Flow properties of dilute solutions of polymers. Part 3: Effect of solute on turbulent field

ARCHEF

_Techn!sche

_oc!icoJ

FLOW PROPERTIES OF DILUTE SOLUTIONS OF

POLYMEI'

PART 111:-EFFECT OF SOLUTE ON TURBULENT FIELD

by

F. W. Boggs

J. Thompsen

December 1967

FINAL REPORT - PART

Contract Nos. Nonr 3120(00) and N000I4-66-0O322

Distribution of this document is unlimited

Prepared for

OFFICE OF NAVAL RESEARCH

UNITED STATES RUBBER COMPANY

(now UNIROYAL,Inc.)

RESEARCH CENTER

WAYNE, NEW JERSEY

i

i

ei

i i

FLOW PROPERTIES OF DILUTE SOLUTIONS OF POLYMERS

PART III - EFFECT OF SOLUTE ON TURBULENT FIELD

by F. W. Boggs J. Thompsen

December 1967

FINAL REPORT - PART III

Contract Nos. Nonr 3120(00) and N00014-66-0O322

Distribution of this document is unlimited.

Prepared for

FOREWORD

This is Part III of the final report on the work performed at the

Re-search Center of the United States Rubber Company under Contracts Nonr

3120(00) and N00014-66-0O322. These contracts were under the direct

super-vision of the Mathematical Sciences Disuper-vision Fluid Dynamics Branch, Code

438, Office of Naval Research with Mr, R. Cooper and Mr, P. Granville acting

as technical monitors.

This program was conducted by the Polymer Physics Department at the

Research Center of the United States Rubber Company (now Uniroyal, Inc.), Wayne, New Jersey. Responsibility for the program resided with Dr. F. W. Boggs and Mr. J. Thompsen. The administrator of Contract Nonr 3120(00)

was Dr. R. H. Ewart, Manager of the Polymer Physics Department. The

follow-on cfollow-ontract (N00014-66-0O322) was administered by his successor, Dr. E. G. Kontos.

ACKNOWL EDGMENTS

Many of the ideas which led to this work were suggested to one of the

authors by Dr. Werner Pfenning of Northrup Aviation. This is particularly

true of the study of the vortex structure. The authors gratefully acknowl-edge this important help. The authors also owe much to Dr. James L. White,

then of U. S. Rubber Research Center, and to Dr. Noboru Tokita of the same

SUMMARY

This third part of the report on the properties of polymeric

solu-tions discusses the effect of the solute on the turbulent field.

The equations for the turbulent fluctuations in a second order fluid

were examined. They are markedly different from those which can be

de-rived from the Navier-Stokes equation. The linear terms show character-istic relaxation effects. The quadratic terms include not only the

Reynolds stress tensor but higher derivatives of it as well as other

quadratic terms.

These terms change the relation between the velocity fluctuation

and the pressure fluctuation; they also change the velocity fluctuations.

Strong damping of vortices close to the wall having axes in the direction

TABLE OF CONTENTS

Pa ge

SUMMARY _iii

LIST OF SYMBOLS _Vi.

1.0 INTRODUCTION . _i

2.0 EFFECT OF ADDITIVES ON HOMOGENEOUS TURBULENCE ₃

3.0 EFFECT OF ADDITIVES ON RELATION BETWEEN

PRESSURE AND VELOCITY FLUCTUATIONS ₆

4.0 AN APPROXIMATE TREATMENT OF THE EFFECT OF

ADDITIVES ON VELOCITY FLUCTUATIONS ₁₂

5.0 _{EFFECT OF ADDITIVES ON VORTICES NEAR WALL 20}

6.0 CONCLUSIONS ₂₈

7.0 REFERENCES ₂₉

APPENDIX - CALCULATION OF VORTEX ARRAY IN SUBLAYER 31

LIST OF SYMBOLS

Cartesian tensor notation is used throughout. Roman lower case

sub-scripts always indicate the components of a tensor except where the subscript

r designates a radial component and is not summed, or when the subscript

is attached to a cylinder function and designates its order in the usual way, or when ít designates a member of a series of coefficients. Greek lower

case subscripts always designate components which are summed over a set of

functions used to develop a solution, except where designates an angle

and the subscript designates an angular component. The summation con-vention is used only for the tensor components.

A. vector potential for flow velocity with solute absent

1A. vector potential for flow velocity with solute present

amplitude of vortex in vortex array

A a vector potential

r

a distance of center of vortex from wall

-foe

iwt

A.e

dt

J Jo J

-oe

aa set of coefficients ure to develop cp.

B. e. e e

¡A

VA

j jrs ruy rum o u,v o m,n

+oe

r' iWt

'Be

dt

J

Ji

b set of coefficients used to develop b.

Q, _J

bn cross stream coordinate of

th

f

o dimensionless frequency

G..(r'-r0) Green's function for the vector Poisson's equation

J(t)

zero order Bessel function

th

m order Bessel function

£ (when not a subscript) characteristic length

characteristic value of

£ set of numbers into which £ is decomposed

lOE

£2 characteristic value of

_2j

£ set of numbers into which £ is decomposed

2OE 2

N..(r) three point velocity correlation due to viscoelastic forces

2 2

-r Lv(rl)vL(r2)V v.(r2) - v.(r2)v(r1)V V (r1)

n concentration of solute (in gm per cc)

P pressure

P pressure fluctuations in absence of solute

o

P1 pressure perturbation

P..(r) velocity pressure correlation

i v.(ri) - _r. P(r1)v.(r2) ) =

.(

P(r)

\ r1 2 3 R Reynolds number

velocity correlation tensor

= v.(r)vj(r)+v.(r)v..(r)

_ii

2

12 ji

r distance between two points in fluid

s y. . V.

1,J

3,1

r '11

si(x) sign integral

J d1

X

T(r)

three point corrtiation of velocities

t time

t, tl-t2

t1 time at instant i

t2 time at instant 2

U free stream velocity

flow velocity perpendicular to plate

y3 flow velocity in cross flow direction

y. average velocity

y., y. velocities of flow

i

.3

velocity fluctuation

vr radial velocity in vortex

v circumferential velocity in vortex

X parameter used in summing a series

x1 position vector at point i

position vector at point 2 vector position of point

Z(r)

th order cylinder function

angular position

F(K)

Fourier Stieltjes transform of T..(r)

Luk

antisymmetric tensor

dimensionless position vector

[TJ intrinsic viscosity of solute

e an angular parameter

characteristic value of fourth order partial differential

equation

X unit vector in direction of flow

j

unit vector perpendicular to direction of flow kinematic viscosity of solvent

kinematic viscosity

v2 coefficient of viscoelastic forces divided by density

rr..(K) Fourier Stieltjes transform of P..(r)

p density of fluid

T relaxation time of velocity field

T relaxation time of polymer molecule

'..(K) Fourier Stieltjes transform of N.(r)

c(K)

Fourier Stieltjes transform of R..(r)

cp(r,t') space time correlation of pressure in absence of solute

p(r,t') space time correlation of pressure in presence of solute set of characteristic functions in which p. is developed portion of vector potential of perturbed flow which satisfies

the equation v21

portion of vector potential of perturbed flow which satisfies

the equation V22cp. ₌

£2 2cpj

vector potential for change in flow induced by solute

function depending on direction perpendicular to direction of

flow

stream function for perturbed flow angular velocity

lj

2j

cpi 0 w

1.0 INTRODUCTION

In Part I of this report we showed how it may be possible to explain, in terms of the second order fluid, the reduction in drag induced in a liquid by the addition of a polymeric solute. The mechanism which we suggested was possible only in a turbulent boundary layer. Modification

.2

in noise level has also been observed in the flow of these fluids . Since

noise should not be expected in a laminar boundarj layer, and since the observed change in noise is a modification of its frequency distribution

rather than its complete elimination, the noise data also indicate modi-fication rather than elimination of turbulence. An analysis of the effect

of the additive on the turbulent field is, therefore, indicated.

It is difficult to explain accurately how the viscoelastic forces would affect the drag or noise in the absence of an adequate theory of

the turbulent boundary layer. However, considerable insight into these phenomena has already been obtained which did not entail a complete solu-tion of the differential equasolu-tions. We have shown that the viscoelastic

coefficient would have a very large effect in the boundary layer regions where the gradients of velocity or the gradients of the Reynolds stresses

are large1. If this is true, one would expect that the greatest

contri-bution of the additive to drag and noise reduction would involve areas

close to the boundary. This possibility has important practical results.

If the effectiveness is confined to a region close to the boundary, the amount of material necessary to provide large improvements may be quite

small. Injection of a viscoelastic fluid near the wall might lead to re-ductions in drag with very small total volumes of material. The results

of these theoretical considerations, may, therefore, have some very

prac-tical results. This part of the report is, therefore devoted to a

dis-cussion of

vortex structure and turbulence.

The discussion considers only the second order fluid employing,

made-quate when the time derivatives of the rate of shear are large. As a con-sequence, we will not consider any flow which, as it proceeds downstream,

suffers an abrupt change in velocity such as is found at leading edges or

stagnation points. The second order fluid may, however, be an adequate representation of the flow along a flat plate at points far removed from

the leading edge; a condition which is approached as the Reynolds number

increases. This treatment is, therefore, confined to the flow of dilute

(1 +

2.0 EFFECT OF ADDITIVES ON HOMOGENEOUS TURBULENCE

The turbulent boundary layer is ess'ntia1ly a shear flow at high rates

of shear. Consequently the turbulent field which it generates must be non-uniform. Despite this fact a brief discussion of a homogeneous turbulent

field will assist our understanding of the more complex inhomogcneous field.

In particular it will show how the behavior of the field is modified and

what the scale of the gradients must be to produce significant changes.

We can rewrite the basic equation in the forni given in Equation (2.1).

)

i + . - 1V2v + J, X.

/

vv =0

- + (2.1) This form has been chosen so as to obtain the maximum possible symmetry.

Now let us consider the correlation functions for the turbulent field. If we calculate them in the usual way (see, for instance, Batchelor, 'The

Theory of Homogeneous Turbulence"3) we obtain Equation (2.2).

R.,.

____ - T (r) - 2 (l

+ V

R . + P. .(r)

t -

ii

2t)

i] 1J

+ V2V2 T. .(r) 2 y N (r)

ij - 2 ii (2.2)

where the function R.., T.., and P.. are defined in Equations (2.3) through (2.5).

R..(r) = v.(r1) v.(r2) + v.(r2) v.(r1)

where r1 and r2 are position vectors of two points, and r = r1 - r2 T. .(r) = a _{v.(r )vk(rl)v.(r2) - v.(r2)vk(rl)v.(rl))} 13

ark\1

i (K) iKr = e dK J ii

i1a

-

a P. .(r) = 1.3 r1 P(r2)v.(r1) - a r. P(r1)v.(r2) .3 = S (K) eiKrdK ii

The function N.. given in Equation (2.2) does not appear in the

usual equation for isotropic turbulence. It expresses the effect of the normal stresses,

N. .(r) a v,(r )vL(r2)V2v.(r2) -

y.

(r2)vL(r )V2v.(rl)1

i

i

J 13

ar2Li

1 = $ ..(K)

e1<dK

_(2.6) (2.4) (2.5)

The tensor T.. which expresses the interchange of energy between wave numbers brought about by the Reynolds stresses appears here in more

complex form. In the special case when V2 is small, which is of

partic-ular interest to us, the term which differs from isotropic turbulence for a Newtonian fluid will be small unless the derivatives of the tensor

T.. or the tensor N. . are large. When the spatial derivative of T. . is

13 13 13

large, the process of interchange of energy between wave numbers will be

changed and indeed there is a possibility of a change in sign. The term in P., is changed by the presence of the viscoelastic effects through influence of the velocity on the pressure (see Equation (3.3.5) of Part

I). The term in V2R1 is modified by the presence of a derivative with

respect to the time. Here again, if V2 is small, this term is unimport-ant unless the time derivatives are large. This term affects the

dissi-pative aspect of the turbulent field. Thus, we see that when the gradient in the Reynolds stresses is large, the transfer of energy between wave

numbers will be affected, and when the derivatives with respect to time are

large, the dissipation will be affected.

The effect of the term N is more difficult to estimate. However, it

13

is clear that when V2 is small, this expression will be small except when

the gradients of the flow are large.

If we take the Fourier transform again in the usual way, we will

obtain Equation (2.7)

- F. .(K) - y K2F. .(K) + 2

(1

+ 2 )K2..(K) 2 ij

+rr..(K) - 2v ..(K)

13 2 ij (2.7)

If we calculate the rate of change of the energy we will find that

the terras in F. . or rr . vanïsh. The other two terms do not vanish. This

13 _1.1

is similar to the behavior of homogeneous turbulence in which the term F.. affects the interchange of energy between wave numbers but not the

total energy. Similarly the effect of ir.. on the total energy is compar-able. The term in cp.., however, is influenced in its effect on the total

energy by the presence of the time derivative The term in is new

and its effect has not been analyzed It will be shown later that terms

of this type play an important role in the effect of the viscoelasticity

on the velocity distribution.

A detailed discussion of this by Singh4 has recently come to our

3.0 EFFECT OF ADDITIVES ON RELATION BETWEEN PRESSURE AND VELOCITY FLUCTUATION

Equation (2.1) covers the rate of change of velocity0 It is quite

compli-cated in structure having two more terms than the conventional equations for

the fluctuations in a turbulent field. It is convenient to try to obtain equa-tions for the pressure fluctuaequa-tions by taking the divergence of Equation (2.1). This procedure has been successfully used for the discussion of the prcssure

fluctuations in a turbulent boundary

layer567.

it does not permit one to

actually make a calculation of the pressure fluctuations in a turbulent

bound-ary layer, but it does give some insight into them. If, for example, the

pressure fluctuations are known, then the velocity fluctuations can be

calcu-lated and vice versa. One of the results is that the pressure fluctuations which have been observed are inconsistent with velocity fluctuations present

in a Newtonian fluid under similar circumstances.

Let us take the divergence of Equation (2.1). This gives us Equation

(3.1).

(

1 +

v2v2)

y.

v

-

2 V2VL .V2v. = .J:_ 2p

3,L

.,,j

j

j,L p

It is convenient to solve for V2vL by differentiating Equation (2.1).

This will lead to Equation (3.2).

i V2v. =

L.

v.

V1 + v2v2)( t

+ Vi

kVk + V.

kvkL)f

vlp X, X j,L

-2v2

[

V2VJL

+ VkV2VJ _kL

+

Vk LV V

(3.2)

If this

is

substituted into Equation (3,1) we will have terms not only in V2 but also in V22. We will neglect the square terms. This will lead to

Equation (3.3). V.

2v

1 2

- V

= -(1 + v2V _(vvz) + y12 L,j 1

\

j,k k

+(v. v)L+) (3.3)

(3.1)

Consider the following identity.

V

+V

(v.vL),L. = ViL

_L,j

_L,j

ViL

Because of symmetry with respect to the indices Equation (3.4) can be

written as

o t j

L'Lj

= 2vLJ

v

:z

(y .v

By substituting Equation(3.5) into (3.3) we get Equation (3.6).

v2p

_{= - (v.vL),2. -}

2V2 +

2v2(

_{1 2P}

\\

+

l

vL

(v

kvk)L +

v

ò x.

xL

(3.6)

If we examine Equation (3.6) we find that all of the terms but the last involve operations on derivatives of the Reynolds stress tensor. We will show that the last term contains third order correlations of the velocity.

That this is true of the first term in brackets at the end of the equation

is obvious. It is not so obvious in the case of the term in the pressure.

Howevr, if we recall that the pressure is usually a second order velocity

correlation, then we see that this must be true We will find, however,

in our discussion of shear flow that when the rate of shear is high this

assumption may not be true for the correlation of the pressure fluctuation.

To strengthen our argument regarding the pressure fluctuation we will

use the following set of substitutions. Consider Equation (3.7) which is obtained by taking the divergence of the Navier-Stokes equation.

El

p1

vivk))k+p

_x I =

iJi

(3.4) (3.5) (3.7)

Since the divergence of the expression in brackets must vanish, the expression in brackets itself must be equal to the curl of a divergence-free vector

field. Consequently, it can be represented as shown in Equation (3.8).

(v.vk)k +-

L. A

p x. jsrr,s

J

where e. = + 1 for even permutations of the indices

j sr

- i for odd permutations of the indices 0 for repeated indices

The vector potential _Ar may also be chosen to be divergence-free, leading to the relationship given in (39)

A = 0 (3.9)

r, r

If we take the curl of (3.8) and use the relationship in (3.9) and some

standard transformations we obtain Equation (3.10)

(3.8)

A

=

cit(vtvk),

ks G. (r-r ) drij o o

- V2A e _(vvk),k

r rst (3.10)

If we use the Green's function G1(r-r) for the vector Poisson's equation, the equation given in (3.10) will lead to Equation (3.11)

(3.11)

This is a function quadratic in the velocities. If we multiply it by the velocity again it becomes cubic. It will be a fourth or higher order correlation when averaged. For the sake of simplicity let us retain it

in the form given in Equation (3,8). This will finally lead to Equation

(3.12) for the pressure fluctuation.

=

-

Ç

1 +

v2V2

-

j

j

y j jrs r,sL

v2)

_{(y y ),} -(vi, .c. A ₎

(3.12)

p

Up to now there has been no averaging process. If we have a homogeneously

turbulent field then we may proceed iunediate1y from Equation (3.12) to

carry out the averaging processes, Usually in the boundary layer we should

make the assumption given in Equation (3.13)

-V.

= V.+V.

.3 .3 .3

(3.13)

Gardner6 and later White7 discussed the pressure fluctuations and the rela-tionships they have to velocity fluctuations by assuming Equation (3.12) for

2 O and assuming that the velocity fluctuations were of the type

postulated by Kolxnogroff8. It will appear that the experimental data on

pressure fluctuations in a non-Newtonian fluid arc inconsistent with this

assumption about the velocity fluctuations. Before we take up this

dis-cussion, however, let us exanine the order of magnitude of the various

terms in Equation (3.12).

The added ternis are multiplied by two different factors. In one case we have

"2

alone and in the other we have V2/l Let us examine the

magni-tude of these terms. The term in V2 multiplies a second derivative with

respect to the spatial coordinates. Its dimensionless coefficient is given by the relationship in (3.14)

"2 vn[TIJT V n[T}T U n[T]

TU

2 £2

£1R

l+n['flJ £R

(3.14)

The expression for in terms of the concentration and the relaxation time

and the intrinsic viscosity is taken from the Section on Properties of

Dilute Solutions in Part I of this report1. This relation is only accurate

at very low concentrations, However, it is apparent that this term will

drop off as the inverse of the Reynolds number and will consequently be

negligible for large Reynolds numbers, It will, however, be proportional

to the ratio of the velocity times the characteristic relaxation time of the molecular species divided by a length which we may suppose is the

boundary layer thickness. For the sake of further calculations it is

conven-ient to substitute the length again in terms of the Reynolds number in which

case Equation (3.14) assumes the form given in (3.15)

= p u1

TU2

nfl]

o

£2 _{(1 n[TJ} )2 V _R2 o

UT

n[1Jf

U2T

ol N[1]

f £ l+n[111 N

Rl+n[TuJ

° (3.17)

This expression does not go to zero as the Reynolds number goes to infinity

because of increasing velocity It will, however, go to zero as £ becomes

large. 'herefore, this term will usually be the most important in the

boundary

layer

for high velocities. Finally, the terms in the third power of the velocity if they are retained can also be shown to go as the

expres-sion in (3.17)

Let us now calculate the expression for the pressure fluctuation. To

simplify let us define po in Equation (3.18)

L

_{= (v.vL),.}

p o

(3.15)

(3.18)

Inserting reasonable values for the velocities and the Reynolds number and

choosing T to be about .001, we find that this term will usually be quite

small. This will not, however, be true of the following terms. If we

suppose that the frequency can be represented in dimensionless form as shown

in Equation (3.16)

Uf

£ = w (3.16)

then we will find that the term in the derivative with respect to time will

This is the relation which would hold in the absence of viscoelastic forces.

Substituting this into Equation (3.12) and assuming velocity correlations of the third order and higher go to zero we obtain Equation (3.19)

(

P =

1+v2v2_"2

If we take the average pressure fluctuation in the usual way we will obtain

the relationship given in Equation (3.20)

2 "2

- i + v2V 2 + v2V

2 't

P (x ,t1)P(x2,t2)

o 1

- 1

1t11

i

We see in this relationship that if we neglect correlations which are fifth and sixth power in the velocity we will obtain the simple expression in

(3.21) between the pressure and the velocity fluctuations.

(r,t')

[

2 2 4

= l+22V +v2V

P o (3.19) v2\2 2 cp(r,t') (3.21)

Equation (3.21) is valid when the rate of shear is small and when the

higher order correlation can be neglected. The case when the rate of shear is large will be discussed later.

It may readily be seen that Equation (3.21) will lead to an increase in

the pressure fluctuation supposing the velocity fluctuation to be unchanged. If an explanation is to be found for the large decrease in pressure

fluctua-tion which has been observed we must suppose a substantial change in the

velocity fluctuation. This will be the subject of the following section of

this report.

P(x1,t1 + t') P (x21,t2)

4.0 AN APPROXIMATE TREATMENT OF THE EFFECT OF ADDITIVES ON VELOCITY FLUCTUATIONS

In a previous section of this report we showed that when the rate of shear is not too great the pressure fluctuations will be increased by the

non-Newtonian character of the fluid if the velocity fluctuations remain unchanged. Here we will show that the velocity fluctuations must be modified in a way

which introduces relaxation phenomena into the pressure fluctuations. For

this purpose we will develop an approximate method for solving the general

equation in terms of the solution of the Navier-Stokes equation. The approach is to separate the non-linear from the linear terms and then calculate a first approximation to the solution of the perturbed equation by substituting the solution of the Navier-Stokes equations into the non-linear terms and solving

the resultant linear non-homogeneous system of equations. This new solution could be substituted into the non-linear terms again and the system could again be solved, resulting in successive approximations to the complete

solu-tion of the perturbed Navier-Stokes equasolu-tion. Actually, only the first step in this process will be taken and even then will not be carried out to

completion. Further, the solution which we will give here is valid only when the rate of shear is low and when the product of the unperturbed and

perturbed fluctuations can be neglected.

It is convenient to introduce a vector potential for the flow. This is in effect a 3-dimensional stream function. If we suppose that 0A. is the vector potential for the flow in the absence of the non-Newtonian materials

and if we take the curl of the Navier-Stokes equation we will obtain the

relationship given in Equation (4.1)

C L

(A

A

)

t i / o j jrs ruy kmn o u,v o m,n ,ks

If we introduce a vector potential for the perturbation again (1A.)

then the first step in our process of successive substitutions will be

given in Equation (4.2)

Using the relationship given in Equation (41) this may be rewritten:

( - VjV2 - v - V2' V,2 _{A. =} (i + v2V2" (-ê- _{- y v2"}

_{Vo j}

2 A.

J i i

+ 2 B.

2j

where the quantity B. if given by Equation (4.4)

B.=c

L L

(A

V A

j jrs ruy ktri \O u,v o xn,ni,ks

f

2

=.

v V V

jrs or

o1 ,ks

Let us now solve for the difference in the two vector potentials 1A.

and A.. This will lead to Equation (4.5)

oj

(-ê- - V V2 - V2

-

v2')v2 A. A.' I ].

j

o j)

\t

i

+ V --

_2t

V4 A

+ V

V2 - V V2 V2

A + 2v

B (4.5)

oj

2 t i J

oj

2 j

Canceling the identical terms on both sides of this equation we finally

obtain the relationship given in Equation (4.6)

- V

-

v y2) y2 p. = y2 (2E. + v1S6 A.'

\t

1 o j)

where cp. is the difference between the vector potentials.

Now this equation alone is sufficient to show the existence of

relaxa-tion times which will occur for funcrelaxa-tions which satisfy the left hand side. However we will go one step further by taking the Fourier transform and

performing a development in terms of characteristic functions.

(4.3) (4.4) (4.6)

vV

L jrs -£ ruy k v) V21A. = (l - y V2) _A A

-2

0u,vom,n

2v

2ou,v

(A

V2

_om,nI,ks

A

Let us therefore take the transform which will lead to Equation (4.7) (. 2 2\ 2 ( '\ 6 -\)1V - iw\)2V ) J = 2 2b1)

+

\)1\2V _a1

where the quantities b. and a1 are defined in Equation (4.8)

Let us now consider the eigenvalue problem given in Equation (4.9)

[

- - wv2 V4 + iwV2

J

cp. = Kcp. (4.9)

The solutions of Equation (4.9) can be expressed in terms of

solutions of the wave equation given in (4.10)

CP1 = a

lj

+ b

1j

2 2 v

lj

= 2i

_lj

'

2j

'2 2j

Carrying out the substitution we find the equation given in (4.11)

(-- 2 + iwL =/ (4.11)

This quadratic equation will have two distinct solutions and L2 given

by the usual rules of algebra and a separate solution of the form (4.10)

will exist for each root. A simple line of reasoning shows that for every value of K it is possible to find values of £1 and £2 which each give

solutions and

2j

which when added together satisfy the boundary

condi-tions. Thus we have a continuous distribution of the characteristic

func-tions which can be used to develop a solution of the differential _equation.

If we choose the functions

lj

and in which the eigenvalues are

distinct, then we can choose arbitrary constants such that a linear

combin-ation of

ij

and

2j

will satisfy the boundary conditions. That this will

be true for rectangular cartesian coordinates may be readily proved in the

dt iwt a

Ae

dt i

_oi

(4.7) (4.8) (4.10)

case of a flat plate by substituting exponential solutions of the wave equation

Each vector wave equation will have three components0 Thus, we will have six arbitrary constants to determine from setting the

veloc-ity eqal to zero on the boundary0 If we choose exponentially decreasing

functi ns in the direction normal to the surface it will suffice to fit the boundary along the plate which will lead to three equations and six

unknowns. In addition, each solution of the wave equation must separately be divergence-free which will lead to two more linear equations. If we impose the wave numbers in the X1 direction and the X2 direction the wave

number in the x3 direction will be determined by the eigenvalue Li or L2

respectively. Since solving for the wave number in the direction

perpen-dicular to the surface involves the extraction of the square root, two possible signs will always exist and we can always choose the solution

in which the real part leads to a decreasing function. Hence for any pair of eigenvalues we may ai'ays find eigen functions which will satisfy

the outer boundary. Since to satisfy the inner boundary and the divergence condition we must satisfy five more linear equations, all arbitrary con-stants will be determined except for an arbitrary multiplier. Thus for

every combination of eigenvalues Li and is2 an eigenvalue of (4.9) will

exist. It should be noted, however, that the function so chosen will

depend on the wave numbers in the directions of the plate and on the

frequency.

Having set down these conditions we may rewrite the characteristic value problem of Equation (4.9) in the form given in Equation (4.12)

[ ('

+ iwv)

V4 + iwv2l p = (v

+ iwv2) 21L2

(4.12)

Now, if the right hand side of Equation (4.7) can be developed in a series

in

the characteristic

functions of

(4.12) it becomes possible to solve Equation (4.7) completely0 Let us suppose therefore that the right hand

I

iwt

A.+2B

b

pe

dw

i

oj

j

'

cYj

and let us further suppose that the solution of (4.7) is given by (4.14)

=

_aCp.

(4.14)

Then we will have the linear relation given in (4.15)

a cp. b cp. (4.15)

+ iwv) L1L2

OE =

L

2

If we assume that and £2 are always distinct then the functions

will also all be distinct and we may identify term by term their coefficients

in Equation (4.15) since they will be linearly independent. This will allow us to solve for the unknown coefficients leading to the equation given in

(4.16)

2

a

-('

+iu)L L

1 2

12

Finally, expressing everything in terms of the relaxation time in (4.17)

T -

(4.17)

vi

we will obtain the solution given in Equation (4.18)

b

C

(p. j

j1±iu

£ L

12Q'

Clearly, this equation introduces a relaxation time T into the perturbed

velocity and hence it will aio lead to this relaxation time in the perturbed

pressure fluctuations resulting from these velocity fluctuations.

A complete analysis of this phenomenon including non-linear terms is

beyond the scope of this outline. Let us, however, use Equation (4.18)

(4.13)

(4.16)

to express the change in the velocity associated with the non-Newtonian

effects. Assuming that the integrations can be carried out this will

lead to equation

iwt cp b

r

íe

jrscr,sa

i J

l+iw

(y

Our next objective must be to express the pressure fluctuations in

terms of this perturbed velocity. In calculating these fluctuations the

double divergence of the Reynolds stress tensor given in (4.20) is

important. In the works of Gardner and White, this is represented by

the quantity S.

y.

y.

. = S (4.20)

1,3

3,1

If we add to the veloity the perturbed velocity given in (4.19) we will

obtain the expression in (4.21)

iwt

.

cp .b r Te

jrs

r,sj a'

S . .

+2

j

l+iu

L L dwl

Li,]

Q.

la'

r

iwt

R.

. + 2 i Te jrs

a' r,si

J

L3'1

-1 iwT L L dw

(4.21)

la' 2

If further we drop the terms which are quadratic in

2 and hence in

T we

will obtain the expression in (4.22)

+

_bTe

iwt

= .

..

.

+

2 dw

. J 1 + iufr irs i,j L1L2a,

1,3 3,1

-lwt

bTe

P,si

+2

$ ' dw e

y

l+iwT

jrs j,i Z L Q'

-Call the quantity added to S' Interchanging i and j in the

second integral makes it identical with the first. Hence

(4.19)

=,\

r y. e.

Teb

S

j i, a r,si rs a dw

(1 + íw'r)

In the treatment of the turbulent field Gardner6 and White7 pass from the

quantity S to the pressure through an integration which uses the Greens

function for Poisson's equation We will suppose that a similar proced-ure can be used here. Let us suppose that pl satisfies the differential

equation given iii (4.24)

y.

p .e. b

v2p1() = i,j a r,si jrs a

L-We can express the pressure in terms of P1 as given in (4.25)

= + TF1dW

l+iwT

iw t S o(x + r,t + t') l(x,w) Te dw P (1 + iarr) iw(t+t _I) -F

5D

l(x + r,w) Te dw o(x,t) (1 + iWT) (4.23) (4.24) (4. 25)

where P is the pressure fluctuation associated with a Newtonian fluid of the

same viscosity. If we calculate the space time correlation of the expression

in (4.25) and again neglect terms quadratic in the term linear in 'V2 will

be the pressure fluctuations associated with change in the velocity

fluctua-dons. To these un.ist of course be added the extra terms appearing in Equation (3.20). Confining ourselves to tha term in

2 we would have only

the term in V2V2cp. The space timc correlation can be expressed in the form

given in Equation (4.26)

(4.26)

Now in carrying out the averaging process it is possible in the space

coordinants in the second expression to substitute x1 for x + r and integrate

over x1 without changing the value of the integral. If this is done the

Furthermore, we may in the first expression, replace w by -w under the integral

sign. The integral will then change sign because of the change in sin w and we will have two terms of the form shown in (4.27)

:Jwt'

_TP

iwt P e i 1 - iwT o(x+r,t) 1(x,-w) dw iw t T P eiwt o(x-r,t) 1(x,w) dw (4.27)

Finally, clearing and collecting terms we will obtain the expression in (4.28) where the quantity Çs satisfies the relation given in (4.29).

(r,w) where .L -I- 1WT ( r,w) - (r,-w) - iwî , , (-r,w)1 T [1 + (WT)2] iw t P P e o(x+r,t) 1(x,w) (r,w) (4.29) (4.28)

Equation (4.29) clearly exhibits the presence of a relaxation time which will strongly affect the amplitude of the pressure fluctuations in certain

frequency ranges.

It has not been established whether this leads to a decrease or an

increase in the level of the pressure fluctuations. This will depend on the nature of the integrals which appear in Equation (4.29). Experimental results indicate that a decrease occurs.

We will try, in the next section of this report, to obtain a physical picture of how the relaxation time affects the vortex structure near the

V. = X

0 +.

i

_i

i

5.0 EFFECT OF ADDITIVES ON VORTICES NEAR WALL

Recent work9'1° shows that the generation of turbulence by a shear flow is associated with the growth and stretching of vortices having

axes in the direction of the stream, It is, therefore, pertinent to

examine how these vortices are affected by the forces induced by the

second-order terms. To evaluate this influence qualitatively, we will consider a simple flow in which the time dependent perturbations are

vortices of this type.

Let us assume that the velocity has the form given in Equation (5.1)

where X. is a unit vector in the direction of the flow and ₀ is a

function depending only on the coordinate perpendicular to the direction of flow and can be written as a function of the contracted product '.x..

-

_3J

Here, X. is a unit vector perpendicular to X.. Substituting Equation

(5.1) into the equation of motion gives

+,.

- _-

_{(X,0 + y2) (X.0}

_{+ y}

)

-

₊

_{l + "2}

_j,LL

j 2

j,L

p

+

v2[(Xk0+k)(Xj0,k +vj,k)_ (Xk0LL+'ku)(X.0

_{j ,k}

+.

_j,k) - 2(XjøkL + _j,kL. )

(X 0

_k,

_{2 +}

(5.1)

(5.2)

If we carry out the differentiation Equation (5,2) becomes, after expanding

lp

= - _{- v,v - - + (y + y}

_)V2(X.Ø

_+.)

- Xj\TL

_{x. j,L p x. 1 2 t j j} j + v2 _{k("j,kLL - ø,LLVj,k -} 20 _LVi +

_{LLvk - 0kL -}

20

Lk

jk

+ kU - 1k,L2'Tj,k -

2k,jk,)i

where all terms which contained products of the form X.X. were dropped 1ecause they were equal to zero due to the orthogonality of the two vectors

X. and X.. The primes on the

0's

denote differentiation.

i i

Noting that all the bracketed terms in the term have the same

form, Equation (5.3) can be written more compactly as:

r-v. + y

vx 0 +

___.l=

_-

--

_-

p x. i j j + y V2v. _- \)2V2(XkØvik + X.

0'

+v.

₎

jk

k

kj,k

2t

j + 2v2(Xkøvj kLL

+ x.5

j k

0''

k

+r-

k'j,kU

'

where the term X.ø O has been dropped.

and

Xv.

=0

k j,k

(5.3)

(5.4)

As a simple first example, we will consider the case where disturbances

do not vary in the streamwise direction; therefore,

XkVk = O

In addition, we will assume that is sufficiently small so that we may neglect square terms and also that the shear flow has no curvature so that

only the first derivative of 0 is non-vanishing. With these assumptions,

Equation (5.4) may be written as

(1 - v2v2) ' = - (1 + v2V2)(X.

Ø') +

v1V.

(5.6)

It is notable that Equation (5.6) represents three components which

must be set equal to zero even when v is two-dimensional. In general,

an equation of this type would, therefore,

flot

have a simple two-dimensional solution of the type postulated, and indeed it will appear that it is only in the case of viscoelastic fluids that this type of solution can exist.

Assuming that the time variation can be represented by the complex exponen-tial and that the vanishes, the three components of Equation (5.6) may be represented as is shown in Equation (5.7)

(1 +

v2V2)2

- o (1 - v2V2) iw' + - - = 2

px2

(1 - \2V2) + = 0 (5.7)

If we introduce the stream function as given in Equation (5.8)

V

= -

V =

2 òx 3

-3

and assume that

0'

is constant, we will obtain the set of equations given

in Equation (5.9)

o

= O X3 (1 - _iWp (1 - \)2V2) + :

If we differentiate the second equation by x3 and the third by x2 and

if we subtract, we obtain the differential equation given in Equation (5.10)

- v2V) V = 0 (5.10)

The first of Equation (5.9) must be solved along with Equation (5.10). Let us start by solving the former. It is clear that any one of its solu-tions must be an eigenfunction of Equation (5.11)

unless the derivative of the stream function with respect to x3 vanishes

which, by hypothesis, we assume is not the case.

If we express the position in terras of the dimensionless variable: X.

i

(5.12)

Equations (5.10) and (5.11) become

(1 - - 'i _{V2) = 0 (5.13)}

iWV2

=

-Substituting Equation (5.14) into (5.13) gives

- _{+ V +}

.1

= O 1WV2 v2

=0

=0

(5.9) (5.11) (5.14) (5.15)

and applying Equation (5.14) once more, we finally get

This equation cannot be satisfied for a non-vanishing stream function

unless the bracketed term is identically equal to zero. This gives

We will proceed to find a solution to Equations (5.13) and (5.14)

subject to the limitation in Equation (5.17). We will first find solutions

representing an isolated vortex which is not close to any wall. For this purpose, we will write Equation (5.14) in cylindrical coordinates. This gives

1 _{Y +.-!.}

+ ' _{= 0 (5.18)}

22

r

where r and are the cylindrical coordinates of the point IL.

The components of flow will be given by

V r i - r y = 0

=

-r = j [rl o vi

(2+.

' )'l = 0 1W V Vi 2v2

iw=

or T

=

-v2

The negative sign assures us that vortices of the type considered in

this analysis die out with time.

v

3r

If we assume that the velocity in the tangential direction is the only one present and that it depends only on the radius, then we will

deduce the conditions given in Equation (5.20)

and Equation (5.18) will be satisfied by the zero order Bessel function,

(5.16)

(5.17)

(5.19)

(5.20)

The Bessel function is regular at the origin and goes to O as r goes to infinity. Consequently, it represents a possible flow. It does not,

however, represent a possible flow in the neighborhood of the wall because

it does not satisfy the wall boundary conditions. A complete solution satisfying the boundary conditions along the wall is given in the

Ap' 'tx.

here,

we will give an approximate solution which will aid us

in tha discussion. The general results will not differ notably from those

obtained by a complete solution. The disadvantage of the latter arises from the analytic complexity of the result.

To find an approximate solution, let us suppose that the center of

the vortex obtained previously is at a distance a from a wall.

Equation (5.21) will then have the form given in Equation (5.22)

The Cartesian components of the velocity will then be given by the two

expressions in Equations (5.23) and (5.24).

2T

_r

2

211

V2 =

{2 - a)2

_TIu/2

(2 -

a) +

3}

1/21 2(112 - a)

v3=-.,

212

o - a) + Îl3

L

-

a)2 +

»5.24)

will be the component of the velocity normal to the plate and y3 will

be the component parallel to the plate. To satisfy the boundary condition,

both these components should vanish when

2 is O. In fact, neither of

them do.

If we add to the expression in Equation (5.21) the Stream function of a similar vortex which is the mirror image with respect to the plate, the

2

[1Ê2

- a)

l/2]

o (5.22)

resulting flow from symmetry considerations can be shown to have a normal

component which vanishes at the boundary. The velocity y2 derived from

such a flow for T2 = O is given in Equation (5.25).

4T3 2

21/2

(a +

Î3)

) 1/21 ji Ç

(-a)

_{- J}

[a2

+

13)'

j

oLt

(5. 25)

It is obvious that this vanishes identically.

The tangential component on the other hand will be given by Equation

(5.27) and, in general, this will not vanish.

JI

r(a2 2 1/

y3

= 1/2

° L

(a2

_{+ T)}

(5.26)

In the Appendix, we discuss an approach to a complete solution by choosing a superposition of solutions of the type given in Equations (5.26) and

(5.27) but in which we have a set of cross stream positions. It is possible,

however, to have e solution which will be small when ₁₂ is equal to

zero. If hc vortex center is sufficiently far from the wall y3 will be everywhere small on the

wall if the

quantity a is a zero of the

derivative of J

o

It is more convenient to substitute derivatives using the well-known

recursion11 relations that we obtain for y3 , the expression given in

Equation (5.28). 4a _[(a2 2 1/21 y3 = 2 1/2 l + T3) _J (5.27) (a

+ 1)

The zeros of the Bessell function appearing in Equation (5.27) are well

vortex will, therefore, be of the order of magnitude given in Equation (5.28) 27 r = 3.8317 = 3,8317

I\)

N[1I]T o o (528)

If, instead of taking one vortex, we had taken two parallel vortices

it would have been possible to assure a better fit of the tangential velocity

along the boundary by the judicious choice of spacings. If this process is

continued, it is possible successively to improve the fit by the addition of

further vortices which alternate in their direction of rotation. This pro-cess is carried out in the Appendix where it is shown that an accurate solu-tion can be obtained in this way.

We find, therefore, that a type of vortex structure which has been found to be associated with the growth of turbulence is highly damped at specific

frequencies and dimensions in a solution of high polymers. Contrary to

what is found in Newtonian flow, a scale factor is imposed through the pres-ence of non-Newtonian coefficients, even in the case of simple shear flows

without curvature. This characteristic radius will go to zero as the concen-tration goes to zero. When the concentration becomes high enough so that the characteristic radius is of the order of magnitude of the thickness of

the sublayer, we would expect a change in the character of the flow.

We also find that the relaxation time of these vortices as defined by Equation (5.17) increases linearly with

2 and, consequently, again,

6.0 CONCLUSIONS

The relationship between the pressure fluctuations and the velocity fluctuations in the flow of a liquid are modified if one dissolves in it a

polymer of sufficiently high molecular weight. For a given level of velocity fluctuations this change is in the direction of an increase in the noise. Therefore, it cannot alone explain the reduction in noise which has been

observed.

The assumption that the additive does not effect the velocity

fluctu-ations is not realistic. At high frequencies we would expect, from

theo-retical considerations, a substantial change in the velocity fluctuations.

This change engenders a decrease in the pressure fluctuations exhibiting

a characteristic relaxation effect. The magnitude of this change and its relaxation time should be proportional to the concentration of the solute

and to its intrinsic viscosity.

In a shear flow vortices with axes in the direction of flow would be

highly damped. Their relaxation times are identical with or proportional to those which appear in the pressure fluctuation and their radii are

pro-portional to the square toot of the concentration. The generation of turbulence has been shown to be associated with this type cf vortex. The

high damping which it suffers in non-Newtdnian fluids probably explains the general reduction in the level of turbulence which leads, not only

to the reduction in noise, but also to the reduction in drag.

All of these effects should be most closely verified if data are

REFERENCES

Boggs, F,W. and Thompsen, J,, "Flow Properties of Dilute Solutions of

Polymers", Part I, "Mechanism of Drag Reduction", Final Report, Contract No. Nonr 3120(00).

Pruett, G.T., and Crawford, H.R, "Investigation of the Noise Reduction

Characteristic of NonNewtonian Fluid (U)", Final Report, Contract

No. NObs 86698 (Confidential).

Batchelor, G.R., "The Theory of Homogeneous Turbulence", Cambridge

University Press, 1953.

Singh, "Theoretical Study of Second Order Fluid", Thesis, Aerospace

Department, Pennsylvania State University (1965),

Kraichnan, R.H., "Pressure Fluctuation in Turbulent Flow over a Flat Plate", J. Acoust. Soc. Amer, 28 (3) 378-390 (1956),

Gardner, S., "Surface Pressure Fluctuations Produced by Boundary Layer

Turbulence", Tech. Research Group Report, Contract Nonr 3208(00), 1963.

White, Frank M., "A Unified Theory of Turbulent Wall Pressure Fluctuations",

USL Report No. 629, AD 457,108,

Kolmogoroff, A.N,, Compt. Rend. Acad, ScL, U.R.S.S. 30, 301 (1941).

Pfenninger, W., "Investigation of Methods for Re-establishment of a

Laminar Boundary Layer from Turbulent Flow", Northrop Norair Report,

Contract No. NOw 63-0762-c (1965).

Bakewell, H., Thesis, Pennsylvania State University Aerospace Dept (1.966).

il. _{Jahnke, E., and Emde, F., "Tables of Functions", Teubnen, 1938.}

12. _{Magnus, W., and Oberhettinger, F., "Formulas and Theorems for the}

CALCULATION OF VORTEX ARRAY IN SUBLAYER

We saw that a stream function of the type

= z

_[(12

+ )2 + T32}

z[(2

- a)2 + lì3

(A-1)

where Z0 is a zero order cylinder function and a is the distance from the

vortex center to the wall, satisfies one of the boundary :onditions, and

that the other one can be satisfied only at one point. Let us consider an

array of vortices which will be given in Equation (A-2):

{2

_[{(112 a) + (3 -

b)2}]

- - a)2

+ (3 -

b)2}}

where b are the dimensionless cross stream positions of the vortices.

n

The condition for the vanishing of y3 on the boundary will be given

by Equation (A-3) APPENDIX (À-2)

Aa

n

_{a2

+ (113 -

b)2

functions11, terms

b)

2m = O

=0

Equation (À-3) (A-4)

L

t2

z1

n-

4a + (T3 - b

n)

If we apply the addition theorem of cylinder (A-3) can be written, after canceling zero

-If we further develop Zi+2m (13 - b ), we will write n

z1(T3-b)=

_Lz

(Tt)J(b)

l+2inp 3 _p n

p

This leads to (-1) 2m

p_

Y

Zl+2mp (3)

A J (b ) = O

np n

If we let i + 2m - p = r and identify to zero the coefficients of

zÇfl)

_{r 3}

, we can write

(-1) (a) A

l+2m-r (bn) = O

if we suppose it possible to write

-foe

A J

(b ) =

e+Zm)

r J (b ) =

e8

z n l+2m-r n

nL n

fl-

fl=_

we must have

(-l)

J (a) e2ime = O (A-9)

2m

It turns out that the series given in Equation (A-9) can be readily

summed. We can write J (x)

(_1)fl

J (x), hence, Equation (A-9) can be written as follows

-foe

2im(9 +)

= J (a) + 2 J (a) cos 2m(ê +) (A-10)

L

2m

m1

In Equation (A-lO) we can use the Jacobi-Anger12 formula which is given

in Equation (A-11)

+ i a sin

e

CJ(a)

+

o Jm(a) cos 2mcp ± 2 i 2m+i (a) sin(2m+i)q

We add the expressions with positive and negative signs to each other,

giving us Equation (A-12)

r ja sin (8+) _{-ia sin(e+)1}

= cos[a sin(& +

+e

= J (a) + 2 J (a) cos 2m(O+) (A-12)

o 2m

m1

As a consequence, we obtain the first condition on our parameters which

is given in Equation (A-13)

T TT T

cos a sin (8 + ) = cos a cos 8 = O (A-13)

L

LJ

L _J

If we change the sign of £ in Equation (A-8) and if L is even, the

left hand side is unchanged. Hence, if £ is even the expression in Equation

(A-8) must be symmetric with respect to a change in sign of L. On the other hand, if L is odd, the left hand side is antisymmetric with respect to a

change in the sign of L. But, if e1mO satisfies Equation (A-8), (_l)me_1m8

will also satisfy it since this is equivalent to multiplying it by -1; hence, we can replace Equation (A-8) by

iriLe

£-iL8l

L

AJL(b) =

- Le + (-1) e _(A-14)

The expression to the right has the proper symmetry for the identity to

be possible. It is convenient to use the integral expression for the Bessel function which leads to Equation (A-15)

IT

-I'A e' cos (

eU

_{d =}

r

iLB

j

Ln

(P

Le +

-1-r

n=-or, from symmetry

(A-15) (A-16) -foe _{i Lrr}

1

S

eT

COS _(P

cos(L(P)d e2 t iLOle

+(-1)e

L -iL8 i L

i

Let cos = X. Then, we have i. -foe

L. j'

A T2(x) dX iL(6-I-) + ( 1)

+iL'

- 6) u n 2

-e

2 - e 2 - n-- l-x cos O +

eL

- 6) T2X = sin O-x 2=1 u

sinO+X

cos(- + 6) - X

Thus, we see that we can obtain a relatively simple closed form expression

for this series In a similar way, we can write the relationship given in

Equation (A-20)

(A-17)

(A-19)

(A-20)

where T2(X) is the Chebychev polynomial. Since the above equation is even

with respect to the interchange of L with -2, we can drop the terms in -L; then, the right-hand side of the previous equation is the coefficient of the development of the left hand side in terms of Chebychev polynomials, whence we may write

oe

L

= i +

e2(

± _{T2(X) +}

(1)L

e2

- 6)

T (X)

2=1 2=1 _(A-18)

The series in Chebychev polynomials which appear on the right-hand side of Equation (A-18) can readily be summed by using the generating functions

for the Chebychev polynomial12. We may write the relationship given in

Equation (A-19) 2i( + O)

l-e

z + + O) _T2(X) = 1 - 2Xe1 + O) 2i( + 6) 2= 1

+e

sin( + 6) -cos O

Combining (A-18), (A-19), and (A-20) we obtain the relatively simple relation-ship given in (A-21)

- _A cos 8

+

e

L

n X - sin 8 X ± sin 8 (A-21)

In writing the equation, we have absorbed multiplying constants into the unknown A's. Since the original equations were linear homogeneous, there will always be a constant multiplier which can be absorbed.

The relationship which is given in Equation (A-21) is valid only when

X is between -1 and +1. Indeed, since X is actually the cosine of an angle, this variable must be confined to ttat interval. In it we can develop (A-21) in a Fourier series. For this purpose, we multiply through

inhiX . .

by e and integrate in the interval -1 to +1. To the left, we will obtain terms of the type given in (A-22).

i i ' i(b +nîr)X - e n dX -1 b = nU n (A-22)

So far, we have not chosen the values _bn which express the crossstream

spacing of the vortices. It is sufficient to choose the b's so that the boundary condition is satisfied. Consequently, any set of values which leads to satisfactory boundary condition is a valid choice. There arises, of

course, a question of uniqueness if one choice is satisfactory. We will not discuss this problem at the present time. If, however, we choose b

according to Equation (A-23)

(A-23)

then, the solution for the coefficients A is immediate. Physically, this

means that the vortices are equally spaced in the cross stream direction. This is a reasonable conclusion and is satisfying to our physical sense. It should be acceptable if this choice leads to a complete solution of the

problem. Since the boundary value problem probably yields a unique solution, the equally spaced one should be satisfactory With this simple assumption, we may write Equation (A-24) for the coefficients of the development

i i

Xf

cosee

dX -A = e d inTTX n J x-sine S

X +sine

-1 -1

We will now show that the integrals to the right of Equation (A-24) can be expressed in terms of sine and cosine integrals, as is indeed apparent. After

some elementary manipulation, we obtain the relationship given in Equation

(À-25)

-

(l+sin e)n

-A = 2i cos e

n

sin :os (nit sin ₎

5

dli (-l+sin e)nrt (A- 24) S1U U1J.)flhl cos -sin (niisine) $ li

li

dli] (À-25) (sin el)nTT

Equation (À-25) expresses the coefficients of the development in terms of

Bessel functions as a function of the angle e, So far, we have established

only one relationship governing e which also involves the quantity a

express-ing the distance of the array of vortices to the wall. We can, therefore, choose a to satisfy extreme conditions for the constant A. Now, it is

apparent that all A's must be fine and this must include those values for

which n vanishes, and for which n+ It is apparent that the second

integral in (À-25) will lead to cosine integrals which have logarithmic singularities at the origin and which will, therefore, lead to values of

A which are not f inite We must, therefore, choose e so that the cosine o

integral terms vanish This can be done by choosing sine = O. This

condi-tion leads to the simple expression in (A-26) for the coefficients

-A = 21 [- r

Sifl

li

_dli +

S

li

Here, again, we can absorb the 4i and the sign into An since the absolute values of these quantities will be obtained by choosing one of them,

pre-sumably A0, arbitrarily. Thus, we may just as well write Equation (A-26) as follows

si(nîT) (A-27)

The sine integral to the right of Equation (A-27) is well-known. It

is equal to rr/2 when the independent variable is O and it vanishes

when the independent variable tends toward . It oscillates between

positive and negative values over an interval of 21T. As a consequence,

the central vortex will be a maximum and the successive vortices on

opposite sides will proceed with decreasing amplitude. Since a reversal

of the sign is equivalent to a reversal of the direction of rotation of the

vortex, the vortices will alternate in the direction of the rotation. Thus,

the vortices will have the general form illustrated in the figure.

This vortex structure has many features in coimnon with what has been observed

in the boundary layer of Newtonian fluids. It has, however, a fundamental

difference. The spacings are determined, not by the curvature, but by the intrinsic properties of the fluid.

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