly by workers

developing boundary-layer calculation methods, that mixing-length and

eddy-viscosity models work considerably better in practice than one has aiy right to expect from Prandtl's original physical model so well described in testbooks.

21

REFERENCES

Batchelor, G.K. 1950 "Note on Free Turbulent Flows, with Special Reference to the Two-Dimensional Wake," J. Aero. Sci., 17, 14141.

Brighton, J.A. and Jones, J.B. ₁₉₆₁₄ "Fully Developed Turbulent Flow in

Annuli," J. Basic E'ig., Trans. ASME, Ser. D,

86, 835.

Cebeci, T. 1970 "Laminar and Turbulent Incompressible Boundary Layers on Slender Bodies of Revolution in Axial Flow," J. Basic E'ig., Trans.

ASZtfE, Ser. D,

92, 5145.

Clauser, F.H. 1956 "The Turbuleit Boundary Layer," Advances in Applied

Mechanics, 14, 1.

Deissler, R.G. 19514 "Analysis of Turbulent Heat Transfer, Mass Transfer and Friction in Smooth Tubes at High Prandtl and Schmidt Numbers," NACA Tech. Rep. 1210.

van Driest, E.R.

₁₉₅₆

"On Turbulent Flow Near a Wall," J. Aero. Sci.,

23, 1007

and

1036.

Head, M.R. and Rechenberg, I.

₁₉₆₂

"The Preston Tube as a Means of Measuring Skin Friction," J. Fluid Mech.,

114, 1.

Keshavan, N.R.

₁₉₆₉

"Axisyiimetric Incompressible Turbulent Boundary Layers in Zero Pressure Gradient Flow," M.Sc. Thesis, Indian Institute of Science, Bangalore, India.

Knudsen., J.G. and Katz, D.L.

₁₉₅₀

"Velocity Profiles in Annuli," Proceedings

of

Midwestern Conference on Fluid Mechanics, 1950.

Landweber, L. 19149 "Effect of Transverse Curvature on Frictional Resistance," David Taylor Model Basin Report 689.

Lawn, C.J. and Elliott, C.J.

₁₉₇₁

"Fully Developed Turbulent Flow Through

Concentriò Annuli," Central Electricity Generating Board, Berkeley,

England, Report RD/B/N1878.

Levy, S. 1967 "Turbulent Flow in an Annulus," J. Heat Transfer, Trans. ASME,

89, 25.

McDonald, H.

₁₉₆₉

"The Effect of Pressure Gradient on the Law of the Wall in Turbulent Flow," J. Fluid Meöh.,

35, 311.

Mellor, G.L.

₁₉₆₆

"The Effects of Pressure Gradients on Turbulent Boundary Layers," J. Fluid Mech.,

24, 255.

Newman, B.G.

₁₉₅₁

"Some Contributions to the Study of the Turbulent Boundary Layer," Aust. Dept. Supply, Report ACA-53.

Owen, W.M.

₁₉₅₁

"Experimental Studj of Water Flow in Annular Pipes," Proc. AScE, 77, Sep. No.

88.

22

Patel, V.C. 1965a "Contributions to the Studr of Turbulent Boundary Layers," Ph.D. Thesis, Cambridge University, England.

Patel, V.C. 1965b "Calibration of the Preston Tube and. Limitations on It's

Use in Pressure Gradients," J.

Fluid

Mech.,

23, 185.

Patel, VC. and head, M.R.

₁₉₆₈

"Reversion of Turbulent to Laminar Fiov,"

J.

Fluid

Mech.,

_{314, 371.}

Patel, V.C.. and Head, M.R. 1969 Velocity Profiles in Fully Mech.,

38, 181.

Quarrnby, A. 1967 "An Experimental Study of Turbulent Flow

Annuli,"

mt.

J. Mech. Sci.3 j, 205.

Rao, G.N.V.

1967

"The Law of the Wall in a Thick Axisy etric Turbulent Boundary Layer," J. Basic Eng., Trans. ASME, Ser. D,

89L, 237.

Réichardt, H.

₁₉₅₁

"Voi1stndige Darsteiling der Turbulenten

Geschwindig-keitsverteilung in Glatten Leitungen," Z.A.M.M., 31,

208.

Rich±ond, R.L.

1957

"Experimental Investigation of Thick Axially Symmetric Boundary Layers on Cylinders at Subsonic and HypersQnic Speeds,"

Ph.D. Thesis, California Institute of Technolor, Pasadena.

RothÍ'û.s, R.R., Monrad, C.C. and Senecal, V.E.

1950

"Velocity Distribution

and Fluid Friction in Smooth Concentric Annuli,"

md and Eng

Chem

,

142, 2511.

Stratford, B..S. 1959 "The Prediction of Separation of the Turbuleirt Boundary Layer," J.

Fluid

Mech.,

5, 1.

Townsend, A.A.

₁₉₆₁

"Equilibrium Layers and Wall Turbulence," J.

Fluid

Mach., 11, 97.

White, F.M.

₁₉₇₀

"Written Discussion of the Paper of Cebeci,

1970," J. Basic

Eng., Trans. ASME.,

92, 550.

Willmarth, W.W. and Yang, C.S.

1970

"Wall-Pressure Fluctuati.ons Bêneath

Turbulent Boundary Layers on a Flat ].ate and a Cylin4er," J.

Fluid

Ñech., 141, 147.

Yasühara, M.

₁₉₅₉

"Experiments of Axisymzaetriç Boundary Layers Along a

Cylinder in Incompressible Flow," Trans. Jqpan Soc. Aerospace Sci.,

?L,

33.

Yu, Y.S.

₁₉₅₈

"Effect öf Transverse Curture on Turbu1et Boundary Layer Characteristics," J. $hip Research, 3, 33.

"Some Observations on Skin Friction and Developed Pipe and. Channel Flows," J.

Fluid

23

L

*

50

40

30

214

50

_y*

lOO

45

40

35 30 25 20 15 IO 5 o 25

-

0.05 0.025 0.005

00

o -1 -0.018 -0.009

-2 -0.04

-0.02

- -3 -0.08

-0.04 i I _{I 11111}

lp

T 0.25 I I 111111 EQUATION (I?) EQUATION (16) I I I I

I lilt

I I I I

i Ill

I -

i liii

4 0.50 3 0.10 2 0.05 I 0.01 to

r

102

Figure 3. The Influence of Pressure Gradient on the Law-of-the-Wall.. A Plòt of

50 45 40 35 30 25 20 15 Io - Equation (16) Equation (14) Equation ('7)

I

Station G F D Equation (15) X X _¿ z X X

o-o

I. I l0 26 102 _io3

Figure )4 _{Velocity Measurements of Newman (1951)} Compared with I"lixingLength Theory.

8 .0094 _.0045

D .0279 .0141

F .1127 .075

Ue

y

w A a y 27

Figure 5. Definition Sketch for the Thick Boundary Layer on a Cylinder.

30

I

II

y1t Figure

6.

The Influence of Transverse Curvature on the Law-o'-the--Wall; A Plot of Equation (26).

i I I

ut

I I i i

ill

I0

o

IO

102

UeY

1/

Figure 7. Determination of Wall Shear

u* 30 20 I0 I0

ye

102 Figure 8.

30 20 I0

l0

io

yR

102

Author(s) Vu _WIIImorth & Yortg

Figure IO. Comparison of the

Measurements of Yu, and Willmarth

and

I I I I

I uil

i i i

i uil

i I I i

liti

io 102

Figure 11. Comparison of Mixing-Length Theory with the

Measuremen-ts of

Lawn and Efliott in Concentric Annuli.

30 c/a I Re xi5 I

Ill

o Theory X 0.088 0.362 120 I 0088 0.577 $80 2 o 0.088 1.200 335 3 + 0.176 0.94$ 567 4 20 5 I

_I,.

F

I0 ____

-Table 1.

Summary of Data on Thick Axisymmetric Turbulent Boundary

Layers

*jndjcates the value of Cf used

to find a* Author(s) a inch x feet R a Cf Author Cf Clauser Plot C. Cebeci Theory a* Richmond .012 16.0 93.8 .0059 .0082* .0i46 6.00 (1957) .012 16.0 253 .0052 .0060* .0101 13.85 .5 10.0 40,200 .0029 .0030* 1555 Yu(1958) 1.0 8.0 15,250 .00318 .0036* 646 1.0 8.0 30,7140 .00282 .0030* 1192 1.0 7.0 145,060 .002614* .0026 1638 Yasuhara .3914 3.28 21,800 .003147* .0035 908 (1959) Keshavan .0625 1.29. 1425 .0068 .0085 .0088* 28.2 (1969) .0625 1.29 825 .0022 .0075 .0072* 49.6 .125 1.29 1320 .0059 .0065 .00614 714.5 .125 1.667 16140 .00147 .006o .0056* 87.6 .25 i.66 39140 .0055 .00514 .0050* 197.0 Wiflmarth & 1.5 24.0 70,200 .00276* .0028 2600 Yang (1970) 1.5 214.0 115,Q00 .00219* . .0022 3800 1.5 24.0 1314,000 .00230* .0023 14550

Table 2.

A Suninary of the Concentric Annuli Data of Lawn

and Elliott

(1971)

Radius Ratio

aia

Reynolds Number U T

i

ft/sec U T o

ft/sec

U T o

a.*

i

-1

p.

i

a *

0 o UT.

.088

.362

0.914 0.714

.7872

120

.0010146

1069

.0021414

.577

1.142

1.12

.7887

180

.000700

i6i14 .0011427

1.20

2.614

2.08

.7879

335

.000375

3000

.000767

2.01

14.30

3.39

.78814 5146

.000230

14892 .0001469

2.37

5.01

3.95

.78814

636

.000198

5702

.00014014

.176

.1496

1.26

1.06

.81413 320

.001003

1530

.00i684

.640

1.59

1.33

.8365

1404

.000787

1920

.00i345

.9141

2.23

1.87

.8386

567

.000563

2700

.000955.

1.38

3.11

2.61

.8392

791 .00014014

3765

.000684

2.19

4.71

3.95

.8386

1198

.000267

5700

.0001i53

.396

.350

1.16

1.08

.9310

662

.001792

1558

.002221.

.613

1.90

1.77

.9316

1085

.0010914

2550

.001353

.854

2.514

2.37

.9331

11452

.000819

31420

.001008

1.24

3.514

3.30

.9322

2022

.000588

14760

.000726

1.61

14.36

4.05

.9289

21492 .0001475 5850

.000593.

Sttirì ( ('1 sst (i(.It Ion

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_{- ...,- j.,..} t

u UNA TI N G AC TI Vt t Y (('..rporate author)

Institute of Hydraulic Research The University of Iowa

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"A Unified View of the Law of the Wall Using Mixing-Length Theory"

4 DESCRIPTIVE NOTES (Type of report and nclusivc ¿atc) IfliR Report No. l37

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II SUPPLEMENTARY NOTES - - 12. SPÖNSORING MILITARY ACTIVITY

Naval Ship Research and Development Center

13, AØSTRACT

It is shown that, if the well-known mixing-length formula is regarded simply as a relationship between the velocity and the stress distri-butions in the wall region of a turbulent flow, then a truly univeral d-istri-bution of mixing length is sufficient to describe the experimentally òbserved

departures of the velocity distribution from the usual law of the wall as a

re-sult of severe pressure gradients and transverse surface curvature. Comparisons

have been made with a wide variety of experimental data to demonstrate the general validity of the mixing-length model in describing the flow close to a smooth wall.

An extension of the relaininarization criterion of Patel and Head, and some xperimental evidence, suggest that the thick axisyetric boundary layer on a slender cylinder placed axially in a uniform stream cannot be majntained in

a fully turbulent state for values of the Reynolds number, based on friction velocity and cylinder radius, below a certain critical value.

D

Uncias sifi ed.

'UtiIV ('1.ui t iet ton

FORM 4

_{i473 (BACK}

.4

--KEY WORDS

-LINK A LINK U . -:

ROLE WT ROLE WT hQL f.

Thrbu.ient boundary 3 ayer s

Law of the

a1l

Mixing-length theöry Pressure gradient Transverse curvature Reiam-inarizatiOfl i-r

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