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Vorticity intensification and transition to turbulence in the three- dimensional Euler equations. (English) Zbl 0755.76062

Summary: The evolution of a perturbed vortex tube is studied by means of a second- order projection method for the incompressible Euler equations. We observe, to the limits of grid resolution, a nonintegrable blowup in vorticity. The onset of the intensification is accompanied by a decay in the mean kinetic energy. Locally, the intensification is characterized by tightly curved regions of alternating-sign vorticity in a \(2n\)-pole structure. After the first \(L^ \infty\) peak, the enstrophy and entropy continue to increase, and we observe reconnection events, continued decay of the mean kinetic energy, and the emergence of a Kolmogorov \((k^{- 5/3})\) range in the energy spectrum.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76F99 Turbulence
76B47 Vortex flows for incompressible inviscid fluids
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[1] Ashurst, W., Meiron, D.: Numerical study of vortex reconnection. Phys. Rev. Lett.58, 1632 (1987) · Zbl 0671.76045
[2] Batchelor, G.K., Townsend, A.A.: Decay of vorticity in isotropic turbulence. Proc. R. Soc. A191, 534–550 (1947)
[3] Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys.94, 61–66 (1989) · Zbl 0573.76029
[4] Bell, J.B., Colella, P., Glaz, H.: A second-order projection method for the incompressible Navier-Stokes equations. J. Comp. Phys. 257–283 (1989) · Zbl 0681.76030
[5] Bell, J.B., Solomon, J.M., Szymczak, W.G.: A second-order projection method for the three-dimensional Euler and Navier-Stokes equations. Preprint 1990
[6] Brachet, M., Meiron, D., Orszag, S., Nickel, B., Morf, R., Frisch, U.: Small-scale structure of the Taylor-Green vortex. J. Fluid Mech.130, 411 (1983) · Zbl 0517.76033
[7] Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp.22, 745–762 (1968) · Zbl 0198.50103
[8] Chorin, A.J.: The evolution of a turbulent vortex. Commun. Math. Phys.83, 517–535 (1982) · Zbl 0494.76024
[9] Chorin, A.J.: Turbulence and vortex stretching on a lattice. Commun. Pure Appl. Math.39, S47-S65 (1986) · Zbl 0595.60098
[10] Chorin, A.J.: Hairpin removal in vortex interaction. LBL-26173, Lawrence Berkeley Laboratory Report, 1988
[11] Chorin, A.J., Akao, J.: Vortex equilibria in turbulence and quantum analogues. Physica D, 1991 (in press) · Zbl 0736.76025
[12] Chorin, A.J.: Statistical mechanics and vortex motion. PAM-500, Center for Pure and Applied Mathematics Report, University of California, Berkeley 1990 · Zbl 0708.76077
[13] Chorin, A.J.: Equilibrium statistics of a vortex filament with applications. LBL-30419, Lawrence Berkeley Laboratory Report, 1991 · Zbl 0734.76015
[14] Colella, P.: Multidimensional upwind methods for hyperbolic conservation laws. J. Comp. Phys.94, 61–66 (1990) · Zbl 0694.65041
[15] Falco, R.E.: Phys. Fluids20, S124 (1977)
[16] de Gennes, P.G.: Scaling Concepts in Polymer Physics, Ithaca, NY: Cornell University Press 1971
[17] Kerr, R.M., Hussain, F.: Simulation of vortex reconnection. Physica D37, 474–484 (1989)
[18] Kim, J., Moin, P.: The structure of the vorticity field in turbulent channel flow. Part 2. Study of ensemble averaged fields. J. Fluid Mech.162, 339 (1986)
[19] Moin, P., Leonard, A., Kim, J.: Evolution of a curved vortex filament into a vortex ring. Phys. Fluids29, 955–963 (1986)
[20] Moin, P., Rogers, M.M., Moser, R.D.: Proceedings of the Fifth Symposium on Turbulent Shear Flows, Cornell University, Ithaca, NY, 1985
[21] Numerical simulations of turbulence. In: Rosenblatt, M., Van Atta, C. (eds.). Statistical Models of Turbulence, pp. 127–147. Berlin, Heidelberg, New York: Springer 1972
[22] Pumir, A., Kerr, R.M.: Numerical simulations of interacting vortex tubes. Phys. Rev. Lett.58, 1636 (1987)
[23] Pumir, A., Siggia, E.D.: Vortex dynamics and the existence of solutions to the Navier-Stokes equations. Phys. Fluids30, 1606–1626 (1987) · Zbl 0628.76033
[24] Pumir, A., Siggia, E.D.: Collapsing solutions to the 3-D Euler equations. Phys. Fluids A2 (3), 220–241 (1990) · Zbl 0696.76070
[25] Shelley, M.J., Meiron, D.I.: Vortex reconnection and smoothness of the Euler equations. In: Anderson, C., Greengard, C. (eds.). Lectures in Applied Mathematics, AMS, 1991 · Zbl 0751.76019
[26] Siggia, E.D.: Collapse and amplification of a vortex filament. Phys. Fluids28, 794–805 (1985) · Zbl 0596.76025
[27] Zabusky, N.J., Melander, M.V.: Three-dimensional vortex tube reconnection: Morphology for orthogonally offset tubes. Physica D37, 555–562 (1989)
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