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Improved solution of steady, viscous, incompressible flow between two rotating spheres
O lhapm~ I b m Lul., 19"/9. lhimd ie (3mr grgain
I M P R O V E D S O L U T I O N OF S T E A D Y , VISCOUS, I N C O M P R E S S I B L E FLOW B E T W E E N TWO ROTATING S P H E R E S DAVID SCI-IULTZt and DONXLS) GZrdSNSPAS~; The Universityof Wisconsin-Milwaukee,Milwaukee,WI 53201,U.S.A. (Received9 January 1978;[orpublication13 May 1978)
Ain~aet--A refined solutionis presented for the analysis of viscous,ittcompress~le,steady flowbetween two ro~tting spheres. A new method, used previously for simpler problems only, is Mapted to this problem. The method allows the use of small grid SlY:ins and thus yields improved accuracy.
I. INTRODUCTION The motion of a viscous fluid inside a rotating container is of interest in engineering and geophysics[l, 2, 6-13, 15]. In this paper, we refine the numerical solution of the steady, non-linear motion of a viscous, incompressible fluid within a spherical annulus presented by Greenspan in 197514]. Let $j and $2 be concentric spheres centered at the origin with respective radii rj and r2 (r2> rt). Let both rotate about the z axis. By rotational symmetry the motion need only be studied in the plane annular sec6on ABCD (see Fig. I). The dimensionless, steady state Navier-Stokes equations of motion are, in the notation of [6] and [10]
I~M + R
D2~ = M
(l.I)
R D~f~ + ~r-~-~ [~,f~,- ~,fl,]= 0
(1.2)
[ll, sin 0 - 11o"cos Ol + ~ [ O , M ,
- ~M,I
+ ~[~,
sin 0 - ~# cos 0)} = 0,
(1.3)
where
~ = a _ O2 _2: + l? a2 u = ~/(r ~
l
.. o2
(|.4) (I.5)
sin 0)
Z c
X
Fi•.I. tDepmment of Mathematics. ~Depanment of Matbematics,Universityof Texas, Arlington,Texas. CAF Vol. 7, No. ~--A
157
158
D. SCHUL'~ arid D. GitEENSPAN
(velocity in the direction of increasing r). v
=
(1.6)
- O~(r sin 0)
(velocity in the direction of increasing 0). w = Q l ( r sin 0)
(1.7)
(velocity perpendicular to the meridional plane). We assume St has angular velocity =0t and $2 has angular velocity ~ , so that the boundary conditions[6, 10] are 0 = 0, = 0, on arcs AD and B C
(1.8)
0 = fl = M = 0, on CD
(1.9)
Q_ [wlrl2sin 2 O, 011AD -- [ ¢02r22Sin2 O, on BC.
(I.I0)
2. THE NUMERICAL METHOD For fixed grid sizes Ar and A0, let A*, B*, C, D be the points (rj, ~2 + A0), (r2, Ir/2 + A0), (rz, 0), (rt, 0), respectively, as shown in Fig. 2. On and within rectangle A*B*CD, construct and number in the usual way the set of interior grid points R, and the set of boundary grid points S,. The special, but extensive, difference equations used are the ones developed in [4]. These equations always yield diagonal dominance and are amenable to solution by SOR (successive over-relaxation). The numerical method to be used is similar to that presented in [5] for an entire class of simpler problems. The essence of the method is that, instead of solving separate systems of equations for each of 0, fl, and M, thus creating a cumbersome problem of triple sequences and their convergence, we solve all of the generated equations simultaneously. In general this cannot be done with a single over-relaxation parameter, but, as recommended in [5], we can accomplish this by using three such parameters r~, rn and rM. The resulting algorithm accelerates convergence and eliminates almost half the memory requirement, thus allowing us to obtain solutions with smaller grid sizes, smaller convergence criteria, and with double precision. The purpose of obtaining more accurate solutions is to check the results in [4], which differ from those of Pearson[11]. 8
8*(r, r/2 , Z~8)
A*(r~, r / 2 +A8)
I
(i', e+A8) ' (r+Z~r,e)
~,, ~, ~,, (roZ~r, #)
D ( r I , O)
, (/.,e.AO)
~, [L,M
C ( r2 ,0)
Fiz. 2. 3. NUMERICAL RESULTS The examples run in [4] were repeated in this paper with a smaller convergence criterion and with a smaller grid spacing in those cases where the results differed from those of [4] and Pearson [11]. For values of R approximately equal to 1000 or less and &0 = 4.5 and &r = 0.05, the method
Improved solution of
steady flow between
two rotating spheres
159
converged rapidly with relaxation parameters of I", = 1.8, r~ = 1.0, ru = 0.9. The results agreed completely with those of [4] (see Figs. 3--5). For R = 3000, the method became unstable and required underrelaxation. The successful procedure for R : 3000 was to start with a smaller ru and run the problem until the difference equations for ¢, and fl were satisfied to, approximately, l0 s . At this point we were able to start increasing ru and run until the difference equation for M were satisfied to, approximately, l0 s . That is, ru would start at about 0.005 or 0.001 and at final convergence would be about 0.2 or 0.3. However, if we started immediately at rM : 0.2 the method would diverge. For all cases the difference equations for M were satisfied to, approximately, l0 s, while those for ¢, and fl were satisfied to much smaller values. Because the results in [4] differed from those of Pearson for large R, the case R = 3000 was chosen as a test problem. That is, the case R : 3000 was run with decreasing values of AV and A0. Figures 6 and 7 contain the results for Ar = 0.025 and A0 = 2.25. The results show the disjoint closed curves as in [4]. No such curves were found by Pearson[10]. However, there are
Fig. 3. Level ~, curves, multiplied by 10' for R : 1000, wt : 0, ,02: !.0, AV : 0.05, A0 : 4.5.
Fill. 4. Level qt curves, multiplied by 10' for R :: 100, ,~ ,. - 1 , ~ = 1.1, Ar = 0.05, A# : 4.5.
160
D. ~CHULTZand D. GREENSPAN
Fig. 5. Level ¢~ curves, multiplied by 10' for R = 100, oJ, = 1.0, ~ = 1.01, A r = 0.05, b0 = 4.5.
Fig. 6. Level ~ curves, multiplied by I0' for R : 3000, ~s = 0, o~ = I, A r : 0.025, AO : 2.25.
only two sets of closed curves and not several, as shown in [4]. To verify the results, the problem was run again with Ar--0.0125 and &O = 2.25 (see Figs. 8 and 9) and also with Ar=0.0125 and AO-- 1.125 (see Figs. 10 and 11). The graphs for all of these cases are very nearly the same and all show the recirculating zone near the equator and the disjoint closed curves (thus verifying the work in [4]). Pearson's Fig. 8 only suggests that such a result is reasonable. As another check, Richardson's extrapolation procedure was applied to our results. Although the validity of application of Richardson's extrapolation method is, as usual, dit~cuit to
Improved solution of steady flow between two rotating spheres
Fig. 7. Level ~ curves, for R = 3000, ~o~: 0, o ~ : I, • r :
161
0.025, 4 0 : 2.25.
Fig. 8. Level ¢ curves, multiplied by 104for R = 3000, oJi = 0, o~2= I, dr = 0.0125, A# = 2.25.
establish, the extrapolation procedure applied to this problem also gives the disjoint closed curves. The method used is described in [16]. The case R = 3000 brought out several interesting techniques that can be used when the solution of the difference equations becomes unstable. If one wishes to go to extreme underrelaxation, one can often obtain convergence. For example, in one case rM = 0.005 yield divergence while r . = 0.001 converged. Once one has convergence for r . = 0.001, one can then obtain convergence for rM = 0.005, etc. That is, one can use a walking technique for the relaxation parameter to obtain the solution.
162
D. SCMULTZ and D. G~ENsP^~
0.02
).O5
Fig. 9. Level ~ curves for R = 3000, w~ = 0, ~ = 1.0, A r = 0.0125, &0 = 2.25.
Fig. I0. Level ,k curves, multiplied by 104 for R = 3000, co= = 0, ~ = 1.0, A r = 0.0125, A# = 1.125.
Improved solution of steady flow between two rotating spheres
0.02
Fill. I 1. Level h curves for R : 3000, oJ~: 0, o ~ : I, A r : 0.0125, 4# = 1.125. REFERENCES I. P. L. Bhatnapr and G. K. Rajeswari, Indian J. Math. $, 93 (1963). 2. K. Bratuhin Ju, J. Appl. Math. Mech. 25, 1286(1%2). 3. D. Greenspan, J. Fluid Mech. ST, 167 (1973). 4. D. Greenspan, Computers and Buids 3, 69 (1975). 5. D. Oreenspan and D. H. Schultz, Topics on Dil. Eqns., Math. Society, Hmos Bolyai, Vol. 15. Budapest (1975). 6. H. P. Oreenspan, The T/~ory of Rotating Fluids, Cambridge (1968). 7. W. L. Habernum, Phys. Ruids $, 1136(1962). S. W. E. Langlois, Quart. Appl. Math. 2, 61 (1963). 9. G. Ovseenko Ju., hr. Vyss. Ucebn. Zaved. Mat. 35, 129 (1963). 10. C. E. Pearson, J. ~ Mech. 211,323 (1967). il. C. E. Pearson, Studies in Numerical Analysis, Vol. I, p. 65. SIAM, Philadelphia (1969). 12. J. Pedlosky, J. Fluid Mech. M, 401 (1969). 13. I. Proudman, I. Buid Mech. !, 505 (19~). 14. A. B. Schubert, University of Wisconsin Computer Sci. Dept. Tech. Rpt. 120 (appendix) (1971). 15. K. Stewartson, J. Flu/d Mech. ~ , 131 (1966). 16. R. Yalamanchili and K. C. Reddy, Numerical/Laboratory Compu:er Methods in Fluid Mechanics, ASME (1976).
163
I M P R O V E D S O L U T I O N OF S T E A D Y , VISCOUS, I N C O M P R E S S I B L E FLOW B E T W E E N TWO ROTATING S P H E R E S DAVID SCI-IULTZt and DONXLS) GZrdSNSPAS~; The Universityof Wisconsin-Milwaukee,Milwaukee,WI 53201,U.S.A. (Received9 January 1978;[orpublication13 May 1978)
Ain~aet--A refined solutionis presented for the analysis of viscous,ittcompress~le,steady flowbetween two ro~tting spheres. A new method, used previously for simpler problems only, is Mapted to this problem. The method allows the use of small grid SlY:ins and thus yields improved accuracy.
I. INTRODUCTION The motion of a viscous fluid inside a rotating container is of interest in engineering and geophysics[l, 2, 6-13, 15]. In this paper, we refine the numerical solution of the steady, non-linear motion of a viscous, incompressible fluid within a spherical annulus presented by Greenspan in 197514]. Let $j and $2 be concentric spheres centered at the origin with respective radii rj and r2 (r2> rt). Let both rotate about the z axis. By rotational symmetry the motion need only be studied in the plane annular sec6on ABCD (see Fig. I). The dimensionless, steady state Navier-Stokes equations of motion are, in the notation of [6] and [10]
I~M + R
D2~ = M
(l.I)
R D~f~ + ~r-~-~ [~,f~,- ~,fl,]= 0
(1.2)
[ll, sin 0 - 11o"cos Ol + ~ [ O , M ,
- ~M,I
+ ~[~,
sin 0 - ~# cos 0)} = 0,
(1.3)
where
~ = a _ O2 _2: + l? a2 u = ~/(r ~
l
.. o2
(|.4) (I.5)
sin 0)
Z c
X
Fi•.I. tDepmment of Mathematics. ~Depanment of Matbematics,Universityof Texas, Arlington,Texas. CAF Vol. 7, No. ~--A
157
158
D. SCHUL'~ arid D. GitEENSPAN
(velocity in the direction of increasing r). v
=
(1.6)
- O~(r sin 0)
(velocity in the direction of increasing 0). w = Q l ( r sin 0)
(1.7)
(velocity perpendicular to the meridional plane). We assume St has angular velocity =0t and $2 has angular velocity ~ , so that the boundary conditions[6, 10] are 0 = 0, = 0, on arcs AD and B C
(1.8)
0 = fl = M = 0, on CD
(1.9)
Q_ [wlrl2sin 2 O, 011AD -- [ ¢02r22Sin2 O, on BC.
(I.I0)
2. THE NUMERICAL METHOD For fixed grid sizes Ar and A0, let A*, B*, C, D be the points (rj, ~2 + A0), (r2, Ir/2 + A0), (rz, 0), (rt, 0), respectively, as shown in Fig. 2. On and within rectangle A*B*CD, construct and number in the usual way the set of interior grid points R, and the set of boundary grid points S,. The special, but extensive, difference equations used are the ones developed in [4]. These equations always yield diagonal dominance and are amenable to solution by SOR (successive over-relaxation). The numerical method to be used is similar to that presented in [5] for an entire class of simpler problems. The essence of the method is that, instead of solving separate systems of equations for each of 0, fl, and M, thus creating a cumbersome problem of triple sequences and their convergence, we solve all of the generated equations simultaneously. In general this cannot be done with a single over-relaxation parameter, but, as recommended in [5], we can accomplish this by using three such parameters r~, rn and rM. The resulting algorithm accelerates convergence and eliminates almost half the memory requirement, thus allowing us to obtain solutions with smaller grid sizes, smaller convergence criteria, and with double precision. The purpose of obtaining more accurate solutions is to check the results in [4], which differ from those of Pearson[11]. 8
8*(r, r/2 , Z~8)
A*(r~, r / 2 +A8)
I
(i', e+A8) ' (r+Z~r,e)
~,, ~, ~,, (roZ~r, #)
D ( r I , O)
, (/.,e.AO)
~, [L,M
C ( r2 ,0)
Fiz. 2. 3. NUMERICAL RESULTS The examples run in [4] were repeated in this paper with a smaller convergence criterion and with a smaller grid spacing in those cases where the results differed from those of [4] and Pearson [11]. For values of R approximately equal to 1000 or less and &0 = 4.5 and &r = 0.05, the method
Improved solution of
steady flow between
two rotating spheres
159
converged rapidly with relaxation parameters of I", = 1.8, r~ = 1.0, ru = 0.9. The results agreed completely with those of [4] (see Figs. 3--5). For R = 3000, the method became unstable and required underrelaxation. The successful procedure for R : 3000 was to start with a smaller ru and run the problem until the difference equations for ¢, and fl were satisfied to, approximately, l0 s . At this point we were able to start increasing ru and run until the difference equation for M were satisfied to, approximately, l0 s . That is, ru would start at about 0.005 or 0.001 and at final convergence would be about 0.2 or 0.3. However, if we started immediately at rM : 0.2 the method would diverge. For all cases the difference equations for M were satisfied to, approximately, l0 s, while those for ¢, and fl were satisfied to much smaller values. Because the results in [4] differed from those of Pearson for large R, the case R = 3000 was chosen as a test problem. That is, the case R : 3000 was run with decreasing values of AV and A0. Figures 6 and 7 contain the results for Ar = 0.025 and A0 = 2.25. The results show the disjoint closed curves as in [4]. No such curves were found by Pearson[10]. However, there are
Fig. 3. Level ~, curves, multiplied by 10' for R : 1000, wt : 0, ,02: !.0, AV : 0.05, A0 : 4.5.
Fill. 4. Level qt curves, multiplied by 10' for R :: 100, ,~ ,. - 1 , ~ = 1.1, Ar = 0.05, A# : 4.5.
160
D. ~CHULTZand D. GREENSPAN
Fig. 5. Level ¢~ curves, multiplied by 10' for R = 100, oJ, = 1.0, ~ = 1.01, A r = 0.05, b0 = 4.5.
Fig. 6. Level ~ curves, multiplied by I0' for R : 3000, ~s = 0, o~ = I, A r : 0.025, AO : 2.25.
only two sets of closed curves and not several, as shown in [4]. To verify the results, the problem was run again with Ar--0.0125 and &O = 2.25 (see Figs. 8 and 9) and also with Ar=0.0125 and AO-- 1.125 (see Figs. 10 and 11). The graphs for all of these cases are very nearly the same and all show the recirculating zone near the equator and the disjoint closed curves (thus verifying the work in [4]). Pearson's Fig. 8 only suggests that such a result is reasonable. As another check, Richardson's extrapolation procedure was applied to our results. Although the validity of application of Richardson's extrapolation method is, as usual, dit~cuit to
Improved solution of steady flow between two rotating spheres
Fig. 7. Level ~ curves, for R = 3000, ~o~: 0, o ~ : I, • r :
161
0.025, 4 0 : 2.25.
Fig. 8. Level ¢ curves, multiplied by 104for R = 3000, oJi = 0, o~2= I, dr = 0.0125, A# = 2.25.
establish, the extrapolation procedure applied to this problem also gives the disjoint closed curves. The method used is described in [16]. The case R = 3000 brought out several interesting techniques that can be used when the solution of the difference equations becomes unstable. If one wishes to go to extreme underrelaxation, one can often obtain convergence. For example, in one case rM = 0.005 yield divergence while r . = 0.001 converged. Once one has convergence for r . = 0.001, one can then obtain convergence for rM = 0.005, etc. That is, one can use a walking technique for the relaxation parameter to obtain the solution.
162
D. SCMULTZ and D. G~ENsP^~
0.02
).O5
Fig. 9. Level ~ curves for R = 3000, w~ = 0, ~ = 1.0, A r = 0.0125, &0 = 2.25.
Fig. I0. Level ,k curves, multiplied by 104 for R = 3000, co= = 0, ~ = 1.0, A r = 0.0125, A# = 1.125.
Improved solution of steady flow between two rotating spheres
0.02
Fill. I 1. Level h curves for R : 3000, oJ~: 0, o ~ : I, A r : 0.0125, 4# = 1.125. REFERENCES I. P. L. Bhatnapr and G. K. Rajeswari, Indian J. Math. $, 93 (1963). 2. K. Bratuhin Ju, J. Appl. Math. Mech. 25, 1286(1%2). 3. D. Greenspan, J. Fluid Mech. ST, 167 (1973). 4. D. Greenspan, Computers and Buids 3, 69 (1975). 5. D. Oreenspan and D. H. Schultz, Topics on Dil. Eqns., Math. Society, Hmos Bolyai, Vol. 15. Budapest (1975). 6. H. P. Oreenspan, The T/~ory of Rotating Fluids, Cambridge (1968). 7. W. L. Habernum, Phys. Ruids $, 1136(1962). S. W. E. Langlois, Quart. Appl. Math. 2, 61 (1963). 9. G. Ovseenko Ju., hr. Vyss. Ucebn. Zaved. Mat. 35, 129 (1963). 10. C. E. Pearson, J. ~ Mech. 211,323 (1967). il. C. E. Pearson, Studies in Numerical Analysis, Vol. I, p. 65. SIAM, Philadelphia (1969). 12. J. Pedlosky, J. Fluid Mech. M, 401 (1969). 13. I. Proudman, I. Buid Mech. !, 505 (19~). 14. A. B. Schubert, University of Wisconsin Computer Sci. Dept. Tech. Rpt. 120 (appendix) (1971). 15. K. Stewartson, J. Flu/d Mech. ~ , 131 (1966). 16. R. Yalamanchili and K. C. Reddy, Numerical/Laboratory Compu:er Methods in Fluid Mechanics, ASME (1976).
163