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A multi-moment vortex method for 2D viscous fluids. (English) Zbl 1408.76210

Summary: We introduce simplified, exact, combinatorial formulas that arise in the vortex interaction model found in [R. Nagem et al., SIAM J. Appl. Dyn. Syst. 8, No. 1, 160–179 (2009; Zbl 1408.76205)]. These combinatorial formulas allow for the efficient implementation and development of a new multi-moment vortex method (MMVM) using a Hermite expansion to simulate 2D vorticity. The method naturally allows the particles to deform and become highly anisotropic as they evolve without the added cost of computing the non-local Biot-Savart integral. We present three examples using MMVM. We first focus our attention on the implementation of a single particle, large number of Hermite moments case, in the context of quadrupole perturbations of the Lamb-Oseen vortex. At smaller perturbation values, we show the method captures the shear diffusion mechanism and the rapid relaxation (on \(Re^{1/3}\) time scale) to an axisymmetric state. We then present two more examples of the full multi-moment vortex method and discuss the results in the context of classic vortex methods. We perform numerical tests of convergence of the single particle method and show that at least in simple cases the method exhibits the exponential convergence typical of spectral methods. Lastly, we numerically investigate the spatial accuracy improvement from the inclusion of higher Hermite moments in the full MMVM.

MSC:

76D17 Viscous vortex flows
76M23 Vortex methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids

Citations:

Zbl 1408.76205
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References:

[1] Agullo, Olivier; Verga, Alberto, Effect of viscosity in the dynamics of two point vortices: exact results, Physical Review E, 63 (2001) · Zbl 1256.76021
[2] Anderson, Christopher; Greengard, Claude, On vortex methods, SIAM Journal on Numerical Analysis, 22, 3, 413-440 (1985) · Zbl 0578.65121
[3] Barba, L. A.; Leonard, A., Emergence and evolution of tripole vortices from net-circulation initial conditions, Physics of Fluids, 19, 1, 017101 (2007) · Zbl 1146.76319
[4] Barba, L. A.; Leonard, A.; Allen, C. B., Advances in viscous vortex methods – meshless spatial adaption based on radial basis function interpolation, International Journal for Numerical Methods in Fluids, 47, 5, 387-421 (2005) · Zbl 1085.76052
[5] Barba, L. A.; Rossi, Louis F., Global field interpolation for particle methods, Journal of Computational Physics, 229, 4, 1292-1310 (2010) · Zbl 1329.76282
[6] Lorena A. Barba, Vortex Method of Computing High-Reynolds number Flows: Increased Accuracy with a Fully Mesh-less Formulation, Ph.D. Thesis, California Institute of Technology, 2004.; Lorena A. Barba, Vortex Method of Computing High-Reynolds number Flows: Increased Accuracy with a Fully Mesh-less Formulation, Ph.D. Thesis, California Institute of Technology, 2004.
[7] Thomas Beale, J.; Majda, Andrew, Vortex methods. II: higher order accuracy in two and three dimensions, Mathematics of Computation, 39, 159, 29-52 (1982) · Zbl 0488.76025
[8] Bernoff, Andrew J.; Lingevitch, Joseph F., Rapid relaxation of an axisymmetric vortex, Physical Fluids, 6, 11, 3717-3723 (1995) · Zbl 0838.76024
[9] Hussaini, M. Y.; Quarteroni, A.; Zang, Th. A.; Canuto, C. G., Spectral Methods: Fundamentals in Single Domains (2006), Springer: Springer Berlin · Zbl 1093.76002
[10] Cerretelli, C.; Williamson, C. H.K., The physical mechanism for vortex merging, Journal of Fluid Mechanics, 475, 41-77 (2003) · Zbl 1048.76500
[11] Chatelain, Philippe; Curioni, Alessandro; Bergdorf, Michael; Rossinelli, Diego; Andreoni, Wanda; Koumoutsakos, Petros, Billion vortex particle direct numerical simulations of aircraft wakes, Computer Methods in Applied Mechanics and Engineering, 197, 13-16, 1296-1304 (2008) · Zbl 1159.76368
[12] Chorin, A. J.; Bernard, P., Discretization of a vortex sheet, with an example of roll-up, Journal of Computation Physics, 13, 423-429 (1973) · Zbl 0273.76022
[13] Cocle, Roger; Winckelmans, Grgoire; Daeninck, Goric, Combining the vortex-in-cell and parallel fast multipole methods for efficient domain decomposition simulations, Journal of Computational Physics, 227, 21, 9091-9120 (2008), (Special Issue Celebrating Tony Leonard’s 70th Birthday) · Zbl 1391.76520
[14] Cottet, Georges-Henri; Koumoutsakos, Petros D., Theory and practice, Vortex Methods (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0953.76001
[15] Eldredge, Jeff D.; Colonius, Tim; Leonard, Anthony, A vortex particle method for two-dimensional compressible flow, Journal of Computational Physics, 179, 2, 371-399 (2002) · Zbl 1130.76393
[16] Gallay, Thierry; Eugene Wayne, C., Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(R}^2\), Archive for Rational Mechanics and Analysis, 163, 3, 209-258 (2002) · Zbl 1042.37058
[17] Greengard, Claude, The core spreading vortex method approximates the wrong equation, Journal of Computational Physics, 61, 2, 345-348 (1985) · Zbl 0587.76039
[18] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, Journal of Computational Physics, 73, 2, 325-348 (1987) · Zbl 0629.65005
[19] Huang, Mei-Jiau; Su, Huan-Xun; Chen, Li-Chieh, A fast resurrected core-spreading vortex method with no-slip boundary conditions, Journal of Computational Physics, 228, 6, 1916-1931 (2009) · Zbl 1280.76028
[20] Josserand, Ch.; Rossi, M., The merging of two co-rotating vortices: a numerical study, European Journal of Mechanics B. Fluids, 26, 6, 779-794 (2007) · Zbl 1152.76445
[21] Le Dizès, Stéphane; Verga, Alberto, Viscous interactions of two co-rotating vortices before merging, Journal of Fluid Mechanics, 467, 389-410 (2002) · Zbl 1062.76014
[22] Leonard, A., Vortex methods for flow simulation, Journal of Computational Physics, 37, 289-335 (1980) · Zbl 0438.76009
[23] Lindsay, Keith; Krasny, Robert, A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, Journal of Computational Physics, 172, 2, 879-907 (2001) · Zbl 1002.76093
[24] Lingevitch, Joseph F.; Bernoff, Andrew J., Distortion and evolution of a localized vortex in an irrotational flow, Physical Fluids, 7, 5, 1015-1026 (1995) · Zbl 1023.76517
[25] Lundgren, T. S., Strained spiral vortex model for turbulent fine structure, Physics of Fluids, 25, 12, 2193-2203 (1982) · Zbl 0536.76034
[26] Majda, Andrew J.; Bertozzi, Andrea L., Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27 (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.76001
[27] Melander, M. V.; Styczek, A. S.; Zabusky, N. J., Elliptically desingularized vortex model for the two-dimensional Euler equations, Physical Review Letters, 53, 1222-1225 (1984)
[28] Melander, M. V.; Zabusky, N. J.; Styczek, A. S., A moment model for vortex interactions of the two-dimensional Euler equations. Part 1: computational validation of a Hamiltonian elliptical representation, Journal of Fluid Mechanics, 167, 95-115 (1986) · Zbl 0602.76026
[29] Meunier, P.; Ehrenstein, U.; Leweke, T.; Rossi, M., A merging criterion for two-dimensional co-rotating vortices, Physical Fluids, 14, 8, 2757-2766 (2002) · Zbl 1185.76255
[30] Meunier, Patrice; DizFs, StTphane Le; Leweke, Thomas, Physics of vortex merging, Comptes Rendus Physique, 6, 4-5, 431-450 (2005), Aircraft trailing vortices
[31] Moeleker, Piet; Leonard, Anthony, Lagrangian methods for the tensor-diffusivity subgrid model, Journal of Computational Physics, 167, 1, 1-21 (2001) · Zbl 1014.76036
[32] Nagem, Raymond J.; Sandri, Guido; Uminsky, David, Vorticity dynamics and sound generation in two-dimensional fluid flow, The Journal of the Acoustical Society of America, 122, 1, 128-134 (2007)
[33] Nagem, Raymond J.; Sandri, Guido; Uminsky, David; Eugene Wayne, C., Generalized Helmholtz-Kirchhoff model for two dimensional distributed vortex motion, SIAM Journal on Applied Dynamical Systems, 8, 1, 160-179 (2009) · Zbl 1408.76205
[34] Rhines, P. B.; Young, W. R., How rapidly is a passive scalar mixed within closed streamlines?, Journal of Fluid Mechanics, 133, 133-145 (1983) · Zbl 0576.76088
[35] Platte, Rodrigo B.; Rossi, Louis F.; Mitchell, Travis B., Using global interpolation to evaluate the Biot-Savart integral for deformable elliptical gaussian vortex elements, SIAM Journal on Scientific Computing, 31, 3, 2342-2360 (2009) · Zbl 1192.65029
[36] Ploumhans, P.; Winckelmans, G. S., Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry, Journal of Computational Physics, 165, 2, 354-406 (2000) · Zbl 1006.76068
[37] Rossi, Louis F., Resurrecting core spreading vortex methods: a new scheme that is both deterministic and convergent, SIAM Journal of Scientific Computing, 17, 2, 370-397 (1996) · Zbl 0848.35091
[38] Rossi, Louis F., Achieving high-order convergence rates with deforming basis functions, SIAM Journal on Scientific Computing, 26, 3, 885-906 (2005) · Zbl 1075.35056
[39] Rossi, Louis F., A comparative study of lagrangian methods using axisymmetric and deforming blobs, SIAM Journal on Scientific Computing, 27, 4, 1168-1180 (2006) · Zbl 1100.35080
[40] Rossi, Louis F., Evaluation of the Biot-Savart integral for deformable elliptical gaussian vortex elements, SIAM Journal on Scientific Computing, 28, 4, 1509-1532 (2006) · Zbl 1122.35099
[41] Rossi, Louis F.; Lingevitch, Joseph F.; Bernoff, Andrew J., Quasi-steady monopole and tripole attractors for relaxing vortices, Physics of Fluids, 9, 8, 2329-2338 (1997) · Zbl 1185.76505
[42] Rossinelli, Diego; Bergdorf, Michael; Cottet, Georges-Henri; Koumoutsakos, Petros, GPU accelerated simulations of bluff body flows using vortex particle methods, Journal of Computational Physics, 229, 9, 3316-3333 (2010) · Zbl 1307.76066
[43] David T. Uminsky, The Viscous N Vortex Problem: A Generalized Helmholtz/Kirchhoff Approach, Ph.D. Thesis, Boston University, 2009.; David T. Uminsky, The Viscous N Vortex Problem: A Generalized Helmholtz/Kirchhoff Approach, Ph.D. Thesis, Boston University, 2009.
[44] van Rees, Wim M.; Leonard, Anthony; Pullin, D. I.; Koumoutsakos, Petros, A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers, Journal of Computational Physics, 230, 8, 2794-2805 (2011) · Zbl 1316.76066
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