Linear stability of flow in a differentially heated cavity via large-scale eigenvalue calculations

International Journal of Numerical Methods for Heat and Fluid Flow

Volumn 14, Issue 6, 2004, Pages 803-822

  Burroughs, Elizabeth A ^a   Romero, Louis A ^b   Lehoucq, Richard B ^b   Salinger, Andrew G ^b  

^a HUMBOLDT STATE UNIVERSITY   (United States)

^b SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

Convection; Finite element analysis; Flow; Stability (control theory)

Indexed keywords

HOPF BIFURCATION; LINEAR STABILITY; STEADY STATE SOLUTIONS;

BIFURCATION (MATHEMATICS); COMPUTATIONAL FLUID DYNAMICS; CONTROL THEORY; EIGENVALUES AND EIGENFUNCTIONS; FINITE ELEMENT METHOD; HEAT CONVECTION; PRANDTL NUMBER; PROBLEM SOLVING;

FLOW OF FLUIDS;

CAVITY; CONVECTION; EIGENVALUE; FINITE ELEMENT ANALYSIS; FLOW STABILITY;

EID: 4344671774     PISSN: 09615539     EISSN: None     Source Type: Journal    
DOI: 10.1108/09615530410544328     Document Type: Article

References (35)

1. Arnoldi, W.E. (1951), "The principle of minimized iterations in the solution of the matrix eigenvalue problem", Quarterly Journal of Applied Mathematics, Vol. 9, pp. 17-29.
2. Brooks, A.N. and Hughes, T.J.R. (1982), "Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations", Computer Methods in Applied Mechanics and Engineering, Vol. 32, pp. 199-259.
3. Burroughs, E.A., Romero, L.A., Lehoucq, R.B. and Salinger, A.G. (2001), "Large scale eigenvalue calculations for computing the stability of buoyancy driven flows", Technical Report, SAND2001-0113, Sandia National Laboratories, Albuquerque, NM.
4. Christodoulou, K.N. and Scriven, L.E. (1988), "Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem", Journal of Scientific Computing, Vol. 3, pp. 355-406.
5. Cliffe, K.A., Garratt, T.J. and Spence, A. (1993), "Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations", Advances in Computational Mathematics, Vol. 1, pp. 337-56.
6. van Dorsselaer, J. (1997), "Computing eigenvalues occuring in continuation methods with the Jacobi-Davidson QZ method", Journal of Computational Physics, Vol. 138 No. CP975844, pp. 714-33.
7. Edwards, W., Tuckerman, L., Friesner, R. and Sorensen, D. (1994), "Krylov methods for the incompressible navier-stokes equations", Journal of Computational Physics, Vol. 110 No. 1, pp. 82-102.
8. Forth, A., Jardak, M., Gervais, J. and Pierre, R. (1997), "Localization of Hopf Bifurcations in fluid flow problems", International journal of Numerical Methods in Fluids, Vol. 24, pp. 1185-210.
9. Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, NY.
10. Hendrickson, B. and Leland, R. (1995), "The Chaco user's guide: Version 2.0", Tech. Rep. SAND94-2692, Sandia National Labs, Albuquerque, NM.
11. Hughes, T.J.R., Franca, L.P. and Balestra, M. (1986), "A new finite element formulation for computational fluid dynamics: circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accomodating equal order interpolation", Computer Methods in Applied Mechanics and Engineering, Vol. 59, pp. 85-99.
12. Hughes, T.J.R., Franca, L.P. and Hulbert, G.M. (1989), "A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective-diffusive equations", Computational Methods in Applied Mechanics and Engineering, Vol. 73, pp. 173-89.
13. Ivey, G.N. (1984), "Experiments on transient natural convection in a cavity", Journal of Fluid Mechanics, Vol. 144, pp. 389-401.
14. Janssen, R.J.A. and Henkes, R.A.W.M. (1995), "Influence of Prandtl number on instability mechanisms and transition in a differentially heated square cavity", Journal of Fluid Mechanics, Vol. 290, pp. 319-44.
15. Le Quéré, P. and Behnia, M. (1998), "From onset of unsteadiness to chaos in a differentially heated square cavity", Journal of Fluid Mechanics, Vol. 359, pp. 81-107.
16. Lehoucq, R. and Salinger, A. (2001), "Large-scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers", International Journal of Numerical Methods in Fluids, Vol. 36, pp. 309-27.
17. Lehoucq, R.B., Sorensen, D.C. and Yang, C. (1998), ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Phildelphia, PA.
18. Maschhoff, K.J. and Sorensen, D.C. (1996), "P_ARPACK: an efficient portable large scale eigenvalue package for distributed memory parallel architectures", in Wasniewski, J., Dongarra, J., Madsen, K. and Olesen, D. (Eds), Applied Parallel Computing in Industrial Problems and Optimization, Lecture Notes in Computer Science, Vol. 1184, Springer-Verlag, Berlin.
19. Mayne, D.A., Usmani, A.S. and Crapper, M. (2000), "h-adaptive finite element solution of high Rayleigh number thermally driven cavity problem", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 10 No. 6, pp. 598-615.
20. Mayne, D.A., Usmani, A.S. and Crapper, M. (2001), "h-adaptive finite element solution of unsteady thermally driven cavity problem", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 11 No. 2, pp. 172-94.
21. Meerbergen, K. and Spence, A. (1997), "Implicitly restarted Arnoldi with purification for the shift-invert transformation", Mathematics of Computation, Vol. 218, pp. 667-89.
22. Mittelmann, H., Chang, K-T., Jankowski, D. and Neitzel, G. (1994), "Iterative solution of the eigenvalue problem in Hopf bifurcation for the Boussinesq equations", SIAM Journal of Scientific Computing, Vol. 15 No. 3, pp. 704-12.
23. Morzyńki, M., Afanasiev, K. and Thiele, F. (1999), "Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis", Computer Methods in Applied Mechanics and Engineering, Vol. 169, pp. 161-76.
24. Paolucci, S. and Chenoweth, D.R. (1989), "Transition to chaos in a differentially heated vertical cavity", Journal of Fluid Mechanics, Vol. 201, pp. 379-410.
25. Ravi, M.R., Henkes, R. and Hoodendoorn, C.J. (1994), "On the high Rayleigh number structure of steady laminar natural convection flow in a square enclosure", Journal of Fluid Mechanics, Vol. 262, pp. 325-51.
26. Salinger, A.G., Bou-Rabee, N.M., Pawlowski, R.P., Wilkes, E.D., Burroughs, E.A., Lehoucq, R.B. and Romero, L.A. (2002a), "LOCA 1.0 Library of continuation algorithms: theory and implementation manual", Technical Report SAND2002-0396, Sandia National Laboratories, Albuquerque, NM.
27. Salinger, A., Lehoucq, R., Pawlowski, R. and Shadid, J. (2002b), "Computational bifurcation and stability studies of the 8:1 cavity problem", International Journal of Numerical Methods in Fluids, Vol. 40, pp. 1059-73.
28. Salinger, A.G., Shadid, J.N., Hutchinson, S.A., Hennigan, G.L., Devine, K.D. and Moffat, H.K. (1999), "Analysis of gallium arsenide deposition in a horizontal chemical vapor deposition reactor using massively parallel computations", Journal of Crystal Growth, Vol. 203, pp. 516-33.
29. Shadid, J. (1999), "A fully-coupled Newton-Krylov solution method for parallel unstructured finite element fluid flow, heat and mass transport", IJCFD, Vol. 12, pp. 199-211.
30. Shadid, J., Tuminaro, R. and Walker, H. (1997), "An inexact newton method for fully coupled solution of the Navier-Stokes equations with heat and mass transport", Journal of Computational Physics, Vol. 137, pp. 155-85.
31. Tukerman, L.S., Bertagnolio, F., Daube, O., Le Quéré, P. and Barkley, D. (2000), "Stokes preconditioning for the inverse Arnoldi method", in Henry, D. and Bayeon, A. (Eds), Notes on Numerical Fluid Mechanics, Vol. 74.
32. Tuminaro, R.S., Heroux, M., Hutchinson, S.A. and Shadid, J.N. (1999), "Aztec user's guide: version 2.1", Technical Report SAND99-8801J, Sandia National Laboratories, Albuquerque, NM.
33. Xin, S. and Le Quéré, P. (1995), "Direct numerical simulations of two-dimensional chaotic natural convection in a differentially heated cavity of aspect ratio 4", Journal of Fluid Mechanics, Vol. 304, pp. 87-118.
34. Xin, S. and Le Quéré, P. (2001), "Linear stability analyses of natural convection flows in a differentially heated square cavity with conducting horizontal walls", Physics of Fluids, Vol. 13 No. 9, pp. 2529-42.
35. Xin, S., Le Quéré, P. and Daube, O. (1997), "Natural convection in a differentially heated horizontal cylinder: effects of Prandtl number on flow structure and instability", Physics of Fluids, Vol. 9 No. 4, pp. 1014-33.