Numerical convergence study of nearly incompressible, inviscid Taylor-Green vortex flow

Journal of Scientific Computing

Volumn 24, Issue 1, 2005, Pages 569-595

  Shu, Chi Wang ^a   Don, Wai Sun ^a   Gottlieb, David ^a   Schilling, Oleg ^b   Jameson, Leland ^b  

^a BROWN UNIVERSITY   (United States)

^b LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

Author keywords

Convergence; Spectral methods; Taylor Green vortex; Weighted essentially non oscillatory method

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; KINETIC ENERGY; LOW PASS FILTERS; PROBLEM SOLVING; VORTEX FLOW;

CONVERGENCE; SPECTRAL METHODS; TAYLOR-GREEN VORTEX; WEIGHTED ESSENTIALLY NON-OSCILLATORY METHOD;

CONVERGENCE OF NUMERICAL METHODS;

EID: 23244456854     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-004-5407-y     Document Type: Article

References (47)

1. Gottlieb, D., and Orszag, S. A. (1977). Numerical Analysis of Spectral Methods: Theory and Application, CBMS-NSF Regional Conference Series in Applied Mathematics Vol. 26, SIAM, Philadelphia.
2. Gottlieb, D., Hussaini, M. Y., and Orszag, S. A. (1984). Spectral Methods for Partial Differential Equations, SIAM, Philadelphia.
3. Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1990). Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York.
4. Canuto, C. (1996). Spectral methods. In Lesieur, M., Comte, P., and Zinn-Justin, J. (eds.), Computational Fluid Dynamics, Proceedings of the Les Hauches Summer School on Theoretical Physics 1993 Session LVIX, North-Holland, Amsterdam.
5. Bernardi, C., and Maday, Y. (1997). Spectral methods. In Ciarlet, P. G., and Lions, J. L. (eds.), Handbook of Numerical Analysis Vol. 5, North-Holland, Amsterdam.
6. Morinishi, Y., Lund, T. S., Vasilyev, O. V., and Moin, P. (1998). Fully conservative higher order finite difference schemes for incompressible flow. J. Comp. Phys. 143, 90-124.
7. Ham, F. E., Lien, F. S., and Strong, A. B. (2002). A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids. J. Comp. Phys. 177, 117-133.
8. Wissink, J. G. (2003). On unconditional conservation of kinetic energy by finite-difference discretizations of the linear and non-linear convection equation. Comp. and Fluids 33, 315-343.
9. Nicoud, F. (2000). Conservative high-order finite-difference schemes for Low-Mach number flows. J. Comp. Phys. 158, 71-97.
10. Majda, A. J., and Bertozzi, A. L. (2002). Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics Vol. 27, Cambridge University Press, Cambridge.
11. Brown, D. L., and Minion, M. L. (1995). Performance of under-resolved two-dimensional incompressible flow simulations. J. Comp. Phys. 122, 165-183.
12. Minion, M. L., and Brown, D. L. (1997). Performance of under-resolved two-dimensional incompressible flow simulations, II. J. Comp. Phys. 138, 734-765.
13. Drikakis, D. and Smolarkiewicz, P. K. (2001) On spurious vortical structures. J. Comp. Phys. 172, 309-325.
14. Batchelor, G. K. (2000). An Introduction to Fluid Dynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge.
15. Beale, J. T., Kato, T., and Majda, A. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94, 61-66.
16. Ponce, G. (1985). Remarks on a paper by J. T. Beale, T. Kato, and A. Majda. Comm. Math. Phys. 98, 349-353.
17. Majda, A. (1986). Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39, S187-S220.
18. Orszag, S. A. (1974). In Glowinski, R., and Lions, J. L. (eds.), Proceedings of the Symposium on Computer Methods in Applied Sciences and Engineering, part II, Springer-Verlag, New York.
19. Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H., and Frisch, U. (1983). Small-scale structure of the Taylor-Green vortex. J. Fluid Mech. 130, 411-452.
20. Mossi, M. (1999). Simulation of Benchmark and Industrial Unsteady Compressible Turbulent Fluid Flows, Ph.D. thesis, École Polytechnique Fédérale de Lausanne.
21. Benhamadouche, S., and Laurence, D. (2002). Global kinetic energy conservation with unstructured meshes. Int. J. Num. Meth. Fluids 40, 561-571.
22. Brachet, M. E., Meneguzzi, M., Politano, H., and Sulem, P. L. (1988). The dynamics of freely decaying two-dimensional turbulence. J. Fluid Mech. 194, 333-349.
23. Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H., and Frisch, U. (1984). The Taylor-Green vortex and fully-developed turbulence. J. Stat. Phys. 34, 1049-1063.
24. Brachet, M. E. (1991). Direct simulation of three-dimensional turbulence in the Taylor-Green vortex. Fluid Dyn. Res. 8, 1-8.
25. Brachet, M. E., Meneguzzi, M., Vincent, A., Politano, H., and Sulem, P. L. (1992). Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional Euler flows. Phys. Fluids A 4, 2845-2854.
26. Morf, R. H., Orszag, S. A., and Frisch, U. (1980). Spontaneous singularity in three-dimensional inviscid incompressible flow. Phys. Rev. Lett. 44, 572-575.
27. Ohkitani, K., and Gibbon, J. D. (2000). Numerical study of singularity formation in a class of Euler and Navier-Stokes flows. Phys. Fluids 12, 3181-3194.
28. Saffman, P. G. (1981). Dynamics of vorticity. J. Fluid Mech. 106, 49-58.
29. Shu, C.-W., and Osher, S. (1988). Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comp. Phys. 77, 439-471.
30. Gottlieb, D., and Hesthaven, J. S. (2001). Spectral methods for hyperbolic problems. J. Comp. Appl. Math. 128, 83-131.
31. Vandeven, H. (1991). Family of spectral filters for discontinuous problems. J. Sci. Comp. 6, 159-192.
32. Kreiss, H. O., and Oliger, J. (1972). Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199-215.
33. Jameson, L. (2000). High order schemes for resolving waves: number of points per wavelength. J. Sci. Comp. 15, 417-439.
34. Liu, X.-D., Osher, S., and T. Chan (1994). Weighted essentially non-oscillatory schemes. J. Comp. Phys. 115, 200-212.
35. Jiang, G.-S., and Shu, C.-W. (1996). Efficient implementation of weighted ENO schemes. J. Comp. Phys. 126, 202-228.
36. Balsara, D., and Shu, C.-W. (2000). Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comp. Phys. 160, 405-452.
37. Shu, C.-W. (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation Laws. In Advanced Numerical Approximation of Non-linear Hyperbolic Equations, Lecture Notes in Mathematics Vol. 1697, Springer-Verlag, New York.
38. Shu, C.-W. (1999). High order ENO and WENO schemes for computational fluid dynamics. In Barth, T. J., and Deconinck, H. (eds.), High-Order Methods for Computational Physics, Springer-Verlag, New York.
39. Harten, A., Osher, S., Engquist, B., and Chakravarthy, S. R. (1986). Some results on uniformly high-order accurate essentially nonoscillatory schemes. Appl. Num. Math. 2, 347-377.
40. Harten, A., and Osher, S. (1987). Uniformly high-order accurate non-oscillatory schemes. SIAM J. Num. Anal. 24, 279-309.
41. Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. R. (1987). Uniformly high order essentially non-oscillatory systems, III. J. Comp. Phys. 71, 231-303.
42. Shu, C.-W, and Osher, S. (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comp. Phys. 83, 32-78.
43. Tadmor, E. (1989). Convergence of spectral methods for nonlinear conservation laws. SIAM J. Num. Anal. 26, 30-44.
44. Tadmor, E. (1989). Shock Capturing by the Spectral Viscosity Method. In Proceedings of ICOSAHOM 89, IMACS, North-Holland, Amsterdam.
45. Maday, Y., Ould Kaber, S., and Tadmor, E. (1993). Legendre pseudospectral viscosity method for nonlinear conservation laws. SIAM J. Num. Anal. 30, 321-342.
46. Ma, H. (1998). Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws. SIAM J. Num. Anal. 35, 869-892.
47. Carpenter, M., Gottlieb, D., and Shu, C.-W. (2003). On the conservation and convergence to weak solutions of global schemes. J. Sci. Comp. 18, 111-132.