Comparison of numerical methods for the calculation of twodimensional turbulence
Authors:
G. L. Browning and H.O. Kreiss
Journal:
Math. Comp. 52 (1989), 369388
MSC:
Primary 65P05; Secondary 7608, 76F99
MathSciNet review:
955748
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Abstract: The estimate derived by Henshaw, Kreiss, and Reyna for the smallest scale present in solutions of the twodimensional incompressible NavierStokes 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 secondorder difference method with twice the number of points as the pseudospectral model, or a fourthorder 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.
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D. G. Fox & S. A. Orszag, "Pseudospectral approximation to twodimensional turbulence," J. Comput. Phys., v. 11, 1973, pp. 612619.
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W. D. Henshaw, H.O. Kreiss & L. G. Reyna, On the Smallest Scale for the Incompressible NavierStokes Equations, ICASE Report No. 888, 1988, 49 pp.
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J. R. Herring, S. A. Orszag, R. H. Kraichnan & D. G. Fox, "Decay of twodimensional turbulence," J. Fluid Mech., v. 66, 1974, pp. 417444.
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HeinzOtto
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)
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D. K. Lilly, "Numerical simulation of developing and decaying of twodimensional turbulence," J. Fluid Mech., v. 45, 1971, pp. 395415.
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J. C. McWilliams, "The emergence of isolated coherent vortices in turbulent flow," J. Fluid Mech., v. 146, 1984, pp. 2143.
 [1]
 D. L. Book, NRL Plasma Formulary, Naval Research Laboratory, 1980, 60 pp.
 [2]
 M. E. Brachet, M. Meneguzzi & P. L. Sulem, "Smallscale dynamics of highReynoldsnumber twodimensional turbulence," Phys. Rev. Lett., v. 57, 1986, pp. 683686.
 [3]
 G. S. Deem & N. J. Zabusky, "Vortex waves: stationary Vstates, interactions, recurrence and breaking," Phys. Rev. Lett., v. 40, 1978, pp. 859862.
 [4]
 B. Fornberg, "A numerical study of 2d turbulence," J. Comput. Phys., v. 25, 1977, pp. 131.
 [5]
 D. G. Fox & S. A. Orszag, "Pseudospectral approximation to twodimensional turbulence," J. Comput. Phys., v. 11, 1973, pp. 612619.
 [6]
 W. D. Henshaw, H.O. Kreiss & L. G. Reyna, On the Smallest Scale for the Incompressible NavierStokes Equations, ICASE Report No. 888, 1988, 49 pp.
 [7]
 J. R. Herring, S. A. Orszag, R. H. Kraichnan & D. G. Fox, "Decay of twodimensional turbulence," J. Fluid Mech., v. 66, 1974, pp. 417444.
 [8]
 H. O. Kreiss & J. Oliger, "Comparison of accurate methods for the integration of hyperbolic equations," Tellus, v. 24, 1972, pp. 199215. MR 0319382 (47:7926)
 [9]
 D. K. Lilly, "Numerical simulation of developing and decaying of twodimensional turbulence," J. Fluid Mech., v. 45, 1971, pp. 395415.
 [10]
 J. C. McWilliams, "The emergence of isolated coherent vortices in turbulent flow," J. Fluid Mech., v. 146, 1984, pp. 2143.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909557480
PII:
S 00255718(1989)09557480
Article copyright:
© Copyright 1989 American Mathematical Society
