on the motion of a compressible fluid in a rotating cylinderon the motion of a compressible fluid in...
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On the motion of a compressible fluid in a rotating cylinder
Citation for published version (APA):Brouwers, J. J. H. (1976). On the motion of a compressible fluid in a rotating cylinder. Universiteit Twente.
Document status and date:Published: 01/01/1976
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A COMPRESSIBLE CY LlNDER
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE
TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL TWENTE, OP GEZAG VAN DE RECTOR
MAGNIFICUS, PROF. DR. J . KREIKEN, VOLGENS BE-
SLUIT VAN HET COLLEGE VAN DEKANEN IN HET OPEN-
AAR TE VERDEDIGEN OP DONDERDAG 24 JUNI 1976
TE 16.00 UUW.
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN
PROF. DR. IR. L. VAN WIJNGAARDEN
PROF. DR. J. LOS (Universiteit van Amsterdam)
Aan mijn ouders
Grote dank ben i k ve r s chu ld igd aan de d i r e k t i e van h e t Reac tor Centrum
Nederland en U l t r a Cen t r i f uge Nederland voor de ge l egenhe id d i e ze m i j ge-
boden hebben i n h e t kade r van mi jn werkzaamheden t e promoveren.
Mijn b i j z o n d e r e e r k e n t e l i j k h e i d g a a t u i t n a a r I r . F . E . T . K e l l i n g en
Ir. P . L . C . Wijnant voor de s t e u n d i e ze mi j ve r l e end hebben b i j de voorbe-
r e i d i n g van d i t p r o e f s c h r i f t . De samenwerking b innen h e t l abo ra to r i um voor
U l t r a Cen t r i f uge Ontwikkel ing heb i k s t e e d s b i j z o n d e r op p r i j s g e s t e l d .
De v e l e d i s c u s s i e s met I r . W.A. E r e s s e r s hebben v e e l b i j g e d r a g e n aan
h e t totstandkomen van d i t p r o e f s c h r i f t .
Veel waa rde r ing heb i k voor de u i t v o e r i n g van h e t typewerk d a t werd
verzorgd door mevr. C . G . M . Bressers -Tie lemans .
D e r ep rog ra f i s c l r e d i e n s t van h e t R . C . N . onder l e i d i n g van de h e e r
E . van Rooy dank Fk voor de d ruk t echn i s che u i t v o e r i n g .
L g r a t e f u l l y acknowledge M r . C . A . Andrews B . S c . , M.Inst.P., f o r c r i t i -
ca l l y r e ad ing t h e Eng l i sh t e x t .
CONTENTS
. . . . . . . . . . . . . . . . . . 2.1. The incompressible fluid 13
2.2. The perfect gas . . . . . . . . . . . . . . . . . . . . . . . 28 . . . . . . . . . . . . . . . . . 2.2.1. l'he basic equations 28
2 .2 .2 . Ths rapidly ro ta t ing heavy gas . . . . . . . . . . . . 40 2.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 54
. . . . . . . . . . . . . . . . . . . 3 THE RAPIDLY ROTATING HEAVY GAS 59
3 . 1 . T h e p r o b l e m s t a t e m e n t . . . . . . . . . . . . . . . . . . . . 59 3.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . 59 3. 2 . 2 . The Ekman suct ion . . . . . . . . . . . . . . . . . . 62
3.2. Theunit cylinder . . . . . . . . . . . . . . . . . . . . . . 67 3.3. The semi-long cylinder . . . . . . . . . . . . . . . . . . . 91 3.4. The long cylinder . . . . . . . . . . . . . . . . . . . . . . 99 3.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 110
LIST OF FREQUENTLY USED SYMBOLS . . . . . . . . . . . . . . . . . . . 120
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
. . . . . . . . . . . . . . . . . . . . . . . . . . . . SAMENVATTING 127
CURRICULUMVITAE . . . . . . . . . . . . . . . . . . . . . . . . . . 130
1. INTRODUCTION
In the last 20 years, the dynamics of rotating fluids have become of
considerable interest, especially in meteorological, oceanographic and
geomagnetic questions involving the earth's rotation. Another interesting
use of rotating fluids is in the gas centrifuge used for the separation
of uranium isotopes. A new object of study in this field is the effect of
compressibility. Since the quantitative experimental investigation of a
compressible gas in a high-speed cylinder is an exceedingly difficult pro-
blem,considerable emphasis has been placed on theoretical efforts to study
this problem.
The first obvious step in a theoretical approach is the assumption
that the motion consists of a small perturbation on the isothermal state
of rigid body rotation. In that case the equations describing the secon-
dary flow
made much
flow in a
can be linearised and, of course, the analytical handling is
easier. Under this assumption two kinds of solutions for the gas
rotating cylinder are known.
The first kind deals with a closed cylinder of finite length and has
been investigated by Matsuda, Hashimoto & Takeda ( l976), Matsuda, Sakurai
& Takeda (1975), Matsuda (1975), Mikami (1973a, 1973b), Nakayama & Usui
(1974), Sakurai & Matsuda (1974) and Sakurai (1975). Assuming a small
Ekman number for the primary rotation, the flow consists of an inviscid
interior and boundary layers near the walls, viz. Ekman layers near the
end caps and two layers of the type first discussed by Stewartson (1957)
near the cylinder wall. In fact the approach is a direct extension of the
boundary layer methods often applied to incompressible fluids in rotating
systems (see Greenspan,l968). Investigations in this field have been pri-
marily motivated by geophysical questions stemming from models of atmos-
pheric and oceanic currents.
The second kind of s o l u t i o n d e a l s wi th a c y l i n d e r of semi- in f in i t e
length. This approach stems from models of gas c e n t r i f u g e s , whose l eng th
i s considerably l a r g e r than t h e i r diameter. So lu t ions f o r t h i s problem
were given by Dirac (l94O), Ging (1962a, 1962b), Parker & Mayo (1963),
Parker (1963) and Steenbeck (1958). The b a s i c assumption c o n s i s t s of a
smal l a x i a l v a r i a t i o n of t h e flow v a r i a b l e s compared t o t h e r a d i a l v a r i-
a t i o n . I n t h i s case t h e flow can be descr ibed by a s e t of r a d i a l eigen-
func t ions which decay exponen t i a l ly i n t h e a x i a l d i r e c t i o n .
Apart from t h i s t reatment Berman (1963) and Soubbaramayer (1961) discus-
sed t h e case of a constant a x i a l flow dr iven by r e s p e c t i v e l y a r a d i a l and
an a x i a l temperature g rad ien t . The ca lcu la ted r a d i a l shape of t h e flow
was approximately equal t o t h e shape of t h e f i r s t e igenfunct ion.
The two kinds of approach l e d t o d i f f e r e n t d e s c r i p t i o n s f o r t h e flow
f i e l d . I n the f i r s t k ind t h e end cap cond i t ions have t h e predominating
in f luence . Corresponding t o t h e Taylor-Proudman theorem t h e flow i n an
i n v i s c i d i n t e r i o r i s constant a long t h e a x i a l coordinate and i s con t ro l-
l e d by t h e pumping of t h e Ekman l a y e r s a t t h e top and bottom. I n t h e se-
cond kind i t i s assumed t h a t , due t o r a d i a l d i f f u s i v e processes end e f f e c t s
decay r a t h e r r a p i d l y , s o t h a t a t a reasonable d i s t a n c e , t h e flow p a t t e r n
i s p r a c t i c a l l y independent of t h e end cond i t ions .
On t h e o t h e r hand, i n t h e f i r s t k ind of approach t h e in f luence of a s t r o n g
d e n s i t y g rad ien t , t y p i c a l i n t h e p resen t day gas c e n t r i f u g e s , was no t con-
s ide red . I n t h e case of i sothermal r i g i d body r o t a t i o n t h e d e n s i t y increa-
s e s exponen t i a l ly wi th t h e r a d i a l d i s t a n c e from t h e a x i s . Due t o t h e high
r o t a t i o n a l speed and t h e l a r g e molecular weight of t h e gas , r a t i o s of lo7
between t h e d e n s i t y a t t h e per iphery and a t t h e a x i s a r e q u i t e normal to-
day. I n t h e case of inc reas ing t h e d e n s i t y g rad ien t Parker and Ging obser-
ved t h a t t h e main p a r t of t h e mass flow i s concentra ted a t t h e per iphery.
A t t h e same time t h e a x i a l decay of t h e e igenfunc t ions inc reases conside-
r a b l y , which i n d i c a t e s t h a t t h e c o n t r o l of t h e Ekman l a y e r s on an i n v i s c i d
i n t e r i o r no longer a p p l i e s .
In o rde r t o know whether o r no t t h e d i f f u s i v e processes desc r ib ing t h e
flow i n t h e main s e c t i o n of t h e c y l i n d e r a r e important , i t i s necessary t o
apply an o rde r of magnitude cons ide ra t ion t o t h e va r ious Germs i n t h e
Navier-Stokes equat ions . Here, the in f luence of ( i ) t h e ]ength-to-radius
r a t i o of t h e cy l inde r and ( i i ) t h e r a d i a l d e n s i t y g rad ien t i s of c e n t r a l
importance. A t f i r s t t h e l eng th e f f e c t i s s tud ied s e p a r a t e l y by cons ide r ing
an incompress ible f l u i d i n a r o t a t i n g c y l i n d e r whose length- to- radius
r a t i o L inc reases from u n i t magnitude t o i n f i n i t y . The Navier-Stokes equa-
t i o n s a r e l i n e a r i s e d wi th r e s p e c t t o r i g i d body r o t a t i o n . The Ekman number
E, based on t h e r a d i u s , i s taken t o be smal l . Three types of flow corre-
sponding t o t h e ranges E 1 j 2 << L << E-'I2, f112 % L << E-I and E-I 6 L
a r e found. I n t h e f i r s t range we i d e n t i f y a geost rophic flow i n t h e i n t e-
r i o r extended by Ekman l a y e r s nea r t h e end caps and an inne r and o u t e r
Stewartson l a y e r nea r t h e c y l i n d e r wa l l ( t h e well-known "geophysical"
f low). When L exceeds t h e upper l i m i t of t h e f i r s t L-range t h e o u t e r
Stewartson l a y e r expands t o t h e i n t e r i o r . For L z E-I t h e inne r l a y e r
a l s o f i l l s t h e whole cy l inde r . Then, t h e r a d i a l d i f f u s i o n of azimuthal
and a x i a l momentum i s important i n t h e e n t i r e c y l i n d e r , a t y p i c a l s i t u a-
t i o n f o r s t u d i e s on flows i n semi- in f in i t e c y l i n d e r s . However, s i n c e B i s
very smal l most conf igura t ions w i l l f a l l i n t o t h e f i r s t L-range.
In t h e case of a p e r f e c t gas i n a r o t a t i n g c y l i n d e r , s p e c i a l a t t e n t i o n
i s focused on s t r o n g r a d i a l d e n s i t y g rad ien t s . The modified Ekman number,
Em, based on t h e d e n s i t y a t t h e pe r iphe ry and based on t h e r a d i a l d e n s i t y
s c a l e he igh t ( i n s t e a d of t h e r ad ius ) i s taken t o be smal l . Inc reas ing t h e
r a t i o of t h e l eng th of t h e cy l inde r t o t h e r a d i a l d e n s i t y s c a l e h e i g h t ,
L-, from u n i t magnitude t o i n f i n i t y t h r e e types of flow a r e i d e n t i f i e d . 111
These correspond t o t h e ranges << L << E -112 -112 $ L < < , - I and m m m y E m m m
E-I L; Lm. However, two e s s e n t i a l d i f f e r e n c e s a r e found: ( i ) An i n v i s c i d m
flow t y p i c a l of t h e f i r s t type i s only observed i n a region of l i m i t e d
th ickness nea r t h e c y l i n d e r wa l l . Due t o t h e decrease of t h e d e n s i t y wi th
d i s t a n c e from t h e w a l l , v i s c o s i t y and conduction a r e important i n t h e core
of t h e cy l inde r .
( i i ) A change of t h e flow type appears when both Stewartson l a y e r s expand
over t h e smal l d e n s i t y s c a l e he igh t . Simultaneously t h e corresponding d i f -
f u s i v e regions i n t h e core come up and jo in . As a r e s u l t , a change of t h e
flow type appears a t r e l a t i v e l y lower va lues of t h e length- to- radius r a t i o .
A l l t h r e e types a r e of r e a l p r a c t i c a l i n t e r e s t .
F i n a l l y , e x p l i c i t s o l u t i o n s f o r t h e v e l o c i t y d i s t r i b u t i o n i n t h e v a r i-
ous parameter ranges of t h e r a p i d l y r o t a t i n g heavy gas a r e c a l c u l a t e d .
Here, we apply p e r t u r b a t i o n techniques and t h e method of matched asympto-
t i c expansions a s given by Van Dyke (1964) . The boundary cond i t ions indu-
cing the secondary flow are: differential rotation of the end oaps, axial
injection and removal of fluid at the end caps and temperature perturba-
tions along the end caps and along the cylinder wall. Thus, the flow of a
heavy gas in a rapidly rotating cylinder, e.g. in a gas centrifuge, is
qualitatively and quantitatively well understood.
2. THE SCALING ANALYSIS
2.1. The incompressible fluid
The motion of an incompressible Newtonian fluid of constant material
properties is considered. The fluid is confined in a circular cylinder
which rotates about its vertical symmetry axis. The plane horizontal top
and bottom caps are assumed to rotate with an angular speed which differs
slightly. Furthermore it is assumed that these end caps are permeable
and that fluid is injected and withdrawn vertically. However, we restrict
ourselves to the case that the total axial net transport is zero.
The equations describing the velocity and pressure field of an incompres-
sible fluid are given by the conservation of mass and of momentum.
We confine ourselves to stationary flows. Both equations, written in a
rotating frame fixed with respect to the cylinder, are (Greenspan, 1968):
+ 3 Here q is the particle velocity and r the position vector in the rotating
-f
frame. R is the angular speed of the cylinder and k is the unit vector
along the rotation axis. P, p and v denote respectively pressure, density
and kinematic viscosity.
The terms on the left hand side of the momentum equation ( 2 . 1 . 2 ) represent
successively the inertia, centrifugal and Coriolis forces per unit mass.
The terms on the right hand side are the pressure, external and shear
forces per unit mass. The external body force is assumed to be conser-
vative e.g.
A static equilibrium is obtained if the velocities in the rotating frame -+
are equal to zero. Setting q = 0 in (2.1.2) the pressure distribution at
rigid body rotation becomes
This solution is used to define a relative pressure
by which (2.1.2) simplifies to
The following dimensionless variables are defined
where a is the radius of the cylinder and where U characterises the
typical particle velocity measured in the rotating frame. Putting (2.1.7)
into (2.1.1) and (2 .1 .6) , and dropping the asterices, the dimensionless
equations of motion become
The dimensionless parameters now entering into the problem are the Rossby
number E and the Ekman number E, defined by
The Ekman number, based on the radius a, is a measure of the magnitude
of the viscous forces relative to the Coriolis acceleration.
The Rossby number is a measure of the relative importance of the non-
linear convective terms with respect to the Coriolis acceleration.
Assuming that the velocities measured in the rotating frame are much
smaller than the circumferential speed of the cylinder, the non-linear
terms in (2.1.9) can be neglected. In the limit of E -t 0, (2.1.9)
reduces to
Since we are concerned with a cylindrical configuration, it is convenient
to apply cylindrical coordinates (r, cp, Lz) with the origin at the centre
of the cylinder and with corresponding velocity components (u, U, w). The aspect ratio L is the ratio of the length of the cylinder to its radius,
L = Zla (2.1.12)
and, at this stage, the axial coordinate has been scaled by the length of
the cylinder. Because of axial symmetry of the boundary conditions at the
end caps it is reasonable to assume an axial symmetric distribution of the
flow. In this case, the continuity equation can be used to define a stream-
function $
Furthermore we define an angular velocity w ( r , z) given by
v = wr (2.1.14)
Applying (2.1.13), (2.1.14) and cross-differentiating the radial and axial
component of the momentum equation (2.1.11) to eliminate p , the equations
of motion become
where (2.1.15) is the azimuthal component of the momentum equation.
The boundary conditions are given by
where
-2 -2 Qb = wbrdr, $t = F 1 wtrdr (2.1.20)
0 0
and
Here, wb, wt, idb and wt are arbitrary functions of r. The functions wb and
wt represent the dimensionless different rotation ratk of bottom and top
end caps and wb and wt are the imposed dimensionless vertical velocities.
Condition (2.1.21) implies that the axial net transport is zero. The no-
slip and no-normal-flow condition at the cylinder wall is expressed by
(2.1.17).
Differential equations (2.1.15) and (2.1.16) with boundary conditions
(2.1.17) - (2.1.21) are the basis for our further investigation. The basic
assumption is a small Ekman number, E << 1 , a condition satisfied in many
practical situations. The aspect ratio is varied from unit magnitude to
in£ ini ty .
Consider now the case that L % 1. Since E << 1 the terms on the left
hand side of (2.1.15) and (2.1.16) are small and can be neglected. However,
since the terms with the highest derivatives, stemming from the viscous
forces, are dropped, the order of the equations is reduced. These simplified
equations do not allow all boundary conditions to be met and one may speak
of a singular perturbation problem (Van Dyke,1964). As is well-known since
the times of Prandtl, this problem is overcome by introducing boundary
layers. These layers are located along the bounding surfaces and are the
narrow regions within which the velocity components are adjusted by the
viscosity, to the prescribed conditions at the walls. A formal method to
develop the boundary layer equations is the stretching of (i) the coordinate
normal to the surface and (ii) the dependent variables with a power of the
small parameter E such that for E -+ 0 the highest normal derivatives are
also significant, which in turn preserves the applicability of the boundary
conditions at the surface. The requirement that at the outer edge of the
layer the boundary layer variables attain prescribed values appropriate to
the inviscid domain is known as the matching principle. It connects the
various flow regions and leads to a consistent description of the flow field.
16
h t r e a t i s e on p e r t u r b a t i o n techniques and t h e method of matched asymptot ic
expansions i s g iven by Van Dyke (1964), t o which t h e r eader i s r e f e r r e d .
Consider now a b a s i c i n v i s c i d flow. I n t h e l i m i t of E -t 0 equat ions
(2.1.15) and (2.1.16) reduce t o
(a/az)$ = o (a/az)w = o
This r e s u l t , however, l eaves t h e magnitude of $ and w wi th r e s p e c t t o L and
E unspec i f i ed . The answer t o t h i s ques t ion i s found from boundary l a y e r
cons ide ra t ions . The i n v i s c i d s o l u t i o n does no t a l low a l l boundary cond i t ions
t o be met a t 2 = 0 and z = 1 . Boundary l a y e r s nea r both end caps w i l l a d j u s t
t h e flow v a r i a b l e s t o t h e p resc r ibed cond i t ions . I n t h e i n v i s c i d r eg ion $
and w a r e cons tan t wi th r e spec t t o z. Therefore we can expect t h a t these
l a y e r s w i l l c o n t r o l 1 t h e flow i n t h e i n t e r i o r ; i . e . t h e s c a l i n g magnitude
of I/J and w i n t h e i n t e r i o r w i l l be t h e same a s i n both l a y e r s . An t i c ipa t ing
t h e outcome of t h e boundary l a y e r ana lyses we s c a l e , f o r t h e i n v i s c i d domain
P u t t i n g (2.1.23) i n t o (2.1.15) one sees t h a t t h e terms on t h e l e f t hand s i d e
a r e of magnitude and L-'E'/~ compared t o those on t h e r i g h t . The terms
on t h e l e f t of (2 .1 .16) a r e of magnitude LE~'~, L - ' E ~ / ~ and LV3e3l2 compared
t o those on t h e r i g h t . A l l t hese terms a r e smal l provided t h a t
Therefore , a b a s i c i n v i s c i d flow i s only observed when t h e a spec t r a t i o
f a l l s i n t h e range (2.1.24);
As a l ready s t a t e d , t h e i n v i s c i d flow does no t a l low a l l boundary condi-
t i o n s a t z = 0 and z = I t o be s a t i s f i e d . We a r e forced t o in t roduce boun-
dary l a y e r s nea r t h e end caps. A balance i n t h e l i m i t of E -+ 0 between t h e
h ighes t z- d e r i v a t i v e s on t h e l e f t hand s i d e of (2.1.15) and (2.1.16) and
t h e terms on t h e r i g h t hand s i d e i s obta ined by in t roduc ing a s t r e t c h e d
boundary l a y e r coord ina te and sca led v a r i a b l e s according t o t h e r u l e s
Putting (2.1.25) into (2.1.15) - (2.1.16) and
compared to unit magnitude, the boundary layer
dropping terms 'L
equations become
E directly
The above layer near the bottom cap is the well-known Ekman layer first
described by Ekman (1905). Solving equations (2.1.26) - (2.1.27) and
applying the boundary conditions at z = 0, a compatibility condition for
the flow outside the Ekman layers can be derived by matching the velocities
to those of the outside flow. This compatibility condition is often referred
to as the Ekman suction condition and allowsthe description of the outside
flow without giving explicit solutions for the Ekman layer itself. The
suction conditions at both end caps are (Greenspan 1968)
where the functions Fb and F are given by t
Fb = W b - E-1/2 'ilb
Note that a combination of the imposed w and $ at the end caps, as expressed
in (2.1.30) and (2.1.31), allows a description of the outside flow without
making a distinction between both types of boundary conditions.
Applying the suction conditions on the inviscid flow one obtains
$, = 4 C F - F b } , wo = 4 I F t + F b l (2.1.32)
In the inviscid region $ and w are constant with respect to z, a result
known as the Taylor-Proudman theorem. Since the axial velocity is constant,
t h e ou t f l ow from t h e Ekman l a y e r a t t h e bot tom i s equa l t o t h e i n f low a t
t h e t op . There i s no i n t e r i o r r a d i a l motion and v e r t i c a l f low i s s o l e l y r e-
tu rned w i t h i n t h e Ekman l a y e r s . The r a d i a l shape of t h e v a r i a b l e s i n t h e
i n t e r i o r depends d i r e c t l y upon t h e r a d i a l shape of t h e imposed c o n d i t i o n s
a t b o t h caps .
Gene ra l l y , t h e s o l u t i o n (2.1.32) does n o t apply n e a r t h e c y l i n d e r w a l l .
Consider , f o r example, t h e c a s e of d i s k s a t z = 0 and z = 1 which r o t a t e a t
an angu la r speed s l i g h t l y d i f f e r e n t from t h e su r round ing c y l i n d e r . Then F b
and F a r e c o n s t a n t w i t h r e s p e c t t o r and s o l u t i o n (2.1.32) does n o t s a t i s f y t t h e c o n d i t i o n t h a t $ and w a r e equa l t o ze ro a t r = 1. To overcome t h i s pro-
blem we a r e f o r c e d t o i n t r o d u c e boundary l a y e r s n e a r t h e c y l i n d e r w a l l .
P u t t i n g
i n (2.1.15) - pa red t o u n i t
one o b t a i n s
(2.1.16) and dropping te rms Q L - ~ E " ~ and Q, LE d i r e c t l y com-
magnitude (which a l l ows L t o f a l l i n t h e range E'/~<<L<<E-~!)
The above boundary l a y e r e q u a t i o n s a l l ow a l l c o n d i t i o n s t o b e met a t = 0.
However, s i n c e f o r t h i s l a y e r w Q, L - ~ ~ ~ E ~ ~ ~ i t s s o l u t i o n cannot be matched
t o t h e i n t e r i o r where w Q I . There fo re , a second l a y e r i s needed which
a d j u s t s t h e angu la r speed p e r t u r b a t i o n of t h e i n t e r i o r t o i t s no s l i p con-
d i t i o n a t t h e w a l l . Consis tency concerning t h e matching of t h e i n t e r i o r and
bo th l a y e r s r e q u i r e s t h a t $ Q, Ell2. A s a consequence we s c a l e
I n t r o d u c i n g (2.1.36) and
i n t o (2.1.15) - (2 .1 .16) and dropping terms ?. L-'E'/~ and Q, LE1I2 d i r e c t l y
compared t o u n i t magnitude i t fol lows t h a t
The above l a y e r s nea r t h e cy l inde r w a l l a r e t h e well-known Stewartson l a y e r s
f i r s t descr ibed by Stewartson (1957). The o u t e r l a y e r of th i ckness L I2E1 l4
a d j u s t s t h e angular speed p e r t u r b a t i o n of t h e i n t e r i o r t o t h e no- sl ip condi-
t i o n a t t h e wa l l . The inne r l a y e r of th i ckness L " ~ E " ~ br ings $ t o zero a t
r = 1 , which impl ies t h a t t h e a x i a l flow i n t h e i n t e r i o r and i n t h e o u t e r
l a y e r r e t u r n s i n t h e inne r l aye r .
The boundary cond i t ions a t z = 0 and z = 1 f o r both Stewartson l a y e r problems
a r e obta ined from t h e Ekman s u c t i o n cond i t ions . I n t h e Ekman l a y e r equat ions
we neglected t h e h i g h e s t r a d i a l d e r i v a t i v e s , These terms w i l l be s i g n i f i c a n t 112 when t h e r a d i a l v a r i a t i o n s t ake p lace on a d i s t a n c e s c a l e t h a t i s ?. E .
This th i ckness s c a l e i s smal l compared t o t h e one of both S t e w a r t s ~ n l a y e r s
i f L >> a cond i t ion which, according t o (2.1.24), is s a t i s f i e d .
Therefore , t h e v e l o c i t y components i n both Stewartson l a y e r s a r e ad jus ted
t o t h e end caps b9 t h e Ekman l a y e r s . A diagram of t h e va r ious flow regions
i n t h e range Ell2 << L << E-'I2 i s given i n f i g u r e I .
A flow a s o u t l i n e d above has been discussed by many i n v e s t i g a t o r s , f o r
example, C a r r i e r (1966), Greenspan (1968) and Stewartson (1957, 1966). A
d e s c r i p t i o n i n terms of an i n v i s c i d flow extended by Ekman l a y e r s and
Seewartson l a y e r s i s only v a l i d when t h e aspect r a t i o f a l l s i n
Ell2 << L << E - " ~ ~ We a r e p a r t i c u l a r l y i n t e r e s t e d i n t h e flow
when L exceeds t h e upper l i m i t of t h i s range. For t h i s purpose
n i e n t t o consider t h e th ickness s c a l e s of t h e Ekman l a y e r L-'E
i nne r Stewartson l a y e r L 'I3E1 l3 and t h e o u t e r Stewartson l a y e r
t h e range
behaviour
i t i s conve-
'I2 , t h e L1/2E1/4
One can s e e t h a t f o r L ?. E-'I2 t h e o u t e r Stewartson l a y e r expands t o t h e
c e n t r e of t h e cy l inde r . For L ?. E-I t h e inne r Stewartson l a y e r a l s o f i l l s
t h e whole cy l inde r . We can a n t i c i p a t e t h a t i n t h e range E-'12 Q L << E-I
t h e r a d i a l d e r i v a t i v e s on t h e l e f t hand s i d e of (2.1.15) become important
desc r ib ing t h e flow i n t h e main s e c t i o n . The same a p p l i e s i n t h e range
E-' % L f o r the r a d i a l d e r i v a t i v e s i n (2.1.16). The Ekman l a y e r r e t a i n s i t s
smal l th ickness s c a l e and i s s t i l l t h e narrow l a y e r w i t h i n which t h e velo-
c i t y components a r e +adjusted a t z = 0 and z = 1.
fnv isc id region
Figure 1: Diagram of the flow regions for
E " ~ < < L << E -112
At first the flow in the range E-I" % L << E-' is considered. Radial
diffusion of azimuthal momentum becomes important by putting
Here, the absolute scaling magnitude of + has been chosen such that it corresponds to the induced flux of the Ekman layers given in (2.1.28) and
(2.1.29).
Substituting (2.1.39) into (2.1.15) - (2.1.16) and dropping' terms 1. L q 2 , -2 2
% L ~ E ~ , % E 2 and % L E directly compared to unit magnitude, we obtain
Applying the Ekman suction conditions & is easily verified from (2.1.40) and (2.1.41) that
+ I = { F ~ - F ~ ~ + 4 IFt%} (z- 1/2) - L - ' E - ~ / ~ ~ , (z- 1/2) (2.1.42)
The problem for o , is given by
and the requirement that w is finite at r = 0. 1 One sees from (2.1.42) and (2.1.43) - (2.1.44) that the velocity distribution
in the interior is influenced by the no-slip condition for o at the cylinder
wall. Furthermore we notice from (2.1.42) that a linear change of the axial
flow along the z-coordinate is possible. The Ekman layers no longer dominate.
main section
Figure 2: Diagram of the flow regions for
E-Il2 fb L << f l .
I n t h e p a r t i c u l a r c a s e t h a t t h e boundary c o n d i t i o n s a t t h e end caps a r e
ant i symmetr ic w i t h r e s p e c t t o t h e mid-plane z = 112, F = - Fb, i t fo l l ows t
from (2 .1 .43) - (2.1.44) t h a t w l = 0. P u t t i n g t h i s r e s u l t i n t o (2.1.42) one
s e e s t h a t t h e i n t e r i o r f low i s equa l t o t h e i n v i s c i d one g iven by (2 .1 .32) .
One would expec t t h i s s i t u a t i o n , of cou r se , s i n c e when Ft = - Fb, wo = 0
and t h e r e f o r e n a o u t e r S t ewar t son l a y e r i s needed and expans ion of t h i s
l a y e r i s i r r e l e v a n t .
Equat ions (2 .1 .40) and (2 .1 .41) do n o t a l l ow t h e boundary c o n d i t i o n s f o r $
a t t h e c y l i n d e r w a l l t o be met. The re fo re t h e i n n e r S tewar tson l a y e r i s
needed. I t s p r i n c i p a l f u n c t i o n i s aga in t h e r echanne l ing of t h e a x i a l f low.
The s c a l i n g r u l e s and equa t ions a r e equa l t o t hose g iven i n (2 .1 .33 ) ,
(2.1.34) and (2 .1 .35) . A diagram of t h e f low r eg ions f o r E - I / ' Q, L << E-'
i s g iven i n f i g u r e 2.
F i n a l l y we s h a l l d i s c u s s t h e f low i n t h e range E-' Q L. I n t r o d u c i n g t h e
s c a l e d v a r i a b l e s
i n (2.1.15) - (2.1.16) and n e g l e c t i n g t h e terms Q L - ~ d i r e c t l y compared t o
u n i t magni tude , t h e e q u a t i o n s d e s c r i b i n g t h e f low i n t h e main s e c t i o n be-
come
These e q u a t i o n s a l l ow a l l boundary c o n d i t i o n s t o be s a t i s f i e d a t r = 1 .
The c o n d i t i o n s a t t h e end caps a r e aga in provided by t h e Ekman l a y e r s . The
l e a d i n g o r d e r of t h e Ekman s u c t i o n c o n d i t i o n s becomes
E l i m i n a t i n g w2 from (2 .1 .47) by means of (2.1.46) i s f o l l ows t h a t
a 2 L2E'{l a a r2}3$2 + 4 - r ar r a r $, = 0 a 2
eigenfunctions
Figure 3: Diagram of the flow regions for
L E-I.
--.-, - y Y u L - A V L I u e bulvea DY separation of variables. As a result, we get
Q2 = kgl {akexp(-hkLEz) + b kexp (+hkLEz) }.fkfk(r) (2.1.50)
where fk satiesfies the differential equation
l-1 L r2}3fk + 4h$fk = o r dr r dr
with the boundary conditions
and the requirement that f is finite at r = 0. The solution of (2.1.51) - k (2.1.52) is then an infinite series of eigenfunctions f with positive real k eigenvalues Xk (k = 1, 2, ....., w , where the eigenvalues are ordered such
that h k - l < Xk < Xk+] ). Expanding Fb and Ft in the eigenfunctions ;F3c the constants ak and b in (2.1.50) can be determined from (2.1.48). A diagram
k of the flow regions for L 1. E-I is given in figure 3.
In the case of a cylinder of semi-infinite length b must be set equal to k zero. Then we retain eigenfunctions decaying only in the axial direction.
This is exactly the situation considered by Steenbeck (1958), who calculated
numerically for the first eigenvalue A , = 41.44 and for the second one
A = 199.2. 2
Summarising, three types of flow can be distinguished when the aspect
ratio increases from unit magnitude to infinity. These types of flow corre-
spond to the parameter ranges E~~~ << I, << E-"~, E-lI2 Q, L << E-I and - 1 E Q, L. In the first L-range the flow consists of an inviscid interior ex-
tended by Ekman layers near the end caps and two Stewartson layers near the
cylinder wall. In the inviscid interior the angular speed and the axial velo-
city are constant along the axial coordinate, a result in agreement with the
Taylor-Proudman theorem. The radial velocity is zero and the in- and outflow
from the Ekman layers controls the vertical flow. Its radial shape directly
depends upon the radial shape of the imposed boundary conditions at the end
caps. The outer Stewartson layer adjusts the angular speed perturbation of
the interior at the cylinder wall. The axial flow in the interior and outer
Stewartson layer returns in the inner Stewartson layer.
I n t h e second L-range r a d i a l d i f f u s i o n of azimuthal momentum i s no t r e s t r i c -
ted t o a small region nea r t h e c y l i n d e r wa l l ( t h e o u t e r Stewartson l a y e r )
but i s important i n t h e main p a r t of t h e cy l inde r . The r a d i a l shape of t h e
flow v a r i a b l e s i s inf luenced by t h e no- sl ip of t h e angular v e l o c i t y a t t h e
wa l l . The Ekman l a y e r s no longer dominate, a s i s t y p i c a l i n t h e f i r s t L-
range, and a l i n e a r change of t h e a x i a l flow along t h e a x i a l coordinate i s
poss ib le .
I n t h e t h i r d L-range r a d i a l d i f f u s i o n of azimuthal and a x i a l momentum a r e
important i n t h e e n t i r e cy l inde r and t h e flow i s s t r o n g l y inf luenced by t h e
condi t ions a t t h e c y l i n d e r wa l l . The v e l o c i t i e s can be w r i t t e n a s a s e t of
e igenfunc t ions which decay exponen t i a l ly from t h e end caps. Higher eigen-
func t ions decay f a s t e r than lower ones and o s c i l l a t e more f r equen t ly along
t h e r a d i a l coordinate . A t a reasonable d i s t a n c e from t h e end caps t h e flow
i s mostly descr ibed by t h e f i r s t e igenfunct ion.
Since E i s very small most conf igura t ions w i l l f a l l i n t o t h e f i r s t
L-range. In o t h e r words, t h e f i r s t type of flow i s of most p r a c t i c a l im-
portance. E x p l i c i t s o l u t i o n s f o r t h e v e l o c i t y d i s t r i b u t i o n i n t h e va r ious
flow regions a r e no t ca lcu la ted . Many of t h e c h a r a c t e r i s t i c phenomena a l s o
occur i n t h e p e r f e c t gas and t h e i r s o l u t i o n s well be presented and discus-
sed i n chap te r 3 .
2.2. The perfect gas
2.2.1. The basic equations
In order to investigate the flow of a compressible fluid in a rotating
cylinder, a new and more comprehensive formulation is needed. The equations
describing the motion and the state are given by the conservation of mass,
momentum and energy, together with an equation of state. It is assumed that
the fluid has constant material properties such as the dynamic viscosity u,
the bulk viscosity 6, the thermal conductivity K and the specific heat at
constant pressure cp. We confine ourselves to stationary flows. The three
conservation equations, written in a rotating frame fixed with respect to
the cylinder are (in the absence of body forces; Greenspan 1968):
-t -?- Here q is the particle velocity and r the position vector in the rotating
frame. C2 is the angular speed of the cylinder and is the unit vector along
the rotation axis. P, p and T denote respectively pressure, density and
temperature. Relaxation processes are not considered. Therefore the bulk
viscosity is set equal tobzero: B = 0.
A relation between the state variables P, p and T is given by the
equation of state. For dilute gases a good approximation is the perfect
gas equation
Here R is the universal gas constant: Ro = 8.314 [kgm2/s2 ~mol], M is 0 the molecular weight of the gas.
conservation equations reduce to
vpe = $ p e n 2 v ( Z G 1 2
A static equilibrium is obtained if the velocities in the rotating frame -+
are equal to zero: q = 0 . Assuming a uniform constant temperature To the
Applying the equation of state
it follows from (2.2.5) that
P e = P o exp{ifi2 1 $ s 1 2 / ~ ~ O )
-+ where P is the pressure at r = 0 .
0
The solution for the state quantities at isothermal rigid body rota-
tion is used to define a dimensionless relative pressure p, density T and
temperature 0
Here el, c2 and c3 are dimensionless parameters which scale the perturbed
state quantities p , T and 8.
The position vector and the particle velocity are made dimensionless as
follows
Note that T does not appear in the conservation equations (2.2.20) - (2.2.22). In other words, the linearised equation of state (2.2.19) is an
extra equation for the extra variable T.
Comparing the conservation equations here to those of the incompres-
sible fluid given by (2.1.8) and (2.1.11), we observe (i) an exponential
function exp A{ (kxrI2 - 1 ) which stems from the density distribution at
isothermal rigid body rotation of the gas, (ii) the extra term ;E 8v[kxrI2 3
in the momentum equation (2.2.21) which represents the perturbation of the
centrifugal force caused by temperature and (iii) an extra equation
(2.2.22), the energy equation, necessary to describe the temperature field.
The term on the left hand side of the energy equation stems from the work
done by compression (U (a/ar)p). The term on the right represents the heat
conduction.
A more concrete opinion concerning the" validity of the linearisation,
the criterion for which is given by (2.2.18), can be formed when E,, E 2
and E~ are known as a function of E; Relations between these scale para-
meters are obtained from balance considerations of the varioup terms in
the linearised equations. It is our intention to consider the flow pro-
cesses for a small Ekman number. This means that the viscous forces in
(2.2.21) can be neglected and hence the pressure gradient must be balan-
ced by the inertia terms. Furthermore since a balance between the Coriolis
acceleration and the centrifugal acceleration causes the motion, we put
A relation between E~ and E is obtained from the linearised equation of state. Putting (2.2.23) into (2.2.19) it follows that
The relation between E~ and E depends on the magnitude of A. Let us first-
ly consider the case that A is very small, a condition fulfilled in many
applications:
(i) A << 1.
From (2.2.24) we conclude
This implies that fluctuations of the density result principally from
thermal effects, which is a basis of the Boussinesq approximation
(see Spiegel & Veronis,l960). Substituting the relations between E 1 ' E ~ , E~ and E given by (2.2.23) and (2.2.25) in (2.2.18) the lineari-
sation criterion becomes
With (2.2.17) the latter criterion becomes
y~/A(y-1) << 1 (2.2.27)
Because A is very small and (y-l)/y is of unit magnitude or smaller,
condition (2.2.27) is in most cases not satisfied. The parameter
y~/A(y-1) scales the heat convection relative to the work done by
compression in the energy equation. In many applications this term is
relatively large. In that case we must include the heat convection
term instead of the compression term and the linearised version of the
energy equation applied here is invalid. In fact convection and con-
duction in the energy equation together with application of the
Boussinesq approximation is the basis of many theoretical investi-
gation on stratification in rotating fluids: see Barcilon & Pedlosky
(1967a, 1967b, 1967c), Greenspan (I968), 'flomsy & Hudson (1969, 1971a,
1971b).
..- - A r \ -VI ILCLII=u W~~~~ sdb cen~r~ruges. Ln tnls application of
rotating fluids the speed parameter is certainly not small:
(ii) A 2 1 .
It is found from ( 2 . 2 . 2 4 ) that
Substituting the relations between E ] , E ~ , c 3 and E given by ( 2 . 2 . 2 3 )
and ( 2 . 2 . 2 8 ) in ( 2 . 2 . 1 8 ) the linearisation criterion becomes
It is clear that for A % 1 and y / A ( y - 1 ) % 1 , the criterion ( 2 . 2 . 2 9 )
is satisfied provided that the velocities measured in the rotating
frame are
for A >>
small compared to the peripheral speed. On the other hand,
we must require
Here EA scales the perturbation of the pressure and density at iso-
thermal rigid body rotation. This can be verified from the pressure
distribution at isothermal rigid body rotation given by ( 2 . 2 . 7 ) .
Introducing a perturbation in the angular speed, it is found that the
perturbation of P is a factor A larger. Nevertheless, we assume that
the Rossby number is so small that ( 2 . 2 . 3 0 ) is satisfied. The linea-
rised conservation equations ( 2 . 2 . 2 0 ) - ( 2 . 2 . 2 2 ) together with the
relations between E, E~ and c 2 given by ( 2 . 2 . 2 3 ) are the basis of our
following investigation.
As we are concerned with a cylindrical configuration it is appropriate
to apply cylindrical coordinates (r, cp, L z ) with the origin at the centre
of the cylinder. As before the aspect ratio L is the ratio of the length
of the cylinder to its radius: i.e.
Because of the symmetry of the imposed boundary conditions, an axially
symmetric distribution of the flow is assumed. The continuity equation
(2.2.20) can then be used to define a streamfunction $,
where u is the radial veloucity and w the axial veloucity. Furthermore
we introduce an angular velocity
v =
Cross-differentiating the radial
equation (2.2.21) to eliminate p
w(r, z) replacing the azimuthal velocity V ,
and axial components of the momentum
we get
LEe I a2 -a(r2-1){' a' a r2 + - -)W = - a $ 2 -
az (2.2.35)
r ar r ar L~ az2
where (2.2.35) is the tangential component of the momentum equation and
where (2.2.36) is the energy equation.
Just as in the incompressible problem it is assumed that the horizon-
tal end caps of the cylinder rotate with an angular speed which differs
slightly and that fluid is injected and withdrawn vertically. In addition,
we assume t e m p e r a t u r e d i s t r i b u t i o n s a l o n g the cylinder wall and along the
end caps. The boundary conditions are now such that the velocities and
temperatures coincide with those of the walls: i.e.
w = $ = (a/ar)$ = 0, e = 0 a t r = 1 23
(2.2.37)
w = wb, $ = e = eb, (a/az)$ = 0 at z = 0 (2.2.38)
wnere
and
$b = at = 0 a t r = 1 (2.2.42)
Here wb, at, wb,ut, Ob aqd Rt are arbitrary functions of r and Ow is an
arbitrary function of z . The functions ob and wt represent the dimension-
less different rotation rates of the bottom and top end caps. The functions
w and w represent the imposed dimensionless vertical velocities at both b t end caps. The functions Rb and Ot are the dimensionless temperature dis-
tributions along the end caps and OW is the dimensionless temperature dis-
tribution along the cylinder wall. Condition (2.2.42) implies that the
axial net transport is zero. The no-slip and no-normal flow condition at
the cylinder wall is given in (2.2.37)
Consider now the equations (2.2.34), (2.2.35) and (2.2.36). In equa-
tion (2.2.34) the two variables $ and 0-it? are present, the latter one
further denoted as x . Multiplying equation (2.2.36) with 1/2 and subtrac-
ting this from (2.2.35) a second equation for $ and x is obtained. We are now in the fortunate position that only these two equations have to be
considered in order to obtain a solution for x and $. However, the proce-
dure is not completely quccessful because the lower radial derivatives on
the left hand side of (2.2.35) and (2.2.36) are not the same. On the other
hand, since E is small we expect a basic inviscid flow, where the terms on
the left can simply be neglected, extended by boundary layers, where only
the highest derivatives are significant. In other words we expect that the
mathematical benefits of the foregoing manipulation are still applicable.
A third equation, necessary for the description of o and 6, is obtained by
eliminating (a/az)$ from (2.2.35) and (2.2.36). As a result we get the
following set of equations
where
)( = (,,-.I 20, + = 0+$r2Brw
Equation (2.2.45) under l i e s a balance between t h e viscous f o r c e s i n t h e
azimuthal momentum equat ion and t h e hea t conduction i n t h e energy equa-
t i o n , terms which were % E i n t h e o r i g i n a l equat ions of motion. There we
dropped terms % E d i r e c t l y compared t o u n i t magnitude whereas i n (2.2.45)
a p a r t of these non- l inear terms compares t o E so t h a t f o r E % E t h e neg-
l e c t i o n i s i n v a l i d . Therefore , t h e terms i n ques t ion , which stem from
convection (w(a/az)v, w(a/az)T) , have been incorporated i n t o (2.2.45).
The boundary condi t ions f o r (2.2.43) and (2.2.44) a r e
The boundary condi t ions f o r (2.2.45) a r e
The coupling between the equations for $ and X, (2.2.43) and (2.2.44),
and the equation for $, (2.2.451, is given by the term in on the left
hand side of (2.2.44). As has already been stated, one hopes that this
term is not of leading order in the various balances of terms to be consi-
dered. This is especially advantegeous with respect to gas centrifuge pro-
blems where one is mainly interested in the distribution of the axial and
radial velocity so that only (2.2.43) and (2.2.44) have to be considered.
The small perturbations of the angular speed and temperature are often of
secondary importance so that (2.2.45) does not need to be solved. Never-
theless, in the next treatment the full set of equations is considered and
for each case the conditions on which a decoupling is allowed will be dis-
cussed.
Compare now equations (2.2.43) and (2.2.44) to those for the incom-
pressible fluid given by (2.1.15) and (2.1.16). Replacing $ by w one sees
that, apart from the term in $ and some lower order derivatives with res-
pect to r and z, both problems are quite similar provided that the speed
parameter and the Brinkman number are of unit magnitude or smaller. For
A s 1 the exponential function stemming from the density distribution at
isothermal rigid body rotation is of unit magnitude and is insignificant
in "an order of magnitude discussion". The same applies to the terms
formed with the Brinkman number, as long as B r S 1. In this case a simi-
lar procedure as has been applied to the incompressible problem seems ap-
propriate.
Consider now a basic inviscid flow. In the limit of E -+ 0 equations
(2.2.43) and (2.2.44) reduce to
which implies that the axial velocity and a combination of the angular
speed and temperature given by w-48, are constant with respect to z, a
result that may be referred to as the compressible Taylor-Proudman
theorem.
The inviscid solution does not allow all boundary conditions to be met
at the walls and similar boundary layers as in the incompressible case
are needed. The conditions at the end caps are provided by layers of the
Ekman type. In these layers of thickness L-'EI'~ the highest z-derivatives
on the left hand side of (2.2.43) and (2.2.44) are comparable in magnitude
with the terms on the right. An inner and outer Stewartson layer, of thick-
ness L ~ ~ ~ E ~ ~ ~ and L ~ ~ ~ E ~ ~ ~ respectively, adjust the inviscid flow to the'
cylinder wall. In the inner layer the highest radial derivatives on the
left hand side of (2.2.43) and (2.2.44) are compareble in magnitude to the
terms on the right, whereas in the outer layer only the highest radial
derivatives of x are significant. Replacing w by $ the scaling rules for
the various flow regions are equal to those given for the incompressible
case.
The thicknesses of the boundary layers are small when the aspect ratio
falls in the range Ell2 << L << E-'I2. Since C is very small most configu-
rations will fall in this range. When, however, the aspect ratio exceeds
the upper limit of this range, both Stewartson layers successively expand
to the interior. Just as in the incompressible case, radial diffusion be-
comes important in the main section of the cylinder.
The distributions of w and €I respectively can be determined by solving
the equation for $ resulting from the differential equation (2.2.45) and
the boundary conditions (2.2.50) - (2.2.52). The coupling between the pro-
blems for x and $ on one hand and for $ on the other is given by the term
in $ on the left hand side of (2.2.44). As long-as the magnitude of $ is
not greater than that of x the term in question is not significant. Then in the inviscid interior the entire left hand side of (2.2.44) can be neg-
lected and in the boundary layers the term is still negligible since
(a/ar)$ n, (a/az)$ $. However, to satisfy the condition O($) L O(X) the
necessary requirements are (i) the magnitude of the imposed boundary con-
ditions for $ is not greater than those for x and (ii) the magnitude of
the non-linear terms on the right hand side of (2.2.45) is not greater
than the magnitude of the terms on the left. The necessity of condition
(i) is illustrated by considering the particular case that Bb = ;ab,
Bt = ;wt, Ow = $b = $ = 0. Omitting the term in @I one would conclude from t the boundary conditions (2.2.47) - (2.2.49) and equations (2.2.43) - (2.2.44) that $ and x are zero whereas the term in $ is now the mechanism
by which a secondary flow is generated. Of course, the magnitude of this
induced flow is small but, nevertheless, the present approach did not con-
sider this situation. If condition (ii) is not satisfied (the magnitude of
the non-linear terms is significantly large) a new scaling analysis would
be needed with, at this stage,incalculable consequences for the flow field.
Anyhow introducing the scaling magnitudes of x and $ of the inviscid inte-
rior, x % I and j~ % (similar to the incompressible case) the non-
linear terms no longer dominate if ~o /LE'/~ 5 1 and E Br(1-o )/LE'/~ I 1. P P
A basic invisdid flow for $ and ~ $ 0 , extended by Ekman layers and
Stewartson layefs,has been investigated in a number, ofirecent Japanese pu-
blications: Matsudq,.Sakurai & Takeda (1975), Matsuda (1975) , ~ikami
(1973a, l973b), ~aka~amd 6 U s d (l974), Sakurai & Matsuda (1974) and
Sakurai (1975). These authors, however, did not consider the effect of an
aspect ratio which increases from unit magnitude to infinity, nor the ef-
fect of large radial density gradients. In present day centrifuges A is
quite large. In this case the exponential density function exp1A (r2- 1 ) 1 in equations (2.2.43) - (2.2.45) varies rapidly in magnitude along the
radial coordinate and a new scaling analysis appears to be necessary. As
will be seen, a fundamental change in the flow phenomena occurs. One also
finds that the effect of an increasing aspect ratio becomes more important.
2.2.2. The papidly rotating heavy gas
Due to the high peripheral speed and the relatively large molecular
weight of the gas, the speed parameter is quite large in present day cen-
trifuges. Values of 17 for A, or more, are today quite normal. Inspection
of (2.2.43) and (2.2.44) shows that the density function eT{~(r2-I) 1 forms, with the Ekman number, a local Ekman number: viz.
El = E expiA(1-r2) I . Although El is small at r = 1 it increases very ra-
pidly with distance from the cylinder wall when A is large: e.g. for
A = 17, exp(~(l-r~)1 is approximately 25 at r =0.9 and 2.4 x lo7 at r = 0. Furthermore, the presence of the density function in equations
(2.2.43) and (2.2.44) requires that all variables depend on exp{~( 1 -r2) 1 : derivatives with respect to r will not be of unit magnitude but, more ex-
plicitly, alar % A. In order to perform an adequate scaling analysis it
is necessary to introduce a coordinate measured from the cylinder wall
which corresponds to a change of unit magnitude of the density function:
Furthermore we define the streamfunction $* by
$* = A$
Putting (2.2.54) and (2.2.55) into (2.2.43) - (2.2.45) and dropping the
asterix we get
with boundary conditions
and
a x a I a2 [ 4 {l+iBr(l- z)} ( I - 5 ) - {I+$BY(]- g)}-] + - -1 4 = A ax A L; az2
with boundary conditions
Note t h a t t h e second p a r t of t h e non- l inear term i n ( 2 . 2 . 4 5 ) has been
neg lec ted i n ( 2 . 2 . 5 9 ) which i s v a l i d when o = 1 . The modified Ekman P
number Em and t h e modified aspect r a t i o Lm a r e based on t h e d i s t a n c e
s c a l e of t h e d e n s i t y decrease i n s t e a d of t h e r a d i u s , and a r e r e l a t e d t o
E and L by
Although A~ i s l a r g e we assume t h a t E << 1 . Furthermore we t ake B r % 1 . m
The modified a spec t r a t i o i s assumed t o vary from u n i t magnitude t o
i n £ i n i t y .
Consider an i n v i s c i d flow. I n t h e l i m i t of Em + 0 equat ions ( 2 . 2 . 5 6 )
and ( 2 . 2 . 5 7 ) reduce t o : ( a / a z ) x = 0 , ( a / a z ) $ = 0. These s o l u t i o n s , how-
ever , do no t s p e c i f y t h e magnitude of x and $ and thus t h e cond i t ion on
which t h e terms on t h e l e f t hand s i d e s of ( 2 . 2 . 5 6 ) and ( 2 . 2 . 5 7 ) can be
neglected. Since $ and x a r e cons tan t wi th r e spec t t o z we expect t h a t
boundary l a y e r s nea r t h e end caps w i l l determine t h e magnitude of x and
$ i n t h e i n v i s c i d domain. An t i c ipa t ing t h e outcome of t h e boundary l a y e r
a n a l y s i s we s c a l e
and t h e i n v i s c i d flow i s descr ibed by
P u t t i n g ( 2 . 2 . 6 2 ) i n t o ( 2 . 2 . 5 6 ) and ( 2 . 2 . 5 7 ) one sees t h a t t h e terms on
t h e l e f t hand s i d e a r e of magnitude L (E Q ~ ) ~ ~ ~ , L i l ( E ~ x ) 3 / 2
~i~ (ErneX) 3 1 2 y L,(E~Q") ' I 2 and L,' (Ernez) lY2 compared t o those on t h e r i g h t .
For x % I t hese terms can be neglected provided t h a t
On t h e o t h e r hand, a t l a r g e r d i s t ances from t h e wa l l x inc reases t o A
which i s supposed t o be l a rge . Consequently ex i nc reases considerably so
t h a t t h e terms on t h e l e f t of ( 2 . 2 . 5 6 ) and ( 2 . 2 . 5 7 ) can become s i g n i f i -
can t . Consider t h e case L m % 1 . Then a l l terms on t h e l e f t w i l l balance
for ~~e~ 2. 1 , which implies that an inviscid region is only observed in - 1 a limited region near the cylinder wall given by x << ZnE . Due to the m
strong decrease of the density the diffusive terms are necessary to des-
cribe the flow in the core of the cylinder. Of course, since x is, at - 1 most A, a viscous core does not occur when ZnE, >> A .
The inviscid solution fails at the end caps. To overcome this non-
uniformity we must introduce boundary layers near z = 0 and z = I . The
scaling rules for the layer at the bottom cap are
- 1 x 112 x 112 z = Lm (Erne Y , X = %0, J) = (Erne ) qO (2.2.65)
x 112 Putting (2.2.65) into (2.2.56) - (2.2.57) and dropping terms 2. (Erne )
compared to unit magnitude we obtain
The above layer of thickness L;~(E,~")'/~ is of the Ekman type and in-
creases in thickness with distance from the cylinder wall. Equations
(2.2.66) and (2.2.67)' are only applicable for x << ZnE-I. At x 2. ZnEm1 m all terms of the original equations of motion become important, and for
L 2. ! the layer will expand and join the previously discussed viscous m core.
Generally, the inviscid solution does not apply near the cylinder
wall, and must be amended by further boundary layer analysis. Two layers
of the type discussed by Stewartson (1957) will adjust $J and x at x = 0.
The scaling rules for the inner layer are
Putting (2.2.68) into (2.2.56) - (2.2.57) and letting L,'E;/~ -+ 0 and
LmEm+ 0 (which allows L t o f a l l i n E l l 2 << 3 << E - I ) t h e equa t ions , m rn rn rn
become
The s c a l i n g r u l e s f o r t h e o u t e r l a y e r a r e
S u b s t i t u t i n g ( 2 . 2 . 7 1 ) i n t o ( 2 . 2 . 5 6 ) - ( 2 . 2 . 5 7 ) and l e t t i n g L - I E ~ ' ~ + 0
and L ~ E ~ ~ ~ -+ 0 (which allows L t o f a l l i n E l l 2 << L << E-m2;"we gee rn rn rn m
The o u t e r Stewartson l a y e r b r ings xO t o ze ro a t t h e c y l i n d e r wa l l .
The inne r l a y e r provides t h e remaining.condi t ions a t x = 0 . Both l a y e r s
a r e terminated a t t h e top and bottom by t h e Ekman l aye r . I n f i g u r e 4 a
diagram of t h e va r ious flow regions of x and ly i s given f o r L I 1. m
The d i s t r i b u t i o n s of w and 0 a r e found from ( 2 . 2 . 5 9 ) . S u b s t i t u t i n g
t h e s o l u t i o n s f o r x and $, a s o l u t i o n f o r c$ can be deduced from t h e d i f -
f e r e n t i a l equat ion ( 2 . 2 . 5 9 ) and boundary cond i t ions ( 2 . 2 . 6 0 ) . U n t i l now
it has been assumed t h a t t h e term i n c$ on t h e l e f t hand s i d e of ( 2 . 2 . 5 7 )
i s no t s i g n i f i c a n t t o l ead ing o rde r i n t h e d e s c r i p t i o n of x and $. It i s
t h e r e f o r e necessary t o r e q u i r e t h a t t h e magnitude of A - ' ~ i s a t most
equal t o one of X . This cond i t ion i s s a t i s f i e d when t h e magnitude of t h e
imposed cond i t ions f o r c$ given i n ( 2 . 2 . 6 0 ) i s no t g r e a t e r than A t imes
t h e magnitude of t h e condi t ions f o r x given i n ( 2 . 2 . 5 8 ) . Furthermore,
t h e p resen t approach is only app l i cab le i f t h e non- l inear terms on t h e
r i g h t hand s i d e of ( 2 . 2 . 5 9 ) a r e a t most of u n i t magnitude compared t o t h e
terms on t h e l e f t : i . e , rAa l L E l l 2 << 1 . P rnrn
I n t h e d i scuss ion of t h e flow i n t h e core of t h e c y l i n d e r a t x ?. z~E-' rn
we took L % 1 . Gas c e n t r i f u g e s have an aspect r a t i o which i s of u n i t m
magnitude o r l a r g e r . Consequently, s i n c e A >> I t h e modified aspect r a t i o
Figure 4: Diagram of the flow regions
of x and 4 for Ln7 a 2 .
i s l a rge . Therefore , we s h a l l r e i n v e s t i g a t e t h e viscous flow i n t h e core
f o r t h e L range I << L << ell2. Note t h a t i n t h i s range t h e previously m m m d iscussed boundary l a y e r s and i n v i s c i d region a r e s t i l l app l i cab le .
Since L - I i s smal l t h e z- der iva t ions of x and $ on t h e l e f t of (2.2.56) m
and (2.2.57) can be neglected compared t o t h e x- der iva t ives . By t h e same
arguments a s used be fo re t h e term i n 4 on t h e l e f t of (2.2.57) i s neglec-
ted. Int roducing a con t rac ted coordinate and sca led v a r i a b l e s according
t o t h e r u l e s
equat ions (2.2.56) and (2.2.57) become
where terms 1. ~-'cx;'~ a r e neglected compared t o u n i t magnitude. It i s
t h e r e f o r e necessary t h a t
where t h e constant a , i s given by
Condition (2.2.76) fol lows from r e q u i r i n g t h a t
which impl ies t h a t t h e d i s t ance s c a l e of t h e above deduced c o r e , measured -112 from t h e r o t a t i o n a x i s J a l , must be l a r g e compared t o A .
The exponen t i a l inc rease of t h e terms on t h e l e f t of (2.2.56) and
(2.2.57) i s r e spons ib le f o r t h e balance a t x z ~ ( L ~ E ~ ) - ' . S i m i l a r l y , f o r
x >> Zn(L E )I t h e terms on t h e l e f t w i l l become much l a r g e r than those m m
on t h e r i g h t and i n t h e l i m i t of n 1 + t h e d i f f u s i v e terms w i l l dominate:
Equation ( 2 . 2 . 5 9 ) becomes
Expanding ( 2 . 2 . 8 0 ) - ( 2 . 2 . 8 1 ) i n t o t h e co- ordinate r and i n t e g r a t i n g
wi th r e spec t t o r we o b t a i n
Requiring t h a t wv and By a r e f i n i t e a t r = 0 i t fol lows t h a t c2 = c4 = 0.
The s o l u t i o n f o r x i s then given by
From ( 2 . 2 . 7 9 ) i t follows t h a t
Since w must be f i n i t e a t r = 0 , c5 and c6 must be s e t equal t o zero. v
Fur the r i n t e g r a t i n g ( 2 . 2 . 8 4 ) with re spec t r we o b t a i n
Requiring t h a t t h e r a d i a l mass flow i s zero a t r = 0 , i t fol lows t h a t
cg = - c Q - ~ / ~ A and $v becomes 7
The unknown cons tan t s e1-?e4 and e7 a r e determined by matching (2.2.83)
and (2.2.86) t o t h e o u t s i d e flow. Expanding (2.2.83) and (2.2.86) i n t o
t h e coordinate 11, i t fa l lows t h a t % l and JI must be constant wi th r e spec t 1
t o n l , provided t h a t (2.2.76) i s s a t i s f i e d . So t h e boundary cond i t ions
f o r (2.2.74) and (2.2.75) a r e
The Ekman l a y e r equat ions given by (2.2.66) and (2.2.67) a r e only app l i-
cab le f o r r << z~E-'. The d i s t a n c e s c a l e of t h e above deduced core , f a l l s m i n t h i s range provided t h a t L >> 1, which i s t h e case according t o our m parameter choice . I n o t h e r words, t h e core i s terminated a t t h e top and
bottom by t h e Ekman l aye r s .
In t roducing x % Zn(L E ) - I i n t o (2.2.62) one sees t h a t t h e s c a l i n g m m
magnitude of $ i n t h e i n v i s c i d r eg ion i s equal t o t h e one of t h e above de-
duced core . The magnitude of X , however., i s an o rde r sma l l e r and hence
t h e core cannot be matched t o t h e i n t e r i o r . Therefore a second core i s
needed. Int roducing t h e con t rac ted coordinate and sca led v a r i a b l e s
t h e equat ions become, t o l ead ing o rde r
provided t h a t
a >> A - ~ 2
where t h e cons tan t a2 i s g iven by
a 2 = I - A-'z~(L~E )-I rn m
Condi t ion (2.2.91) imp l i e s t h a t t h e d i s t a n c e sca le ,measured from t h e ro- - 1 12 -
t a t i o n a x i s , d o ,must be l a r g e compared t o A . For q2 -t a x2 must match 2 t o i t s pu re v i s c o u s v a l u e : (a /an )? = 0 . The s c a l i n g magnitudes a r e 2 2 equa l t o t h o s e of t h e i n v i s c i d i n t e r i o r s o t h a t and T2 can match
smoothly t o xO and $O a s q2 + - m. As b e f o r e , t h e c o r e i s t e rmina t ed a t
t h e t o p and bot tom by t h e Ekman l a y e r s .
The above d i s c u s s e d c o r e s a r e denoted a s t h e i n n e r c o r e a t
x -L Z ~ ( L ~ E ~ ) - ' and t h e o u t e r c o r e a t i % Z R ( L ~ F ) - I . I n t h e o u t e r c o r e m m
x i s a d j u s t e d t o i t s pu re v i s cous v a l u e . The i n n e r c o r e b r i n g s $ of t h e
i n v i s c i d r e g i o n and of t h e o u t e r c o r e t o i t s pure v i s cous v a l u e a t t h e
a x i s . J u s t a s i n t h e i n n e r S tewar tson l a y e r t h e x- d e r i v a t i v e s of x and
$ a r e s i g n i f i c a n t i n t h e c o r e w h i l e , a s i n t h e o u t e r S tewar tson l a y e r ,
on ly t h e x- d e r i v a t i v e s of x a r e impor t an t i n t h e o u t e r co re . On t h e o t h e r
hand, i n bo th Stewar tson l a y e r s t h e e x p o n e n t i a l d e n s i t y g r a d i e n t i s neg-
l i g i b l e whereas i n bo th c o r e s t h i s g r a d i e n t i s t h e mechanism by which t h e
i n v i s c i d f low i s coupled t o a pu re v i s c o u s f low a t t h e r o t a t i o n a x i s .
Cond i t i ons (2 .2 .76) and (2 .2 .91) a r e needed i n o r d e r t h a t a d i s t i n c t i o n
i s p o s s i b l e between a r e g i o n where d i f f u s i o n dominates and a r e g i o n where
d i f f u s i o n and i n e r t i a a r e bo th impor t an t (both c o r e s ) . I f t h e s e c o n d i t i o n s
a r e n o t s a t i s f i e d , e i t h e r t h e " region of compet i t ion ' ' i s l o c a t e d i n t h e
immediate v i c i n i t y of t h e r o t a t i o n a x i s r 2. A - " ~ o r i t does n o t appear
a t a l l . I n t h e l a t t e r c a s e one obse rves an i n v i s c i d r e g i o n up t o t h e r o t a-
t i o n a x i s .
As a l r e a d y s t a t e d , i n t h e Ekman l a y e r a l l te rms become impor t an t a t
r 2. z ~ E - ' , where t h e t h i c k n e s s of t h e l a y e r i s % L - l . So t h e compl ica ted m
" a l l te rms r eg ions" i d e n t i f i e d i n t h e c o r e f o r L 1 has c o n t r a c t e d t o a m l a y e r of t h i c k n e s s L - I n e a r t h e end caps . It w i l l become c l e a r , i n t h e
range 1 << L << E;ly2, r e f e r r e d t o a s t h e u n i t c y l i n d e r , more a n a l y t i c a l m p r o g r e s s can be made t o d e s c r i b e t h e f low o u t s i d e t h e Eknian l a y e r s .
F i g u r e 5 i l l u s t r a t e s t h e v a r i o u s f low r e g i o n s of x and $ i n t h e u n i t cy-
1 i n d e r .
A d e s c r i p t i o n i n te rms of an i n v i s c i d f low, boundary l a y e r s and v i s -
cous c o r e s i s on ly v a l i d when L f a l l s i n t h e range I << L << iT-lI2. m m m
Since L can become l a r g e w e a r e p r i m a r i l y i n t e r e s t e d i n t h e f low beha- m v i o u r when L exceeds t h e upper l i m i t of t h i s range . For t h i s purpose we
cons ide r t h e d i s t a n c e s c a l e s of bo th c o r e s and t h e s c a l e h e i g h t s of bo th
Figure 5 : Diagram o f the flow regions of x and
+ i n the un i t cy Zinder.
Stewar tson l a y e r s . For Lm 2. E-'I2 t h e o u t e r S tewar tson l a y e r expands o v e r m t h e d e n s i t y s c a l e h e i g h t (x % 1 ) . S imul taneous ly t h e o u t e r c o r e comes up
- 1 and j o i n s t h e l a y e r . Fo r Lm 2. Em t h e i n n e r l a y e r and i n n e r c o r e expand,
r e s p e c t i v e l y , come up and j o i n . One can expec t t h a t f o r E 2. L~ << em1 m t h e x- d e r i v a t i v e s of x i n (2.2.57) become impor t an t te rms d e s c r i b i n g t h e
f l ow i n t h e main s e c t i o n . For Lm 2. E i l t h e x- d e r i v a t i v e s of 11 i n (2.2.56)
w i l l do t h e same. The Ekman l a y e r r e t a i n s i t s sma l l s c a l e h e i g h t and i s
s t i l l t h e nar row r e g i o n w i t h i n which t h e f low i s a d j u s t e d t o t h e end cap .
At f i r s t t h e f low i n t h e range B - ' I2 2. L << E - I , r e f e r r e d t o a s t h e rn m m semi- long c y l i n d e r , i s d i s c u s s e d . The x- d e r i v a t i v e s of x become impor t an t
by s c a l i n g
Here, t h e a b s o l u t e s c a l i n g magnitude of I,, has been chosen such t h a t i t
corresponds t o t h e induced f l u x of t h e Ekman l a y e r s . I n o r d e r t o show
t h a t t h e t e rm i n $ i n (2.2.57) can be n e g l e c t e d , we f i r s t c o n s i d e r t h e
problem f o r 4 given by (2.2.59) and (2 .2 .60) . S u b s t i t u t i o n of (2 .2 .93)
i n t o (2.2.59) shows t h a t t h e non- l inea r te rms can be dropped provided
t h a t EA << 1 , a cco rd ing t o (2.2.30) s a t i s f i e d . The z- d e r i v a t i v e s can be
neg l ec t ed s i n c e q2 i s sma l l and we o b t a i n
Th i s s i m p l i f i e d problem does n o t a l l ow t h e boundary c o n d i t i o n s f o r $ a t
z = 0 and z = 1 t o "be s a t i s f i e d . The re fo re l a y e r s of t h i c k n e s s L-' n e a r m
t h e end caps a r e needed.
I n t e g r a t i n g (2.2.94) w i t h r e s p e c t t o x and app ly ing t h e c o n d i t i o n t h a t
$ must be f i n i t e a t x = A (r = 0 ) , we g e t an e q u a t i o n which i s used t o
e l i m i n a t e t h e t e rm i n 4 on t h e l e f t of (2 .2 .57) . F u r t h e r i n t r o d u c i n g - 2
(2.2.93) i n t o (2.2.56) and (2.2.57) it i s s e e n t h a t t h e term i n $ i s Q A
compared t o u n i t magnitude and can t h u s be dropped. The l e a d i n g o r d e r of
(2.2.56) and (2.2.57) becomes
The n e g l e c t e d te rms a r e % , % g2 and 2. A - I compared t o u n i t mag-
n i t u d e , which i s v a l i d i n ou r pa rame te r s e t t i n g . One s e e s t h a t (2.2.95)
and (2.2.96) a l l o w t h e no- s l ip c o n d i t i o n f o r x a t r = I . For x + X , 1
must t end t o i t s pu re v i s c o u s v a l u e : (a/ax)x = 0. The remaining condi- 1
t i o n s a t t h e c y l i n d e r w a l l and t h e pu re v i s c o u s v a l u e f o r $ a t t h e a x i s ,
however, a r e n o t a p p l i c a b l e . The re fo re t h e i n n e r S tewar tson l a y e r and t h e
i n n e r c o r e a r e needed. The s c a l i n g r u l e s and e q u a t i o n s a r e equa l t o t h o s e
g iven b e f o r e . The Ekman l a y e r s p rov ide t h e c o n d i t i o n s a t t h e end caps . I n
f i g u r e 6 we have g iven a d iagram of t h e f low r e g i o n s f o r x and $ i n t h e
semi- long c y l i n d e r .
F i n a l l y t h e f l ow i n t h e range Lrn I. E;', r e f e r r e d t o a s t h e l ong cy-
l i n d e r , i s d i s c u s s e d . I n t h i s range a l s o t h e i n n e r c o r e and t h e i n n e r
S tewar tson l a y e r come up, r e s p e c t i v e l y , expand and j o i n a t x 2. 1 . I n t r o-
ducing t h e s c a l e d v a r i a b l e s
112 112 $ = Ern @2, X = Ern X2 (2.2.97)
e q u a t i o n s (2.2.56) and (2.2.57) become, t o l e a d i n g o r d e r
[E al2,-x aX2 m m ax ax $2 = F (2.2.98)
The n e g l e c t e d te rms a r e % Li2 and 2. A = ' . As b e f o r e , t h e te rm i n 4 on t h e
l e f t of (2.2.57) can be dropped, which i s c l e a r when one per forms t h e same
p rocedu re a s b e f o r e w i t h r e s p e c t t o ~ h ; ' ' ~ % Lrn << E: . Equat ions (2.2.98)
and (2.2.99) a l l ow a l l boundary c o n d i t i o n s t o be met a t t h e c y l i n d e r w a l l .
For x -+ x and $ must t end t o t h e i r pu re v i s c o u s v a l u e s a t t h e a x i s : 2 2 ( a / a ~ ) ~ = ( a / a ~ ) $ ~ = 0. The c o n d i t i o n s a t t h e end caps a r e provided by
2 t h e Ekman l a y e r s .
Figure 6 : Diagram o f the flow regions of x and
+ i n the semi-long cyl inder.
2.3. Discussion
In the present chapter we investigated the secondary flow of an incom-
pressible fluid and of a perfect gas in a rotating cylinder, applying a
linearised analysis to a small perturbation on the (isothermal) state of
rigid body rotation. For the incompressible problem we identified three
types of flow, when we increased the length-to-radius ratio L from unit
magnitude to infinity. These types correspond to the ranges
<< L << E-"~, E-'I2 % L << E-I, E-' % L , where E is the Ekman number
based on the radius and taken to be small. In the first range the flow
consists of an inviscid region, where the motion is governed by the
Taylor-Proudman theorem, extended by Ekman layers near the end caps and an
inner and outer Stewartson layer near the cylinder wall. In fact this flow
type is well-known from literature and is of relevance to geophysical pro-
blems stemming from models of oceanic and atmospheric currents. In the se-
cond L-range the outer Stewartson layer expands to the interior and radial
diffusion of azimuthal momentum becomes important in the main section of
the cylinder. In the third range also the inner Stewartson layer fills the
entire cylinder. As a result, radial diffusion of azimuthal and axial mo-
mentum are both significant, a situation that is also characteristic in
studies on flows in semi-infinite cylinders. Since E is very small most
configurations will fall in the first range. The first type of flow is of
most practical importance.
For the problem of a perfect gas in a rotating cylinder two extra di-
mensionless parameters enter into the linearized conservation equations.
These are: the speed parameter A and the Brinkman number Br. The speed
parameter appears in the exponent of the exponential density distribution
at isothermal rigid body rotation: i.e.
where A is the square of the ratio of the circumferential speed to the
most probable molecular speed. In gas centrifuges A is at least of unit
magnitude. The Brinkman number appears in the linearised energy equation
where Br/E scales the work done by compression u(a/ar)p relative to the
heat conduction. In gas centrifuges Br is of unit magnitude.
The Ekman number E based on the radius and on the density at the cylinder
wall is taken to be small. The length-to-radius ratio L is varied from
unit magnitude to infinity.
result, namely
constant along
where $ is the
In the case of non-diffusive motion, E = 0, the azimuthal component
of the momentum equation and the energy equation both lead to the same
and axial momentum equation it follows that
a zero radial velocity and thus an axial velocity which is
z: i.e.
streamfunction. Eliminating the pressure from the radial
where w is the angular speed perturbation and 8 the temperature pertur-
bation. The foregoing result is referred to as the compressible Taylor-
Proudman theorem. The distributions of w and t3 individually are found
from a lower order balance with respect to E, viz. a balance between the
viscous forces in the azimuthal moventum equation and the heat conduction
in the energy equation. In fact, it is possible to separate two equations
for the streamfunction $ and the composite variable w-18(= x), which can
be treated independently, to leading order, from the equation determining
w and 8 individually. This is especially advantegeous with respect to gas
centrifuges where one is mainly interested in the distribution of the
axial and radial velocity so that only the problems for $ and x have to be considered.
Taking a speed parameter of unit magnitude and replacing x by w the
problems for x and $ are quite analogous to those for w and $ in the in-
compressible case. For E ' / ~ << L << E-''~ the flow consists of an inviscid
....--- rllr u L v L L v l L L D guverrlea DY cne compressible Taylor-Proudman
theorem, extended by Ekman l a y e r s n e a r t h e end caps and Stewar tson l a y e r s
n e a r t h e c y l i n d e r w a l l . When t h e l eng th- to- rad ius r a t i o exceeds t h e upper
l i m i t of t h e L- range bo th Stewar tson l a y e r s s u c c e s s i v e l y expand t o t h e
i n t e r i o r .
Taking A >> I , t h e d e n s i t y dec rea se s s t r o n g l y w i t h d i s t a n c e from t h e
c y l i n d e r w a l l and t h i s i s i n f a c t t h e unde r ly ing r ea son f o r a d r a s t i c
change of t h e f low phenomena. The parameters now s p e c i f y i n g t h e motion
a r e t h e modif ied Ekman number E which i s based on t h e s c a l e h e i g h t of t h e rn
d e n s i t y d e c r e a s e i n s t e a d of t h e r a d i u s and t h e modif ied a s p e c t r a t i o & which i s t h e r a t i o of t h e l e n g t h t o t h e d e n s i t y s c a l e h e i g h t . These modi-
f i e d pa rame te r s we r e l a t e d t o E and L by: E = E A ~ , Lm = AL. Taking rn
E << 1 f o u r t ypes of f low f o r x and j~ can be d i s t i n g u i s h e d when L i n - m rn c r e a s e s from u n i t magnitude t o i n f i n i t y . For L % 1 we i d e n t i f y a b a s i c
m i n v i s c i d f low governed by t h e compress ib le Taylor-Proudman theorem i n a
l i m i t e d r e g i o n n e a r t h e c y l i n d e r w a l l g iven by x << 2nE-I where x=A( l - r2 ) . m
Two Stewar tson l a y e r s of t h i c k n e s s E:l3 and E;l4 n e a r t h e c y l i n d e r w a l l
and Ekman l a y e r s of t h i c k n e s s (E e x ) 1 1 2 n e a r t h e end caps a d j u s t t h e i n - m v i s c i d f low t o t h e bounda r i e s ( s e e f i g u r e 4 ) . Due t o t h e e x p o n e n t i a l de-
c r e a s e of t h e d e n s i t y t h e Ekman l a y e r s expand away from t h e end caps w i t h
d i s t a n c e from t h e c y l i n d e r w a l l and form a v i s c o u s c o r e a t x I. t n ~ m l ,
where t h e f u l l e q u a t i o n s a r e needed t o d e s c r i b e t h e f low.
For 1 << L << E - ' / ~ t h e " a l l terms" c o r e c o n t r a c t s t o a l a y e r of t h i c k - rn m
n e s s L" n e a r t h e eGd caps ( s e e f i g u r e 5 ) . On t h e o t h e r hand, i n t h e c o r e m
of t h e c y l i n d e r we can d i s t i n g u i s h two r eg ions where on ly a p a r t of t h e
v i s cous f o r c e s i s impor t an t . I n an o u t e r c o r e a t x 2. ~n(L$?3~)- ' t h e r-
d e r i v a t i v e s of x ba l ance w i t h t h e i n e r t i a te rms j u s t a s i n t h e o u t e r
S tewar tson l a y e r of t h i c k n e s s L ~ I ~ E " ~ . In an i n n e r c o r e a t x I. 2n(L,Em)-I m m a l s o t h e x- d e r i v a t i v e s of $ ba l ance j u s t a s i n t h e i n n e r S tewar tson l a y e r
of t h i c k n e s s Provided t h a t A - Z ~ ( L ~ ~ ) - ' >> I and A - Z ~ ( L ~ E ~ ) - ' > > l rn m
t h e v i s c o u s te rms of t h e o u t e r c o r e and of t h e i n n e r c o r e , r e s p e c t i v e l y ,
w i l l dominate compared t o t h e i n e r t i a te rms a t t h e r o t a t i o n a x i s , a r e s u l t
of which i s t h a t $ and x a r e c o n s t a n t w i t h r e s p e c t t o x . The o u t e r c o r e
a d j u s t s x t o t h e pu re v i s c o u s v a l u e a t t h e a x i s and i t s s c a l i n g magnitude
i s such t h a t i t matches t o t h e i n v i s c i d r e g i o n . The i n n e r c o r e a d j u s t s $
of t h e i n v i s c i d region and of t h e o u t e r c o r e t o i t s pure v i s c o u s v a l u e a t
t h e a x i s .
For E i 1 I 2 % L, << E i l t h e o u t e r S tewar tson l a y e r expands ove r t h e sma l l
d e n s i t y s c a l e h e i g h t . S imul taneous ly t h e o u t e r c o r e comes up from t h e i n-
t e r i o r and j o i n s t h e o u t e r S tewar tson l a y e r a t x % I , s o t h a t t h e v i s c o u s
x- d e r i v a t i v e s of x a r e impor t an t t o d e s c r i b e t h e f l ow f i e l d i n t h e main
s e c t i o n ( s ee f i g u r e 6 ) . Fo r E,] Q, Lm t h e i n n e r c o r e and t h e i n n e r
S tewar tson l a y e r come up, r e s p e c t i v e l y , expand and j o i n , s o t h a t t h e v i s -
cous x- d e r i v a t i v e s of IJJ a r e a l s o impor t an t . As a r e s u l t , we o b t a i n a v i s -
cous f low i n t h e main s e c t i o n which i s s t r o n g l y i n f l u e n c e d by t h e c y l i n d e r
w a l l , a s i t u a t i o n t h a t i s a l s o t y p i c a l i n s t u d i e s on f lows i n s e m i- i n f i n i t e
gas c e n t r i f u g e s .
The l eng th- to- rad ius r a t i o of a gas c e n t r i f u g e i s of u n i t magnitude o r
1arger:Sinee i n t h e p r e s e n t day c e n t r i f u g e s A >> I , we can t a k e Lm >> 1 ,
which means t h a t t h e f low type co r r e spond ing t o Lm % 1 i s n o t of p r a c t i c a l
i n t e r e s t . On t h e o t h e r hand, t h e t ypes co r r e spond ing t o 1 << Lm << - 1 /2 - 1 r e f e r r e d t o a s t h e u n i t c y l i n d e r , Em % Lm << Em , r e f e r r e d t o a s t h e
semi- long c y l i n d e r and E ~ I % L,, r e f e r r e d t o a s t h e l ong c y l i n d e r , a r e a l l
of impor tance and, a t t h e same t ime , t h e whole scope of c e n t r i f u g e f lows
i s covered . I n t h e n e x t c h a p t e r we w i l l p r e s e n t e x p l i c i t s o l u t i o n s f o r t h e
f l ow f i e l d i n t h e s e t h r e e r anges .
A t a b l e of t h e f low r e g i o n s o u t s i d e t h e Ekman l a y e r s f o r t h e most im-
p o r t a n t parameter r anges i s g iven on t h e n e x t page.
i ncompres s ib l e
f l u i d ,
p e r f e c t
gas A 5 1,
p e r f e c t
ga s A >> 1
u n i t
c y l i n d e r
p e r f e c t
ga s A >> 1
semi- long
c y l i n d e r
p e r f e c t
gas A >> 1
l ong
cy 1 i n d e r
pu re
v i s c o u s
r e g i o n
i n n e r
c o r e
no
o u t e r
c o r e
o u t e r inner i n v i s c i d
Stewar t son
l aye r
Yes
3. THE RAPIDLY ROTATING HEAVY GAS
3.1. The problem statement
3 . 2 . 2 . Introduction
During the last decade a lot of emphasis has been placed on efforts to
develop countercurrent gas centrifuges used for the separation of uranium
isotopes. The configuration of a countercurrent gas centrifuge consists of
a circular cylinder with plane horizontal end caps and rotating about its
vertical symmetry axis. The working fluid in the cylinder is a mixture of
gaseous uranium hexafluoride (UF ) of molecular weight 349 and 352. Due to 6
the rotation of the mixture a radial separation of both components appears.
An axial countercurrent flow superimposed on the primary rotation causes a
vertical concentration gradient. The in- and oupput of UF at various po- 6 sitions allows a continuous utilization of the separation process.
From a fluid dynamical point of vieuw the physical constants of both
components in the mixture are practically equal which allows a homogeneous
approach to the flow problem. A different rotation, a different temperature
and an imposed axial flow at the end caps, and an axial temperature gra-
dient along the rotor wall are the possible means by which the counter-
current is established. The above mentioned induced perturbations on the
state of isothermal rigid body rotation are often small, at least in those
regions where the real separation occurs. Therefore a linear treatment as
applied in the previous chapter appears reasonable.
The most probable molecular speed 4- of OF6 is approximately
equal to 120 [m/s] at room temperature. This quantity is relatively small
due to the large molecular weight of UF6 (air: 4- 480[m/sl). The 0 0
separative performance of a centrifuge increases with the circumferential
speed (Cohen 1951 , Los 1963) and speeds of 500 [m/s] and more are today quite normal. Taking Ra = 500 [m/sl the speed parameter A becomes :
A M 17.36. The viscosity of OF is equal to 1.694 x [kg/msi . 6 2 Assuming a radius of 10-'[ml and a wall pressure of 8000 [ ~ / m 1 the modi-
-4 fied Ekman number be comes:^ 0.92 x 10 . For UF the heat conductivity m 6
is approximately equal to 6.68 x [~/mkl and the Brinkman number be-
comes: Br M 2. AS a consequence the assumptions A >> 1, E <.: I and Br % 1 m are reasonable and a treatment of the secondary flow as discussed in 2.2.2
is applicable.
The governing equations for the variables X, $ and $ are given by
(2.2.56), (2.2.57) and (2.2.59). As already seen, equations (2.2.56) and
(2.2.57) contain only one term in $, namely on the left hand side of
(2.2.57). But this term is not of leading order in the force balances
under consideration and can therefore be omitted. As a result, equations
(2.2.56) and (2.2.57) form a complete set determining x and $. Equation
(2.2.59) may be considered as an extra equation for the extra variable 4 . In gas centrifuge problems one is mainly interested in the distribution of
the radial and axial velocity determined by $. Therefore we restrict our-
selves to the equations (2.2.56) and (2.2.57). Omitting the term in @
these are
1 a2 + - - I x = - 2 il+f~r(l- 2)) az (3.1.2)
L~ az2 m
with boundary conditions
where
Here, wt and wb represent the different rotation rates of the top and
the bottom. The functions Wb and ut represent the vertical velocities
and Ob and Ot the temperature distributions at the bottom and at the top.
According to (3.1.10) we restrict ourselves to the case that the axial
net transport is zero. The function is the temperature distribution
along the cylinder wall. The no-slip and no-normal-flow conditions at the
cylinder wall are expressed in (3.1.3).
The aspect ratio of a gas centrifuge is at least of unit magnitude.
Since A >> 1 the modified aspect ratio is large. For L >> 1 three types m of flow can be distinguished corresponding to the ranges 1 << L << E
-112
-112 m m y
the unit cylinder, Em c Lm << E;', the semi-long cylinder and E-I % L m m' the long cylinder. In all three cases Ekman layers form near the end caps.
In the next subsection we will investigate the flow in these layers. In
particular, the influence of the imposed boundary conditions at z = 0 and
z = 1 on the flow outside the Ekman layers i.e. the Ekman suction. In sec-
tions 3.2, 3.3 and 3.4 we will successively investigate the flow outside
the Ekman layers in the unit cylinder, the semi-long cylinder and the long
cylinder.
6.1.2. The Ekman s u c t i o n
The Ekman layer equations near the bottom cap are obtained by intro-
ducing the stretched coordinate
and the perturbation expansion
Putting (3 .1 .11) - (3 .1 .13) into ( 3 . 1 . 1 ) - ( 3 . 1 . 2 ) and letting E e x + 0
m we obtain
The terms neglected in these equations are " Erne x , As a consequence we
must require: x << I ~ E ; ' . At x % InE-I , where the layer is of thickness - 1 m
Lm , all terms of the original equations become important. Furthermore, the X-derivatives are neglected which indicates that the dependent vari-
ables may not vary too fast along x. This can be illustrated near x = 0
b y introducing the coordinate 5 = E - ' / ~ x . Then we obtain a cross-section m at the corner of the cylinder where both the highest x- and z-derivatives
are significant. In general, we can conclude that the local x-variations
of the dependent variables iJ and 2, must take place on a distance scale x 112
that is large compared to (Erne ) .
Differentiating ( 3 . 1 . 14 ) with respect to y and eliminating ( a 2 / a y 2 ) 2 0
by means of ( 3 . 1 . 15 ) we get
t h e s o l u t i o n of which i s g iven by
a J i o f Y 1 +Y - - a~
- e cos y l , e ' s i n y 1
where
Here, ( a / a y ) f J O may n o t t end t o i n f i n i t y a s y -+ -, where y + - means f o r
from t h e bot tom cap on t h e sma l l t h i c k n e s s s c a l e of t h e Ekman l a y e r b u t
s t i l l w i t h i n t h e c y l i n d e r . As a r e s u l t , ( 3 . 1 . 1 7 ) can be s i m p l i f i e d t o
ago -y - - - e ' s i n y I , eY1cos y a y l 1
The boundary c o n d i t i o n f o r t h e r a d i a l v e l o c i t y
b Y
s o t h a t ( 3 . 1 . 1 9 ) becomes
a% -Y - - - f e ' s i n y aY1 1
a t t h e bot tom cap i s g iven
where t h e i n t e g r a t i o n c o n s t a n t f i s an a r b i t r a r y f u n c t i o n of x , excep t
t h a t ( a / a x ) f << ( E ~ Q ~ ) - " ~ . I n t e g r a t i n g ( 3 . 1 . 2 1 ) w i t h r e s p e c t t o y and
app ly ing t h e c o n d i t i o n t h a t
i t i s found t h a t
I n t e g r a t i o n of ( 3 . 1 . 1 4 ) w i t h r e s p e c t t o y and a p p l i c a t i o n of t h e c o n d i t i o n
t h a t
go = wb - ieb a t y 1 = 0 ( 3 . 1 . 2 4 )
y i e l d s
-2 2, = f ( l + k ~ r ( l - $ ) 1 ~ / ~ ( 1 - e l c o s y ) + ub - heb
1 ( 3 . 1 . 2 5 )
The unknown f u n c t i o n f i n t h e s o l u t i o n s ( 3 . 1 . 2 1 ) , ( 3 . 1 . 2 3 ) and ( 3 . 1 . 2 5 ) i s
de termined by matching t h e v e l o c i t i e s of t h e Ekman l a y e r s and t h e o u t s i d e
f low. Here, o u t s i d e f l ow means t h e i n t e r n a l f low i n t h e c y l i n d e r o u t s i d e
t h e Ekman l a y e r . The dependent v a r i a b l e s of t h e o u t s i d e f low a r e denoted
by t h e s i g n ^. They v a r y a long t h e z - coo rd ina t e on a s c a l e t h a t i s l a r g e
compared t o L - I ( ~ ~ e ~ ) ' I 2 , t h e t h i c k n e s s s c a l e of t h e Ekman l a y e r . The m match c o n d i t i o n s a r e g iven by
S u b s t i t u t i n g ( 3 . 1 . 2 5 ) i n t o ( 3 . 1 . 2 6 ) and l e t t i n g E e x -+ 0 we o b t a i n m
P u t t i n g ( 3 . 1 . 2 1 ) i n t o ( 3 . 1 . 2 8 ) one s e e s t h a t an o u t s i d e r a d i a l v e l o c i t y
can on ly be a d j u s t e d t o t h e end cap by t h e second o r d e r c o n t r i b u t i o n of
t h e Ekman l a y e r e x t e n s i o n s . As f o l l o w s , we conclude t h a t
P u t t i n g t h i s i n ( 3 . 1 . 2 7 ) we observe t h a t t h e second o r d e r c o n t r i b u t i o n s
a r e sma l l compared t o t h e l e f t hand s i d e i f
which i s t h e case a s long a s t h e v a r i a t i o n s of t h e o u t s i d e flow take
p lace on an a x i a l d i s t a n c e s c a l e t h a t i s l a r g e compared t o t h e one of t h e
Ekman l a y e r . I n o t h e r words, t h e adjustment of t h e r a d i a l v e l o c i t y t o i t s
no- sl ip cond i t ion induces a secondary flow which i s an o rde r of magnitude
smal l e r than t h e one induced by t h e coupl ing of and 2 t o t h e i r proper
va lues a t t h e end cap and can t h e r e f o r e be omit ted.
S u b s t i t u t i n g (3.1.23) i n t o (3.1.27) and applying (3.1.29) we ob ta in , t o
l ead ing o rde r
The compa t ib i l i ty condi t ion (3.1.32) f o r t h e ou t s ide flow i s r e f e r r e d t o
a s the Ekman s u c t i o n cond i t ion and a l lows us t o desc r ibe t h e ou t s ide flow
without an e ,xp l i c i t d i scuss ion of t h e Ekman l a y e r s themselves. The f i r s t
o rde r r e p r e s e n t a t i o n of t h e Ekman l a y e r s o l u t i o n s can now be w r i t t e n a s
Analogous t o t h e above procedure we can de r ive a suc t ion cond i t ion a t
t h e top cap. Without f u r t h e r proof we put
The f i r s t o rde r r e p r e s e n t a t i o n of t h e Ekman l a y e r s o l u t i o n s near z = 1
i s given by
* -Y
io = - { ~ ( Z = I ) - ut + 1 0 i ( e lcos y* -1) + ,a+ - $ 0 t I t (3.1.36)
y = tl+fBr(l- ~ ) l " ' L (E e") ""(1-z) 1 rn m (3.1.38)
We are primarily interested in the flow outside the Ekman layers. For this
purpose it is convenient to join the imposed boundary conditions at the
end caps as
Fb
Ft
The suction
l+t~r(I-
C l+tBr(l-
follows
= wb - :eb - 2(~~e~)-'/~11+fBr(l-- :)}3/4~ b
= w t - :€lt + 2(~~e~)-'~~{l+:~r(l- 2)}3/48 t
conditions then simplify to
x 314 ) - 5 - i(Emex)1/2P = - 1 .(Erne) I / 2 Fb a t z = O
314 + : ( ~ ~ e ~ ) ~ ~ ~ ~ = + :(Erne ) -$I P " 1 1 2 ~ t at z = I
One sees that a mathematical description of the outside flow is
(3.1.39)
(3.1.40)
(3.1 -41)
(3.1.42)
pqssible
without making a distinction between the imposed different rotation, dif-
ferent temperature and axial flow at both ends. Only the combination of
these three boundary conditions as expressed by (3.1.39) and (3.1.40) is
essential. The governing equations for the flow outside the Ekman layers
are obtained by dropping the z-derivatives on the left hand side of
(3.1.1) and (3.1.2)
x a x 4~ E e {I+:BP(I- S$-U+:BP(I- ~)I-'(I- z)2 = TI m ax A ax A
x a$ = - 2{ l+i~r(l- -) I - A az
The boundary conditions are given by the suction conditions and
Equations (3.1.43) - (3.1.44) with boundary conditions (3.1.41), (3.1.42) and (3.1.45) describe the flow outside the Ekman layers in the unit, semi-
long and long cylinder.
3.2. The unit cylinder
In the present section we investigate the flow outside the Ekman
layers in the unit cylinder whose L -range is given by m
As was seen in section 2.2.2 five flow regions can be distinguished. These
are: the inner and the outer Stewartson layer, the inviscid region and the
inner and the outer core(see figure 5 ) . Firstly, we discuss the flow in the
inviscid region.
In the inviscid region x and $ are constant along z . Therefore the
Ekman layers at: the top and the bottom will control the flow, which means
that the scaling magnitudes of the dependent variables correspond to the
induced flux expressed in the Ekman suction conditions. As a consequence
we scale for the inviscid flow
x I12 rn=c~,e) $ o , n = x o
Substituting (3.2.2) into (3.1.43) and (3.1.44) we obtain
X L E3/2{4 ii- ex a (1- g))2ji0e-112x = 2 - m m ax ax az
a 2 - 4L E ~ ~ ~ Q ~ ~ ~ ~ -?- {l+!Bp(l- 5)))-{ ]+t~p(l- :)]-'(1- --)Xo - M m ax A ax
x = - 2 {l+$Br(l-
The terms on t h e l e f t hand s i d e of ( 3 . 2 . 3 ) and ( 3 . 2 . 4 ) can be n e g l e c t e d
i f , i n t h e f i r s t p l a c e , Lm f a l l s i n t h e r ange ( 3 . 2 . 1 ) . Secondly, t h e te rms X i n q u e s t i o n i n c r e a s e w i t h a power of e by which we must add t h e c o n d i t i o n
t h a t x << Z ~ ( L E E ~ ) - ] . For x % z ~ ( L ~ E ~ ) - ' t h e te rms on t h e l e f t of ( 3 . 2 . 4 )
become of l e a d i n g o r d e r and t h e r e s u l t i n g ba l ance i s one of t h e o u t e r
co re . I n t h e i n v i s c i d r e g i o n we g e t
Applying t h e s u c t i o n c o n d i t i o n s ( 3 . 1 . 4 1 ) and ( 3 . 1 . 4 2 ) , t h e s o l u t i o n s f o r
X o and become
A p p l i c a t i o n of ( 3 . 1 . 7 ) g i v e s f o r t h e r a d i a l and a x i a l f low
where po(= e - x ) i s t h e d imens ion l e s s d e n s i t y d i s t r i b u t i o n a t i s o t h e r m a l
r i g i d body r o t a t i o n . One s e e s t h a t i n t h e i n v i s c i d r e g i o n t h e a x i a l f l ow
i s c o n s t a n t a long z . The ou t f l ow from t h e Ekman l a y e r a t t h e bottom i s
equa l t o t h e i n f low a t t h e t op . There i s no i n t e r i o r r a d i a l motion and
t h e a x i a l f low i s on ly r e t u r n e d w i t h i n t h e Ekman l a y e r s . Both l a y e r s a r e
i n d i r e c t communication and c o n t r o l t h e f low i n t h e i n t e r i o r , a conse-
quence of t h e compress ib le Taylor-Proudman theorem. Once aga in we must
p o i n t ou t t h a t such a f low i s on ly observed i n a l i m i t e d r e g i o n n e a r t h e
c y l i n d e r w a l l g iven by r << Z ~ ( L ; E ~ ) - ] .
Genera l l y , t h e s o l u t i o n s ( 3 . 2 . 6 ) and ( 3 . 2 . 7 ) do n o t
l i n d e r w a l l . To overcome t h i s problem we must i n t r o d u c e
l a y e r s . At f i r s t we c o n s i d e r t h e o u t e r S tewar tson l a y e r
L ' / ~ . E ~ / ~ . In t roduc ing t h e s t r e t c h e d c o o r d i n a t e m m
apply a t t h e cy-
t h e Stewar tson
of t h i c k n e s s
and t h e p e r t u r b a t i o n expansion
1 / 2 E 1 1 4 - 1 2 = x2 + Lrn rn x2 + .... ( 3 . 2 . 1 0 )
112 Q = Ern lip2 + L A / ~ E ; ~ ~ ~ $ + . . . . 1 (3.2.1 I )
i n t o ( 3 . 1 . 4 3 ) - ( 3 . 1 . 4 4 ) and l e t t i n g L - ' E ' / ~ rn rn -+ 0 and L ~ E A / ~ -+ 0 we ge t
a x 2 - - a~ - 0 ( 3 . 2 . 1 2 )
D i f f e r e n t i a t i n g ( 3 . 2 . 1 3 ) with re spec t t o z and applying ( 3 . 2 . 1 2 ) we g e t
The boundary cond i t ions a t z = 0 and z = 1 a r e obta ined from t h e Ekman
suc t ion . I n s e c t i o n 3.1.2 we pointed out t h a t t h e x- var ia t ions of t h e de-
pendent v a r i a b l e s i n t h e Ekman l a y e r s must t ake p lace on a d i s t a n c e s c a l e
t h a t i s l a r g e compared t o The d i s t a n c e s c a l e s of t h e inne r and t h e rn o u t e r Stewartson l a y e r a r e r e s p e c t i v e l y ~ i ~ ~ E i ~ ~ and L ~ / ~ E ~ / ~ . These
th ickness s c a l e s a r e l a r g e compared t o a s L >> a cond i t ion m rn m s a t i s f i e d i n the u n i t cy l inde r . Both Stewartson l a y e r s a r e terminated a t
t h e top and t h e bottom by t h e Ekman l a y e r s .
S u b s t i t u t i n g ( 3 . 2 . 9 ) - ( 3 . 2 . 1 1 ) i n t o ( 3 . 1 . 4 1 ) - ( 3 . 1 . 4 2 ) and l e t t i n g
L " ~ E " ~ -+ 0 we o b t a i n rn rn
General ly , t h e imposed boundary cond i t ions a t both end caps a r e discon-
t inuous a t t h e corners of t h e cy l inde r ( x = 0 , z = 0 , 1 ) . On t h e o t h e r
hand, f o r x > 0 F and F a r e smooth wi th r e spec t t o x. Therefore b t
in the limit of L ~ ~ ~ E ~ ~ ~ + 0. In other words, the outer Stewartson m m layer does not see the discontinuity at the corner, in contrast to the
inner Stewartson layer, as will be seen later in this section.
Having in mind that g 2 is constant with respect to z it follows from
( 3 . 2 . 1 4 ) , ( 3 . 2 . 1 5 ) and ( 3 . 2 . 1 6 ) that
i/j2 = i ( l + i ~ r ) - ~ / ~ [ R 2 - Fb(O+) + z iFb(O+) + Ft(O+) - 2;Y2]]
substitution of ( 3 . 2 . 1 9 ) into ( 3 . 2 . 1 3 ) gives
A t the outer edge we must require that g 2 matches the inviscid solution:
The solution of ( 3 . 2 . 2 0 ) is now given by
where the unknown constant a follows from the matching to the inner
Stewartson layer. Putting ( 3 . 2 . 2 2 ) into ( 3 . 2 . 1 9 ) the solution of iJ2 be-
comes
a solution that also matches the inviscid solution.
The equations and boundary conditions for the inner Stewartson layer
problem are obtained by introducing the stretched coordinate
and the perturbation expansions
where b is an unknown constant that fixes the absolute magnitude of the
dependent variables. Its value depends upon the function of the layer and
will be discussed in the subsequent treatment. Because of algebraic con-
venience we have also added the outer Stewartson layer contributions in
expansions (3.2.25) and (3.2.26). Putting (3.2.24) - (3.2.26) into
(3.1.43) - (3.1.44), applying (3.2.22) - (3.2.23) and dropping the terms
- 1 - 1 112 - 1 1/6E5/12 1/3E1/3 compared to unit magnitude, the % b LmEm , Q J b Lm 9 Q L m
first order representation of the inner Stewartson layer equations becomes
Substituting (3.2.24) - (3.2.26) into the suction conditions (3.1.41) -
(3.1.42) and dropping terms , L-1/3E1/6 one obtains m m
& 1 = - b - 1 ( l + i ~ r ) - 3 1 4 e : / 3 ~ b ( ~ + ) ~ ~ ( ~ 1 ) - 1 ) a t z = O (3.2.29)
&, = + b-I ( I + ~ B ~ ) - ~ / ~ E : / ~ F , ( ~ + ) { H ( < ~ ) - ~ } a t ~ = 1 (3.2.30)
where E * = L~'E;/~ and is small.
As already stated, the imposed boundary conditions are discontinuous at
the corners of the configuration. To illustrate this behaviour we repla-
ced ~ ~ ( i i f i / ~ ~ A ~ ~ < l) and F ~ ( L A / ~ E A ~ ~ < ]), 5 2. 0, by respectively Fb(O+)H(< I )
and F (@+)H(< ) , where H(cl) is Heaviside's unit function. Of course, a t 1 discontinuity in Fb and F will be smoothed out to thickness E:" in a t crass-section in the Ekman layer. But this thickness scale is much smaller
than the one of the inner Stewartson layer. Therefore, the inner layer
still "sees" a jump. At the outer edge of the inner Stewartson layer the
variables must match to those of the outer Stewartson layer. Due to the
addition of the outer Stewartson layer contributions in the expansions
( 3 . 2 . 2 5 ) - ( 3 . 2 . 2 6 ) , however, matching is provided if
The boundary conditions at the cylinder wall are given by ( 3 . 1 . 4 5 ) . Prom
( 3 . 2 . 2 5 ) - ( 3 . 2 . 2 6 ) and ( 3 . 2 . 2 2 ) - ( 3 . 2 . 2 3 ) one finds that
Inspection of the conditions ( 3 . 2 . 2 9 ) - ( 3 . 2 . 3 0 ) and ( 3 . 2 . 3 2 ) - ( 3 . 2 . 3 4 )
shows that the flow induced in the inner Stewartson layer by condition
( 3 . 2 . 3 2 ) is largest with respect to the small parameter E * Therefore we
Put
(i) b = 1. Letting E * -t 0 the conditions reduce to
5, = 0 a t z = O a n d z = I ( 3 . 2 . 3 5 )
Differentiating ( 3 . 2 . 2 7 ) twice with respect to and eliminating
( a 2 / a r 2 ) g by means of ( 3 . 2 . 2 8 ) one gets 1 1
Separation of variables and application of the homogeneous boundary con-
ditions ( 3 . 2 . 3 5 ) gives
$ I = ngl xn sin nrrz ( 3 . 2 . 3 9 )
where X s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n
Applying (3.2.31) t h e s o l u t i o n of (3.2.40) becomes
where 6' i s g iven by
Applying (3.2.36) i t fo l l ows t h a t
1 Bn = - An, C = -d3A
n 3 n (3.2.43)
From (3.2.27) we deduce
P u t t i n g (3.2.44) i n t o t h e boundary c o n d i t i o n (3.2.37) one f i n d s t h a t
? e w + a + ~ F ~ ( O + ) + ? F ~ ( o + ) =
=4(l+:Br) A 6 cos ~ n 2 n=l n n (3.2.45)
Along t h e end caps and a long t h e c y l i n d e r w a l l we took an a r b i t r a r y tem-
p e r a t u r e d i s t r i b u t i o n . Up t o now we have n o t de f ined t h e uni form c o n s t a n t
temperature TO at isothermal rigid body rotation. For this purpose we
write the temperature along the cylinder wall T as W
In other words, T is the zero order term of the expansion of TW in 0
cos nrz. A s a result OW is given by
where
Inspection of the right hand side of ( 3 . 2 . 4 5 ) shows that the zero term of
the cos nrz expansion of x l , is missing, which is in fact due to the re- quirement that x 1 tends to zero as < + a. Therefore the boundary condi-
tion ( 3 . 2 . 4 5 ) can only be satisfied if
A s a result. the inner Stewartson layer cannot bring xO of the inviscid region to zero at the cylinder wall. Therefore the outer Stewartson layer
is needed! Substituting ( 3 . 2 . 4 9 ) into ( 3 . 2 . 2 2 ) and ( 3 . 2 . 2 3 ) the outer
Stewartson layer solutions become
By means of ( 3 . 2 . 4 7 ) - ( 3 . 2 - 4 9 ) it follows from boundary condition
( 3 . 2 . 3 4 ) that
and the inner Stewartson layer solution becomes
The above solution is obtained by putting b = 1 in conditions (3.2.32) - (3.2.34) and implies that the inner Stewartson layer requires an 0 (I)
term in x to balance the temperature perturbation at the wall. The in- duced flux is o(L;/~E;/~) and represents a closed circulation within the
layer. In figure 7 we have plotted ( l + ~ ~ r ) ~ / ~ ( a / a C ~ ) $ ~ versus ( l + f ~ r ) 1
1 ( l + f ~ r ) ~ ' ~ -
a5 1
Figure 7: ( l + ~ ~ r ) ~ / ~ ( a / a i ~ ) $ ~ versus (l+fBr) 116c1 at z=0.05, z=0.25, z=0.50 for Ow = -z+:.
a t z = 0.05, 2 = 0.25, 2 = 0.50 f o r a l i n e a r temperature grandtent a long
t h e w a l l , €I = - z + 6 .
The requirement, however, t h a t t h e a x i a l flow i n t h e i n v i s c i d r eg ion
and i n t h e o u t e r Stewartson l a y e r r e t u r n s i n t h e inne r Stewartson l a y e r ,
expressed by (3.2.33), i s no t g e t s a t i s f i e d . Taking 8 = 0 and W
( i i ) b = E:'~, t h e l ead ing o rde r wi th r e spec t t o E* of (3.2.29) - (3.2.34)
becomes
One s e e s t h a t t h e r i g h t hand s i d e s of (3.2.55) and (3.2.56) a r e ze ro when
5 , > 0. We t h e r e f o r e r ep lace (3.2.55) - (3.2.57) by
a t 2 = 0 and z = 1 (3.2.59)
and t h e d i s c o n t i n u i t i e s a t t h e co rne r s of t h e c y l i n d e r a r e now mathemati-
c a l l y discounted i n t h e a x i a l boundary i n s t e a d of t h e r a d i a l one. From t h e
equat ions (3.2.27) (3.2.28) and t h e boundary cond i t ions (3.2.58) - (3.2.59) i t follows t h a t
$ = D s i n nnz e-"riel s i n (1 J ~ S ~ C , + + ) 1 n=l n (3.2.61)
where t h e unknown D fol lows from (3.2 .60) . Therefore we expand t h e r i g h t
hand s i d e of (3.2.60) i n t h e e igenfunc t ions s i n n r z , n 2 1 , according t o
where
The expansion in sin nITz requires that
satisfies the same boundary conditions
the left hand side of (3.2.62)
at z = 0 and z = 1 as sin n n z .
This requirement is provided by both Heaviside's functions. Therefore,
the incorporation of the discontinuities in the Ekman suction conditions
was necessary. In contrast to the inviscid region and the outer Stewartson
layer, the inner layer indeed sees the jumps at the corners, a result
also identified by Moore & Saffman (1969). Substituting (3.2.61) into
(3.2.60) and applying (3.2.62) it follows that
After some algebraic manipulation the inner Stewartson layer solutions
become
w e r e y ana D a r e glven by
The above s o l u t i o n f o r t h e inne r Stewartson l a y e r r ep resen t s t h e rechan-
n e l i n g of t h e a x i a l flow moving from t h e bottom t o t h e top Ekman l aye r .
I n f i g u r e 8 we have p l o t t e d ( l + f ~ r ) ~ / ~ ~ ( a / a 5 ~ ) $ ~ versus ( l + ! ~ r ) ~ ' ~ ~ ~ a t
z = 0.05, z = 0.25, z = 0.50 f o r F (O+) = - Ft(Oe) = 1 and 0 = 0. b w L e t t i n g z -t 0 one sees t h a t t h e a x i a l flow p r o f i l e tends t o a Dirac
func t ion and thus j o i n s t h e d i s c o n t i n u i t y a t t h e corner of t h e cy l inde r .
Figure 8: ( l + f ~ r ) ~ / ~ ~ ( a / a c ~ ) $ ~ versus ( I + f ~ r ) 1 1 6 c 1 at z=O.O5, 2=0.25, z= 0.50 for
Fb(O+) = - F (O+) = I and Ow = 0. t
Summarising, in the outer Stewartson layer QO is brought to zero at
the cylinder wall. The inner Stewartson layer serves a dual purpose. First,
it requires an O(1) term in x to balance the temperature perturbation at the wall. The induced flux is O ( L ; ~ ~ E ~ / ~ ) and represents a closed circula-
m tion within the layer. Secondly, it requires an O ( E I / ~ ) term in Q in order
rn to return the axial flow moving from the bottom to the top Ekman layer.
Whereas the imposed conditions at the end caps induce a secondary flow in
the entire cylinder, the temperature perturbation along the cylinder wall
only produces an internal circulation within the inner Stewartson layer.
A similar situation was observed by Homsy & Hudson (1969), Matsuda (1975),
Nakayama & Usui (1974) and Sakurai & Matsuda (1974).
The axial flow behaviour in the inner Stewartson layer is illustrated in
figures 7 and 8. In figure 7 we have plotted (I+!Br)(a/a<l)~I versus
(~+fBr)l'~<~ at z = 0.05, z = 0.25, z = 0.50 for B?,, = - z + 4 and Fb(O+) = Ft(O+) = 0. We clearly observe the closed circulation in the
layer. In figure 8 we have plotted ( l + f ~ r ) ~ / ~ ~ ( a / a < ~ ) i $ ~ versus ( l + f ~ r ) ~ ' ~ < 1
at z = 0.05, z = 0.25, z = 0.50 for Fb(O+) = -Ft(Oc) = I and 0 = 0. W
One sees that the inner layer rechannels the vertical flow. Letting z -+ 0
the axial flow profile tends to a Dirac function and thus joins the discon-
tinuity at the corner of the cylinder.
In the discussion of the suction conditions applied to the outer
Stewartson layer we have already noticed that the radial shape of Fb and
F is subjected to certain restrictions. This is best illustrated by put- t ting the inviscid solutions (3.2.6) and (3.2.7) into the original equations
of motion (3.2.3) and (3.2.4). In this case, we observe that the terms on
the left hand side can only be neglected if the imposed functions Fb and
F satisfy the requirement t
1/2e1/2x)-l F-I ( a 2 / a x 2 ) ~ << (LrnEm (3.2.69)
Condition (3.2.69)
must take place on
the distance scale
implies that for I << Z~(L;E~)-' the x-variations of F
a distance scale that is large compared to L ~ / ~ E ~ / ~ rn r n '
of the outer Stewartson layer. In the case of a discon-
tinuity in F at x << Zn(L2E )-I free vertical Stewartson layers will be rn rn
formed. In these layers the discontinuity is smoothed out by viscous forces.
Such a situation is observed in the split-disk configuration e.g. discussed
by Baker (1967), Moore & Saffman (1969) and Stewartson (1957). Of course,
L V ~ ~ ~ ~ ~ ~ ~ ~ ~ LJ.L.WY) aoes no t apply t o t h e co rne r s of t h e cy l inde r . The
jumps a t x = 0 a r e , i n f a c t , r e spons ib le f o r t h e appearance of t h e s i d e
The jumps a t x = 0 a r e , i n f a c t , r e spons ib le f o r t h e appearance of t h e
s i d e w a l l l a y e r s . As a l r eady s t a t e d , t h e inne r Stewartson l aye r "sees"
t h e d i s c o n t i n u i t y : t h e o u t e r l aye r and i n t e r i o r do n o t .
As a l r eady no t i ced , an i n v i s c i d flow i s only observed i n a l imi ted
region nea r t h e cy l inde r wa l l . Due t o t h e exponent ia l decrease of t h e
d e n s i t y viscous fo rces become important i n t h e core of t h e cy l inde r .
Two types of v iscous cores can be d i s t ingu i shed : v i z . an inne r core and
an ou te r core . A t f i r s t we d i scuss t h e inne r core . In t roducing t h e con-
t r a c t e d coordinate
and t h e p e r t u r b a t i o n expansion
i n t o ( 3 . 1 . 4 3 ) - ( 3 . 1 . 4 4 ) and l e t t i n g A-lay1 -+ 0 t h e f i r s t o rde r represen-
t a t i o n of t h e inne r core equat ions becomes
where
a I = I - A - ' P ~ ( L E )-I rn m (3.2.75)
The cond i t ion t h a t ~ - ' a ; l << I impl ies t h a t t h e d i s t ance s c a l e of inne r
core measured from t h e r o t a t i o n a x i s must be l a r g e compared t o
i . e . r2 >>' A - I .
The unknown constant a i n expansions (3.2.71) and (3.2.72) determines t h e
abso lu te magnitude of and 9 . I t s va lue depends upon t h e func t ion of t h e
inne r core and w i l l be discussed i n t h e subsequent t r ea tmen t . -
For ri + x and must tend t o t h e i r pure viscous va lues , g iven by I 1
(a/anl);l = ( a / a ~ ~ ) T ~ = 0 , as r i ~ + r n (3.2.76)
The boundary cond i t ions a t the end caps a r e obta ined from t h e Ekman suc-
t i o n cond i t ions . P u t t i n g (3.2.70) - (3.2.72) i n t o (3.1.41) - (3.1.42) and
dropping terms % ~ 1 7 1 " ~ and % A - I ~ ; ' compared t o u n i t magnitude we o b t a i n
- = + l{l+!Bra } -314 - I L -
I " m
Provided t h a t
boundary cond i t ions (3.2.77) and (3.2.78) reduce t o
Now i n t e g r a t e (3.2.74) wi th r e spec t t o z from 0 t o 1 and apply (3.2.80)
I n t e g r a t i n g (3.2.81) wi th r e spec t t o r i l i t fol lows t h a t
i s cons tan t wi th r e spec t t o r i I and, consequent ly , cannot tend t o two d i f -
f e r e n t f i n i t e va lues a s ri -+ t rn. As a r e s u l t , t h e inne r core cannot ad-
j u s t a a /ax)x which i s an o rde r of magnitude l a r g e r than E 1 /2e 1 12s m Fb'
E ~ / ~ ~ ~ / ~ ~ F ~ . Since x I. Fb, F i n t h e i n v i s c i d r eg ion a second c o r e , t h e m t o u t e r co re , i s needed t o a d j u s t (a /ax)x t o i t s pure viscous va lue .
0
I n t r o d u c i n g t h e c o n t r a c t e d c o o r d i n a t e
and t h e s c a l e d dependent v a r i a b l e s
-2 i n t o ( 3 . 1 . 4 3 ) - ( 3 . 1 . 4 4 ) and l e t t i n g L + 0 , t h e o u t e r c o r e e q u a t i o n s m become
where
D i f f e r e n t i a t i n g ( 3 . 2 . 8 6 ) w i t h r e s p e c t t o z and appl-ying ( 3 . 2 . 8 5 ) one g e t s
The boundary c o n d i t i o n s a t z = 0 and z = 1 a r e ob t a ined from t h e Ekman
s u c t i o n c o n d i t i o n s . P u t t i n g ( 3 . 2 . 8 3 ) - ( 3 . 2 . 8 4 ) i n t o ( 3 . 1 . 4 1 ) - ( 3 . 1 . 4 2 )
one f i n d s t h a t
Having i n mind t h a t F2 i s c o n s t a n t w i t h
( 3 . 2 . 8 8 ) - ( 3 . 2 . 9 0 ) t h a t
r e s p e c t t o z i t fo l l ows from
Substituting this result into (3.2.86) we get
For a2 s > A-l, which implies that the distance-scale of the outer core,
measured from the rotation axis, must be large compared to A-"~,
r2 >> A - I , equation (3.2.92) can be simplified to
L
- For n2 -+ - m x must match the inviscid solution: i.e.
2
For n2 -t m (a/an2)x must match the inner core solution. However, as we 2 have previously shown, the inner core is not capable adjusting (a/ax)x to
its pure viscous value at the axis and, consequently, the outer core
directly brings (alax)~ to zero: i.e.
Equation (3.2.93) with boundary conditions (3.2.94) and (3.2.95) appears
more familiar when we introduce the coordinate
Putting (3.2.96) into (3.2.93) - (3.2.95) one finds
This problem can be solved by means of t h e Hankel t ransform of zero
o rde r : s e e Erdglyi e t a l . ( 1 9 5 4 ) and Sneddon ( 1 9 7 2 )
where J i s t h e ze ro o rde r Bessel func t ion . The s o l u t i o n of ( 3 . 2 . 9 7 ) - 0
( 3 . 2 . 9 9 ) i s given by
provided t h a t
- 1 Consider t h e case: 4(Ft + F ) = v 2 . Then Fk = k-I and t h e s o l u t i o n f o r - b x2 becomes
where r i s t h e Gamma func t ion
- I n f i g u r e 9 we have p l o t t e d X2 and v 2 x 2 versus v 2 f o r ?(Ft + F ) = v - I
- b 2 " Note t h a t f o r v
2 - + O 0 v 2 x 2 tends t o 1 which i s e x a c t l y t h e i n v i s c i d
s o l u t i o n .
The o u t e r co re a d j u s t s x0 t o i t s pure viscous va lue i n t h e r o t a t i o n
a x i s . U n t i l now, however, (a/ax)$ i s not brought t o zero. Therefore t h e
I 0 ' 1
I
2 I
3 4 5 6 7 8 - v2
F i g u r e 9: z2 and v2X2 v e r s u s v 2 for 4 (Fb+F )=v-' t 2 '
i nne r co re i s needed. The s c a l i n g r u l e s f o r t h i s r eg ion a r e obta ined by
p u t t i n g a = L - " ~ i n expansions ( 3 . 2 . 7 1 ) - ( 3 . 2 . 7 2 ) . The equat ions a r e m given by (3 .2 .73) - (3 .2 .74) and t h e boundary cond i t ions a t z = 0 and
z = 1 , given i n ( 3 . 2 . 7 7 ) - ( 3 . 2 . 7 8 ) , now become
For 111 -+ - t h e dependent v a r i a b l e s must match those of the o u t e r core .
Let us t h e r e f o r e i n v e s t i g a t e t h e behaviour of t h e o u t e r core s o l u t i o n s
f o r 112 expanded i n t o 111:
Furthermore i t i s appropr ia t e t o r e s c a l e t h e dependent v a r i a b l e s of t h e
o u t e r core such t h a t t h e i r magnitudes correspond t o those of t b e inne r
core: i . e .
P u t t i n g (3.2.1 11) - (3.2.112) i n t o (3.2.91) - (3.2.92) and dropping terms
Q, a ; ' ~ - ' compared t o u n i t magnitude, we o b t a i n
Since L i s l a r g e i n t h e u n i t c y l i n d e r , (3.2.115) - (3.2.116) can be sim- m p l i f i e d t o
- $12 = (l+;Bra1)-314einl 1- Fb + z(Fb + Ft )} (3.2.117)
As a consequence, t h e inne r core no t only must a d j u s t $ bu t a l s o
(a2/ax2)X. Although no t shown e x p l i c i t l y i n t h e p resen t work, t h e same
a p p l i e s t o t h e inne r Stewartson l a y e r . Here, $ and ( a 2 / a x 2 ) ~ a r e ad jus ted :
see Stewartson(I966) . I n t e g r a t i n g (3.2.118) wi th r e spec t t o n l one f i n d s
t h a t
The o u t e r core b r i n g s ( a l a x ) ~ t o ze ro i n t h e r o t a t i o n a x i s . Therefore t h e
i n t e g r a t i o n cons tan t , ( d l d r ~ ~ ) ; ~ ~ ( ~ -+ a), must be s e t equal t o ze ro and
t h e match cond i t ions f o r t h e i n n e r core become
The inne r core problem i s now s p e c i f i e d by d i f f e r e n t i a l equat ions
(3.2.73) - (3.2.74) and boundary cond i t ions (3.2.76), (3.2.109), (3.2.110)~
(3.2.120) and (3.2.121)-
Int roducing
(I+iBral) 112 y = Zn - Zn v 1
4 a y 2
and
- $ ] =
1
4a:14(1+:~ral) 112
we ob ta in
2 a x ; {V lyl = -
1 av2 1 I az 1
More a n a l y t i c a l progress can be made by t h e s u b s t i t u t i o n
- -112 $; = q + v , i- % + z(Fb + F,)} (3.2.131)
It i s q u i t e e v i d e n t t h a t t h e r e should ho ld
- a a ax;'
v l y {vl -I2vl [p; + v ~ ~ ~ ~ ~ - F ~ + z ( F ~ + F ~ ) I I = az V 1 - av, (3.2.133) av2
1
- $; = 0 a t z = 0 and z = 1 (3.2.135)
~,(a/av,)?; = o a t v 1 = o (3.2.136)
v 1 (a/av,)Vi = v 1 ( a l a ~ , ) ~ ; ' ~ ~ { F ~ - Z ( F ~ + F ~ ) } a t v l = o (3.2.137)
vl(a/av1)?; = ij7; = o a s v -t (3.2.138) 1
If it i s assumed t h a t F and F s a t i s f y t h e o r d i n a r y d i f f e r e n t i a l equa t ion b t
one deduces from (3.2.133) and (3.2.134) t h a t
t h e s o l u t i o n of which i s g iven by
- 11 - a, J I 1 - Zl f sin nnz
where f s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n
n b e - N
m V
+ LO- 3 N N a
+ m-
3- N
m
+ - 3 0 N
m
II
The inne r core problem i s , i n genera l , t oo d i f f i c u l t f o r a s a t i s f a c t o r y
t reatment and no f u r t h e r a t tempts will. be made. A t l a s t we s h a l l i n v e s t i-
g a t e t h e r e s t r i c t i o n s sub jec ted t o F and Ft a s v l -+ 0. From ( 3 . 2 . 1 3 7 ) one b
f i n d s t h a t
{ F - z ( F b + F t ) 1 a s v I
P u t t i n g t h i s i n t o ( 3 . 2 . 1 3 4 ) and i n t e g r a t i n g wi th
-+ 0 ( 3 . 2 . 1 5 3 )
r e spec t t o v we g e t
In o rde r t h a t condi t ion (3 .2 .136) i s s a t i s f i e d we must r e q u i r e
F ~ , F~ % v a l , a l > - 4 1 a s v l -+ 0 ( 3 . 2 . 1 5 5 )
o r , i n terms of t h e coordinate x,
In f a c t t h e same requirement a p p l i e s t o t h e o u t e r core . For small v 2
( 3 . 2 . 9 7 ) reduces t o
I n t e g r a t i n g t h i s wi th r e spec t t o b2dv2 and r e q u i r i n g t h a t t h e cond i t ion
( 3 . 2 . 9 8 ) ho lds , we f i n d t h a t
which i s , i n terms of x, see ( 3 . 2 . 9 6 ) and ( 3 . 2 . 8 3 ) , e x a c t l y t h e same re-
quirement a s given by ( 3 . 2 . 1 5 6 ) .
3.3. The semi-long cylinder
In the present section we investigate the flow outside the Ekman
layers in the semi-long cylinder whose L -range is given by m
Three flow regions can be distinguished. These are: the main section, the
inner Stewartson layer and the inner core (see figure 6). Firstly, the
flow in the main section is discussed.
The x-derivatives of x become significant by scaling 5 and 2 as
putting ( 3 . 3 . 2 ) - (3.3.3) into (3.1.43) - (3.1.44) and dropping terms
2. L ~ E ~ compared to unit magnitude we obtain m m
Differentiating (3.3.5) with respect to z and applying (3.3.4) we get
The boundary cond i t ions a t z = O and z = 1 a r e obta ined from t h e Ekman
s u c t i o n cond i t ions . P u t t i n g (3.3.2) - (3.3.3) i n t o (3.1.41) - (3.1.42) i t
fol lows t h a t
Having i n mind t h a t x 1 i s constant a long z we deduce from (3.3.6) - (3.3.8) t h a t
P u t t i n g (3.3.9) i n t o (3.3.5) one f i n d s t h a t
For x 1. 1 and s i n c e A - I << 1 d i f f e r e n t i a l equat ion (3.3.10) can be s impl i-
f i e d t o
For x + O x must match t h e inne r Stewartson l aye r . For x + (a/ax)xl I must match t h e inne r core . J u s t a s i n t h e u n i t c y l i n d e r , however, t h e
inne r Stewartson l a y e r and t h e inne r core a r e no t capable of a d j u s t i n g
x r e s p e c t i v e l y (a/ax)x t o zero. As a consequence, x I and ( d l d x ) ~ ~ a r e
brought t o ze ro i n the main s e c t i o n i t s e l f : i . e .
The analytical handling becomes easier by putting
Differential
now become
where
equation (3.3.11) and boundary conditions (3.3.12) - (3.3.13)
F = - :L,E;/~(F~ + Ft), a3 = J~(I+!BP) 1/8L-1/2E-1/4 , , (3.3.18)
The solutions of (3.3.15) with (3.3.16) - (3.3:17) can be constructed in
terms of Green's function: see Courant & Hilbert (1953). Without further
proof we put
where I and K are the modified zero order Bessel functions: see 0 0 Abramowitz & Segun (1968). From solution (3.3.19) one deduces that
which i s e x a c t l y t h e same requirement a s i d e n t i f i e d f o r t h e o u t e r and t h e
i n n e r core i n t h e u n i t cy l inde r .
One s e e s from (3.3.9) and (3.3.19) t h a t t h e v e l o c i t y d i s t r i b u t i o n i n
t h e main s e c t i o n i s inf luenced by t h e no- sl ip cond i t ion f o r x a t t h e cy-
l i n d e r wa l l . Furthermore we n o t i c e from (3.3.9) t h a t a l i n e a r change of
t h e a x i a l v e l o c i t y a long t h e z-coordinate i s poss ib le . The Ekman l a y e r s
no longer dominate, a s i s t y p i c a l i n an i n v i s c i d domain.
I n t h e p a r t i c u l a r case t h a t t h e boundary cond i t ions a t t h e end caps
a r e antisymmetric wi th r e spec t t o t h e mid-plane z = 112, F = - t Fby
i t follows from (3.3.19) t h a t x 1 = 0. P u t t i n g t h i s r e s u l t i n t o (3.3.9)
one f i n d s t h a t t h e i n t e r i o r Slow i s equal t o t h e i n v i s c i d one given by
(3 .2 .7) . One would expect t h i s s i t u a t i o n , of course , s i n c e when F t ' -Fb 9
x0 = 0 and t h e r e f o r e no o u t e r Stewartson l a y e r and no o u t e r core a r e nee-
ded and expansion of these regions i s i f r e l e v a n t . The flows i n t h e u n i t
and t h e semi-long c y l i n d e r a r e t h e same.
General ly , t h e p rev ious ly given s o l u t i o n s do no t apply nea r t h e cy l in-
der wa l l . Therefore t h e inne r Stewartson l a y e r ex tens ions a r e needed. The
equat ions and boundary cond i t ions f o r t h i s l a y e r a r e obta ined by int rodu-
c ing t h e s t r e t c h e d coordinate
and the p e r t u r b a t i o n expansions
where b i s an unknown constant t h a t determines t h e abso lu te magnitude of
t h e dependent v a r i a b l e s . I t s va lue depends upon t h e func t ion of t h e l a y e r
and w i l l be discussed i n t h e subsequent t reatment . Because of t h e conve-
nience i n a l g e b r a i c manipulation we have a l s o added t h e "main s e c t i o n so-
lutions" in expansions (3.3.24) - (3.3.25). Putting (3.3.23) - (3.3.25)
into (3.1.43) - (3.1 .44) and dropping the terms u b - 1 ~ - 1 / 3 ~ 1 / 6 , b-IL E ~ / ~ , m m m m ?. L A ~ ~ E A ~ ~ compared to unit magnitude, the first order representation of
the inner Stewartson layer equations becomes
a2ji, 2 - = - I (l+fBr) -
az (3.3.27) a<;
substituting (3.3.23) - (3.3.25) into the suction conditions (3.1.41) - (3.1.42) and dropping terms u L-' l3E1 I6
m m one obtains
= - b-I ( ~ + ! B ~ ) - ~ / ~ E : / ~ F ~ ( O + ) { H ( ~ ) - I} at z = 0 (3.3.28)
where E * = L " E ~ / ~ and is small. m m
Also here we have introduced Heaviside's unit function to express expli-
citly the discontinuities of F b and F at the corners of the cylinder. t At the outer edge of the layer the variables must match those of the main
section. Due to the addition of the main section contributions in expan-
sions (3.3.24) - (3.3.25) matching is provided if
The boundary conditions at the cylinder wall are given by (3.1.45). From
(3.3.24) - (3.3.25) and applying the main section solutions one finds that
Just as in the unit cylinder, we assume that the temperature distribution
along the cylinder wall can be expanded into cos nnz where the uniform
constant temperature at isothermal rigid body rotation corresponds to
n = 0. As a result CO
8 = Z 0 cos nnz w n=I wn (3.3.34)
Comparing the inner Stewartson layer equations and boundary conditions,
here, to those identified in the unit cylinder, one sees that both pro-
blems are equal. Putting b = 1 we obtain the solutions (3.2.53) - (3.2.54). Taking b = E fI3 we get the solutions (3.2.65) - (3.2.66). In
other words, the inner Stewartson layer again serves a dual purpose.
Firstly, it balances the temperature perturbation at the cylinder wall,
inducing a closed circulation within the layer. Secondly, it requires an
O(E;/~) term in I/J in order to return the axial flow moving from the bottom
to the top Ekman layer.
Also in the semi-long cylinder the radial shape of F and Ft is sub- b jected to certain restrictions. It is necessary that
This condition implies that for r << Zn(L E )-I the i-variations of F m m
must take place on a distance scale that is large compared to L ~ ~ ~ E ~ ' ~ , . .
the distance scale of the inner Stewartson layer. In the case of a dis-
continuity in F free vertical inner Stewartson layers will be formed,
smoothing out the discontinuity. Of course, condition (3.3.36) does not
apply to the corner of the cylinder. Again, the inner layer "sees" the
jump; the main section does not.
Finally we shall discuss the flow in the inner core. Introducing the
contracted coordinate
q l = x - Z~(L,E~)-' (3.3.37)
and the perturbation expansions
i n t o (3.1.43) - (3.1.44) and l e t t i n g ~-lct; l + 0 t h e inne r core equat ions
be come
For 01 + x and '4 must tend t o t h e i r 1 1
(a/an,)Yl = (a/an1)Vl = 0
The boundary cond i t ions a t z = 0 and z =
pure viscous va lues , given by
a s n l + w (3.3.42)
1 a r e obta ined from t h e Ekman
suc t ion cond i t ions , which a r e , t o l ead ing o rde r
For n1 -+ - t h e dependent v a r i a b l e s must match those of t h e main s e c t i o n .
W e t h e r e f o r e i n v e s t i g a t e t h e behaviour of t h e main s e c t i o n f o r x expanded
i n t o n1: i . e .
z = n 1 + tn(L E )-I rn rn (3.3.45)
Rescal ing the dependent v a r i a b l e s of t h e main s e c t i o n such t h a t t h e i r
magnitudes correspond t o those of t h e inne r co re , i . e .
one deduces from (3.3.9) - (3.3.10) t h a t
where we dropped terms a;'~-l compared to unit magnitude. Since L is m large (3.3.47) - (3.3.48) can be simplified to
Integrating (3.3.50) with respect to n 1 and applying the condition
(a/anl)xI2 = 0 as n l - f C O (3.3.51)'
since ( a / a x ) ~ is brought to zero in the main section, we obtain 1
The match conditions for the inner core now become
The inner core problem is now specified and appears to be the same as one
given for the unit cylinder. Therefore we shall not investigate this pro-
blem any further.
3.4. The long cylinder
In
i n t h e
Radial
the p resen t s e c t i o n
long cy l inde r whose
we i n v e s t i g a t e t h e flow ou t s ide t h e Ekman l a y e r s
L - range i s given by rn
d i f f u s i o n becomes an important f a c t o r i n t h e flow i n t h e main
s e c t i o n . Let
Here, t h e abso lu te s c a l i n g magnitude of 5 has been chosen such t h a t i t
corresponds t o t h e induced f l u x of t h e Ekman l a y e r s . S u b s t i t u t i n g ( 3 . 4 . 2 )
i n t o (3.1.43) - ( 3 . 1 . 4 4 ) we o b t a i n
a x a x 2 L E 14 - - e x & ( I - ;i)i2$2e-X = 2 - m rn ax a z
For x % 1 t h e terms % x/A can be neglected compared t o u n i t magnitude,
l ead ing t o
sor x -t r;ne alrruslve terms on the Lert of: (3.4.3) and (3.4.4) wlll do-
minate. As a result, x and $ must tend to their pure viscous values 2 2 given by
The boundary conditions at the cylinder wall are given by ( 3 . 1 . 4 5 ) . In
terms of q2 and x2 these are
The conditions at the end caps are obtained from the Ekman suction condi-
tions. Putting ( 3 . 4 . 2 ) into ( 3 . 1 . 4 1 ) - ( 3 . 1 . 4 2 ) and dropping terms % E 112 m
and % A-I we get
The problem for the main section is now specified by differential equa-
tions ( 3 . 4 . 5 ) - ( 3 . 4 . 6 ) and boundary conditions ( 3 . 4 . 7 ) - ( 3 . 4 . 1 0 ) .
Because of the convenience in algebraic manipulation we substitute
Differential equations ( 3 . 4 . 5 ) - ( 3 . 4 . 6 ) now become
ax; 8 L E {L L 3 2 e - x 1 -
m m ax ax $2 - ZT
The boundary conditions (3.4.7) - (3.4.8) become
The solution of (3.4.15) can be written as
~liminating X' from (3.4.13) by means of (3.4.14) it follows that 2
where the eigenfunctions X k are solutions of the eigenvalue problem
Putting (3.4.18) into (3.4.15), multiplying this with e - 9 where 7 is j' i
the adjoint eigenfunction, integrating this with respect to x from O to
and applying the orthogonality relation
m
o J e-x~kK.dx J = o for k + j
we get for Z k
d2Zk - - 16 ( I + ! B ~ ) - ' L ~ E ~ A 7, =
dz m m k k
The solution of ( 3 . 4 . 2 3 ) is given by
where Z is a particular solution of the differential equation ( 3 . 4 . 2 3 ) . i? k
In order to determine Z p k it is necessary to know the axial distribution
of Ow. In the subsequent treatment we restrict ourselves to the case that
0 i s a l i n e a r f u n c t i o n o f z , Then Z is equal to zero and the solution W ~ k for $ is given by
2
- ~ ~ I ~ L ~ E ~ ( ~ + ~ B P ) - ~ ~ + ~ J A L E ( I + : B P ) - ; ~ q 2 = kZl iake + b e k } X k
k m m
The six basic solutions of the differential equation ( 3 . 4 . 1 9 ) are al-
ready given by ( 3 . 2 . 1 4 6 ) - ( 3 . 2 . 1 5 1 ) . Therefore we only have to replace
v by e-x. Applying the boundary conditions ( 3 . 4 . 2 0 ) we retain 1
Putting ( 3 . 4 . 2 6 ) into ( 3 . 4 . 1 9 ) one gets recurrence relations for the co-
efficients Bin, (32n and B3n . These relations express 6 I n ' B 2 n 9 B 3 n 9 2 in terms of the basic coefficients B B20 and B 3 0 . Application of the
boundary conditions
gives BZ0 and 830 as a function of B I w As a result the coefficients can
be written as follows
Table I :
k = 1
JX 2.411
BIO 1.0000
a , , -0 .4844
3 2 0.0013
B I 3 0.0000
B20 - 0.5948
B 2 ] 0 .0780
az2 - 0.0001
B 2 3 0 .0000
respect to x rrom u to and applying the orthogonality relation
(3.4.22) we get
where
Since yk 2 1 as x 4 00, the integrals
m - ix- I Fbe Xk& and ~ ~ F ~ e - ~ ~ ~ d x 0 0
are only finite if
This is exactly the same requirement
outer core!
Summarising, the imposed boundary
x and I) in the main section which are
as identified for the inner and the
conditions at the end caps induce a
" E 1 1 2 ~ b and (I. E:l2Ft. The variables m
can be written as a set of radial eigenfunctions which decay exponentially
with the distance from the end caps. Higher eigenfunctions decay faster
than lower ones and at a reasonable distance from the ends the flow is
mainly described by the first eigenfunction.
A linear temperature gradient along the cylinder wall induces a flow
" (~~E,)-l(d/dz)e,. The radial shape of the streamfunction is almost equal
to that of the first eigenfunction. The particular solution for $2 given
in (3.4.25) also indicates that $ is constant along z. However, due to the
fact that the induced flow is not accepted by the Ekman layers, the de-
caying eigenfunctions (for the most part the first eigenfunction) must
bring $ to zero at z = 0 and z = 1: see the solutions for ak and b given k
in (3.4.34) - (3.4.36). As a result the rate of decay of the first eigen-
function and thus the magnitude of 4 J h L E (l+!~r)-' mainly determines 1 m m
the axial behaviour of $. Consider the base that L >> E-l. All eigen- m m functions now decay rapidly with distance from the ends. A linear tem-
perature gradient induces an axial flow being constant along z. This flow
is brought to zero at z = 0 and z = 1 by the first eigenfunction in a
region of thickness L;'E~'. The x induced here is adjusted at the end caps by the Ekman layers.
Parker & Mayo (1963) and Parker (1963) considered a semi-infinite cy-
linder. The Navier-Stokes equations were linearised taking a small pertur-
bation on the isothermal state of rigid body rotation. Furthermore, they
assumed eigenfunction solutions of the type
for the dependent variables. Putting (3.4.41) into the complete set of
linearised equations, the resulting eigenvalue problem was solved numeri-
cally. Since the eigenvalues and the radial shape of the eigenfunctions
were calculated for various magnitudes of A, their results can be com-
pared with our asymptotic solution for large A. In figure 12 we have plot-
ted il and the numerically calculated smallest eigenfunction for the an- gular speed perturbation of Parker versus x for A = 9 and A = 16. Here
one should notice that in the long cylinder $ q~ / l ' - l X (see equation
(2.2.59))so that from (2.2.46) it follows that
I n o r d e r t o compare t h e graphs i n f i g u r e 12 we took w l R. 1 a s x -+ One
s e e s t h a t t h e agreement between t h e numerical and t h e asymptot ic s o l u t i o n
i s a l r eady good f o r A = 16. Without showing t h i s e x p l i c i t l y , t h e same con-
c l u s i o n a p p l i e s t o t h e magnitude of t h e s m a l l e s t e igenvalue and t h e co r re-
sponding r a d i a l shape of t h e temperature p e r t u r b a t i o n , t h e a x i a l v e l o c i t y
and t h e r a d i a l v e l o c i t y .
Figure 12: First eigenfunction of the angular
speed perturbation versus x.
Ging (1962a) a l s o s t u d i e d t h e semi - - i n f in i t e c y l i n d e r applying an a-
symptot ic s o l u t i o n f o r l a r g e A , s i m i l a r t o t h e one used by Dirac (1940),
bu t i nc lud ing e f f e c t s due t o h e a t conduction. I n t h e p resen t work Ging's
r e s u l t i s obta ined by s e t t i n g 0 = 0 and bk = 0 i n t h e s o l u t i o n (3.4.25). W
Ging (1962b) a l s o showed t h a t t h e d i f f e r e n t i a l equa t ion (3.4.19) wi th t h e
va lues .
must be
( i ) He
boundary cond i t ions (3.4.20) and (3.4.21) on ly y i e l d s p o s i t i v e r e a l eigen-
He c a l c u l a t e d f o r t h e f i r s t e igenvalue: JX = 2.408. Two remarks 1 made, however, concerning Ging's asymptot ic s o l u t i o n .
used t h e coord ina te
x = 2A(I - r ) g
(3.4.43)
and t h e r e s u l t i n g e q u a t i o n s were equa l t o t h o s e a p p l i e d i n t h e p r e s e n t
work. I n t h e above approximat ion terms % x2/A a r e dropped whereas i n t h e g
p r e s e n t work we n e g l e c t e d te rms % x/A compared t o u n i t magnitude. We t h e r e
f o r e expec t t h a t f o r x L 1 t h e s t r e t c h e d c o o r d i n a t e a p p l i e d h e r e g i v e s a
somewhat b e t t e r approach a t r e l a t i v e l y s m a l l e r v a l u e s of A. An i l l u s t r a -
t i o n i s g iven i n f i g u r e 12 where we a l s o p l o t t e d o v e r s u s x ob t a ined i n 1
t h e c a s e of Ging ' s c o o r d i n a t e f o r A = 9 . Of c o u r s e , i n t h e l i m i t of A -+
bo th a sympto t i c s o l u t i o n s become t h e same.
( i i ) Ging d i d n o t i n v e s t i g a t e t h e a sympto t i c behaviour of t h e e q u a t i o n s
f o r l a r g e x and assumed t h a t w, 8, W and u a r e c o n s t a n t w i t h r e s p e c t t o r
a s r -* 0 . From t h e s o l u t i o n s (2 .2 .82) , (2.2.84) and (2.2.86) one f i n d s t h a t
t h i s assumpt ion a p p l i e s t o o , 0 and w b u t n o t t o u . For example, when
r I/J g iven by (2.2.86) and t h u s u d e v i a t e s c o n s i d e r a b l y from v t h e a sympto t i c v a l u e a s x -+ Without showing t h i s e x p l i c i t l y , Pa rke r e t
a 1 ob t a ined t h i s d e v i a t i n g behaviour f o r u a s r -+ 0.
component of t h e momentum e q u a t i o n was used t o de termine t h e r a d i a l
of w, be ing e q u a l , f o r l a r g e A , t o t h e p a r t i c u l a r s o l u t i o n g iven i n
(3 .4 .25) . It i s c l e a r t h a t t h i s s o l u t i o n i s o n l y a p p l i c a b l e a s l ong
L >> E-] m m '
Berman (1963). Lotz (1970), Olander (1972) and Soubbaramayer (1961)
cons ide red t h e c a s e of an a x i a l f low be ing c o n s t a n t a long z . The a x i a l
shape
a s
3.5. Discussion
I n t h i s chap te r we have developed e x p l i c i t s o l u t i o n s f o r t h e s t ream
func t ion $ and t h e composite v a r i a b l e x (X = w - i 8 , where w i s t h e angu-
l a r speed p e r t u r b a t i o n and 0 t h e temperature p e r t u r b a t i o n ) . The modified
Ekman number was taken t o be smal l , Em << 1 , t h e speed parameter t o be
l a r g e , A >> 1 , and t h e Brinkman number t o be of u n i t magnitude, BY % 1 .
Furthermore we have considered t h r e e success ive parameter ranges f o r t h e
modified a spec t r a t i o L given by 1 << L << E ; ~ ' ~ , r e f e r r e d t o a s t h e m m u n i t cy l inde r , E i 1 I 2 % Lm << E i l , r e f e r r e d t o a s t h e semi-long cy l inde r
and s Lm, r e f e r r e d t o a s t h e long cy l inde r . The boundary cond i t ions
inducing t h e secondary flow a r e : t h e d i f f e r e n t i a l r o t a t i o n of t h e end
caps , a x i a l i n j e c t i o n , removal of f l u i d a t t h e end caps , temperature
pe r tu rba t ions a long t h e end caps and a long t h e cy l inde r wa l l .
I n a l l t h r e e Lm-ranges Ekman l a y e r s form nea r t h e end caps. The in f luence
of t h e imposed boundary cond i t ions a t z = 0 and z = 1 on t h e flow o u t s i d e
the Ekman l a y e r s can be expressed i n terms of t h e Ekman suc t ion cond i t ions
given by (3.1.41) and (3.1.42). These cond i t ions allow us t o desc r ibe t h e
ou t s ide flow a s a func t ion of a c e r t a i n combination of t h e imposed boun-
dary cond i t ions a t t h e end caps expressed i n terms of t h e func t ions F b and Ft: see (3.1.39) and (3 .1 .40) .
I n t h e u n i t cy l inde r we i d e n t i f y a b a s i c i n v i s c i d flow i n a l imi ted
region near t h e cy l inde r wa l l . I n t h i s region x and $ a r e gi-ven by
where
The streamfunction and the composite variable are constant along z . There
is no radial motion and the axial flow is only returned within the Ekman
layers, a result referred to as the compressible Taylor-Proudman theorem.
Generally the imposed conditions at the end caps are discontinuous at the
corners of the cylinder ( z = 0.1; x = 0). The inviscid solution (3.5.1)
does not "see" the discontinuity at the corner so that (3.5.1) does not
satisfy the condition that $ and x are zero at the cylinder wall. To over- come this non-uniformity, inner and outer Stewartson layer extensions are
needed. The outer layer of thickness L A ~ ~ E ~ / ~ brings x to zero at x = 0. m
The inner layer of thickness adjusts $ at the cylinder wall. m rn Due to the exponential decrease of the density with distance from the cy-
linder wall the diffusive terms will become much larger than the inertia
terms near the rotation axis, resulting in that x and $ are constant with
respect to x. Two regions, referred to as the inner and the outer core,
adjust the pure viscous flow near the rotation axis and the inviscid flow
near the cylinder wall. In the outer core at x % z~(L~E~)-~ the r-deriva-
tives of x balance the inertia terms just as in the outer Stewartson layer. In the inner core at x u tn(L i? )-I the x-derivatives of $ are also impor- m m tant, just as in the inner Stewartson layer. On the other hand, in both
Stewartson layers the exponential density gradient is negligible whereas
in both cores this gradient is the mechanism by which the inviscid flow
and the pure viscous flow are coupled. For both the inner and the outer
core problem we must require that
The inner Stewartson layer also serves a second purpose. It requires an
O ( 1 ) term in x to balance the temperature perturbation at the cylinder wall. The induced $ is % L A ~ ~ E A ' ~ and represents a closed circulation
within the layer.
In the semi-long cylinder the outer Stewartson layer expands over the
radial density scale height and joins the outer core which comes up from
the centre of the cylinder. As a result, radial diffusion of x becomes an important factor describing the flow in the main section. The flow field
is influenced by the no-slip condition for x at the cylinder wall. A line- ar change of the axial flow along z is possible. The Ekman layers no lon-
ger dominate which is typical for an inviscid flow. The inner Stewartson
layer of thickness ~ " ~ 2 " ~ brings $ of the main section to zero at the m m
cy l lnde r wa l l . The inne r core a t x Z ~ ( L ~ E ~ ) - ' a d j u s t s $ t o i t s pure
viscous va lue nea r t h e r o t a t i o n a x i s . As be fo re , t h e imposed cond i t ions a t
t h e end caps must s a t i s f y t h e requirement (3.5.3).
Also i n t h e semi-long cy l inde r a temperature pe r tu rba t ion along t h e cy l inde r
wa l l only induces a c losed c i r c u l a t i o n wi th in t h e inne r Stewartson l a y e r .
S imi la r ly , i n t h e long cy l inde r t h e inne r Stewartson l a y e r
t h e r a d i a l dens i ty s c a l e he igh t and j o i n s t h e inne r core which
from t h e c e n t r e of t h e cy l inde r . As a r e s u l t , r a d i a l d i f f u s i v e
expands over
comes up
processes
s t r o n g l y in f luence t h e flow ou t s ide t h e Ekman l a y e r s . The imposed condi-
t i o n s a t t h e end caps induce a flow which can be descr ibed by an i n f i n i t e
s e r i e s of r a d i a l e igenfunc t ions which decay exponen t i a l ly wi th d i s t ance
from t h e end caps. Higher e igenfunct ions decay f a s t e r than t h e lower ones and
a t a reasonable d i s t a n c e from t h e ends t h e flow i s mostly descr ibed by t h e
f i r s t e igenfunct ion. Also i n t h e long cy l inde r t h e imposed condi t ions must
s a t i s f y t h e requirement (3.5.3).
A l i n e a r temperature g rad ien t along the cy l inde r wa l l induces a constant
a x i a l flow along z , whose r a d i a l shape i s p r a c t i c a l l y equal t o t h a t of t h e
f i r s t e igenfunct ion. This flow, however, can only be ad jus ted t o t h e ends
by mainly t h e f i r s t e igenfunct ion. In t h e case t h a t 10 L E >> 1 t h e f i r s t m m e igenfunc t ion decays r ap id ly . As a r e s u l t a l i n e a r temperature g rad ien t
produces an a x i a l flow which i s constant a long z over t h e g r e a t e s t p a r t
of t h e cy l inde r .
The developed s o l u t i o n s allow us t o desc r ibe t h e secondary flow i n
p resen t day gas cen t r i fuges . A convenient s i t u a t i o n f o r t h e sepa ra t ion i n
a countercurrent gas c e n t r i f u g e i s t h e one i n which t h e a x i a l f 1 o w . i ~
cons tan t a long z . The t h e o r e t i c a l maximum s e p a r a t i v e power of t h e c e n t r i-
fuge 6U . i s then given by (Cohen 1951, Los 1963): max
where p i s t h e d e n s i t y of the mixture , D t he d i f f u s i o n cons tan t , Z t h e
l eng th of t h e c y l i n d e r , AM t h e d i f f e r e n c e between t h e molecular weights
of bo th components, M t h e molecular weight of t h e mixture (AMIM << 1 ) and
A t h e speed parameter. To f i r s t o rde r t h e product pD i s constant a t con-
s t a n t temperature .
The influence of the secondary flow on the separative power is expressed
in the flow parameter m and the flow pattern efficiency E given by f '
In terms of the coordinate x appropriate for the rapidly rotating heavy
gas and having (2.2.55) in mind, we get
The flow efficiency E depends upon the radial shape of the streamfunction f $. Let us therefore consider the flow in the various parameter ranges. As
already stated for optimal separation the flow must be constant along z .
In the unit cylinder such a situation is observed in the inviscid region.
In the semi-long cylinder one can also create a constant axial flow in an
inviscid region by taking anti-symmetrical boundary conditions with res-
pect to the mid-plane z = 112, Fb = - Ft. In this special case, only, an
outer Stewartson layer and an outer core are missing and the flows in the
unit and the semi-long cylinder are equal. As a consequence, for Fb = - Ft and L << E;' we observe an inviscid region upto the inner core. In this
m region x = 0 and I/J is described by
provlded t h a t
x << 2n(LmEm)-I (3 .5 .10)
At x n, z~(L,E,)-', which i s t h e l o c a t i o n of t h e i n n e r c o r e , t h e i n v i s c i d - 1
s o l u t i o n no l onge r a p p l i e s . For x >> 2n(LmEm) , t h e d i f f u s i v e te rms do-
mina t e , r e s u l t i n g i n t h a t $ i s c o n s t a n t a l o n g s. Th i s i m p l i e s t h a t + e-x
approaches ze ro . At t h e c y l i n d e r w a l l IJJ i s brought t o z e r o i n t h e i n n e r 1 /3E1/3 Stewar tson l a y e r of t h i c k n e s s Lm m
For Lm 2. E i l t h e f low i s mainly d e s c r i b e d by t h e f i r s t e i g e n f u n c t i o n
b e i n g a lmost c o n s t a n t a long z . A r ea sonab le d e s c r i p t i o n f o r $ i s t h e n
where a de t e rmines rhe magnitude of +. A good approximat ion f o r a i s
g iven by
-x a n, ;Fbe-'"(l - e - ~ e - ~ (3.5.12) 0
The imposed c o n d i t i o n s a t t h e end caps on ly de t e rmine t h e magnitude of
t h e c o u n t e r c u r r e n t f low; t h e r a d i a l shape i s no t i n f luenced .
For L >> E - I a c o n s t a n t a x i a l f low a l o n g z can be c r e a t e d by imposing m m
a l i n e a r t empera tu re g r a d i e n t a long t h e c y l i n d e r w a l l . I n t h i s c a s e I) i s
de sc r ibed by
The r a d i a l shape i s a g a i n t h a t of t h e f i r s t e i g e n f u n c t i o n .
Concerning t h e e f f i c i e n c y t h r e e parameter ranges can now be d i s t i n -
guished.
( i ) L E 5 The i n n e r c o r e i s l o c a t e d i n t h e immediate v i c i n i t y of m m
t h e r o t a t i o n a x i s , r % A - " ~ , o r i t does n o t e x i s t s a t a l l . One obse rves
an i n v i s c i d r e g i o n almost up t o t h e r o t a t i o n a x i s . As a r e s u l t t h e most
opt imal r a d i a l shape f o r I) y i e l d i n g t h e h i g h e s t e f f i c i e n c y can be genera-
t e d . From Los and Kistemaker (1958) i t i s known t h a t
gives the highest E equal to 1 . Of course, Heaviside's unit function H
is smoothed out to thickness L ~ ~ ~ E ~ ~ ~ in the inner Stewartson layer, but rn rn
since L << E i l the net result for Ff is almost unef fected. rn
Another important profile in present day gas centrifuges is known as the
two-shell profile:
In this case the axial mass flow (eYXw) consists of two Dirac pulses, the
inner one at x = b and the outer one near the cylinderwall and rechanne-
ling the axial flow. The efficiency is now given by
which has a maximum equal to 0 .814 for b = 0 .7A . In other words, the most
optimal position of the inner shell is r = 0 . 5 5 . For b < 0.7A E decrea- f ses with decreasing b. Of course the inner and outer pulse are smoothed
out to thickness L ~ ~ ~ E ~ ' ~ by respectively a free vertical inner Stewartson rn rn
layer at x = b and an inner Stewartson layer at x = 0 .
(ii) << LmEm << I . The inner core is located between the rotation
axis and the cylinder wall. For x << I n ( L E ) - I the most optimal radial rn rn shape for $eeX can be generated. On the other hand, for x >> h ( L r n E r n ) - '
$eWX approaches zero which means that the pure diffl~sive region does not
contribute to E ' If the inner core is located at r 2 0 .55 an optimum E f' f is given by the two shell profile. The minimum radial position of the
inner shell, thus yielding the highest Ef, is now the location of the
inner core. The corresponding Ef is obtained by putting
into (3.5.16). Since the distribution of I) in the inner core is not sol-
ved the value of the constant number b l is unknown.
(iii) L E,' 2 1. In this case one can generate an almost constant + along rn rn z whose radial shape is equal to that of the first eigenfunction:
I - e-x-xe-x. The corresponding Ff is equal to
An es t lma t lon f o r t h e con t r ibu t ion of t h e O(A ") terms i n (3.5.18) cannot
be given s i n c e t h e 0(AV1) s o l u t i o n s f o r I/J a r e not presented. On t h e o t h e r
hand, Parker & Mayo (1963) ca lcu la ted numerical ly , E f o r t h e f i r s t eigen- f
func t ion and obta ined: E = 0.67 f o r A = 9 (asymptotic E = 0.80) , f f
E = 0.42 f o r A = 16 (asymptotic E = 0.45) and E = 0.29 f o r A = 25 f f f (asymptotic E = 0.29) . One s e e s t h a t f o r A L 16 t h e r e l a t i v e dev ia t ion
f of E given by (3.5.18) i s a l r eady l e s s than 7%.
f
I n t h e range ( i ) E can be a maximum dependent upon t h e r a d i a l shape f
of Fb. Due t o t h e appearance of t h e inne r co re , i n t h e range ( i i ) E de- f
c reases wi th inc reas ing L E and A . I n t h e range ( i i i ) Ef i s a minimum m m and p ropor t iona l t o A - I .
I n t h e case of t h e two- shell p r o f i l e a continuous d e s c r i p t i o n of E, f o r
t h e t h r e e parameter ranges can be approximated a s fol lows:
( i i )
( i i i ) l 2 ~ L ~ E ~ ( I + & B ~ ) - - ~ ' ~ , b = 1815
where
b E = - 2b2/A22n (I- ;i) f
Although Em i s smal l , most p resen t day c e n t r i f u g e s w i l l f a l l
ranges ( i i ) and ( i i i ) , which i s due t o t h e l a r g e value of e0 '7A and L m . As a r e s u l t , e f f i c i e n c i e s considerably lower than t h e maximum one obta ined
(3.5.22)
i n t o t h e
i n t h e range ( i ) , E = 0.814, can be expected. f
The dependence of 6 U on t h e magnitude of t h e flow occurs through m. max Inspec t ion of (3.5.4) shows t h a t 6 U reaches i t s l a r g e s t value a s m -+
max However, t he l i m i t i n g va lue i s approached very r ap id ly : when m = 3 ,
m2/(l+m2) = 0.9 and when m = 5 , m2/(l+rn2 = 0.96. Therefore i n a c t u a l coun-
t e r c u r r e n t gas c e n t r i f u g e s m i s of u n i t magnitude and from (3.5.8) one can
t e s t t h e v a l i d i t y of a l i n e a r t reatment . For L E 5 1 t h e opt imal flow i s m m generated a t t h e end caps. I n a i n v i s c i d region one r e q u i r e s t h a t $ ex.
I n t h e pure d i f f u s i v e region $ e-X approaches zero. As a r e s u l t , t h e maxi- - 1 mum ~ e r t u r b a t i o n of yl occurs i n t h e inne r core a t x % ~ E ( L ~ E ~ ) . In o rde r
t o induce a I) eqX t h a t i s constant a long x i n t h e i n v i s c i d r eg ion , it i s
necessary t h a t F % LA12E112e'x: s e e (3.5.9). Here we sca led F such t h a t m b
a t x % In(L E I-', where t h e pe r tu rbed $ i s a maximum, Fb n 1 . Taking m m
rn % 1 one f i n d s from (3.5 .8)
For t h e working f l u i d i n a gas c e n t r i f u g e (YF6): p D / u % 1 . Since L and A m a r e l a r g e and s i n c e z ~ ( L ~ E ~ ) - * v a r i e s i n magnitude from A ( t h e inne r core
near t h e r o t a t i o n a x i s ) t o 1 ( t h e inne r core nea r t h e cy l inde r w a l l ) E i s
small and a l i n e a r t reatment seems reasonable . O f course , a more concrete
opinion concerning t h e o v e r a l l v a l i d i t y of t h e l i n e a r i s a t i o n can be formed
by consider ing the magnitude of the non- l inear terms i n t h e va r ious flow
regions . For ins t ance , i t i s well-known t h a t i n t h e Stewartson l a y e r s a
s t ronger l i n e a r i s a t i o n c r i t e r i o n a p p l i e s , due t o t h e f i r s t o rde r der iva-
t i v e s wi th r e spec t t o r i n t h e non- l inear convective terms. Never theless ,
one hopes t h a t a l i n e a r approach i s s t i l l app l i cab le f o r an Ef cons ide ra t ion .
For L E >> 1 a cons tan t a x i a l flow i s induced by a l i n e a r temperature m m g rad ien t a long t h e cy l inde r wa l l . Taking (dldz)Ru I. 1 and m % I , and apply-
ing (3.5.13) one f i n d s from (3.5.8)
Although E ~ I A " ~ i s sma l l , E can become of
LmEm can reach considerable magnitudes and
u n i t magnitude o r l a r g e r , s i n c e
t h e A?' necessary t o d r i v e t h e
secondary flow becomes t o l a r g e f o r a l i n e a r approach.
Due t o t h e r a p i d decrease of t h e d e n s i t y wi th d i s t ance from the cy l in-
der w a l l , t h e dens i ty nea r t h e r o t a t i o n a x i s can be q u i t e low with corre-
spondlngly long mean f r e e pa ths . Thus, i n many p r a c t i c a l c e n t r i f u g e app l i-
c a t i o n s , t h e flow a t t h e c e n t r e of t h e cy l inde r w i l l no t be continuum flow.
An opinion about t h e a p p l i c a b i l i t y of continuum flow theory can be formed
by determining t h e region where t h e Knudsen number K n i s , o f u n i t magnitude.
Here, K n i s t h e r a t i o of t h e mean f r e e pa th A t o t h e c h a r a c t e r i s t i c l eng th
s c a l e of t h e gas flow. The mean f r e e pa th i s given by
Near t h e r o t a t i o n a x i s two reg ions can be d i s t ingu i shed ( see f i g u r e s 5
and 6 ) . These a re : t h e pure d i f f u s i v e region a t t h e i n n e r s i d e of t h e
inne r co re and t h e Ekman l a y e r s of th i ckness L - I near t h e end caps. The m c h a r a c t e r i s t i c r a d i a l l eng th s c a l e i n t h e pure d i f f u s i v e region i s t h a t
of t h e d e n s i t y v a r i a t i o n : a/A. The c h a r a c t e r i s t i c a x i a l l e n g t h s c a l e i n
t h e Ekman l a y e r s i s L~'Z, which i s equal t o a/A. As a r e s u l t , f o r both
regions t h e Knudsen number can be approximated by
One s e e s t h a t r a r e f i e d gas flow occurs a t x 2 z~(E~/A~'~)-'. For t h e
u n i t , t h e semi-long and t h e long cy l inde r t h i s region appears wi th in t h e
pure d i f f u s i v e r eg ion , a region t h a t does no t c o n t r i b u t e t o t h e separa-
t i o n . I n o t h e r words, r a r e f i e d cons ide ra t ions a r e not needed.
LIST OF FREQUENTLY USED SYMBOLS
speed parameter Q 2 a 2 ~ / 2 ~ O ~ 0
radius of the cylinder
Brinkman number pQ2a2/~~0
specific heat at constant pressure
diffusion constant
Ekman number v/Qa2, u/Qa2pw
flow pattern efficiency
modified Ekman number E A ~
combination of the imposed boundary conditions at the bottom cap
combination of the imposed boundary conditions at the top cap
Heaviside' s unit function
unit vector along the rotation axis
aspect ratio, length-to-radius ratio
modified aspect ratio AL
length of the cylinder
Mach number of the primary rotation
molecular weight
difference between molecular weights
flow parameter
pressure
equilibrium pressure at isothermal rigid body rotation -+
pressure at r = 0
wall pressure
pressure perturbation
particle velocity in the rotating frame
universal gas constant
position vector in the rotating frame
radial coordinate
temperature
uniform constant temperature
temperature along the cylinder wall
typical particle velocity in the rotating frame
radial velocity
perturbed azimuthal velocity
axial velocity
imposed axial velocity at the bottom cap
imposed axial velocity at the top cap
radial eigenfunction
adjoint radial eigenfunction
radial density coordinate
stretched axial boundary layer coordinate
axial coordinate
square of the distance between inner core and rotation axis
square of the distance between outer core and rotation axis
bulk viscosity
Gamma function
ratio of specific heats at constant pressure and volume respectively
6Umax maximum separative power
E Rossby number U/Qa
dimensionless parameter scaling the pressure perturbation
E~ dimensionless parameter scaling the density perturbation
E~ dimensionless parameter scaling the temperature perturbation
C stretched radial boundary layer coordinate
11 contracted radial core coordinate
0 temperature perturbation
Bb temperature perturbation along the bottom cap
Bt temperature perturbation along the top cap
OW temperature perturbation along the cylinder wall
p dynamic viscosity
v kinematic viscosity
K heat conductivity
hk eigenvalue
P density
P, equilibrium density at isothermal rigid body rotation
pw density at the cylinder wall
a Prandtl number u c /K P P
uerislrry perturbation
8 + i r 2 ~ r w
u - i e streamfunction
imposed streamfunction at the bottom cap
imposed streamfunction at the top cap
angular speed of the cylinder
angular speed perturbation
imposed angular speed perturbation at the bottom cap
imposed angular speed perturbation at the top cap
W EFERENCES
Abramowitz, M. and Stegun, I . A . Handbook of Mathematical Functions,
Dover P u b l i c a t i o n s New York (1968)
Baker, D.J. J . Fluid Mech. 2, 165 (1967)
Ba rc i l on , V. and Pedlosky, J . J. Fluid Mech. g, 1 (1967a).
Ba rc i l on , V . and Pedlosky, J . J. Fluid Mech. 29, 609 (1967b).
B a r c i l o n , V. and Pedlosky, J . J. FZuQd Mech. 29, 673 ( 1 9 6 7 ~ ) .
Berman, A.S. A Simplified Model for the Axial Flow i n a Long
Countercurrent Gas Centrifuge, Union Carbide Nuclear Company,
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Summary
The secondary flow of an incompressible fluid and of a perfect gas in
a rotating cylinder is considered, by applying a linearised analysis to a
small perturbation on the isothermal state of rigid body rotation.
In chapter 2 we apply an order of magnitude consideration to the line-
rised dimensionless conservation equations. In section 2.1 the incompres-
sible fluid is treated assuming a small Ekman number E based on the radius
of the cylinder. The importance of the various viscous terms is investiga-
ted increasing the length-to-radius ratio L from unit magnitude to infini-
ty. As a result, three types of flow can be distinguished which correspond
to the L-ranges: E ~ / ~ << L << E-"~, E-''~ % L << E-' and E-I 5 L. In the
first L-range we identify a flow which underlies a balance between the
Coriolis forces and the pressure gradients. This inviscid flow is adjusted
to the boundaries by Ekman layers near the end caps and two Stewartson
layers near the cylinder wall. In fact, this flow type is well-known from
the literature and is of importance to models of oceanic and atmospheric
currents. In the second L-range the outer Stewartson layer expands to the
interior which means that radial diffusion of azimuthal momentum becomes
an important parameter describing the flow in the main section of the
cylinder. In the third L-range also, the inner Stewartson layer fills the
entire cylinder. Radial diffusion of azimuthal and axial momentum are both
important, a situation that characterises flows in semi-infinite cylinders.
In section 2.2 the perfect gas is considered, special attention being
focussed on rapidly rotating heavy gases. In this case the density decrea-
ses strongly with the distance from the cylinder wall, a situation that
occurs in present day gas centrifuges. The modified Ekman number Em, based
on the density at the cylinder wall and on the density scale height
LIIL. la^^^^ J L V L a k c L i LU ut: bluall auu m e DrlnKman numDer DP Is
assumed to be of unit magnitude. The secondary flow is investigated for the
case that the ratio of the length to the radial density scale height Lm
increases from unit magnitude to infinity. Again three types of flow can
be distinguished which correspond to the L -ranges g1I2 << Lrn << E;''~, m m E-"~ << and E-' m Q L m rn rn 2 Lrn. However, compared to the incompressible fluid two essential differences are found.
(i) An inviscid flow characteristic for the first range is only observed
in a limited region near the cylinder wall. Due to the strong density de-
crease diffusive processes are important in the core of the cylinder.
(ii) A change of the flow type appears when both Stewartson layers succes-
sively expand over the small radial density scale height. Simultaneously
corresponding diffusive regions come up from the centre of the cylinder
and join. As a result, a change of the flow type appears at relatively
small values of the length-to-radius ratio. In contrast to the incompres-
sible fluid where the first type is of most practical importance, here all
three types are of strong interest.
In chapter 3 explicit solutions for the flow in the various regions
of the rapidly rotating heavy gas are presented. Here we apply perturba-
tion techniques and the method of matched asymptotic expansions. The im-
posed boundary conditions are: temperature differences along the end caps
and along the cylinder wall, the differential rotation of the end caps and
axial injection and removal of fluid at the end caps.
Sarnenvatting
De secundaire stroming van een onsamendrukbare v l o e i s t o f en van een
i d e a a l gas i n een ro te rende cy l inde r wordt beschouwd, waarbi j de Navier-
Stokes ve rge l i jk ingen z i j n g e l i n e a r i s e e r d t en opzichte van isotherme s t a r r e
r o t a t i e .
I n hoofdstuk 2 wordt een orde- groot te beschouwing toegepast op de ge-
l i n e a r i s e e r d e dimensieloze behoudswetten. I n paragraaf 2.1. behandelen we
de onsamendrukbare v l o e i s t o f en nemen aan d a t h e t Ekman g e t a l E, gebaseerd
op de s t r a a l van de cy l inde r , k l e i n i s . Vervolgens onderzoeken we de be-
l a n g r i j k h e i d van de ve r sch i l l ende viskeuze termen wanneer de l e n g t e- s t r a a l
verhouding van de cy l inde r L toeneemt van eenheidsgroot te t o t oneindig.
Het b l i j k t dan d a t d r i e typen stromingen onderscheiden kunnen worden welke
overeenkomen met de L-gebieden l?'I2 . < L .< E-I 12, % L E-' en E- 1 s L. In h e t e e r s t e gebied vinden we een stroming waaraan een evenwicht
tus sen de C o r i o l i s krachten en de drukgradienten t e n grondslag l i g t . Deze
nie t -viskeuze stroming wordt aangepast aan de randen van de cy l inde r door
Ekman l a g e n b i j de e inddeksels en twee Stewartson lagen langs de cy l inde r
wand. D i t type stroming i s e r g bekend u i t de l i t e r a t u u r en i s van belang
voor modellen van stromingen i n de at.mosfeer en i n de oceaan. In h e t twee-
de L-gebied expandeert de b u i t e n s t e Stewartson l aag naa r binnen hetgeen
betekent d a t r a d i a l e d i f f u s i e van azimuthale impuls b e l a n g r i j k wordt voor
de beschr i jv ing van de stroming i n h e t hoofdgedeel te van de cy l inde r . In
h e t derde L-gebied v u l t ook de b innens te Stewartson l a a g de h e l e cy l inde r .
Radiale d i f f u s i e van azimuthale en a x i a l e impuls z i j n be ide b e l a n g r i j k ,
een s i t u a t i e d i e stromingen i n half -oneindige cy l inde r s k a r a k t e r i s e e r t .
I n paragraaf 2.2. behandelen we h e t i d e a l e gas waarbi j we vooral onze
aandacht r i c h t e n op s n e l roterende zware gassen. Hie rb i j neemt de d ich t -
he id s t e r k exponent ieel af met a f s t and van de cylinderwand, een s i t u a t i e
welue voorlcomt ln de huldlge gas centr.~tuges. We nemen aan dat het gemo-
dificeerde Ekman getal Em, dat gebaseerd is op de dichtheid aan de wand en
op de radiale dichtheidsschaal (in plaats van de straal) klein is en dat
het Brinkman getal B r van eenheidsgrootte is. We onderzoeken de secundaire
stroming voor het geval dat de verhouding van de cylinderlengte tot de ra-
diale dichtheidsschaal L toeneemt van eenheidsgrootte tot oneindig. m We nemen weer drie typen stromingen waar die overeenkomen met de L -gebie-
I * den E:l2 << Lm << E-'I2, << E-' en E-'
% L m m m % Lm. Echter, vergeleken m m
met de onsamendrukbare vloeistof vinden we twee essentiele verschillen.
(i) Een niet-viskeuze hoofdstroming karakteristiek voor het eerste L - m gebied treedt alleen op in een gebied van beperkte dikte langs de cylinder-
wand. Tengevolge van de sterke dichtheids-afname zijn diffusie processen
belangrijk in het hart van de cylinder.
(ii) Een verandering van het type stro~ning vindt plaats wanneer beide
Stewartson lagen achtereenvolgens over de radiale dichtheidsschaal expan-
deren. Tegelijkertijd komen overeenkomstige gebieden uit het centrum van de
cylinder op. Hieruit voortvloeiend zal een verandering van het type stro-
ming optreden bij relatief kleine waarden van de lengte-straal verhouding.
In tegenstelling tot de onsamendrukbar~e vloeistof waar het eerste type van
het meeste praktische belang is, zijn nu alle drie de typen belangrijk. I
In hoofdstuk 3 presenteren we expliciete oplossingen voor de stromingen
in de verschillende gebieden van het snel roterende zware gas. , Hierbij maken we gebruik van een mathematisch formeel proces, de z.g. sin-
guliere storingsrekening en "the method of matched asymptotic expansions".
De opgelegde randvoorwaarden die de secundaire stroming veroorzakkn zi jn:
temperatuurverschillen over
schillende rotatiesnelheden
en extractie van gas bij de
de einddeksels en langs de cylinderwand, ver-
van onder- en bovendeksel en axiale inspuiting
deksels.