zbMATH — the first resource for mathematics

Singularity formation for complex solutions of the 3D incompressible Euler equations. (English) Zbl 0789.76013
Summary: Moore’s approximation method [D. W. Moore, Proc. R. Soc. London, Ser. A 365, 105-119 (1979; Zbl 0404.76040)], first formulated for vortex sheets, is generalized and applied to axi-symmetric flow with swirl and with smooth initial data. The approximation preserves the forward cascade of energy but neglects any backflow of energy. It splits the Euler equations into two sets of equations: one for $$u_ +=u_ +(r,z,t)$$ containing all non-negative wavenumbers (in $$z$$) and the second for $$u_ -=\bar u_ +$$. The equations for $$u_ +$$ are exactly the Euler equations but with complex initial data. Traveling waves solutions $$u_ +=u_ +(r,z-i\sigma t)$$ with imaginary wave speed are found numerically for this problem. The asymptotic properties of the resulting Fourier coefficients show a singularity forming in finite time at which the velocity blows up.

MSC:
 76B47 Vortex flows for incompressible inviscid fluids 35Q35 PDEs in connection with fluid mechanics
MPFUN
Full Text:
References:
 [1] Abramowitz, M.; Stegun, I.A., () [2] Anderson, C.R.; Greengard, C., The vortex ring merger problem at infinite Reynolds number, Commun. pure appl. math., 42, 1123-1139, (1989) · Zbl 0689.76011 [3] Bailey, D.H., Automatic translation of Fortran programs to multi-precision, RNR technical report RNR-90-025, (1991) [4] Bailey, D.H., Mpfun: A portable high performance multiprecision package, RNR technical report RNR-90-022, (1991) [5] Baker, G.R.; Caflisch, R.E.; Siegel, M., Singularity formation during Rayleigh-Taylor instability, J. fluid mech., (1993), to appear · Zbl 0791.76027 [6] Bardos, C.; Benachour, S., Domaine d’analycite des solutions de l’equation d’Euler dans un ouvert de $$R$$_n, Annali Della scuola normale superiore di Pisa IV, 4, 647-687, (1977) · Zbl 0366.35022 [7] Beale, J.T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. math. phys., 94, 61-66, (1984) · Zbl 0573.76029 [8] Bell, J.B.; Marcus, D.L., Vorticity intensification and transition to turbulence in the three-dimensional Euler equations, Commun. math. phys., 147, 371-394, (1992) · Zbl 0755.76062 [9] Bhattacharjee, A.; Wang, X., Finite-time vortex singularity in a model of three-dimensional Euler flows, Phys. rev. lett., 69, 2196-2199, (1992) · Zbl 0968.76528 [10] Brachet, M.E.; Meiron, D.; Orszag, S.; Nickel, B.; Morf, R.; Frisch, U., Small-scale structure of the Taylor-Green vortex, J. fluid mech., 130, 411-452, (1983) · Zbl 0517.76033 [11] Caflisch, R.E.; Ercolani, N.; Hou, T.Y., Multi-valued solutions and branch point singularities for nonlinear hyperbolic systems, Commun. pure appl. math., 46, 453-499, (1993) · Zbl 0797.35007 [12] Caflisch, R.E.; Li, Xiaofan; Shelley, M.J., The collapse of an axi-symmetric swirling vortex sheet, Nonlinearity, (1993), to appear · Zbl 0796.76022 [13] Caflisch, R.E.; Orellana, O.F., Long time existence for a slightly perturbed vortex sheet, Commun. pure appl. math., 39, 807-838, (1989) · Zbl 0603.76039 [14] Caflisch, R.E.; Orellana, O.F., Singularity formulation and ill-posedness for vortex sheets, SIAM J. math. anal., 20, 293-307, (1989) · Zbl 0697.76029 [15] Caflisch, R.E.; Orellana, O.F.; Siegel, M., A localized approximation for vortical flows, SIAM J. appl. math., 50, 1517-1532, (1990) · Zbl 0712.76026 [16] Childress, S.; Ierley, G.R.; Spiegel, E.A.; Young, W.R., Blow-up of unsteady two dimensional Euler and Navier-Stokes solutions having stagnation point form, J. fluid mech., 203, 1-22, (1989) · Zbl 0674.76013 [17] Chorin, A., Estimates of intermittency, spectra and blowup in developing turbulence, Commun. pure appl. math., 24, 853-856, (1981) [18] Chorin, A., The evolution of a turbulent vortex, Commun. math. phys., 83, 517-535, (1982) · Zbl 0494.76024 [19] Constantin, P.; Kadanoff, L.P., Dynamics of a complex interface, Physica D, 47, 450-460, (1991) · Zbl 0713.76107 [20] Grauer, R.; Sideris, T., Numerical computation of 3D incompressible ideal fluids with swirl, Phys. rev. lett., 25, 3511-3514, (1991) [21] Greene, J.M.; Percival, I.C., Hamiltonian maps in the complex plane, Physica D, 3, 530-548, (1981) · Zbl 1194.37071 [22] Howison, S.D.; Ockendon, J.R.; Lacey, A.A., Singularity development in moving-boundary problems, Quart. J. appl. math., 38, 343-360, (1985) · Zbl 0591.35087 [23] Kerr, R.M., Evidence for a singularity of the three-dimensional, incompressible Euler equations, (1992) · Zbl 0800.76081 [24] Krasny, R., Desingularization of periodic vortex sheet roll-up, J. comput. phys., 65, 292-313, (1986) · Zbl 0591.76059 [25] Krasny, R., A study of singularity formation in a vortex sheet by point-vortex approximation, J. fluid mech., 167, 65-93, (1986) · Zbl 0601.76038 [26] D.I. Meiron and M.J. Shelley, personal communication (1992). [27] Moore, D.W., The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, (), 105-119 · Zbl 0404.76040 [28] Moore, D.W., Numerical and analytical aspects of Helmholtz instability, (), 629-633 · Zbl 0667.76075 [29] Pugh, D.A., Development of vortex sheets in Boussinesq flows—formation of singularities, () [30] Pugh, D.A.; Cowley, S.J., On the formation of an interface singularity in the rising 2D Boussinesq bubble, J. fluid mech., (1993), to appear [31] Pumir, A.; Siggia, E.D., Collapsing solutions to the 3D Euler equations, Phys. fluids A, 2, 220, (1990) · Zbl 0696.76070 [32] Pumir, A.; Siggia, E.D., Development of singular solutions to the axisymmetric Euler equations, Phys. fluids A, 4, 1472-1491, (1992) · Zbl 0825.76121 [33] Rotunno, R., A note on the stability of a cylindrical vortex sheet, J. fluid mech., 87, 761-771, (1978) · Zbl 0389.76039 [34] Saffman, P.G., Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell, Quart. J. appl. math., 12, 146-150, (1959) · Zbl 0087.21203 [35] Shelley, M.J., A study of singularity formation in vortex sheet motion by a spectrally accurate vortex method, J. fluid mech., 244, 493-526, (1992) · Zbl 0775.76047 [36] Shelley, M.J.; Meiron, D.I.; Orszag, S.A., Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex reconnection of perturbed anti-parallel vortex tubes, J. fluid mech., 246, 613-652, (1993) · Zbl 0781.76028 [37] Siegel, M., An analytical and numerical study of singularity formation in the Rayleigh-Taylor problem, () [38] Siggia, E.D., Collapse and amplification of a vortex filament, Phys. fluids, 28, 794-805, (1985) · Zbl 0596.76025 [39] Stuart, J.T., Nonlinear Euler partial differential equations: singularities in their solution, () · Zbl 0332.35019 [40] Sulem, C.; Sulem, P.L.; Frisch, H., Tracing complex singularities with spectral methods, J. comput. phys., 50, 138-161, (1983) · Zbl 0519.76002 [41] Tanveer, S., Evolution of Hele-Shaw interface for small surface tension, Philos. trans. R. soc. London, (1992), to appear · Zbl 0778.76029 [42] Tanveer, S., Singularities in the classical Rayleigh-Taylor flow: formation and subsequent motion, (), to appear · Zbl 0789.76031 [43] Wang, X.; Bhattacharjee, A., Is there a finite-time singularity in axisymmetric Euler flows?, (1991)
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.