243
Flow in a Narrow Gap Along an Enclosed
Rotating Disk with Through-Flow*
Junichi KUROKAWA** and Masato SAKUMA**
Flow in a narrow gap along an enclosed rotating disk superimposed with through-
i
flow is studied theoretically and experimentally. When the axial gap is narrow, or a
large outward through-flow is imposed, the boundary layers on the rotating and the
stationary walls interfere with each other. The present study proposes an analytical
model for such interference of gap flow and gives a theoretical analysis which is easily
applicable to various boundary conditions. For non-interference of gap fiow, a theory
giving better results than conventional theories is also presented and the critical gap
between these two flow patterns is determined theoretically. It is shown that the radial
pressure distribution is dependent mainly upon wall shear stress and radial expansion
of the sectional area in the case of gap interference, and on the centrifugal force of the
core flow in the case of gap non-interference. The coefficient of the disk friction
moment is shown to be proportional to Re 1/4 in the case of gap interference, and to
be Re 1/5 in the case of gap non-interference.
Key Words : Boundary Layer, Enclosed Rotating Disk, Through-flow, Narrow Gap,
Disk Friction Moment, Pressure Distribution, Critical Gap
1. Introduction
and the flow model adopted in the theory is no longer
applicable to real flow conditions.
i
In the previous report(1), one of the present
The analyses of disk flow in a narrow gap so far
authors determined the flow characteristics in a gap
reported are the laminar flow analysis(4) and the turbu-
along an enclosed rotating disk superimposed with
through-flow, theoretically and experimentally, and
established that the distributions of pressure and
velocity in the gap are influenced siguificantly by
lent flow analysis(5) based on the finite difference
radial through-flow. The theoretical method was further applied to radial flow turbomachinery, and was
shown to give good predictions of axial thrust, Ieakage loss and disk friction torque with sufiicient accuracy(2)
However, with a decrease in the width of an axial
gap (narrow gap) or with an increase in the volume of
method of the equations of motion in the case of
radially outward through-flow ; however, the case of
inward through-flow has not been treated yet. In
addition, these theoretical studies do not properly
consider the boundary conditions which give the initial value of the calculation, so that it is still difflcult
to apply these methods to actual turbomachinery. In
turbomachinery there are various cases of both the
direction of leakage and the boundary conditions of
narrow gap flow, and accordingly, a theoretical
radially outward through-flow, an agreement of the
calculated results with the measured ones diminishes
and in some special cases the numerical calculation
method easily applicable to various boundary condi-
diverges(3). In such cases, the flow pattern must change
narrow gap and determines a theoretical method
*
**
Received 22nd September, 1987. Paper No. 86-0942 A
Faculty of Engineering, Yokohama National Univer-
sity, 156 Tokiwadai, Hodogaya, Yokohama, 240,
.
apan
JSME Ir terr ati01 al Jourual
tions is still in strong demand.
The present study proposes a new flow model in a
easily applicable to various boundary conditions and
directions of through-flow.
2. Theoretical Analysis
The flow model shown in Fig.1 has been widely
Series II, Vol. 31, No.
2, 1988
244
adopted for the analysis of an enclosed rotating disk
flow(1)(6). It is supposed that there exists a core region
in which the radial velocity v is equal to O, and the
peripheral velocity is constant in the axial direction
between the boundary layers on the rotating disk and
the stationary wall. A more detailed flow model has
also been proposed(7). However, with a decrease in the
denotes kinematic viscosity.
2. I Theory of the interference gap fiow
In the interference gap, the boundary layers on
both walls interfere and no longer grow in the radial
direction, while the radial velocity changes largely
with r. A Iarge difference between interference and
non-interference gap lies in the fact that the pressure
axial gap or with an increase in the through-flow rate,
in interference gap is largely influenced by wall shear
the core region diminishes and the boundary layers on
stress and radial expansion of the gap sectional area,
both walls interfere with each other. Such a narrow
gap flow is encountered, for example, in axial thrust
mainly by the centrifugal force of the core.
balancing equipment or in the cooling mechanisms of
gas turbine rotors. In the following chapter, we call
such a narrow gap the 'interference gap', and a wide
gap with a core reglon the 'non-interference gap'.
The present study treats mainly the interference
gap flow, and also re-examines the conventional theory of the non-interference gap flow in order to obtain
a general method of analyzing gap flow.
The equations of momentum and the continuity
equation in the cylindrical coordinates r and z are
expressed as follows in the axially symmetrical flow
field showri in Figs. I and 2 :
aar {Jfr r2uvdz} p2
r ( rR. HL rs,)
(1)
r
}- f'u2dz=
a {f
r 'v2dz
aps-Ji(zl
.+ r*.)
(2)
2lcr fo 'vdz= Q ( 3 )
where u , v and p denote the velocity components in
the tangential and radial directions and the pressure,
respectively, and p is fiuid density. Q is the through-
flow rate and is positive for the radially outward
direction and negative for the inward one. Subscripts
r and 6 show the radial and tangential components,
respectively. The wall shear stress rR on the rotating
disk, and z:s On the stationary wall are expressed as
follows, referring to Daily and Nece(8).
TR = cp[ U2rel( v/U.etz') l/4].,_o,
while that in non-interference gap is determined
Now we consider the flow model in the interference gap as follows :
( a ) the boundary layer thickness on both walls is
the same and is equal to half of the gap width.
( b ) radial velocity v is expressed as a linear
combination of through-flow component vl and the
circulatory flow component which is induced even in
the case of zero through-flow.
The velocity distribution in the gap is assumed as
follows (Fig. 2) using the tangential velocity Kra; at
the gap centre :
velocity in the disk boundary layer [O
where c'=0.025 6(1), Urel and U are the fluid velocities
E
u'=rco{1-(1-K) l/7}, v'={vl+v2(1- } ln
(5)
velocity in the stationary wall boundary layer
[O V 1,
Ez/(s/2)] :
u=Kra) 1/7, v={vl v2(1- )}V1/7 ( 6 )
Here, vl is determined from the continuity equation
( 3 ) and v2 can be expressed in the same manner as in
the non-interference gap case(1) .
vl=4Q/7lrrs, v2=arw ( 7 )
Introducing Eqs. ( 4 ) -( 7 ) into Eqs. ( I ) -( 3 )
leads to the ordinary differential equations for the
tangential velocity ratio K and the non-dimensional
pressure P as a function of the radius ratio R :
8 C; RdK_ 1+16K C
T R2S dR
9 R2S
( dR)+c(
e RSida
)1/4[(1
1449
40 4a+R
5 _ K)
T*= cp[ U2( //Uz)1/4]._o ( 4 )
ielative to the rotating and the stationary wall, and Ll
1,
z'/(s/2)] :
x{(1_ K)2+Fj
K(K2+F2)3/8]
( 8 )
* 3/8
_ s
;!a(3a+2R d drRa)
1 dP _ 56K2+7K+1
36
Cas i ng va I l
Control volume
Rotating disk
Fig.
1 Flow model of non-interference gap
Series II. Vol. 31, No. 2, 1988
45
Rotating disk
Fig. 2 Flow model of interference gap
JSME 11 tematior al Journal
)2
245
+16238(RC 2c- (2R35)ll4
lowing boundary equation :
ReS
1+16K2C .+ 49
a2S+ C
18 q 1440
x[F {(1 K) +F2}3/8+F(K +F2)3/8] ( 9 )
-cK2( 2 )1/4(K; +F.2,)3/8(E+S)=0 (15)
where
ReS
FR=8C /7R2S+a, Fs=8C;/7R2S-a (10)
p 2p/pr22w2. Re = r22col /, C sQl2lcr23a,,
R
r/r2, S :s/r2 (11)
where the subscript 2 indicates the value at r = r2 and
E = e/r: . C
is the angular momentum coefficient of
and r2 and (o denote the disk radius and the angular
through-flow and is given as follows using the pre-
velocity.
rotational velocity K.r2(v of the through-flow :
The parameter a in Eq. ( 7 ) represents the circulatory flow rate and is closely related to the limiting
streamline angle in the vicinity of the wall. The tan-
C
E AM/2 frpr25(v2= - K.C ( C < O) ( 16)
In the case of outward through-fiow, the calculated results have shown that the radial pressure
gent of the limiting streamline angle a on the station-
distribution changes little, even if the inner boundary
ary wall can be calculated from Eq.( 6 ) , as K 0.5
value is varied to some extent, so that it is enough to
over the whole radii in a narrow gap with zero
give a small boundary value at the inner edge, such as
through-flow :
K=0.1 at R =0.1 and to advance the numerical calculation from the inner edge to the outer radii. This
a =.-oIim( - v/u).= cv/K= 2a ( 12)
According to the flow visualization measurements
shown later, a is expressed as a function of the local
In the case of zero through-flow ( C =0), Eq. ( 8 )
a=ao(ReR2/105+2)"', a0=0.064 6S- ,
o . 693
0.028 5S-0.693
influenced by the centrifugal force in the interference
ga p.
Reynolds number Rel= r2a,/v = ReR2 :
al =
implies that the pressure distribution is hardly
(13)
2. 2 Numerical calculation of the interference
gap flow
Equations ( 8 ) and ( 9 ) can be calculated numeri-
cally, if the gap ratio S and the through-flow rate
coefiicient Cq' are given and the boundary values K2
and P2 are properly determined. In turbomachines
there are various boundary conditions of configuration
and through-fiow direction, so that it is important to
establish a general method of determining the boundary values.
In the theoretical studies thus far performed(4)(5), a
very small but non-zero value is given to the boundary
value at the inner boundary for the outward throughflow case, but this method is not adequate for the
inward through-flow case because the boundary value
at the inner boundary becomes very large. In order to
determine the boundary value correctly, it is neces-
sary to consider the angular momentum balance at the
boundary at which through-flow comes into the gap.
In the case of radial inward through-flow, the
f 2
angular momentum balance in the control volume
is not needed.
The boundary value of pressure is, in general,
dependent upon the operation conditions of turbomachines, and here the differential pressure coef :cient is
defined as follows :
Cp = P - P2= 2( p - p2) Ipr2za,2
o ..
+ (r2+
e)2 IC'r.,dzf + AM (14)
J,
) p
(17)
2. 3 Theory of the non-interference gap flow
For the case of a relatively large gap between a
rotating disk and a stationary wall, the theory developed in the previous report(1) is applicable. However,
the application of this theory for the prediction of
axial thrust of many types of actual turbomachines
has revealed that the prediction becomes less satisfac-
tory in the case of a large outward through-fiow rate.
This is mainly due to the change in flow pattern from
the non-interference of gap flow to the interference,
and also to the accuracy of the velocity assumption.
Here, we improve upon the previous theory by
introducing more accurate assumptions in regard to
the velocity profile and boundary layer thickness. The
radial velocity assumption in the boundary layers on
the disk and the stationary wall is
v'=a*(1-K)r(v(1-
) In, v=-aKra,(1-V)Vl/7
( 18)
p
shown in Fig. 2 can be expressed as
[27rr2f uvdzJ = 27r f.."+'r r.,dr
becomes an algebraic one and the boundary equation
where
=z'/ ,
= z/
and
and 8 denote the bound-
ary layer thickness on the rotating and the stationary
where rb denotes shear stress on the outer cylindrical
wall, respectively, and are expressed as follows, refer-
surface, AM denotes the angular momentum which
ring to Daily et al.(8).
through-flow takes into the fiow field at the outer
boundary and (; is the radial gap around the disk tip.
Introducing the velocity and wall shear stress assumptions of Eqs.( 4 ) ;
( 7 ) into Eq. (14) Ieads to the fol-
JSME International Journal
8=b(1-K) r/(a;r2/2/)1/5, 8=fr/(a,r2/v)l/5
b=0.54, m=2.5 (19)
Here, f is determined from the equation of continuity
( 3) as
Series II, Vol. 31, No. 2 1988
/ ,
246
f={a*b(1-K)"+1 120 Cq aK
- 49 R13/5
3. Experimental Apparatus and Procedure
Cq (Q/27rr23a')Rel/5 (20)
The measured results shown later have revealed
that the velocity factors a and a* in Eq. (18) change
largely with the local Reynolds number Rel' and the
following empirical equations are obtained :
a = I .03(ReR2/ 105 + 2) -0.387
gap at the disk tip is e=0.5 mm. The axial gap
between the disk and the stationary wall can be varied
from O to 8.5 mm. The rotational speed of the disk is
a* = I .18(ReR2/ 105 + 2) -0.490
(21)
For the tangential velocity, the same profiles are
assumed as in Eqs. ( 5 ) and ( 6 ) using the non-dimen-
sional distances
The rotating disk apparatus enclosed in a cylindrical casing is the same as that used in the previous
study(rs). The disk radius is r:z=150 mm and the radial
and v.
n= 750 rpm, corresponding to the Reynolds number of
Re = r22w/1/ = 1.73 x 106. The setting accuracy of the
axial gap is 0.1 mm.
The measurement of velocity in a narrow gap is
difficult, so only the radial pressure distributions were
measured in order to confirm the validity of the the-
{- -
}
Introducing these equations into Eq. ( I ) results in
the following ordinary differential equation for the
velocity ratio K of the core as a function of nondimensional radius R :
(m+1)a*bll-Kl
56 Rrs/5
C 49
720
=T
-
R dK
dR
5 Cql51127a*b(1-K)"+1
R13
K- 3 600
4290 b(1 -K)"+1R ddaR*
ory. The axial thrust coefficient CT and the disk
friction torque coefficient C
defined as follows are
also used to compare the theory with the measured
results.
CT=f3CpRdR, C =2M/pr25w2
(26)
The momentum theory requires a velocity
{ (1 + a*2)3/8 (7_ )/4
+ c (1 - K)
b u4
f
K3 1/4
(1+a2)3/8K(
) } (22)
assumption, so that it is important to determine the
velocity factors a and a* in Eqs. (12) and (18) as
correctly as possible. The flow visualization technique
using an oil-film method was used to obtain these
In the above equation, the signs of the terms (1
factors, as they are closely related to the tangent of
K) +1 and (1 K)(7- )/4 are to be IF, depending upon
the limiting streamline in the vicinity of the wall in the
K<> 1. In the large outward through-flow case, the
calculation gives a negative K-value in the inner
radii, so that the K-value should be treated as O if it
case of zero through-flow.
4. Results and Discussions
becomes negative.
The pressure distribution is to be determined
4. I Strealnline visualization measurements
One of the big problems associated with the oil-
from Eq. ( 2 ) , but in the non-interference gap it can be
film method lies in the fact that the limiting stream-
simply calculated by means of the following equation,
as the radial pressure gradient mainly balances with
line indicated by the oil-film streak does not give the
the centrifugal force of the core fluid.
correct streamline on a rotating wall because of the
density difference between the fiuid and the oil-film.
The boundary value equation is obtained from Eq.
(14) in the same manner as that in the interference
Here, we propose a method for correcting the limiting
streamline by using a known velocity profile under the
assumption of uniform oil-film density and thickness.
gap case :
74290 a* b(1 - K2)"+1 + CqK2 + C.
the tangent of the correct fiow angle y and that of the
dP/dR = 2RK2 (23)
( 2) 2-
-c(1+E)2(S+E)(a2+1)3/8 K_f2 3 l/4J ; -O (24)
In the non-interference gap, the difference between
measured oil-film streak angle 7 is nearly equal to
AF.lFeoc Cf/roc(wr2/v)1/5/r, as shown in Fig. 3, where
AF. and Fe are the difference of centrifugal force and
where Can is the inlet angular momenturn coefficient
wall shear force, respectively, and Cf is friction
given by :
coefficient. The proportionality constant can be determined from a known velocity profile(1), and finally the
C
= (AM/ 2 ;Tpr25 a'2)Re 1/5
=
-(5K2+1)Cq/6 (Cq >0)
In the non-interference gap case, the basic equa-
AFC=FC( oi I )-Fc (vater)
tions (22) and (23) can be numerically calculated
from the outer edge to the inner radii using the bound-
ary value K2 given from Eq. (24) at the outer edge for
both the inward and the outward through-flow cases.
Sene 11 Vol. 31, No. 2, 1988
Flg. 3 Correction of the limiting streamline on the rotating wall
JSME Irtternatior al Journal
24 7
of S ;
correct angle is given by
tan r=tan 7 - 0.006 25(Re/R3)1/5
In the interference gap case, the correction is
unnecessary, as the present analysis assumes the same
velocity profiles in both boundary layers for the case
of zero through-flow, that is, a=a*.
4. 2 Radial pressure distribution
The radial pressure distributions in three
different gaps are shown in Figs. 6 ( a ) - ( c ) for the
case of ihward through-flow, and in Figs. 7 ( a ) - ( c )
for the case of outward through-flow. Figure ( a )
The measured and corrected results of a and a*
are shown in Figs. 4 (in the case of S
0.013 to the interference gap. The empirical
equations of a and a* are given in Eqs. (13) and (21) .
(27)
shows the case of a relatively large gap, Fig. ( b ) an
0.026) and 5 (S
0.013)for the varlation of the local Reynolds number
Rel' The values are seen to change largely with S in
the range of S 0.013, but little in the range of S;
0.026, which indicates that the range of S 0.026
corresponds to the non- interference gap and the range
intermediate gap and Fig. ( c ) a narrow gap. The
theoretical results are also compared.
Radially inward through-flow takes larger angular momentum of the outer radii to the inner radii and
considerably increases the tangential velocity in the
inner radii, so that the pressure decreases siguificantly
toward the disk centre. On the contrary, the outward
through-flow decreases the tangential velocity of the
whole region and the pressure distribution becomes
relatively uniform over the whole disk, as seen in the
1 .O
,,
O
0.5
figures.
A comparison of the theory with the measurements reveals that the theory of non-interference gap
O. 1
1 Os I oe Ret l07
( a ) on the rotating wall, a*=tan rR
1.0
o.
5
1.0
o
o
O. 5
o. 3
0.1
0.05
0.1
1 Os
1 Oc
Re[
1 07
0.02
1 Os I Oc Re 1 1 O
( b ) on the stationary wall, a=tan 7s
Fig. 4 Velocity factors of non-interference gap flow
R
-O . 1
-o . 2
-o . 2
-o. 3
-o. 3
-o . 4
-0.4
-O . 5
-O . 5
0.2 0.6
0.4
0.8 1.0
oo
0.2 0.4 0.6
8
oo
R
o
O 0.2 0.4 O 6 R
-o . l
Fig. 5 Velocity factor of interference gap flow
o.
1.0
-O . 1
-o . 2
-o . 3
-0.4
-O . S
a
ua
u
-0.6
-O . 6
-o. 7
-o. 7
-0.8
( a ) S=0.0556.
Fig. 6
JSME Ir terTtattonal Jourl al
a
u
-O . 6
-o . 7
-o . 8
( b ) S=0.0113 ( c )
S=0.0050
Radial pressure distribution in the inward through-flow case
Series II, Vol. 31, No 2 1988
248
R
R
o
0.4
0.6
0.8
1 .O
i ir -
O 6 R 0.8
1.0
l
1
l
ld ,
I
I
v v v
,: /
i
-o . 1
0.4
/
d
,_._lJ 'I /"
-o . l
0.2
r
I
･L.--I
o
o
o
o 0.2
/
-O. 1
L
Theory
Interf.
Noo-iot.
O
-0.2
-o . 2
,,
u
o.
y
L
o
OO
S - 0.0113
(a) S=0.0556
Cq
0146
･c
,
u
,,
u
0.0063
-o . 2
(c)
( b ) S=0.0113
Fig.
S=0.0050
7 Radial pressure distribution in the outward
through-flow case
shows good agreement with the measurements in the
60
case of a relatively large gap (Figs. 6 ( a ) and 7 ( a ) ) .
However, in the case of an intermediate gap, good
agreement with the non-interference gap theory is
only seen in the inward through-flow case (Fig.
6 ( b ) ) ; in the outward through-flow case, the theory
gap theory gives good agreement (Figs. 6 ( c ) and
x
N:L 40
¥
L
Q
CL
1
of the interference gap gives much better results (Fig.
7 ( b ) ) . In the case of a narrow gap, the interference
o
c
20
7 ( c ) ) except for the extremely large inward through-
flow case, for which the non-interference gap theory
.4
0.6
pressure distribution
0.2
(a)
8
o
ior is recoguized in the measured data and agreement
is not good. This is because of a deformation of the
chamber caused by too high a pressure in the case of
large inward through-fiow. Though the original gap
o
gives better agreement. In Fig. 7 ( c ) , irregular behav-
O.
･R
1,0
2'o
was set to 0.7mm over the disk, the gap width was
1.5 mm at the centre after the measurements.
For the case of a narrow gap with large outward
l:)
¥
:)
through-flow, Bayley et al.(5) performed theoretical
analyses and measurements, with which the present
1.0
theory is compared in Fig. 8 ( a ) . The present theory
3
is seen to give better results, though agreement is not
good in the inner radii because of the large diameter
L
¥s
suction pipe connected to the inner radii of their
apparatus .
From the above-mentioned, it is suggested that
for the case of large inward through-fiow, the noninterference gap theory should be applied even if the
gap is very narrow, and for the case of large outward
through-flow, the interference gap theory should be
05 z/s 1.0
( b ) velocity distribution
Fig. 8 Comparison of theory with the results by Bayley
et al.(5)
applied even if the gap is relatively large. This is also
Such remarkable influences of through-flow as
verified from the velocity data measured by Bayley et
described above are essentially due to the fact that the
al. shown in Fig. 8 ( b ) , in which the present theory is
inward through-flow is an accelerated- flow and the
also compared. The gap ratio of S = 0.03 is relatively
outward one a decelerated flow. Ito et al.(9) established
large, but the direction of radial velocity is seen to be
by experiment that the disk boundary layer is much
more stable than the stationary wall boundary layer
for the case of zero through-flow. If radial inward
outward in the whole section, so that the gap is
obviously the interference one.
Series II, Vol. 31, No. 2, 1988
JSME Ir tematiortal Journal
249
through-flow is superimposed on it, the stationary
Cq than the points shown as A V in Fig. lO, the non-
wall boundary layer becomes thinner due to a
interference gap theory fails to give solutions, so that
significant increase in the tangential velocity ; and in
this range is to be treated as the interference gap.
this thin layer, a large volume of through-flow comes
inward as an accelerated flow, which makes the stationary boundary layer very stable. Accordingly, in
the case of inward through-flow, both boundary layers
become stable, and being very thin, they realize noninterference gap flow. On the contrary, in the outward
4. 3 Disk friction torque
In this section, the theoretical resutls are compar-
ed with the measured data by others to confirm the
validity of the theory.
Figure 11 shows a comparison with the data
obtained by Yamada et al.(ro) and Bayley et al.(5) for
through-flow case, both boundary layers develop rapidly to generate interference gap, because the tangential velocity decreases in the whole region and the
the case of zero through-flow. The present theory is
seen to be in good agreement both in the interference
gap and the non-interference one. In the zero through-
through-fiow comes outward as decelerated fiow in
flow case it is also recognized that the C
the disk boundary layer.
The above-described pressure distribution is integrated to give the axial thrust coefiicient CT, defined
interference gap is proportional to Re 1/4, and that of
of the
in Eq.(26) , and its variation is shown in Fig. 9 for the
the non-interference gap to Re 1/5, which agrees with
the well-known results by Soo(11), Schultz-Grunow(12)
and Daily et al.(6) In the case of outward through-flow
variation of S. If the inward through-flow rate is not
too high, the variation of CT is very small and the non-
illustrated by the thick line, good agreement is also
seen. For comparison, the variation of C is shown in
interference gap theory gives good agreement over
the outward and the inward through-fiow cases.
the wide range of S, but over the range of small gap
ratio S, the interference gap theory is seen to give
better agreement.
Application of the present theory to actual tur-
24
Non-interference
e- 20
bomachines requires the boundary between the noninterference gap theory and the interference one.
9a p lyf'l :A" -' -'A-'
/ / !
;LL
,,
/
6
'l
Taking the intersection point of CT-curves of the noninterference and the interference gap as the boundary,
the solid line is obtained in Fig. 9. We call this point
/
L
u
v'
l.
/
12
1
O Re = 1.73xl06
l
l
the <'critical gap", for which variation is shown in Fig.
10 with the variation of Cq. The curves are different
for different Re numbers, and the range above this
5.0xl06
107
I
8 'T7'
'v
' _ "'/
4 ': _ r l--
f
A
V
ll
Interference
curve is to be treated as the non-interference gap, and
gap
below as the interference gap. In the range of larger
Cq . I = 12
Flg. 10 Critical gap ratio as a function of Cq
Thoory
h
u
Neasurements
O S=0.004 c t_O
E
cq ' =0 ' 005
u
O . 005
_ 'I-* I!
I ' -I,,.I l¥
:
o . 06
¥ ¥'
o . 002
'¥ '¥
¥ ,¥
¥
¥ ¥ ,¥
S=0 . 08
Cq I Re=1 590 t s )
'
=
j ,=
O . 04
Cq O
04
i
o . 04
_ .¥1 5c
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o . ool
Cq ' =-O . 005
o . 02
o . ooos
cl o)
e S=0.04 q -
0.01
O . 08
}
0.02
0.10
Il
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5xl,Os 10e 2xloe 5xl06 107 2xl07
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Sxl07
Re
002
o . 005 o . ol o . 02
o.05 s 0.1
Fig. 9 Axial thrust coefficient as a function of S
JSME htternational Jourr al
Fig. 11 Comparison of theory with measurements of disk
friction moment coefiicient C
Series II. Vol. 31, No.
2, 1988
250
0.01
E
u
O . 005
o . 002
0.001
O.
Fig. 12
002 o.005 o.ol o 02 o.05 s 0'1 o 2
Disk friction moment coeff. as a function of S
-2 -1 o 1
C xl03 2
Fig. 13 Disk friction moment coeff. as a function of Cq
that the experimental data vary smoothly from the
theoretical curve of a non-interference gap (shown by
solid lines) to that of an interference gap (dotted
is easily applicable to various boundary conditions.
For the case of non-interference gap flow, the theoret-
lines) with a decrease in S. This transition gap ratio
ical analysis was also presented, giving better results
agrees well with the critical gap in Fig. lO. These
than the conventional theories.
tendencies are similar for different Reynolds numbers,
( 2 ) The critical gap which gives the boundary
but as indicated in Fig. 12, the present theory gives the
between the interference and the non-interference gap
optimum gap ratio at which the disk friction torque
takes the minimum value, and this value is nearly
equal to the critical gap ratio. The reason why there
was clarified. It becomes very small in the case of
exists an optimum gap is as follows. The disk friction
torque of a non-interference gap is proportional to (1
disk fiow tends to be an interference gap flow. It was
case of outward through-flow. In the latter case, the
also shown that such fiow behavior is partly due to the
decreases in proportion to
fact that the outward through-flow is a decelerated
S-1/5 in the range of a very small gap because K is
nearly constant. In a larger S range, however, the C
value increases with S, because the decrease in K
fiow and the inward one an accelerated flow, and
partly because the tangential velocity change is
-
)/(r2a,s/v)1/5, so that C
radially inward through-fiow, but is very large in the
becomes larger than the S-1/5 effect.
Lastly, the influence of Cq' upon C is shown in
Fig. 13 for the case of a relatively wide gap (S=0.04)
significant, depending on the through-fiow direction.
( 3 ) The radial pressure distribution is dependent
mainly on the centrifugal force acting on the core fiuid
changes little at C <0, but
in the non-interference gap, while it is dependent
mainly on the radial expansion of the sectional area
and the wall shear stress in the interference gap.
increases largely with C at C > o, For the case of S
=0.04, both the interference (shown by solid line) and
( 4 ) The theory of interference gap shows that
there exists an optimum gap in which disk friction
the non-interference gap theory (dotted line) are
compared. As the disk flow tends to be interference
torque takes a minimum value, and that it is close to
the critical gap. It also shows that the disk friction
fiow in the outward through-flow case, the real phenomena is considered to move along the dotted line
torque coefiicient is proportional to Re 1/4 in the inter-
at C <0 and changes smoothly to the solid line at
gap flow.
( 5 ) In the flow visualization technique using oil
and a narrow gap (S=0.004). It is recoguized that in
the case of S=0.004, C
C
> o.
5. Conclusions
The flow along an enclosed rotating disk superimposed with through-flow was studied theoretically
ference gap flow and Re 1/5 in the non-interference
film, a method of eliminating the influence of wall
rotation was proposed from the measured flow angle
obtained by the oil-film streak lines, and the empirical
equations of the correct flow angle were presented.
and experimentally in order to determine the influence
of an axial gap and radial through-flow. The conclusions are sammarized as follows.
( I ) An analytical flow model was proposed for
the gap flow in which the boundary layers on the
rotating and the stationary wall interfere with each
other, and a theoretical method was presented which
Series II, Vol. 31, No. 2, 1988
Ref erences
(1)
Kurokawa, J. and Toyokura, T., Study on Axial
Thrust of Radial Flow Turbomachinery, Proc. 2
nd JSME Symp., (Tokyo) , Vol. I (1972-9) , p. 31.
(2)
Kurokawa, J. and Toyokura, T., Axial Thrust,
Disk Friction Torque and Leakage Loss of Radial
JSME Ir terl atior al Joumal
/
The variation of C with the variation of the gap
ratio S is shown in Fig. 12. Comparing the results of
measurements with the theory at Re = 106, it is seen
251
Flow Turbomachinery, Proc. Pumps and Tur(3)
lino, T., Sato, H. and Miyashiro, H., Hydraulic
Axial Thrust in Multistage Centrifugal Pumps,
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(9)
Muller, U., Uber die laminare Str6mung in einer
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(1985) , p. 452.
(10)
Spaltweiten und radialen Massenstrom, Acta
Mechanica, Bd. 11 (1971), S. 99.
Bayley, F. J. and Owen, J. M., Flow between a
Vol. 20 (1969), p. 333.
(7)
Yamada, Y. and Ito, M., On the Frictional Resistance of Enclosed Rotating Cones (2 nd Report) ,
Bull. JSME, Vol. 19, No. 194 (1976), p. 943.
(11)
Rotating and a Stationary Disc, Aeronaut. Quart.,
(6)
Ito, T., Yamada, Y. and Nishioka, K., Trans. Jpn.
Soc. Mech. Eng., (in Japanese) , Vol. 51, No. 462, B
inkompressiblen Fltissigkeit zwischen einer rotier-
(5)
Daily, J.W., Ernst, W. D. and Asbedin, V. V.,
Enclosed Rotating Disk with Superimposed
Trans. ASME, Ser. I, Vol. 102 (1980), p. 64.
(4)
(8)
bines Conf. (NEL, Glasgow) , Vol. I (1976-9) .
Soo, S. L. and Princeton, N.J., Laminar Flow
Over an Enclosed Rotating Disk, Trans. ASME,
Vol. 80, No. 2 (1958) , p. 287.
Daily, J. W. and Nece, R. E., Chamber Dimension ,
Effects on Induced Flow and Frictional Resistance
of Enclosed Rotating Disk, Trans. ASME, Ser. D,
(12)
Schultz-Grunow, F., Der Reibungswiederstand
rotierender Scheiben in Gehausen, Z. AMM, Bd.
Vol. 82, No. I (1960), p. 217.
(13)
Kurokawa, J., Toyokura, T. and Shinjo, M.,
15, Hf. 4 (1935), S. 191.
Hayami, H. and Senoo. Y.. An Analysis on the
Transient Flow along an Enclosed Rotating Disk
Flow in a Casing Induced by Rotating Disk Using
a Four-Layer Theory, Trans. ASME, Ser. I, Vol.
at the Start-Up, Proc. 6 th. Conf. on Fluid Machinery (Budapest) , Vol. 2 (1979) , p. 655.
98, No. 2 (1976), p. 192.
JSME Irbterr ational Jourrtal
Series II, Vol. 31, No. 2, 1988
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