Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Comparison of numerical methods for the calculation of two-dimensional turbulence


Authors: G. L. Browning and H.-O. Kreiss
Journal: Math. Comp. 52 (1989), 369-388
MSC: Primary 65P05; Secondary 76-08, 76F99
MathSciNet review: 955748
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The estimate derived by Henshaw, Kreiss, and Reyna for the smallest scale present in solutions of the two-dimensional incompressible Navier-Stokes equations is employed to obtain convergent pseudospectral approximations. These solutions are then compared with those obtained by a number of commonly used numerical methods.

If the viscosity term is deleted and the energy in high wave numbers removed by setting the amplitudes of all wave numbers above a certain point in the spectrum to zero, the "chopped" solution differs considerably from the convergent solution, even at early times. In the case that the regular viscosity is replaced by a hyperviscosity term, i.e., the square of the Laplacian, we also derive an estimate for the smallest scale present. If the coefficient of hyperviscosity is chosen so that the spectrum of the hyperviscosity solution disappears at the same point as for the regular viscosity solution, the hyperviscosity solution is also completely different from the convergent solution. If we "tune" the hyperviscosity coefficient, then the solutions are similar in amplitude or phase, but not both.

The solution obtained by a second-order difference method with twice the number of points as the pseudospectral model, or a fourth-order difference method with the same number of points as the pseudospectral model, is essentially identical to the convergent solution. This is reasonable since most of the energy of the solution is contained in the lower part of the spectrum.


References [Enhancements On Off] (What's this?)

  • [1] D. L. Book, NRL Plasma Formulary, Naval Research Laboratory, 1980, 60 pp.
  • [2] M. E. Brachet, M. Meneguzzi & P. L. Sulem, "Small-scale dynamics of high-Reynolds-number two-dimensional turbulence," Phys. Rev. Lett., v. 57, 1986, pp. 683-686.
  • [3] G. S. Deem & N. J. Zabusky, "Vortex waves: stationary V-states, interactions, recurrence and breaking," Phys. Rev. Lett., v. 40, 1978, pp. 859-862.
  • [4] B. Fornberg, "A numerical study of 2-d turbulence," J. Comput. Phys., v. 25, 1977, pp. 1-31.
  • [5] D. G. Fox & S. A. Orszag, "Pseudospectral approximation to two-dimensional turbulence," J. Comput. Phys., v. 11, 1973, pp. 612-619.
  • [6] W. D. Henshaw, H.-O. Kreiss & L. G. Reyna, On the Smallest Scale for the Incompressible Navier-Stokes Equations, ICASE Report No. 88-8, 1988, 49 pp.
  • [7] J. R. Herring, S. A. Orszag, R. H. Kraichnan & D. G. Fox, "Decay of two-dimensional turbulence," J. Fluid Mech., v. 66, 1974, pp. 417-444.
  • [8] Heinz-Otto Kreiss and Joseph Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), 199–215 (English, with Russian summary). MR 0319382 (47 #7926)
  • [9] D. K. Lilly, "Numerical simulation of developing and decaying of two-dimensional turbulence," J. Fluid Mech., v. 45, 1971, pp. 395-415.
  • [10] J. C. McWilliams, "The emergence of isolated coherent vortices in turbulent flow," J. Fluid Mech., v. 146, 1984, pp. 21-43.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65P05, 76-08, 76F99

Retrieve articles in all journals with MSC: 65P05, 76-08, 76F99


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1989-0955748-0
PII: S 0025-5718(1989)0955748-0
Article copyright: © Copyright 1989 American Mathematical Society