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Transcript of Pohlhausen_Laminar Flow Between Two Parallel Rotating Disks
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ARL 62-318 2 5-
LAMINAR FLOW BETWEEN TWO PARALLELtI ROTATING DISKS
MARK G. BREITERKAR. POHLHAUSEN
APPLIED MATHEMATICS RESEARCH BRANCH
AERONAUTICAL RESEARCH LABORATORY
OFFICE OF AEROSPACE RESEARCHUNITED STATES AIR FORCE
rrI
ii
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ARL 62- 318 March 1962
ERRATA - July 1962
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AERONAUTICAL RESEARCH LABORATORIESOFFICE OF AEROSPACE RESEARCH
UNITED STATES P IR FORCEWRIGHT-PATTERSON AIR FORCE BASE, OHIO
ADI 275562
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ARL 62-318
LAMINAR FLOW BETWEEN TWO PARALLELROTATING DISKS
MA.9K C. BREITERKARL POHLHAUSEN
APPLIED MATHEMATICS RESEARCH BRANCH
"MARCH 1962
Project No. 7071Task 70437
AERONAUTICAL RESEARCH LABORATOBYOFFICE OF AEIROSPACE RESEARCH
UNITED STATES AIR FORCEWVRIGHT-PATUERSON AIR FORCE BASE', OHIO
300 - May 1962 - 31.366 9 1367
FURE WORD
This report was prepared in the Applied Mathematics Research 13)
Aeronautical Research Laboratory, Office of Aeroispace ResearC-., Wr
Patterson Air Force Base, Ohio~.
The authors extend grateful acknowledgement to Dr. K. G. Guderlt
the Applied Mathematics Research Branch, for encouragement and mu-
advice.
The numerical calculations were carried out partly with the Recon
graciously made available by Professor R. T. Harling of the Air Forc,
of Technoulogy, and partly with the 1103A computer of the Digital C-Friip
Branch, Systems Dynamic Analysis Division. Captain R. V. Hendon of
Computation Branch rendered valuable assistance in programming.
The work was carried out un. ?roject 7071, "Methods of Mathev
Physics", Task 70437.
BESTAVAILABLE COPY
FOREWORD
i
This repo t was prepared in the Applied Mathematics Research Branch of the
Aeronautical Research Laboratory, Offlce of Aerospace Researc'., Wright-
Patterson Air Force Base, Ohio.
The authors extend grateful acknowledgement to Dr. K. G. Guderley, Chief of
the Applied Mathematics Research Branch, for encouragement and much helpful
advice.
The numerical calculdtions were carried out partly with the Recomp computer,
graciously made available by Professor R. T. Harling of the Air Force Institute
of Technology, and partly with the 1103A computer of the Digital Computation
Branch, Systems Dynamic Analysis Division. Captain R. V. ilendon of the Digital
Computation Branch rendered valuable assistance in programming.
The work % L , .a, ried out under Project 7071, "Met. of Mathematical
Physics", Ta k 7043
ii
ABSTRACT
The report inrestigates the viscous flow between two parallel disks rotating in
the same direction with the same velocity. The fluid enters the space between the
two disks at a certain radius In the radial direction. fcause of the shear forces,
it assumes a rotating motion with about the velocity of the disks. The centrifugal
forces then build up a pressure increase in the radial direction. The arrangement
corresponds to a centrifugal fluid pump, which may be advantageous if cavitation
is ai problem.
The general equations of viscous flow are simplified by the assumption that
the pressure difference normal to t..le disks is negligible (boundary layer assump-
tions). One obtains a system of parabolic partial differential eq,,ations. For large
radii the deviation from rigid body rotation (with the angular velocity of the disks)
is smail. The lsi, _:- i equations which then result are s. , analytically. The
velocity profile. dePenc! ipon a parameter containing the kinematic viscosity, the
angular velocity and the distance of the disks, but not the radius.
The non-linearized parabolic differential equations are approximm.Led by a
difference scheme and solved numerically. The results are given in non-dimen-
sional form with the entrance velocity and the distance of the disks as parameters.
Furthermore, the efficiency of the pump is computed from the gain of the total
pressure and the torque at the shaft of the rotating disks,
iii
TABLE OF CONTENTS
Section Page
1. Introduction I
11. Statement of the Problem 3
111. Linearized Treatment and A symnptotic Solution 5
IV. Numerical Solution of the System of Equations
(The Inlet Problem) 10
V. Results and Conclusions 14
VI. Appendix 17
iv
LIST OF ILLUSTRATIONS
Page
Figure
1. Schematic diagram of the flow between two rotating disks 21
2. k I and k2 versus ad 22
3. g(ad) versus ad 23
4. Radial velocity u and relative tangential velocity V for
various values of ad 24
5. Radial velocity 5 and relative tangential velocity V for
ii0= 1.0 d=0.5 25
6. Radial velocity 5 and relative rangential velocity Vfor
0 =o. 5 d=0.5 26
7. Radial ve jct,.. a ard relative tangential velocity N or
d 0 = 0 , .5 C ..5 27
8. Radial velocity d anid relative tangential velocity V for
di0= 0. 1 a = 0.5 28
9. Radial velocity il and relative tangential velocity V for
ao = 1.0 a=l.O 29
10. Radial velocity ai and relative tangential velocity V for
i0 = 0. 5 d=l. 0 30
11. Radial velocity i and relative tangential velocity V for
5 0' 0. 25 = l.0 31
12. Radial velocity 5 and relative tangential v,-locity V for
0o= 0. 1 =l.0 32
v
LIST OF ILLUSTRATIONS (Continued)
PageFigure
13. Radial velocity i and relative tangnetial veloc*.Ly V for
a 0= 1.0 d- 1.5 33
14. Radial velocity i and relative tangential velocity V for
f0 =0.5 d=l.5 34
'A 5. Radial velocity d and relative tangnetial velocity V for
10- 0.25 d =1.5 35
16. Radial velocity 5i and relative tangential velocity V for
a 0=0.1 j= 1.5 36
17. Radial velocity 5 and relative tangnetial velocity V for
r,0 = 0= n j - i7
18. Radin. velocity i and relative tangential velocity V for
u0 =0.5 d=2.a 38
19. Radial velocity 5 and relative tangnetial velocity V for
i0=0.5 d=2.0 39
20. Radial velocity 5 and relative tangential velocity V for
5 o= 0. d 2=2.0 40
21. Radial velocity 5i versus r/r. for d=0. 5 -&1ni 1
22. Radial velocitv 5 versus r/r. for d=l.0 41m 1
23 Radial velocity 5i versus r/r. for d=.5 42
24. Radial velocity 5G versus r/r. for d= 2 42
25. Fi() for various values of 6 0 versus r/. for d=0.5 430v1
vi
LIST OF ILLUSTRATIONS (Continued)
Figures Page
26. T"(Q) for various values of 50 versus r/r. for d= 1.0 44
27. F(0 for various values of -0 versus for d:.5 44
28. P(r) for various values of a0 versus r/ri for d=2.0 45
29. Moment, pressure, and efficiency versus r/r i for
d =0.5 46
10. Moment, pressure, and efficiency versus r/ri for
d= 1.0 47
31. Moment, pressure, and efficiency versus r/ri for
S=1.5 48
32. Moment, pressure, and efficiency versus r/r i fo-
d=2. 0 49
I. INTRODUCTION
Investigated in this report is the viscous flow between two parallel disks,
which rotate with the same constant angular velocity in the same direction. The
fluid enters the space between the disks in radial direction through a cylindrical
surface (Figure 1). It is set into a spiral motion whose tangential and radial
velocity components are due to the friction forces and to the centrifugal forces
respectively. The pressure will increase with the radius because of the centrif-
ugsl forces, thus the arrangement can serve as a centrifugal pump. One might
suspect that the absence of blades leads to low values of the efficiency, but if the
blades are spaced very closely and the mass flow is low, the efficiency may be
quite acceptable. On the other hand, such a pump would not encounter the problem
of cavitation and also its characteristics over a wide operatin ° -ange might be
more favorable than in i conventional turbo pump. These advd,,tages were recog-
nized by Mr. S.: . Hasinrer, Mr. L. G. Kehrt and Dr. J. P. von Ohain of the
Thermomechanics Research Branch of this laboratory. Technical details and
experimental results will be published by them in a future report.
The present mathematical analysis will determine the velocity distribution
between the disks, the pressures, the torque applied at the shaft and the resulting
efficiency.
There exists an extensive literature on the single disk rotating in an infinite
medium*. Assuming that the disk extends from radius zero to infinity, it is
*Advances in Applied Matheratics, Academic Press, Inc.. New York, Vo) IV,
1956, p. 166. F. K. Moore: Three-Dimensional Boundary Layer Theory.
1
possible to reduce the partial differential equations governing this problem to a
sy3tem of ordin:'y differential equations by means of a similarity bypothesis. But
an approach of this kind is not feasible in the present case because of the presence
of the seccid disk and the fact that the flow enters the space between the disks at a
radius different from zero. However, another simplification can be made here.
For any practical application the distance between the disks must be very small,
for otherwise the friction would not be sufficiently effective to produce a tangential
f1jv,. But then the approximations made in boundary layer theory apply reducing
the original system of Navier-Stokes equations, which is elliotic, to a system of
parabolic differential equations. They are derived in Section II.
A first insight into the basic characteristics of the proble n is obtained if one
assumes that the velocities of the fuid relative to the disks are small so that
quadratic ternr. , -.- gligible. This is a condition whici 'ell satisfied for
large radii; F a idea ii carried out in section Ill. It gives valuable insight into
the general character o" the velocity profiles; these profiles are governed by one
dimensionless pararmeter which contains the angular velocity, the vi-cosity and
the distance, but not the mass flow and the radius. Naturally this parameter is
important for the practical design. The solutions thus obtained can be considered
as an asymptotic expression for the solution of the actual non-linear system.
In the • icinity of the entrance radius the linearization is not justified, in par-
ticular the tangential velocity relative to the disks is certainly not small. This
region is most critical from the point of view of cavitation, and also the principal
losses are encountered there. Therefore, the system of differential equations has
been integrated numerically. The methods and the results are shown in section
IV and V and in an appendix.
2
fl. STATEMENT OF THE PROBLEM
We use a systrn of cylindrical coordinates r and z (Figure 1), where the
z axis coincides with the axis of symmetry and r is the distance from it. We
denote by
ai the component of the velocity in the radial direction
v the component of the velocity in the tangential direction
measured in a system of coordinates fixed in space
w the component of the velocity in the axial direction
2d the distance between the two disks
ri the radius of the entrance hole
p the density
i the viscosity, v = the kinematic viscos'"
p th. pru sure
w he commant anguldr veloci 'y.
An incompressible axi-symmetric steady viscous flow is described by the Navier-
Stokes equations
Ov +wav + u. v 4v- a + 8Vur wT: r 2z z -Or r 2 28z 8r
w 8w .! + V __ + . --. + - (2.1)
ur 8z p Oz 8z r Or 8r2
+ &u A.. u 8 uu au v z I Ip + V 8 L !
+r a-: r p Or 8z 2 Or r r
and the equation of continuity,
Ou + a__2w =0 (2.2)
Or r 8z3
By applying the boundary layer approximations, this system of equations is reduced
toOU 2 au 2 2u
_U M- V+ w 2-=.-iu -r O z p dr Oz2
(2. 3)
u V u. .v + w w K Xv
Pr r z 2--
uu 8w+r 2 + L X 0 (2.4)Or r' r 8z3
It is convenient to introduce instead of v, which is the tangential velocity in a
fixed system of coordinates, the tangential velocity V relative to the disks
V1, v - rw (2.5)
One then obtains
u2 2 u 1 dp 2uOu V Vrr w * =- 3
U r iz Pdr O(2.6)
8V u[V+,rw] + V av (2.7)r z = az 2
Along the disks, given by plane z - d, one has the boundary cnditions
u (r, *d) =0
(2.8)
V(r, *d)= 0
4
To determine the boundary value problem com.pletely, one must also prescribe
the velocity disti ioution in the entrance cross section. One has
u(ri, z)a u0
(2.9)
V(r i ,z)-rw
IMI. LINEARIZED TREATMENT AND ASYMPTOTIC SOLUTION
Omitting quadratic terms in u, V and w one obtains from Eqs (2. 6) and (2. 7)
V u : d (3.1)0 8z2 p dr
u a z (3.2)
In these equations does not depend upon z becauL che boundary layer
p&w dr
assumption thai rhe pre sure does not depend upon z. The unknown w does not
appear, and so it is possible to solve these equations separately; w is determined
from Eq (Z. 4) afterwards. It is ecmarkAble that in these equations .!,ere appear
no parlai derivatives with respect to r. This means that, except for the expres-
sion
rw = F (r) (3. 3)p'aw dr
which depends upon r a nd is undetermined so far, the equations can be solved
for each value of r independently. The expression (3. 3) is the only inhomogeneous
term that occurs in the equations and in the boundary conditions; therefore, it will
enter the solution for u and V a a a factor which depends only on r. This factor
5
is determined by the condition that the mass flow between the disks is constant for
every cross section r =const. Thus u and V as well as - can be found.p dr
After elimination of u one obtaiv, from Eqs (3. 1) and (3. 2)
2 4VP d V + 2V z F(r) (3.4)2rZ dz 4
Setting V = one has finally
dV + 4a 4V 24F(r) (3.5)dz
4...
An the most general solution of Eq (3. 5) that satisfies the symmetry conditions at
the plane z v 0, a ac .,: 1
V=A sin!, (az) sin (oz)+A cosh(az)cos(az)+ F(r) (3.6)2 2
where A I and A2 are two constants which will be determined presently. The
boundary conditiona (2.8) together with Eq (3.2) give fcbr z =d
V=0
(3.7)
1 d2 V
2 2 dz 2
6
Hence, by inserting Eq (3. 6)
A, F(r) k A -F(r) k2 381 2 1 2 2 (38
with
k 2 asi a). sin (ad)I cosh'.r.)+cos42ad)
(3.9)
k2 2 cosh(ad).cos (ad)2 cosh (2ad)+cos(2ad)
Figure 2 shows kI and k2 as a function of ad
The shear force at the wall is proportional to dV/dz. One finds
dV F(!) ! (k I , ) cosh(az) sin(az)+(kl +k 2 )einh(az)coo(az) (3.10)dz *a2 (320
Let t)he flow of liquid (measured by volume) be given by 0. The flow through
a surface r = const extending between the disks is given by
+d 2 +dQ= 27r f udz= I df dz = dV (3.11)
-d a -d dz a -d
Hence ith Eq (3. 10)
Q . .. F(r). g (ad) (3.121a
where:
g(ad) a sinh(Zad)- sin (2ad)cosh(Zad)+coo (2ad) (3.13
Figure 3 shows g(ad) as a function of ad
irrom Eq (3. 13) we find the function F(r)
IF W 4 eQw- r.g(ad) r (3.14)
where
q -w. g(ad) (3.15)
One thus obtain, finally tor the velocity components, expressed in terms of the
mass flow
v U3 [ k , ,inh(az) sin (, z)+k. cosh(z) cos(az)- 1 (3. 16)(3. 16)
U a -[kl cosh(az)cos(aL.-k 2 sinh(a-) sin (az)]
Inserting these expressions into the continuity equation Eq (2.4) one finds w = 0.
The velocities in the middle plane between the two disks are found by setting z = 0.
Z, =0 r(3. 17)
5U klzZO = r
8
Figure 4 shows the radial velocity component for different values of ad as a func-
tion of z/d for q/r = I. For small values of ad, the profiles have a near para-
bolic shape with maximum velocity in the middle (z = 0). For ad = w/Z the profile
is flat in the rviddle. for higher valie of ad the flow is moro and more concentrated
in the neighborhood of the disks.
The tangential velocity V is represented in Figure 4 for q/r = 1. For small
values of ad it also has a parabolic shape, but with increasing ad the slope in the
vici;:ity of the disks becomes steeper and the relative velocity in the middle plane
ajroaches -1.
Actually the profiles for high values of ad must be viewed with considerable
caution. One sees that in this case the velocity u may become n 1gative in the
middle; i. e. for such cases one has a radial outflow close to the disks with a
radial inflow in t'k roddle. Obviously it would take a rathe -ial arrangement
to produce such • o, pattern physically, Theoe results ought to be disregarded
for technical aFplications. Important in any case is the role played by the dimen-
sionless parameter ad which determines the character of the profiles. This
character is neither inflaenced by the mass flow Q nor by the radius r. For a
given w and a given v, I. e. . a given fluid, the only quantity that can be influencid
in thi3 parameter is the distance of the disks. If one wants to use high angular
velocities, one must try to lower the value of d by skillful design.
The pressure distribution is obtained from the function F(r) in Eq (3. 5).
Inserting Eq (3. 14) into (3. 3).
one finds d pw [r -2q (3.18)
dr r
9
and by integration
2
p = [ r. . 2qlnr +const] (3.19)2
The momentum required at the ahaft of the rotor can be determined in two
different ways, either from the moment of the shearing forces on the inner side
of two disks, or by applying the law of conseration for the moment of momentum.
One has in the first case
r rz I dVdM =-2- z. ' f r '-dr (3.20)
ri iz zd
in the second
2 dl4 u2,rr p f u.vdz (3.21)
-d
For the numerical ealuation the second one is preferable.
IV. NUMERICAL SOLUTION OF THE SYSTEM
OF EQUATIONS (THE INLET PROBLEM)
The approximation shown in the previous section is unsatisfactory in the
vicinity of the entrance, where the velocity components are certainly not small.
Moreover, the lineariration causes all r-derivatives to vanish, i. e. it introduces
such i strong change of the character of the flow equations, that it becomes
impossible to satisfj the boundary conditions at the inlet. But the inlet is im-
portant for it accounts for most of the losses.
10
As was mentioned above, the introduction of the boundary layer assumption
gives us a parabolic system of differential equations where lines r = const are the
characteristics, therefore, it is possible to proceed in the numerical solutions
from one line r = const Lu the next one without any need to go haclk. Thus, we may
adopt a numerical scheme for the integration of the parabolic differential equations.
For our computations we assumed that the rotating disks lie in horizontal
planeR, and introduce a new system of coordinates x, y, whose origin lies in the
entrance cross section at the lower disk. Thus the x-axis coincides with the
loder disk. We have
x~rr
y = z+d
0
uO
ri r
We introduce dimensionless variables in the system of ,'Iferential equations
(2.6), (2.7), and (2.4)uo +
r. r.1 1
r. Y=Y"; r~ d1
-m W
11
One thus obtains
a " -[ + + +2 !: +
u + -+ ] = 8 (4.2)
where for abbreviation
F'(r) a r. 1 d (4. 3)142 r Z dr
The functions 5 ind 7 and v are given at the entrance (i01. The channel
extends from j = 0 to -=d. The boundary condition at y = Zd can bd replaced
by a symmetry coiidition at i = d. Thus, one has as boundary conditions
0) 0 a,! (q(,a) = 0
0)=0 aV(x,d) = 0 (4.4)
V,(; 0) =0 V(;, a) = 0
The natural method of solving such a system numerically is the introduction
of a difference scheme. Here caution is needed from the point of view of stability.
12
The coefficients of the derivatives with respect to ;., in Eq (4. 2) may become very
sma)l for large values of i and it also is small in the vicinity of y d, but then it
is necessary to use a very fine mesh size in the ; direction, if one applies a
'"iirect" difference scheme where the derivatives with respvct to are computed
along the line I = const which is already known. Otherwise the procedure would
become unstable.
For this reason it was decided to use an ";nverse" difference procedure where
the Ierivatives are formed along the line ; a const for which the state is to be
-Imputed. This brings about a rather severe complication, for the stat. along
the new line ; = const is determined by a system of simutaneous equations.
Since the original differential equations are non-linear, this system of equations
is also non-linear. At first the non-linear system was solved directly by an itera-
tion method. Sirw this method proved to be inconvenient, ?proach vras modi-
fied in the follo, ng manner: In going from one line = const to the next one we
computed the chanies of i, V and Z rather than the values themselves. If a
difference procedure is admissible at all, these changes ought to be small and
second order terms in these changes are negligible. In other words we use a
linearization which considers as a basic approximation the values at the line
= const which has Leen computed previously.
The state along the new line can then be obtained by a suitable super-position
of particular solutions of the linear equatioa which arises in this manner. The
linear system of equation which arises in this process has a rather simple struc-
ture insofar as the matrix of systems has elements that are different from zero
on only a few lines that are paraUel to the main diagonal. This brings about
certain simplifications in the inversion of the matrix. For further details of the
13
numerical analysis see the appendix.
V. RESULTS AND CONCLUSIONS
The numerical procedure described above gives a nearly complete description
of the flow pattern. Assuming constant velocity u0 at the entrance, we give the
results for a set of four initial radial velocities 1i-- 1. 0, 0. 5, 0. 25 and 0. 1. The
initial tangential velocity V 0 is always V0=-I because of Eq (2. 5). For the
dimensionless distance between the disks we choose d = 0. 5, 1. 0, 1. 5, and 2.
Figures 5 thru 20 show the dimensionless velocity profiles versus z/d with
r/r i as parameter. These p:ofiles have a common characteristic: at the wall the
inlet velocities 0 and ;00 wb ch are constant over the cross section, are immedi-
ately reduced to zero because oz the boundary conditions. The reduction of the
radial velocities at the wall causes the velocities in the middl. *.o increase. There-
fore the radial ve)c -.iE , & the profiles close to the entrance c. us section over-
shoots the entra, "e prnf;,-, in the middle. The effect decreases with increasing r
because of the increase of the available cross section with r, finally the profile
approaches the form given by the linearized theory. Figures 21 thru Z4 show this
behavior again; here the radial velocities V in the median plane of the i41.m
(z = 0) are plotted versus r /r.. The average radial velocities are obviously de-I
termined by the condition of continuous flow.
The average tangential velocity is directly connected with the moment M of
the shear forces and thus with the work that must be performed to drive the pump.
Further.more it gives the main contribution to the dynamic pressure of the fluid
particles as they leave the pump, thus it influences rtrungly the efficiency. As
mentioned before shear forces can be obtained from the slope of the velocity
14
profiles at the wall Eq (3. 18); but it appears to be preferable (from the numerical
point of view) to compute it from the radial and tangential velocities Eq (3. 19).
The pressure distribution is computed from the function F() which is found
in the numerical procedure as a function of r/r i . Here an integration in the
direction of r is required. In our figures the contribution to p due to the term
in Eq (4. 3) is shown separately, (as a straight line). The pressures are given by
the difference of the curve ;(M) from this straight line.
Properly speaking the dynamic pressure varies from streamline to streazn-
Ii-.. For technical purposes only the average velocity rark be utilized. The
dynamic pressure for the average velocity is slightly lower than the a-erage of
the dynamic pressures; in other words in defining the dynamic pr :ssure with
average velocities one takes into account the mmxing losses. Thus, we have accord-
ing to Bernoulli'p e-,tion
PLI 2 U v av
The efficiency ?I of the process in I.he pump is best defined by
Ps + P d
11= - . O
where Q = Zirri . 2du0 denotes the amount of fluid entering at r.. Figures 29 thru
31 show, for - p = L, the moment M, the total pressure p8+ Pd and the efficiency
versus for the chosen values of 40 and d.r1 0
According to the linearized approach the profiles at large values of r do not
depend upon the flow field at smaller radii, thus dynamic pressure and torque are
determined only by the exit cross section. However the stz.tic pressure arises by
15
an integration over the pressure gradient in the radial direction. The losses
encountered appear in this analysis as Icsse in static pressure.
The exit cross section does not play a special role in the computation; in
other words, each value of .r can be considered as the other radius of the rotor.
In the vicinity of the erncrance, one has always a slight pressure drop, for
large volume flow the pressure drop is more pronounced, and this fact is of
interest from the point of view of computations. High mass flows are also detri-
nr ental from the point of view of efficiency.
16
APPENDIX
For the numarical solution of the system of equations (4. 2) we set up a net-
work of lines x-constant and y constant with a spacing of h and I respectively
along the x and ; axes. The length I is chosen as uzi twentieth of the distance
from the wall to the median plane of the disks. The derivatives in Eqs (4.2) are
replaced by difference quotients, e. g.
-u - w - -
u i+l,kmUi, k rv - i+l,k+l 1 i+l, k-I
(A 1)
a2 j i+l, k+l l2 I+l, k + 5I+1, k-I
8j2 t 2
Here i and i+ I correspond to successive lines i = constav, and k - 1, k, and
k+ 1 to success4 e linei = conqtant with k- 0 at the wall. Starting with given
values of fi, V, 7 at che line x = const (i = 0) corresponding to the inlet, we
determine the unknown values of a. V, 7 on the (+ 1 )st line from the known
values on the i-th line.
It is convenient to introduce as new ,.wikowna Lho- differences Aa, k'
Arvk , defined by
Ak 'ui+ lk "ui,.k
and similarly for AVk , and &Ck' and neglect all terms in the equations of higher
order in the A's. (Naturally these A's will also depend on i, but only one value
of i is needed at a time).
17
Equations (4.2) then lead to the following linear difference equations in the
unknown differences ,ii, AV, and A and the unknown pressure function f(i).
- - 1,k ) +I 1k+l I2 Wi,k) + Ui, I i, k-1(1+ -" Wi,k)
2 - i,k -. k +a)+ (2 + 4L rl,) A-k
-( L k k-+ (ik+li, k ;wk
-
Lz k
2(l-w. )A wik+-i-+ (1+Z)
2 ik VkI zx 2 k h ik(
v itkl (+ IZVk- I ) x, k-
2~ ~ itk~ Itk1
itk x k2i,kk-i
A2t I h z-I a v + VkI &d k- I -h X Auk_ 1 ik i,k+l iLk-l
In addition to Eqs (A2) the complete specification of the problem requires
Eqs (A3), (A4), and (A5) resulting from boundary conditions.
18
Since W = = = 0 along the wall, one has
A 50 = A *0 = A 0 = 0. (A 3)
Since and V are symmetric and v antisymmetric with respect to the
median plane of the disks,
AGz - AG1 =0,A ,21 Ai19 0
AV -21 = 0, (A4)
a Iv20 0 .
In the continuity equation there appear only first derivatives with respect to
', whiie the otl . r .':' "ns contain second derivatives. 0, .11 notice that in
Eq (Al) the fir. derivtive of 'v with respect to j has -iot been formed from
values wv at adjacent points. This was done in order to obtain a central difference
a a2- ;-formula for ; - jsa for andB2fom , just as a-- As a consequence or. equation
is lost. Therefore, the following approximation is made in the vicinity of the wall.
In the last of equations (4.2) the term a(I is zero at the wall; a may be approxi-
mated by its linear term. Thus
5= f(4 ~
= f',( ). ,
(A5)8w,- f,(i) K -I ) =A
-27v =A(;) y--2
19
From the last equation one finds
-~1
wi,l 4 - 1.w,2
and so
- 1 -
J. 4 2
An alternate form i.s
I i , 2 -
A*,' i "A' + ( "1 wt, ) )
Here the second term ought to be zero, but by proceeding in this fashion we are
more sure th.L C '-: equation of (AS) is not violated bec of error accumu-
lation.
By numerically integrating the radial velocity a with respect to j for any
fixed value of ;, we obtain the total fluid volume passing through thf- cylindrical
surface characterized by ;, and comparison with input volume provides some
check of the numerical results.
20
r
-c-d 0 +d
r.r
Fig 1. Schematic diagram of the flow between two rotating disks
21
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Fig. 11 Radial velocity Z and relative tangential velocity Vfor 0. 25 ;,ilit
31
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