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.


76M20 Finite difference methods applied to problems in fluid mechanics
76F99 Turbulence
76B47 Vortex flows for incompressible inviscid fluids
Full Text: DOI


[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.