On the numerical solution of the regularized Birkhoff equations

Börgers, Christoph

doi:10.1090/S0025-5718-1989-0969481-2

On the numerical solution of the regularized Birkhoff equations

Author: Christoph Börgers
Journal: Math. Comp. 53 (1989), 141-156
MSC: Primary 76C05; Secondary 76-08, 76D25
DOI: https://doi.org/10.1090/S0025-5718-1989-0969481-2
MathSciNet review: 969481
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Birkhoff equations for the evolution of vortex sheets are regularized in a way proposed by Krasny. The convergence of numerical approximations to a fixed regularization is studied theoretically and numerically. The numerical test problem is a two-dimensional inviscid jet.

[1] Frederick H. Abernathy and Richard E. Kronauer, The formation of vortex streets, J. Fluid Mech. 13 (1962), 1–20. MR 138296, https://doi.org/10.1017/S0022112062000452
[2] H. Aref & E. D. Siggia, "Evolution and breakdown of a vortex street in two dimensions," J. Fluid Mech., v. 109, 1981, pp. 435-463.
[3] Garrett Birkhoff, Helmholtz and Taylor instability, Proc. Sympos. Appl. Math., Vol. XIII, American Mathematical Society, Providence, R.I., 1962, pp. 55–76. MR 0137423
[4] D. R. Boldman & P. F. Brinich & M. E. Goldstein, "Vortex shedding from a blunt trailing edge with equal and unequal external mean velocities," J. Fluid Mech., v. 75, 1976, pp. 721-735.
[5] Russel E. Caflisch and John S. Lowengrub, Convergence of the vortex method for vortex sheets, SIAM J. Numer. Anal. 26 (1989), no. 5, 1060–1080. MR 1014874, https://doi.org/10.1137/0726059
[6] Russel E. Caflisch and Oscar F. Orellana, Long time existence for a slightly perturbed vortex sheet, Comm. Pure Appl. Math. 39 (1986), no. 6, 807–838. MR 859274, https://doi.org/10.1002/cpa.3160390605
[7] Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, https://doi.org/10.1017/S0022112073002016
[8] C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
[9] Robert Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech. 167 (1986), 65–93. MR 851670, https://doi.org/10.1017/S0022112086002732
[10] R. Krasny, "Desingularization of periodic vortex sheet roll-up," J. Comput. Phys., v. 65, 1986, pp. 292-313.
[11] Robert Krasny, Computation of vortex sheet roll-up, Vortex methods (Los Angeles, CA, 1987) Lecture Notes in Math., vol. 1360, Springer, Berlin, 1988, pp. 9–22. MR 979557, https://doi.org/10.1007/BFb0089767
[12] E. Meiburg, "On the role of subharmonic perturbations in the far wake," J. Fluid Mech., v. 177, 1987, pp. 83-107.
[13] D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. London Ser. A 365 (1979), no. 1720, 105–119. MR 527594, https://doi.org/10.1098/rspa.1979.0009
[14] C. Sulem, P.-L. Sulem, C. Bardos, and U. Frisch, Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability, Comm. Math. Phys. 80 (1981), no. 4, 485–516. MR 628507