A Study of Motions in a Rotating Liquid

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A STUDY OF MOTIONS IN A ROTATING LIQUID

A Thesis Presented to the Faculty of the Graduate School of Cornell University for the degree of

Doctor of Philosophy

By

George Walter Morgan

June 1950

ProQuest Number: 10834649

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BIOGRAPHICAL

SKETCH

The author was horn in Vienna, Austria in 19S4.

He left his native city in 1938 after having

completed eight years of primary and secondary education* After a brief stay in France he was admitted to Canada.

He attended the Highschool of Montreal for

two years and graduated in 1940.

Upon completion of

a one-year Pre-Engineering course he entered McGill University and was graduated with Honours In May 1945 as a Bachelor of Mechanical Engineering, From June 1945 to September 1947 he was in charge of a textile mill. In September 1947 he came to Cornell University upon being awarded a Fellowship in Engineering Research to take up studies for an advanced degree and in June 1948 he received the degree of Master of Mechanical Engineering* He continued his graduate studies at Cornell University toward the Ph. D. with a Major In Fluid Mechanics.

ACKNOWLEDGEMENT

The. author wishes to express his sincere gratitude to Professor W.R. Sears of the Graduate School of Aeronautical Engineering for his valuable advice, his encouragement and his sympathetic and lasting Interest throughout the preparation of this thesis.

PREFACE

The motion of solids in a perfect, incom­ pressible fluid when the flow is not irrotational was studied in a series of papers, published in the decade between 1916 and 1926, by J. Proudman, S.F. Grace, and G.I. Taylor.

The chief Interest in this subject lies

in various startling experimental results obtained by G.I. Taylor and in the fact that, in spite of the simplifying assumptions of an incompressible and inviscid fluid, only very little progress has been made in attempting to explain some of these results theo­ retically.

The subject thus presents one of the few

problems in "classical11 hydrodynamics which is not yet understood. It is the purpose of this paper to review some of the work which has been done and to obtain solutions to several problems which, it is hoped, will throw a little more light on this curious subject.

CONTENTS

Page I II III

Introduction ...................... ..........

1

Equations of Motion ...................... ... 22 Linearization of the Equations of Motion ... 52 Equations Referred to a Frame Fixed in a Moving Body

IV V

Forced Oscillations of a Rotating Fluid .... 40 Forced Oscillations in a Tank - Elliptic Regime

VI

......

49

Forced Oscillations in a Tank - Hyperbolic Regime ...............................

VII

Forced Oscillations of W Velocity Component along z Axis -Hyperbolic R e g i m e

VIII

53

57

Non-oscillatory Disturbance Imparted to a Rotating Fluid Initially at Relative Rest

IX

..............

60

Similarity Law for "Broad" Bodies Oscillating with Frequencies ^ > £ R ... 74

X

Flow due to the Movement of a Disc which Starts from Rest and Ultimately Moves with Constant V e l o c i t y ............

84

Page XI XII

Power Series Solution

92

Summary and Suggestions for Further Work ...,............. Appendix

........

B i b l i o g r a p h y ...............

106 109 114

-1-

I

INTRODUCTION

In the general case of incompressible, inviscid, rotational flow the vorticity will be a function of posi­ tion and time.

In studying any such problem it is there­

fore necessary to make an ..ssdmption regarding the ini­ tial distribution of vorticity.

The assumption made in

previous investigations and to be nic.de also in this paper is that the fluid initially rotates about a vertical axis,

(the z axis), with constant angular velocity R

like a solid body.

This results in a uniform initial

vorticity vector 2Rk where k is the unit vector in the z direction.

This situation of solid body rotation has

the advantage of being easily obtained in an experi­ mental setup. The first paper on the subject was published by J. Proudman

in 1916+ .

He first considered the

two-dimensional case, that is the case in which all derivatives with respect to z vanish.

It can be shown

that the assumption of two-dimensionality results in

+ Proc. Roy. Soc. London: Series A, Vol. 92, p

. 403

-2-

constant uniform vorticity*

The equation for vorticity

transport in an incompressible, inviscid fluid ,is given by

= sa-

I (!) D

where to- curl £ , ^ being the velocity vector, and — — is the convective derivative, given in Cartesian co­ ordinates by

-jeL + ix -3 - 4- v -S - 4- w^2L

3t

Vj

W

3* '

3y

being the Cartesian components

of ^ .

Refer the motion to axes which rotate: with con­ stant angular velocity R about the z axis;

(so that there

is no initial motion relative to these axes). Let (x,y,z) be the coordinates of a point referred, to the moving frame.

At each instant let all velocity

vectors be resolved in directions parallel to the axes of the moving frame.

Let u(x,y,z,t), v(x,y,z,t) ana w(x,y,z,t)

be the Cartesian components of the absolute velocity (i.e. the velocity as seen in a fixed frame) of the fluid at the point (x,y,z) at time t, the components being measured parallel to the Cartesian axes of the rotating frame. Let u*(x,y,z,t), v*(x,y,z,t), w t (x,y,z,t) be the Cartesian components of the relative velocity (i.e. the velocity as seen in the rotating frame) of the fluid at the point (x,y,z) and time t, the components again being measured parallel to the Cartesian axes of the rotating frame.

-3-

Then

u — u 1 - yR ,

v ^ v*

and

h^-curl £* -- curl £ - 2Rk - on - 2Rk

4

xR ,

w — wf

1 (£) I (3)

Write the transport equation (1) in terms of relative velocity and vorticity vectors.

Ds&> vt '

V i

Since R is constant and k does not change due to the rotation

V’Oj Vt

D dj Dt

Therefore & dk _ r> t Now expanding

. grad

in components in terms of the

primed quantities and rearranging terms, gives

^ oJ • & rod Th i

t

*Jjjf r dK

j (4)

3 k Consider the case of two-dimensional motion

with initial uniform vorticity

-

r»'i *

_ L

?

A t *■ 2.R U.' ~ j ^ y ,t a.-r c? t and may be -written

^

4

E

1* X.

J_. O p ? L 3 =

_ " '

e

d !

P , • o ^

-V A.

l..

;v.v .

, .

4.

£. h U

^

•>t

c* h

3y _

"S t

I (5)

_0 '3

y:

where k

*

^

- i R “ i* >>•*)

Proudman noticed that any small motion which is steady (i.e. ;_~p *G) must be two-dimensional; since zLu-Q* He proceeded to set up the solution of the motion D J: ' over an interval of time, under the assumption that the above equations remain validj in the form of a power series in

2

Rt and showed how to derive the coefficients

of this series as functions of x,y,z for any type of motion of a body from that of constant translation parallel to each of the axes of the rotating system, provided only that the initial motion be irrotational and that it be generated suddenly from relative rest. This method was later adopted by S.F. Grace to solve

several problems for the motion of a sphere.

Before

discussing these, however, some attention will be de­ voted to further theoretical and particularly experimental work by G.I. Taylor. First Taylor repeated his previous experiments with an apparatus which permitted him to take photo­ graphs and verified very accurately the theoretical predictions regarding the motion of a two- and a three-dimensional body respectively through a rotating fluid.'*’ Taylor believed that the close agreement between theory and experiment lay in the fact that the phenomenon depends only to a minor extent upon the condition of slip at the boundary, which is usually the cause for disagreement between theory and experiment in the study of irrotational flow of an inviscid fluid.

In the

present case the all-important difference between the motion of a long cylinder and a sphere is the fact that the former is two-dimensional, the latter three-dimen­ sional, and that this is independent of boundary conditions with respect to slip. Next Taylor considered the fact, previously noted by Proudman, that small, steady, relative motions of a rotating fluid must be two-dimensional.

He

Proc. Royal Soc. London: Series A, Vol. 100, p.114

-9-

demonstrated this by imparting a small disturbance to

a tank of water initially rotating with uniform angular velocity and by insering some coloured liquid.

The

movement of the fluid drew the coloured portion into thin sheets which always remained parallel to tne axis of rotation.

The boundaries of the fluid in these ex­

periments were two-dimensional. In a later p a p e r +

Taylor obtained a family

of solutions of the non-linear equations for the case in which a sphere moves with steady velocity along the axis of rotation of the fluid.

He assumed that the

motion had been set up in such a way that it has become steady, i.e. that all time derivatives vanish.

He ob­

tained an infinite number of possible solutions all of which have the property that the relative velocities vanish at infinity;

Apparently there are not enough

conditions to obtain a unique solution.

Taylor noticed

that It is possible to obtain solutions for which the velocity of the fluid vanishes at the surface of the sphere.

This represents r, solution for which there is

no slip at the surface and for which the sphere is surrounded by a non-rotating sheatn of fluid*

This

behaviour was demonstrated by an experiment described

+ Proc. Hoy. Soc. London: Series A, Vol. 102 p.130

-10-

in the same paper.

The apparatus was set up so that a

celluloid hall could be drawn along the axis of rotation of the fluid? contained in a cylindrical tank, by means of a string.

The apparatus was rotated uniformly for

some time until the ball and the liquid rotated as a solid boay with the tank.

The ball was then given a

uniform velocity along the axis ana it was found that it stopped rotating immediately, indicating that it was surrounded by a non-rotating sheath of fluid.

As

soon as the ball was made to stop moving along the axis it quickly picked up the rotation of the tank. To ensure this behaviour, It was observed

that the

ball had to move with a speed greater than about one diameter per revolution of the system.

For slower

rates it was found that a column of liquid of the s^me diameter as the sphere was apparently pushed along in front of the sphere.

This result seems to agree

with the theoretical observation that for slow steady motions the flow has to be two-dimensional.

The ex­

planation suggested in the case in which the ball was moved at a greater speed was that the streamline which lies along the axis of rotation away from the sphere kept to its surface, that is, that the fluid particles, originally near the axis of rotation, which had no angular momentum moved along the surface of the sphere.

-li­

lt was further discovered that "if the sphere was suddenly stopped when half-way up the cylinder a mass of liquid appeared to detach itself from the sphere and to continue moving along the axis of rotation -with the same velocity as that with which the sphere had been moving." The above considerations and experiments suggest a number of questions: First: why is Taylor*s solution for the steadily moving sphere not unique? Does the indeterminacy lie in the manner of starting the motion? If so, will any of his possible solutions represent the ultimate con­ dition when one starts a sphere from rest, or will the motion never become steady, or will it perhaps not vanish at infinity? Secondly: It seems that if a small steady twodimens onal disturbance is imparted to a uniformly ro­ tating fluid, the resulting motion will be two-oimensional.

This was verified by the experiment In which

some coloured portions of the fluid were drawn into sheets parallel to the axis of rotation.

The equations

show that if the motion is slow and three-dimensional it cannot be steady.

The question now arises as to

what will occur if an attempt is made to produce a slow, steady motion by introducing a boundary condition which

_1

is not two-dimensionul,

such as the motion of any

hody, except an infinite cylindrical body with its generators parallel to the axis of rotation, with small uniform velocity relative to the rotating axes. If a coordinate system is chosen which rotates with uniform angular velocity h and is fixed in the moving body then the equations of small motion can be shown to be: cka; Jt c? w'' -ht

(see Section II, p. 56) _

f k v>

^

^ p_ y ,jf--1

s K ijJ' T ^

~ ~

a I: where u T(x”, y ”, z %

I (6 ) .111

‘i t) etc. are the velocities of

the fluid with respect to the rotating but non-trans­ lating frame at a point in space given by the co­ ordinates

(xn, y u, z") of the frame which moves with

the body; Now if the body moves with uniform velocity then the boundary conditions on the velocities as functions of the coordinates

(xM, y n, z ”) of the ro­

tating and translating system are not functions of time,

vie are therefore tempted to assume that the

whole motion will eventually not be a function of time.

But this would mean that the motion would have

-13-

to be two-dimensional even though the boundary con­ dition is a three dimensional one. This question was considered by Taylor in a paper published in 1923

,

He argued that if such a

motion is attempted three possibilities present them­ selves * " a) The motion in the liquid may never become steady, however long the body goes on moving. b) The motion may be steady but it may not be small in the neighbourhood of the body. c) The motion may be steady and two-dimensional,” Taylor reasoned that a) and b) are not very likely. If c) were true, then, if the liquid were contained between parallel planes perpendicular to the axis of rotation, a cylinder of fluid would have to move as if It were fixed to the body.

The boundary of this cylin­

der would have to act on the fluid outside it as if It were a solid body and hence no fluid could cross it. This was tested by towing a short cylinder along the bottom of a rotating tank and by observing some coloured liquid which was discharged at a point above and in front of the body.

It was found that the coloured

liquid "flowed straight towards the imaginary cylinder

^ Proc, Hoy. Soc. London; Series A, Vol. 104, p.213

-14-

to a point vertically above the foremost point on the body.

At that point the stream divided as though it

had struck a solid obstacle."

A further experiment

showed that coloured liquid discharged from a point inside the imaginary cylinder did in fact move with it. Thus possibility c) seems to be the one which actually takes place and this verifies the previously recorded observation that when a sphere Is moved slowly with uniform velocity along the axis of rotation the motion tends to become two-dimensional. The first attempts to find some theoretical explanations as to how such motions could be established were made by S.F. Grace who solved some problems con­ cerning the motion of a sphere of the same density as the rotating fluid by Proudman*s method of expansion in powers of 2Rt. small motion.

Grace started with the equations of

He assumed that u*, v*, w» and p could

be represented by power series of the form

a 1(x", y ,i",t)« 2.u.s kJitr v,» V Vj ^

vi Ai^

f* •> •

. J

- v'*Z_v$-AA-

p ■- > f p p Ah;VC s r I S-o ^ !

Arguing that the initial disturbed motion will be irrotational because the effects of rotation take time to develop, he assumed that u*, v^, w£ are the com­ ponents of the gradient of a velocity potential.

He

-15-

then showed that the coefficients of any power of 2Bt can be determined by the solution of a boundary value problem once the coefficients of the lower power are known.

Proceeding in this manner he obtained the

general term of the expansion. Using this method Grace solved the problem of a moving sphere which, from a position of relative rest, is suddenly projected in a direction parallel to the axis of rotation and is then free to move without any external forces other than the fluid pressures

.

Grace found that the velocities tend to

zero at infinity.

In general the disturbance in the

neighbourhood of the sphere is greater in the region above and below the sphere than anywhere else. discussing this result Taylor suggested

In

that this

might possibly have some bearing on the question of the motion becoming two-dimensional.

The solution

breaks down because the velocity gradients continue to increase with time so that the differential equations are no longer applicable. The same analysis was then applied to the case of the free motion of a sphere projected at right | 1I■

angles to

the axis of rotation

J

Hoy. Soc. London? Roy. Soc. London? Roy. Soc. London?

Proc. + Proc. -HH-proc.

.Again all velocities

Series A, Vol. 102, Series A, Vol. 104, Series A, Vol. 104,

p. 89 p.218 p.278

-16-

tend to zero at infinity.

On the axis of the sphere,

i.e. the line through the center of the sphere parallel to the axis of rotation, the motion is wholly perpen­ dicular to the axis. Combining the results of the two problems the small general motion of a sphere projected in a rotating fluid with any given initial velocity can be obtained.

While the actual motion remains small,

the velocity gradients increase with time and hence the solution can only be expected to give the true state of the system for a restricted time. In his last paper published in 1927 Grace made use of the same method to investigate the general slow motion of a sphere in a rotating fluid. It was again assumed that the sphere is initially at rest relative to the rotating axes and that a sudden small motion is imparted to it.

The general motion

was shown to depend upon two special types, namely uniform translation of the sphere parallel to the axis of rotation, and uniform translation (relative to the rotating axes) at right angles to the axis of ro­ tation.

These two motions were considered in detail,

the resulting power series being summed numerically

^ proc. Roy. Soc, London: Series A, Vol. 113, p, 46

■17-

for several points in space and time. in the first case Grace found that the re­ sultant force on the sphere is opposite In direction to the motion of the sphere, that it increases from zero to a maximum and then oscillates with continually diminishing amplitude about a constant value, the oscillation being inappreciable after one complete rotation of the undisturbed liquid,

(fig. I (1) )

Along the axis of the sphere the motion is in the direction in which the sphere is moving and tends to zero with increasing distance from the sphere (fig. I (2) ). Across the equatorial plane (the plane through the center of the sphere at right angles to the axis) the only velocity component is in the direction of the axis and the disturbance is small except in the neigh­ bourhood of the sphere where it continues to increase with time:

(fig. I (3) ).

Figures I (2) and I (3)

show the velocities on the axis and on the equatorial plane at four successive timesj r is the spherical coordinate, a the radius of the sphere. ¥

Rt

2

In)

tl/a * P kq X U)

o--

z

a/©

v

-18-

Similar results were obtained for the motion at right angles to the axis of rotation.

Here again

the motion in the immediate neighbourhood of the sphere tends to become violent with time. Grace concluded that his results indicated that for the case of a sphere moving in a rotating • liquid with small uniform velocity relative to the axes which rotate with the liquid, the ultimate physical state would be one in which the motion would be steady but not small in the neighbourhood of the sphere. It would seem that this conclusion is in contradiction with the experimental results of Taylor which Indicate that when a small motion is started by moving a three-dimensional body with uniform velocity the motion tends to become steady and two-dimensional* Taylor does not make any mention of the motion b e - • coming increasingly violent in the neighbourhood of the body. The last effort, which the author was able to find in the literature, to make further progress in the theoretical study of the flow in rotating fluids is a paper by H. Gortler

published in 1944t

G&rtler

Zeitschrift angew. Math, u, Mech. Band 24, Nr. 5 u. 6

-19-

considered solutions of the equations of small relative motion in the form of an oscillation with time, i.e. he tried solutions

,

\l

\/ r"

i

/

This results in the same differential equation in (x,y,z) for each of the “unknowns TP, V 1, W f, P*.

G&rtler noticed that this equation is elliptic, parabolic or hyperbolic, depending on whether

respectively.

Hence, he argued, entirely different

flow patterns may be expected in each of these cases. In the hyperbolic case there will be real characteristic surfaces which will be visible throughout the fluid as locations of a disturbance.

The solutions may not

be analytic on these surfaces and it is therefore possible that they will appear to divide the fluid into regions of varying disturbance*

These discon­

tinuities cannot be expected in a real viscous fluid, but may show up as regions of sharp gradients. Gortler claimed that this consideration may

-20-

throw some light on the two-dimensional behaviour observed by Taylor.

For the case of a body oscillating

with very small frequency

the situation at the time

when the body passes through its position of equili­ brium resembles the motion of a body moving with uni­ form velocity.

Now as ^

approaches zero the charac­

teristic surfaces tend to become cylindrical with generators parallel to the axis of rotation.

It

therefore seems feasible, Gortler claimed, that a cylinder of the same diameter as the sphere and en­ closing the latter should appear as the physical separation of two disturbed regions in Taylor*s ex­ periments.

Gortler believed that a more general

solution of the form he considered which satisfies the boundary conditions and which consists of a super­ position of solutions for an infinite number of frequencies may well lead to further results. After studying the work reviewed above, it seemed to the author that one of the main points which needed further investigation was the question of how a steady two-dimensional motion can be set up when the disturbance is three-dimensional.

Gortler»s comparison

of the limiting hyperbolic case of an oscillating body with uniform motion can only be taken as an in­ dication, not as a real explanation of the problem,

-21-

because the past history of the flow for an oscillating body and that for a body starting from rest and moving with uniform velocity are entirely different. Another point of interest seems to be the apparent disagreement between Grace*s solution and Taylor>s experimental results.

Taylor»s indetermincay

in the solution for a sphere moving along the axis of rotation with constant velocity, based on the assumption of steady flow* appears to indicate that some of the difficulties may lie in the question of how the motion is started.

Proudman's and Grace is investigations to

some extent avoid this question by assuming that the flow immediately after sudden creation of the disturbance is irrotational, and this irrotational flow, satisfying the boundary conditions, really is their initial con­ dition on which the solution of the problem is based. The author first follows G&rtleris method to investigate the flows resulting from forced oscillations of the elliptic and hyperbolic regime.

An attempt is

then made to gain more insight into the phenomenon of the two-dimensional steady flow with three dimensional boundary conditions by considering a problem from the moment at which the disturbance is created from relative rest with the only initial assumption that the fluid is rotating uniformly like a solid body.

-22-

II

EQUATIONS

OF

MOTION

Throughout this work it was found convenient to use equations of motion referred to rotating axes, bhile the general equations referred to moving axes can be found in several standard books of reference, the author thought it might be useful to give a care­ ful derivation here.

Definitions Consider a fixed and a moving frame.

Let

(x,y,z) be the coordinates of a point referred to the moving frame.

Let U, V, W, be the components of the

vector velocity V(t) of the origin of the moving frame and let P, Q , H

be the components of its

angular

velocity vector CO (t), all velocity vectors being re­ solved at each ihstant into components parallel to the instantaneous directions of the axes of the moving frame.

To avoid repetition let it be stated here that

this scheme of resolving vectors into components the directions of which are parallel to the instantaneous directions of the axes of the moving frame will be applied to all vectors throughout this paper.

-23-

Let the vector £(x,y,z,t) and its components u, v, w represent the "absolute” velocity, i.e. the velocity with respect to the fixed system, of the fluid at time t and at a point in space given by the co­ ordinates

(x,y,z) of the moving system.

(Note that to

specify a point in'fixed space it is necessary to give not only (x,y,z), but also the position of the moving frame), Finally let

(x,y,z,t) and its components

u » , v f, w* denote the "relative” velocity, i.e. the velocity with respect to the moving frame, of the fluid at time t and at a point in space given by (x,y,z). (Notes for a fluid particle which always has co­ ordinates, say (Xj, y , z (),

is zero at all times).

Relation between Absolute and Relative Velocities. In vector form this relation is given by

£ l= V . »• a 1 +- oj x _ n where

XI (1)

is the displacement vector from the origin

of the moving frame to the point (x,y,z).

Written

in terms of components the relation becomes: u . ~ u. +• u - - y R

+

1

IX (2) ! W

/ ~YI

r /1 — + W

1

- x Q

'r X X -t-y

-24-

Derivatives. is the rate of change of the u component at of velocity of the fluid at a point which is rigidly attached to the moving frame.

It is the measure of the

combination of two rates of change.: the first is due to the change with time of the velocity field;

the second

is due to the fact that u(x,y,z,t) and u(x,y,z,t-hft ) refer to the velocities of the fluid not only at different times, but also at different points in fixed space owing to the movement of the relative frame in time L it

. •%

Bx

is the rate of change of the velocity com-

ponent u at a given instant of time t with a change in x (parallel to the instantaneous position of the moving axes). Keeping in mind the definitions, the following relations are derived from II (2): iLi- _ X

'1 X

'

"37

c)v

S’irk m U. + V/ t£^) —

f v ‘ 3-

Also, using the previous definition of the partial derivative with respect to time

Thus the u component of the particle at t -eat is given by

11(7) the quantities in bracket being evaluated at x,y,z,t. Similar expressions can be derived for the v and w components. It is now possible to write an equation similar to I I (6) for the new K component of velocity. Re­ calling that during the time interval at and n(t) will have changed to

d £{-h

l(t), m(t)

e^c *:

-28-

at

'( ' ' '

c3x

L

y

, ( 0+ No k writing the difference quotient for K(t) =* u. ^jS~ +■ dt

(t

n relative to the moving

frame. Then jiti » £*

If the vector

remains fixed

,

in space

and the system

rotates with angular velocity OJ ~ (ip + j_Q

kR) then the

apparent rate

of change of the vector r relative to the

moving system

will he —

will he -

1

Kjt .

CO

In

time cf'£the change

)—

2* t

A

0

X( Kc a)

d»/ For z ^ 0 the sign in the exponent and the sign of the entire expression must be changed. space the solutions for z y

0 only

The boundary condition

will be considered

at z = 0 is:

f b/ S trtfit

^ j

To conserve

co < a

0

co > a

^ x Using Dini*s expansion

where

A ( , A T«-*are the positive roots (arranged in

ascending order of magnitude) of the function

X X where H

(T)+

and V are real constants and

V + i >/0 .

The coefficients As are determined by the formula:

If H=*0 and .nd

V - 0 ? then Am are the roots of JtLT] as 2

/

I

^ '

Watson: Bessel Functions, p. 596

m)j ^

V(l)

-51-

The expansion ‘has a constant term which can be ob­ tained by starting with s -

0

and putting

° 0 and

^ * °Va ^ I

so that the asymptotic form is applicable.

lt~ h GJ, ^ C,

and

( __

*a H- K gt *. = C z

|

Let Then

,Aj

\JGJ, W(CCT» jit,) t ,lGTz W(a3z ,-£*) —

x t?

*£

b K

C3' i a ( ?r W , ) * £ ) }

V I(4 ) rj

' ®SJIt iL0St ^

+1 Now if

i*c,isrtt (iv-wg ]}J

■£, - ^ ’ 0 , the relations ?, — hUJt ~ C\

j 'i.i -k'h Bj, =■ - C |

Z 1+/iGTx - C t_

j

can be substituted in the previous expression, giving* a p .

/ro, vj(art}o) + joh. w

-

j®v./+

V I (5)

+ i i B S -*{

s; &

h c o s /^ c 1 1 ]+ 2 c ^ and

Let their intersection be (ca0?to ).

?//1 *■*' are straight lines.

-55l **

T ank

•' ~D T- '- ^•----- C q / * C? r"“ i i '* '*'Tx.[ W vTT.)°/“

Since are zero.

,

co>ca

both

w^o)

- ftL I! ~T- ^ 1rV ;Lj

and

W (

i o)

Therefore;

\

*

| W Q1-

T-

I^

__

!~ ;

J

I '-v^o * JCET, ■"S1&?«,.J

b

L j&v

This gives the approximate value of in terms of S3, yCfe which are the values of ta at z « 0 given by the characteristics through (Qo^io) of the differential equation for w, provided both

V/icr, ^o)

and

w

, 0) are zero.

Expression V I (7) shows that if Sj^i.e.

is nearly

is small, or if the characteristics are

very steep, i,e, h large or (I very small, then

VI (7)

-

and

W(£T0

56

-

is approximately zero.

This consideration seems to indicate that there will be a zone of nearly undisturbed fluid outside the characteristic cone generated by the straight line

a — ho. ,

This cannot be expected

to be true everywhere outside this cone because of the influence of the walls of the container from which the characteristic lines will be reflected carrying the disturbance with them.

Wave Form of Asymptotic Solution By combining the trigonometric functions in ¥1(5)

w can be written in the following form;

This represents a superposition of travelling waves, each wave having a length of ^ The velocity of the wave would be

2?r

a ir

VII

FQBCED OSCILLATION OF V VELOCITY COMPONENT ALONG Z AXIS:

HYPERBOLIC CASE.

To permit a closer investigation of the zones of undisturbed fluid which were indicated in the last section it was deemed advisable to try to obtain a solution in a form which could be evaluated exactly. To accomplish this a problem will be considered which is probably of academic interest only because the boundary conditions which.will be imposed cannot be realized in an ordinary experimental setup. To avoid the effects due to the walls of a container an infinite fluid is assumed and the following boundary conditions are imposed: V I I (1)

Using the elementary solutions IV(10) the velocities are expressed in the form of Fourier integrals.

VII (2 ) h

-

58-

After satisfying the boundary conditions these become: s'

VI-

yu.

‘T c

y

Vb — Yfj- 5 1tyf’f’

-t y fit , \ ^ [ ~ q ~ hyu) dyU

7|

‘ gfoto'fttjStop

A/uj d/vt

V I I (3)

S < V 1 -3,C % u j

Calculate w from the integral by means of the formula4' X U js(_xf)c>irix °/x ^

fz 0l

V I I (4)

The integrals in Vl(3) can easily be changed to the form V I I (4) by -writing the product of sine and cosine as the sines of the sum and difference of the angles. Carrying out the integration gives the following results for w for z > 0 .

The results for

z < 0 are anti-symmetrical. ;Qk

as*o

^

2 =0 3»-a

Whittaker and Watson: Modern Analysis, Chap.XVII Ex,27

-

59

-

In region:

© 2 -a ;'Z:+AnTa.

@

h -> g

@

^ > Q. y 'Z-hVZ > - a . ; *2r- h £o - ^ :

W — ^ ©

1£

> cl ,

;

fCU'C

vi-=0

J -o^c

V I I (5)

sir? -^~L]

■£ - f? Co < — c l ^

W = W ^r?/\/' lure c i n ^ i ^ - o r ’c stn^:~( T

''

-

bro

be:]

These results show a very peculiar behaviour indeed.

Except at Q = o ^ :i^ia> w is everywhere continuous

and tends to zero with increasing radius.

The forced

oscillation at ed=Gj2 o ,tj =

$ ( I-e

The exponent is dimensionless.

Suppose

R t ) V o ( K co) This simply means that

time is measured in terms of the number of revolutions of the undisturbed fluid.

The graph of this function

is shown in figure VI I I (1).

O Rt It may be desirable later to compare the solution with that of Grace who assumes that the uni­ form motion of the sphere is started suddenly from rest; i.e. that the sphere has its ultimate velocity at t « 0 ,

Suppose the following boundary condition is

chosen:

vj(G5}o,t)** W ( i - e “ b ^ t )o*(Kco) where b is a constant.

VIII(5)

The larger b is taken the

sooner does the disturbance reach its ultimate value. For b-*'^ V I I I (5) results in the impulse function which Grace uses. To determine a (s ) in V I I I (4) find the trans­ form of the boundary condition.

and

This is

-66-

Hen ce the transforms of the three velocities are;

f c G w )

-

V'etai.s)—

77= 7 ^

e

1& * * U K * )

^ ^ 5 - T t w

^

VIII(6)

c k «7)

Inversion of the Laplace Transforms A search in tables for some known transforms which would help invert V I I I (6) failed.

Recourse

was therefore taken to the complex inversion formula; - 2 R

» In view of the difficulties encountered in in trying to satisfy the boundary conditions in the case of a moving body it would be very useful if a class of bodies could be found for which it would, be permissible to satisfy approximate boundary conditions only. Refer the equations of motion to a frame which rotates with angular velocity a about its z ^.xis ciiu. which is fixed in the body, the velocities oeing measured with respect to the rotating but non-tr >ns~ lating frame; i.e. equations III(10) in cylindrical coordinates with the primes omitted.

St 7 w

7)t where

7

IX (1) ,Q P

7)1

are the cylindrical coordinates' in the

frame which is fixed in the body.

-75-

Failure of *Blender" Body Approximations First consider a "slender" body moving in the z direction with velocity $ defined as one for which

, a "slender" body being

(if

I'M being the contour *c of the body. The boundary condition is v ! 5-N ; \ V 53 bod A’ — ( vJ W

where ^ is the angle between the z axis and a tangent to the surface of the body. This could be simplified to involve one una* known only if W , as would be expected Tor a slender body.

Since the flow is expected to become

twodimensional, however, w will approach

W

.

Hence

this approximation breaks clown, or rather using it would certainly eliminate the possibility of obtaining a solution which becomes two-dimensional, and this violates experimental evidence.

Approximation for a "Broad" Body Suppose

, i.e. a "broad" body.

The boundary condition is:

^ . g

V S(■dd *f**/./'Cb1J ” VsI mOOQf where °is is the angle between the z axis and a normal to the surface of the body.

In this case V^r can be

expected to be very small everywhere except near the Hence the boundary condition can

edge of the body. be simplified to

W-

VI

and it may be hoped that the

solutions will be violated near the edge only.

Similarity Law for Broad Oscillating Body The above result may not only prove useful in the study of the flow caused by moving bodies for which the approximate boundary condition is applicable, but it also suggests the idea of developing a similarity law for the oscillation of a broad body with freT , i.e. in the elliptic regime, which

quency

will permit one to write down the solution for the rotational case provided the solution for the motion of a similar body in the irrotational case can be found. Suppose a body oscillates with frequency in a rotating fluid.

jl

The fluid^velocities will be of

the form Vt3 (svi,)eu^

V& ,kl

and Vin;

)

} w-lVfGJ,^)ed^

must satify the differential equations

-77-

Let an axial

represent the surface of the hody in

plane and let 1 and n be the direction

cosines of the normal to the surface of the body. If

denotes the velocity of any point on

the body normal to its surface, then the boundary condition on Var and

W

is;

£Vro + n W = Expressing 1 and n in terms

of the slope of the body

surface this boundary condition becomes: —

.11'. - V/sj,?ccj)+

A—

W(c3;i(C!j)= C t f a l m )

jr+2

i x (s)

Hence the solution of the problem must satisfy equations IX(2) and IX(3).

For oscillation inthe

direction of the z axis c(w))« - - = = - W >N * where the constant W is the maximum velocity of the body. Now suppose a similar problem is solved in the irrotational case.

When irrotational!ty is assumed

it is immaterial if the body oscillates or moves in any other manner because the equations of motion for the velocities are independent of time, i.e, any instantaneous flow pattern of an unsteady flow is also a possible *****^ steady flow pattern. Let (VcsjWj be the velocities ,

in the irrotational case.

Then the differential

equations to be satisfied are:

and the boundary condition is: -

--— I — - 1/(3( ® > 3 ( 3 3 ; ) r * y Jl + Z ^ ''! + ? -

where 55,1

are ^/ ^

body and Q

3 fro,2/S) ■ ix (5)

the coordinates, the surface of the """T r1 \ ? ^ &(£)')/ the velocity of points on the

surface of the body in a direction normal to that sur­ face.

For motion in the direction of the z axis

'u?. 2/S?;}-

where A is the velocity of the body.

Try the following forms of solution in the rotational case: V'fitf(kj l^)SK Xj Vcj(^)^ / ~ X | '/si [C o

^ J .% )

ix(6) where A n A ^ u a r e functions of u3; 3determined.

Hence:

or constants to be,

£T =* 65 ; £ ®

Substituting the proper relations in I X (1) gives the following condition on Vti7 and W 7r,,T , _L 'jfL4a* m

_ JXe. 4 . ,-7“ > 3 © ’ txr'*' ‘“/

_ JT^

o

Because of IX(4) the above equations will be satisfied provided C^tx •* / .

Hence, if

j o then

'

1^(6)

-

79-

satisfies the differential equations IX(S). It is now necessary to satisfy the boundary condition IX(3).

Since the irrotational flow problem

has been solved, the relation I X (5) is known to be true.

Rewriting this equation in terms of the proposed

solutions for the rotational case IX(6), it becomes: . v & y

j x /

1

M

•'

Xi.

u / m i ® i L s ( r o £(«»)) ‘ I 7 IX(V)

In order that this last expression may represent the boundary condition in the rotational case similar to I X (3) the following two relations must be true: via) i + r -

I! + T f o

X,

Solving the first one for

I X ( 8)

A ( gives:

7c^ +'2')^

i + t 7Hence, in general, Aiwould have to be a function of itf . Suppose now that a class of problems is con­ sidered for which

! , i.e. the "broad" bodies dis­

cussed in the previous section. Since 0 a . This is due to the condition of antisymmetry about the * plane z = 0. The velocities tend to zero at infinity. At the edge £3 - a w and Vs* become infinite. This, of course, is due to the sharp edge where the solution cannot be expected to represent the actual physical conditions.

^Watson: Bessel Functions, Chap.

XIII , Art. 42

-84-

X

FLOW DUE TO THE MOVEMENT OF A DISC WHICH STARTS

FROM REST AND ULTIMATELY MOVES WITH CONSTANT VELOCITY.

In section VIII the flow due to a disturbance which started gradually and approached a steady state was discussed by making use of Laplace Transforms. It was pointed out at the end of the section that there did not yet appear to be a possibility of applying the method to the study of the motion of a disc because the w velocity component on the plane of the disc and at a radius greater than the radius of the disc was not know. In this section it will be shown how the Similarity Law developed in section IX, though applicable only to the case of a broad body which oscillates with frequency

suggests a method

of obtaining the solution to the problem in question, i.e. to a disc which accelerates from rest and whose velocity then approaches a constant. The elementary solutions to the equations obtained after introducing the Laplace transforms into the equations of motion IX(l) are given by V I I I (4). They are:

-

85

-

i h£

Vv ~

A

(s, K)

6

Jo[li

W" — A A A ) -b~ 'VcJ- ^ r — L &

—t , ! / - "» J.lftoo; X(l)

i>., = — V cr

7 p fl/,

-.ill, >?-.). ^ ^ —fj / , S 003 S -3J u I)

.. by Usti - ms -

=0

o XI(£)

c)

Vj)-! tils

~0 OPs _

dj Ws+i

u

From these the following relations are obtained:

s+l t 1-^s-l Us+i s-i

+ Vs-i Wtn

-

3 PS

-,53-

= - l , ' sl d'CO

XI (5)

+ Ws-i r-t

where Wo is defined by

Wo-'-'

-93-

Differentiating these and using the continuity equation XX(2a) gives: vzg - i ^ : , 4 5 4 ^cn*^ 4)c^ For

,-ZB 4)

s.?,v 4)

2

'1"

XI(4a)

1 V

->r^> ^ Wo r[ * 7)T

Also for s ® 0 >

by XI (3d)

XI(4b)

using X I (2) XI(4c)

% rs o'V'o ( , ^ ^ ^ 4)CD ^ ^ These are the differential equations which must he satisfied by the coefficients H

Ps,

are known, the coefficients

be obtained from the equations

W-stVj X I (2).

.

j Ws

If the can

It is now

necessary to find the boundary conditions on the functions

id

.

To avoid complicating this investigation with extremely cumbersome expressions the simple boundary condition employed in section VIII will be made use of again.

Let XI (5)

V i f e , 0,t) ■**

This represents an injection and withdrawal of the fluid at the plane

0.

Any conclusions that will

be drawn for this problem can probably be extended to a superposition of elementary solutions in terms of a Fourier-Bessel integral which would represent the

-94-

solution for the problem of a moving disc. Following Grace it is assumed that the d i s t u r ba n ce is started suddenly at t * 0 flow given by

b U ; tt,

and that the initial

is irrotational.

Hence*

XI (6)

W&(cb,o)-0 Let R,

S “ /, 2 * -.

be the velocity potential of the initial

irrotational flow, so that ) v'0 ^'cd

"c j

X I (7)

The differential equations for the Fs V*PS - -

cd C and the boundary conditions are

then become:

,V - R - 7 ^ , - 0

X I (8)

XI (9) bzl*o =“ W IK®)

.

From these it is seen that for all even s

Fs-0

since they must satisfy Laplacefs equation and their normal derivatives vanish on the boundary. The Fs for odd s are found by determining the particular integral of X I (8) and then adding harmonic functions to satisfy the boundary conditions.

A w a 0 i' ■nYr— ^ w j' P-, Y v.. ^ , y ^^ rP — b t \; K p1 ^ 7' / -° l r' J ’J h ' /-7 _ K Y2- . '■/ h oz. ==• xL r U 1 y'' p. y\

k -2lR ;

Define where:

-I