Spin-up of a contained fluid

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OFFICE OF NAVAL RESEARCH _{Technische Hogeschool} DEPARTMENT OF THE NAVY

Deift

CONTRACT NONR-220(35)

SPINUP OF A CONTAINED FLUID

BY

GIULIO VENEZIAN

DIVISION OF ENGINEERING AND APPLIED SCIENCE

CALIFORNIA INSTITUTE OF TECHNOLOGY

PASADENA, CALIFORNIA

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Office of Naval Research Department of the Navy

Contract Nonr 220(35)

SPIN-UP OF A CONTAINED FLUID

by

Giulio Venezian

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

This document has been approvied for public

release and sale; its distribution is unlimited.

Division of Engineering and Applied Science

California Institute of Technology Pasadena, California

'Present Address - Univ. of Hawaii

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When a container filled with fluid rotates uniformly, the fluid ultimately reaches a state of rigid body rotation with the same angular velocity as the container. The process whereby the fluid changes its angular velocity from an initial value to a final one when the walls of the

container are given a corresponding change is called up (or

spin-down if the final angular speed is less than the initial one). In a more

general sense, the spin-up process is involved in the approach to steady

state of all contained fluid motions in which the final state is dominated

by rotation. We shall adopt this wider view here.

In all spin-up phenomena, the steady state is reached not by viscous diffusion but by an internal circulation produced by the Ekman layer suction. It was shown in the first of these seminars(U that the

time constant associated with the decay of motion in a contained fluid is

proportional to Ez, where E = vh2 is the Ekman number.

The characteristic time for viscous diffusion, on the other hand, is

pro-portional to E and since in most applications E is very small, a time

scale of order E2 is much shorter than one of order E. Spin-up is

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and

-2-The Ekman layer and its formation

In 1905, Ekman2 investigated the effect of the motion of a rigid

boundary on the flow of a rotating fluid. In particular, he considered the

motion that is produced in a fluid adjacent to an infinite flat plate when the

plate, as viewed ìn a rotating reference frame, is moved rectilinearly in

its own plane.

The simplest example of such a motion is the following. A homo-geneous fluid occupies the space above a rigid plane located at z = 0.

Far away from the plane the fluid is in a state of rigid body rotation with an angular velocity 2, parallel to the z axis. Relative to a reference

frame rotating with the same angular velocity 2, the plane is moving

with a constant velocity U along the x axis. We are interested in the

motion of the fluid.

The equations of motion of a fluid relative to a rotating frame are:

+v.Vv +2X

=-Vp+vV2v

(1)

Vv

= O . (2)

Let u,v,w be the components of y along the x,y,z axes.

For the problem at hand we seek a solution of Eqs. (1) and (2) subject to the

con-ditions

u=U

,

v=w=0

on z=0

and

u=v=w=0

at z=

(5)

which are functions of z only. In that case Eq. (2) states that

8z

and, in view of the boundary conditions, w = O throughout. This

elimi-nates the non-linear advection term, and the equations for u and y are

simply d2u

V-+ZV=O

, (3) dz2 d2

y-

2Çu=0 (4) dz2

The linear combination u u + iv then satisfies

dk

v__±+2iu =0

dz2 + so that -(l+i)zf u+ = ¡Je

where we have made use of the boundary conditions at z = O

and z

=

The real and imaginary parts of this expression then give u and y: - z

u = ¡Je cosXz , (6)

->.z

y = -tJe sinXz (7)

where X = (Wv)2. When y is small,

u and y drop rapidly to zero

and are significant only in a thin layer of thickness (v/2)2 . This layer

is called the Ekman layer.

As the plate drags fluid along, the Coriolis force deflects the fluid to the right so that there is a mass transport in the -y direction as well as in the x-direction. The two mass fluxes are equal, since

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so that and i 1 v U(l -i)

udz

= o

i

Q_x _=Ç

_j

a=('12

o Q y

rc

vdz o

-4-The net transport of fluid is thus at 45° to the motion of the plate. With slight modifications, these results hold for a uniform flow

over a stationary flat plate. A flow with velocity U in the x direction

in the bulk of the fluid can be maintained by a pressure field p = 22Uy. The velocity then has to drop to zero at the wall and this is accomplished in the Ekman layer. The expression for u in this case is

-(l+i)X z

u = TJ(l-e

with corresponding expressions replacing Eqs. (6) - (9).

One important question that arises, and which was discussed by

Ekman in his paper, is how the steady state solutions given by Eqs. (6) and (7) are attained when the plate is initially at rest and at t = O it is

impulsively given the velocity U. For this problem, Eq. (5) is replaced by

au₊ a2u

-4-V

-+ (10)

subject to u₊ = O for t < O,

and u, = U at

₁

z = O for t> O.

This problem can be solved by Laplace transforms. Let

= iT(s) + i(s) be the Laplace transform of _u+ then

(8)

(7)

V ₌ _(s+2i)u+ a Z

with =

U/s at z

= O. Therefore

Is + 2 i2 -z'

'1

e

The inverse transform is found in tables of Laplace transforms, with the result + (1+i)f j =

s:

dz = I Uy2 sjs+2i2 (12) (14)

The inverse transform of this expression is

=

()2U(l-i)erf{(l+i)]

, (15)

u = U

-

e-(l+i)Xzerfcii z

2/

+ U e(l+i)Xz erfci z

2f

The error function complement is defined as

erfcu = Ç

edx

(13)

and it is equal to 0, 1, or

2 when u is equal to

, O, or

Equation (12) thus tends to the correct values for t =

O and t

= oo The

details of how the final state is achieved can be obtained from the asymp-.

totic behavior of the error function. Briefly, the results are that the layer forms in a time of order i

/,

that is, in a few revolutions,

al-though the inertial oscillations generated by the impulsive motion of the boundary last much longer. These results can also be obtained by ignoring the detailed structure of the flow and looking only at the mass

flux. From Eq. (11),

U u+ _s

(8)

-6-I which may be separated into real and imaginary parts in terms Fresnel

integrals.

For large values of x,

i -X2

erfx'l

e so that

i

1 vi2

()

U(ii)[i

_[zt

and

For small values of x,

1 31

erfxr.x

-x

so that

Q

-UJic2t

y 3 Tr

The time in which the mass flux builds up from zero at t = O to the final

value can be estimated by extrapolation of these equations. This gives a

time which is a fraction 1/2. What happens in fact is that there is a considerable overshoot followed by oscillations of frequency Z2 about the final value which decay in amplitude as

For our purposes, it is sufficient to note that the Ekman layers are established very rapidly. If one is willing to forego a detailed

(18) (19) or Q

u[l

+ sin 22t (16) J2iî2t7

cos2t

X and Q y

2)

U _{/2 Tr2t} (17)

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examination of their formation, they can be regarded as formed instant-aneously, as compared to the other time scales involved. At the same time one should keep in mind that in the case of the earth, a few revolu-tions means a few days. It may seem unreasonable to regard such a

period as short, but it is if the other time scales are measured in months or years.

Ekman layer suction

The flows we have just discussed are exact solutions of the Navier

Stokes equations. In them, rapid spatial variations in the fluid velocity occur only in a layer of thickness ô = (v/2)2. If this thickness is small

when compared to the geometrical dimensions of the system being con-sidered, such as the curvature of the wallor separation between walls, it can be expected that a good approximation to the flow may be obtained from equations similar to Eqs. (3) and (4).

Two requirements have to be met.

If h is a characteristic

dimension parallel to 2, the Ekman number is E v/2hZ, and thus

the condition that ô « h means that E « 1. The second condition is that the non-linear advection terms must be negligible. The flows considered

in the previous section were parallel, and the non-linear terms were

exactly zero. Since we can no longer expect this to be the case we must

require that in some sense vVv should be small compared to, say

Z X y. If 13 is a characteristic velocity (relative to rigid body

rota-tion) this means that the ratio E = U/22L should be small, where L is

some typical dimension perpendicular to 2. This ratio is called the

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-8-If these conditions are met, a set of equations which are correct

up to terms of order E and

can be derived for the flow adjacent to

the wall.

Let y

_{= VT + VN} be the velocity relative to the interior fluid where VT is tangential to the wall and _VN is normal to the wall. Let n be a coordinate normal to the wall and = nIó. Then the tangential part of the momentum equation becomes

I

+ 2X

=

SV

at

while the normal component simply gives the pressure, and is not

important for our purposes. The equatiön of continuity reads

SVN

= - o V. VT

These are the boundary layer equations, except that, as it was discussed in the preceding section, the first term can be omitted in Eq. (20) if the

time required for the boundary layer to form can be neglected.

By assumption y exists only in the boundary layer and is zero for large values of . At the wall, y added to the velocity V of the

interior flow must be equal to the wall velocity. _{Note that VN is of}

order E2 smaller than

_VT

To illustrate the use of these equations, consider the steady flow that exists in a rotating fluid over a flat plate, when the plate is given an

additional angular velocity i&2. Let the plate be at

z = O and let u,v,w

be the velocity components in cylindrical coordinates.

The interior flow (for zero Ros sby number) satisfies the inviscid

(11)

and But so that and

ia(s)

_+-C

r 8r

ar

From Eq. (23) and the boundary condition u+ = iM2r at r = o,

= ir

el+

Similarly from Eq. (24) and the condition that W = O at = O,

=-W(r)-ó

Ç 1 a

- (ru)d

₍₂₆₎

Now, w(oc) O, and hence

/b OC W(r) = 1 a

(r)d

o

(1+i)2r

Z

d= -S2r

- 22V = - _8r 22U = O (22) ap - az

where we have made use of the axial symmetry. The equation of continuity is simply 3W/Sz = O since ¡J = O. Thus the interior flow can be satis-fied by any flow field (O, V(r), W(r) ). The angular velocity of the fluid

at z = has been specified as zero, so what remains is W(r).

The boundary layer equations are:

ar +

(12)

-lo-W(r) = - . (27)

Thus there is a uniform suction of fluid into the boundary layer. The suction is proportional to y2 and to the excess vorticity of the wall

velocity as compared to that of the fluid.

Non-linear effects in the boundary layer

The problem we have just considered has a similarity solution which reduces the full non-linear equations to ordinary differential

equations, which can be solved numerically. The similarity solution was

found by von

Karman3,

who also solved the resulting equations by

ap-proximate methods. A numerical solution was later given by Cochran4.

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More recently Rogers and Lance gave numerical results for the full

range of relative rotation rates between the fluid and the plate when the

angular velocities are in the same direction. Cochran had considered only the case where the plate is rotating and the fluid at infinity is not.

Bodewadt6 had considered the opposite case.

The key to the von Karman solution can be seen from the ap-proximate treatment of the preceding section. The velocity components

parallel to the plate are both proportional to

r, while w is only a

function of z _{Under these conditions, the components of the non-linear}

term v.Vv, while non-zero,have the

same dependence on r as the corresponding components of y _{The linearized equations are not an}

exact solution, but there is an exact solution with the same r dependence.

Moreover, z and y appear only in the combination _zIó, so that 6 is

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is often used to introduce the notion of boundary layers, and is much more familiar to fluid dynamicists than Ekman flow. For example, Schlichting's book on boundary layer theory7 treats Karman flow but

does not mention Ekman flow. The details are found in any of the papers cited as well as in many textbooks.

The result that concerns us is the dependence of the suction on

the Rossby number. Figure 1 is plotted from the numerical data in

Rogers and Lance.

It shows the value of -w()/2ó plotted versus the

ratio w/2, where w is the angular velocity of the fluid at infinity and

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-12-represents the linearized approximation given by Eq. (27).

It can be seen that the linearized approximation is quite good over

a large range of Rossby numbers. Even when the fluid at infinity is at rest a fairly good estimate is obtained. The correction for small Rossby numbers is found in Rogers and Lance. It is

-w(cc)/2ó =

- 0. 3+

0.0875E3 + 0(é) , (28)

wh e r e

The corrections are small, and if necessary can be carried in the analysis

in a form equivalent to Eq. (28), Alternatively Eq. (28) may be used to

estimate the error involved in the linearization.

A much more difficult error to estimate is the error incurred in

the local nature of the linearized Ekman layer. In the analysis given in the preceding section, the flow at any point in the boundary layer is deter-mined exclusively by the difference be tween the velocities of the wall and interior flow over that point. The suction is determined from the diver-gence of the Ekman layer transport and depends only on the first

deriva-tives, since it is proportional to the difference between the wall vorticity

and interior vorticity over that point. For moderate Ros sby numbers, and particularly over regions in which the fluid vorticity changes rapidly,

the boundary layer is definitely non-local. We shall return to this point

later,

Spin-up between parallel planes

The earliest theoretical study of spin-up appears to have been done by Bondi and Lytt1eton8 in their investigation of the role of the

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liquid core in the dynamics of the secular retardation of the earth. They

concluded that the motion of the core lagged behind that of the mantle by a time essentially equal to the spin-up time. Later a thorough study of linearized spin-up was made by Greenspan and

Howard9 for

a general

axially symmetric container. More recent publications include those of

Greenspan0 and Pedlosky and Greenspan('U for non-ially symmetric

containers.

The spin-up of a fluid contained between parallel planes was con-sidered in detail by Greenspan and Howard. It is probably the simplest example of spin-up, although it lacks some essential features because

of the absence of lateral walls. The simplicity derives from the fact that,

as in Karman flow, the r dependence is trivial,

u and y are

proportion-al to

r,

while w does not depend on r. Under the circumstances, the

equations of motion involve only

z and t,

and the linearized problem

can be solved exactly, although not in closed form. The non linear problem also involves functions of z and t only and can be solved

numerically. This was done by

Pearson2.

We shall only give the

approximate results, using boundary layer theory.

The problem is that of a fluid contained between the planes

z = -h/Z and z = h/2.

At t = O, the fluid is rotating at an angular speed

- M2, and the angular velocity of the plates is impulsively increased to 2. We want to find the subsequent motion of the fluid. The equations which

govern the interior flow are

3V

Z2V-at_

-=0 (29)

aW_

ap

(16)

-

-14-where we have made use of the axial symmetry. Now, W is produced

i

by the Ekman layer suction so it is of order E2,

and since U is

con-nected to W by the equation of continuity, U is of the same order. Thus the term au/at is certainly negligible compared to 22V, In the

second of these equations, both terms must be of the same order since

V must change in time and [J cannot be zero if fluid is to be conserved.

It follows that 8/at is of order E2.

The third expression then equates

8P/Oz to a quantity of order E,

and since 3P/8r is of order

1, the

left hand side may be put equal to zero. Equations (29) are therefore replaced by

22V=

+22U=0

(30)

0-Except for the presence of the av/at term these are the same as Eqs.

(22).

The interior flow is thus satisfied by functions U, V, P which

depend only on r and t, and W which depends linearly on z (to satisfy

continuity) This flow must be connected to the boundary conditions by Ekman layers of the type described in Eqs. (23) - (27).

At z = -h/2,

the Ekman suction requires that

In both cases, since the boundary is rotating faster than the interior which is the analog of Eq. (27). Similarly, at z = h/2, the condition is

W=--(rV)

(32)

'

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(V < O), fluid is ejected radially along the boundary layer and must be

supplied to the layer from the interior.

This requires a negative W at

the lower boundary and a positive one at the upper boundary.

Since W is linear in

z,

W=-z--(rV)

The equation of continuity then gives

---- (rU)=---(rV)

r8r

hrar

so that

U=rV

This indicates that for V < O a radial influx is needed in the interior

fl"w to make up for the fluid that is leaving the boundary layer.

Finally, the equation for V becomes

(35)

and since V = -A2r at

t = O,

2Th t

V=-A2re

h (36)

The corresponding expressions for U and W can then be obtained from

Eqs. (33) and (34). The spin-up time is the decay time in Eq. (36), that

1.

is, h/ZQ5 = E2/22.

The physical mechanism for the spin-up process is therefore the following: since the interior fluid is initially rotating at a slower rate

than the boundaries, fluid is ejected by the walls along both Ekman layers. The fluid transported outward flows with a velocity of order 1 in a layer

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-16-i

i

of thickness proportional to E2,

so the mass transport is of order E2.

This requires an equal and opposite mass transport, this time in a layer

with a thickness of order 1, so the radial velocity is proportional to

E. The incoming fluid comes from regions of higher angular momenftm, and since the interior flow is essentially inviscid, the angular momentum

is conserved. In this way the interior flow is speeded up until it rotates at the same angular velocity as the plates and the Ekman suction ceases.

Non-linear spin-up in a cylinder

Linearized spin-up in a cylinder was considered by Greenspan

and Howard. The non-linear case can be treated with equal ease,

pro-vided one assumes that the flow in the Ekman layers may be linearized. This approach was used by

Wedemeyer3

and is presented here with some extensions by Ingersoll and

Venezian4,

Wedemeyer considered only spin-up from rest. We shall look at

the general case when spin-up is from an initial angular velocity w to a

final angular velocity 2, with the axis of rotation parallel to the axis of

the cylinder. Since the Rossby number is no longer small, there is no advantage in referring the flow to rotating coordinates, so we shall use an inertial frame of reference.

The equations of motion for axially symmetric flow, including

r

non-linear terms are:

au 8u y2

+u+w---

+ u (37)

8r 8z r - 8r _r2

8v

+u +w+ =v

8v 8v uy _(Va _- ₍₃₈₎

(19)

In view of Eq. (42), however, V is not a function of z so that the last term of Eq. (41a) is zero and that equation is replaced by

SV

_{+ -- (rV)=0}

US

Moreover, it follows that U must also be a function of

r and

t only,

and hence W is again linear in z

The Ekman layers on the flat lids of the cylinder at z = ± h/2

are treated in the same manner as before. The results are the same as Eqs. (31) - (33), except that they must be referred to the inertial frame.

W,

of course, is the same, but V must be replaced by V -

2r.

Cor-responding to Eqs. (33) and (34) we now have

zó 1 8

(rV-r2)

--

-7i:-and Sw 3w _- _{+ v2w}

- +u - +w - =

_St _Sr _Sz _Sz

For the reasons discussed in the previous section, we expect that

in the interior flow U, W and S/St are of order E2.

To leading order,

so that Eq. (41) becomes

U=(V-2r)

(39)

Eqs. (37) - (39) then simplify to

+

U/-y

+ V2 SP 0 (40) (41a) (42) r -W Sr = St Sr

r)

0-Sz - Sz

(20)

-18-8V

-+--2(rV)=O

6 V 8

. (45)

This will be called Wedemeyer's equation. It contains within it the entire physical description of the spin-up process.

Wedemeyer's equation simplifies somewhat if the angular momentum density G rV is used instead of the azimuthal velocity.

The resulting equation is

8G 6 _{(GÇ2rz)8G_}

-This equation can be solved by the method of characteristics. Indeed,

when the equation is written in this form it is clear G is constant along

a characteristic, i e. a line in the r-t plane such that

dt h r

dr -

_Ç2r2-G

Since G is constant along such a line, Eq. (47) can be integrated directly

to give

t=

h log(2rZG) _{+ F(G)} ₍₄₈₎

where the constant of integration may vary with G.

On the initial line t = O, G = wr2.

This value of G is carried

along each characteristic curve, so that

F(wr2) = log(2w)rZ

or

F(G) =

log - G

O)

Hence, Eq. (48) becomes

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In the case of spin-up, when < 2, Eq. (49) carries only the values

O G wa2. The characteristic that carries the value G = wa2

starts

at r = a at

t = O and is subsequently given by r = r0(t), where

i

rw 2-w -Z62t/h1 2 (50)

r

=aI+---- e

J

o

For t> O,

r0 is less than a and thus Eq. (49) represents the value of

G only for

r < r0. For r > r,

all the values of G between wa2 and

QaZ have to be filled up, by a fan of characteristic lines all of which

start at r = a.

In this case, F(G) = log(a2-G) so that 2r2 _ÇZ -Zó2t/h G= _{- Z 6 Qt / h} 1 -e

The motion described by these equations has several interesting

properties. The flow is divided into two regions by a moving front located at r = r0(t). This front starts at the outer wall, and gradually

moves in to the position r/a = (/2). The flow inside the front rotates

rigidly with an angular velocity that increases from w to 2, as it can

be seen from Eq. (49).

If the spin-up is from rest, G is zero in the

entire region

r < r0 for all time.

This region, however, shrinks to

zero radius as

t - oc, so that the entire fluid eventually spins up.

In the region between the front and the wall, the azimuthal velocity consists of two terms, one proportional to r and one to i Ir.

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-20-These are such that the azimuthal velocity is 2a at the wall, and o.a2/r

at r. The latter value is the same as that reached at

r by the velocity

on the other side of the front.

Thus U and V are continuous across

the front. The derivatives, however, are not continuous so the front is

a moving shear layer. U at the shear layer is equal to dr0/dt,

its

velocity of advance, so that no fluid crosses it. The fluid that pushes the layer inward is the fluid that has been sucked into the Ekman layers

in the region r < r, has been ejected

outward in the layers acquiring

angular momentum in the process, and is _{ttxen ejected from the Ekman}

layers in the region r > r0.

In the case of flow between parallel plates, the front does not exist, since fluid of higher angular momentum is always available at

larger and larger r This feature is a consequence of the infinite

extent of the plates. In the case of spin-up from rest, however, the

model breaks down unless time dependent Ekman layers are used, since

there is no reservoir of angular momentum. No such breakdown occurs

in the finite cylinder since the side walls act as constraint forcing the fluid

sucked into the Ekman layer to return to the interior.

The behavior of the flow during spin-down is markedly different from that during spin-up. If w > , the flow no longer divides into two

regions and Eq. (49) describes the entire flow. The dividing characteristic

moves towards increasing r, and thus it has no physical significance.

In this case there is only one shear layer, which remains attached to the

wall.

In the discussion of the cylinder given by Greenspan and Howard, they found no difference between spin-up and spin-down. The reason for

this is that the shear layer moves only from r

a to

r = a(w/2)2 =

(23)

E = O, _{the shear layer does not move away from the wall at all.}

Non-linear spin-up in an axially symmetric container

Wedemeyer's theory can be extended to the case of an axially

'symmetric container. A brief account of this will be given here. Further details appear in Ref. 14. Let the walls be at

z = f(r) and z = -g(r),

so that the height of the container is given by h(r) = f(r) + g(r). The

linearized Ekman layers for this configuration were treated by Greenspan and Howard. When they are taken into account, Eq. (46) is replaced by

+ _(G-2r2)

9

₌ O at r where i

i

t '4

(1+f)+(1+g

) o = 2(f+g)

G is constant along the characteristics, which have a slope dt

dr -

r

The initial conditions can be incorporated into this expression by writing

t as a definite integral.

For r < r(t),

t= -

r r dr (54)

while for r > r(t),

ca

_rdr

r

(r)(rZG)

o

where a is the radius of the container.

The characteristic r (t) is the o

(24)

-22-one that carries the value G = waZ and is thus given by

t=

rdr

(56)

r

For the special case in which f = g = h/2, Eqs. (55) - (57) are equivalent to Eqs. (49) - (51).

The character of the flow is substantially the same as in the case

of the cylinder. For spin-up, there are two distinct regions separated by a shear layer.

A non-linear theory of spin-up for axially symmetric containers was also developed by Greenspan and

Weinbaum5.

It consists of a

systematic expansion in powers of of both the interior flow and the Ekman layers. This is in contrast with the theory just discussed, in which the Ekman layers are treated as linear, while the resulting interior

equations are then solved exactly. To a certain extent this could be

remedied by allowing ,u to be a function of G as well as r, according

to an expansion similar to Eq. (28). The main error, however, is that ¡L is really non-local and must depend not only on G but on its

deriva-tives as well. This may well be the source of serious error near the shear layer.

On the other hand, the approach of Greenspan and Weinbaum

suf-fers from its drawbacks as well. A moving shear layer is an essentially non-linear feature, which cannot appear in a power series expansion. Moreover, the expansion fails near h = O. The Wedemeyer approach

has the advantage that the physical situation is more readily appreciated and is not hidden in interminable algebra. The real test of the theory, however, rests on the presence or absence of the moving shear layer. In this respect, experiments favor the non-linear theory discussed here.

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References

1. Venezian, G. Seminar of 4/9/68.

Z. Ekman, V.W.,, Arkiv. Math. Ast. Fys. Vol. Z, No. 11 (1905).

von Karman, T,, ZAMM1, 244 (1921)

Cochran, W,G,, Proc. Camb. Phil. Soc. 30, 365 (1934).

Rogers, M.H. and Lance, G.N., J. Fluid Mech. 7, 617 (1960).

Bodewadt, IJ.T., ZAMMZO, 241 (1940).

Schlichting, H, Boundary Layer Theory, (McGraw Hill Book Co., New York, 1960) P. 83.

Bondi, H. and Lyttleton, R.A., Proc, Camb. Phil. Soc. 44, 345

(1 948).

Greenspan, H.P. and Howard, L.N., J. Fluid Mech. 17, 385 (1963). Greenspan, H.P., J. Fluid Mech. 20, 673 (1960).

Pedlosky, J. and GreenspanH.P., J. Fluid Mech. 27, 291 (1967).

12, Pearson, C.E., Sperry Rand Res. Center Report RR-64-41 (1964).

Wedemeyer, E.H. , J. Fluid Mech. 20, 383 (1964).

Ingersoll, A. and Venezian, G., unpublished.

Greenspan, H. P. and Weinbaum, S,,, J. Math, and Phys. 44, 66 (1965).

(26)

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DDIFORTMNOV 65

1473

I (PAGE 1) _{Uncias s if je d}

DOCUMENT CONTROL DATA - R & D

Secur,ty classification of rUle, body of abstract and indexing' annot8lic,rl n,ust be entered s'hen the overall report is classified) ORIGINA TING ACTIVITY (corporate author)

California Institute of Technology

Division of Engineering and Applied Science

2e. REPORT sEcuRITY CLASSIFICATION

Unclassified

2b. GROUP

Not applicable

3. REPORT TITLE

SPIN-UP OF A CONTAINED FLUID 4. OESCRIPTIVENOTES(TtpeofreportandinC1usivedateS)

Technical Report

5. AU THORIS) (First name, middle initial. last name)

Venezian, Giulio

6. REPORT DATE

March 1969

78. TOTAL NO. OF PAGES

23

7b. NO. OF REPS

15

Sa. CONTRAC T OR GRANT NO. Nonr 220(35) b. PROJECT NO.

C.

d.

Sa. ORIGINATORS REPORT NUMBER(S)

97-17

Sb. OTHER REPORT NO(S) (Any other numbers that may be assigned this report)

10. DISTRIBUTION STATEMENT

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

I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY AC TI VITY

Office of Naval Research

3. ABSTRACT

In all spin-up phenomena, the steady state is reached not by viscous

diffusion but by an internal circulation produced by the Ekman layer suction. It

was shown in the first of these seminars(U that the time constant associated with

the decay of motion in a contained fluid is proportional to E2, where E = vh2

is the Ekman number. The characteristic time for viscous diffusion, on the other

hand, is proportional to E and since in most applications E is very small, a

time scale of order E2 is much shorter than one of order E. Spin-up is

there-fore the more effective of two mechanisms.

(36)

Uncias sified Security Classification

D D _{I NOV 6S J}FORM 1473 (BACK)

(PAGE 2)

Uncias sified

Security Classific atior

KEY WORDS LINK A LINK B LINK C

ROLE WT ROLE Wr ROLE WT

Viscous Flow - unsteady

Rotating flow Boundary layer