zbMATH — the first resource for mathematics

From Navier–Stokes to Stokes by means of particle methods. (English) Zbl 1088.76054
Summary: The viscous flow around a circular cylinder was investigated by means of a particle method over a wide Reynolds number range, from 0.0001 to 1000. A special care was devoted to the satisfaction of the no-slip condition which was expressed through a fourth-order partial differential equation for the stream function according to the method initially proposed by Y. Achdou and O. Pironneau [SIAM J. Numer. Anal. 32, No. 4, 985–1016 (1995; Zbl 0833.76032)]. This equation was solved by a boundary integral method which simultaneously satisfied a Dirichlet and a Neumann condition. The algorithm was immersed within a particle method framework and results in a versatile method which can deal with relatively high Reynolds numbers as well as Stokes flows. The numerical results were analysed and compared to those obtained by others numerically, experimentally and even theoretically for the low Reynolds number limit. The behaviour of the method for two extreme cases was specially investigated.

76M28 Particle methods and lattice-gas methods
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Achdou, Y.; Pironneau, O., Integral equations for the generalized Stokes operator, CR acad. sci. Paris, t. 415, Série I, 91-96, (1992) · Zbl 0754.76057
[2] Achdou, Y.; Pironneau, O., A fast solver for Navier-Stokes equations in the laminar regime using mortar finite element and boundary element methods, SIAM J. numer. anal., 32, 4, 985-1016, (1993) · Zbl 0833.76032
[3] Batchelor, G.K., An introduction to fluid dynamics, (1967), Cambridge University Press Cambridge · Zbl 0152.44402
[4] Beale, J.T., A convergent 3D vortex methods with grid free stretching, Math. comput., 46, 401-424, (1986) · Zbl 0602.76024
[5] Beale, J.T.; Majda, A., Rates of convergence for viscous splitting of the Navier Stokes equations, Math. comput., 37, 243-259, (1981) · Zbl 0518.76027
[6] Bouard, R.; Coutanceau, M., The early stage of development of the wake behind an impulsively started cylinder for 40<re<104, J. fluid mech., 101, 583-607, (1980)
[7] Bowman, F., Introduction to Bessel functions, (1958), Dover New York · JFM 64.1087.01
[8] Buzbee, B.L.; Dorr, F.W., The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions, SIAM J. numer. anal., 11, 753-763, (1974) · Zbl 0294.65059
[9] Chaplin, J.R., History forces and the unsteady wake of a cylinder, J. fluid mech., 393, 99-121, (1999) · Zbl 0954.76019
[10] Choquin, J.P.; Huberson, S., Particles simulation of viscous flow, Comput. fluids, 17, 397-410, (1989)
[11] Chorin, A.J., Numerical study of slightly viscous flow, J. fluid mech., 57, 785-796, (1973)
[12] Chorin, A.J., Vortex sheet approximation of boundary layers, J. comput. phys., 27, 428-442, (1978) · Zbl 0387.76040
[13] Collins, W.M.; Dennis, S.C.R., The initial flow past an impulsively started circular cylinder, Quart. J. mech. appl. math., XXVI, Pt. 1, 53-75, (1973) · Zbl 0267.76016
[14] Cottet, G.H., A new approach for the analysis of vortex methods in two and three dimensions, Ann. inst. H. Poincaré, 5, 227-285, (1988) · Zbl 0688.76017
[15] Cottet, G.H.; Koumoutsakos, P., Vortex methods theory and practice, (2000), Cambridge University Press Cambridge
[16] Cottet, G.H.; Mas-Gallic, S., A particle method to solve the Navier-Stokes system, Numer. math., 57, 805, (1989) · Zbl 0707.76029
[17] Cottet, G.H.; Poncet, P., Advances in direct numerical simulations of 3D wall-bounded flows by vortex-in-cell methods, J. comput. phys., 193, 136-158, (2003) · Zbl 1047.76092
[18] Daube, O., Resolution of the 2D Navier-Stokes equations in velocity-vorticity form by means of an influence matrix technique, J. comput. phys., 103, 402-414, (1992) · Zbl 0763.76046
[19] Degond, P.; Mas-Gallic, S., The weighted particle method for convection-diffusion equations, part 2: the anisotropic case, Math. comp., 53, 509, (1989) · Zbl 0676.65122
[20] Eldredge, J.D.; Leonard, A.; Colonius, T., A general treatment of derivatives in particle methods, J. comput. phys., 180, 686-709, (2002) · Zbl 1143.76550
[21] A. Gharakhani, A.F. Ghoniem, in: 3D Vortex Simulation of Flow in An Opposed-Piston Engine, Vortex Flows and Related Numerical Methods III ESAIM: Proceedings, vol. 7, 1999, pp. 161-172. · Zbl 0945.76063
[22] Graham, J.M.R., Computation of viscous separated flow using particle method, (), 310-317
[23] Huberson, S.; Jollès, A., Correction de l’erreur de projection dans LES méthodes particles/maillage, Rech. Aérosp., 4, 1-6, (1990) · Zbl 0711.76070
[24] Huberson, S.; Jollès, A.; Shen, W., Numerical simulation of incompressible viscous flows by means of particle methods, Lect. appl. math. ser., 28, 369-384, (1991) · Zbl 0751.76054
[25] Huner, B.; Hussey, R.G., Cylinder drag at low Reynolds number, Phys. fluids, 20, 8, 1211-1218, (1977)
[26] Kaplun, S., Low Reynolds number flow past a circular cylinder, J. math. mech., 6, 595-603, (1957) · Zbl 0080.18502
[27] Koumoutsakos, P.; Leonard, A., High resolution simulations of the flow around an impulsively started cylinder using vortex methods, J. fluid mech., 296, 1-38, (1995) · Zbl 0849.76061
[28] Lange, C.F.; Durst, F.; Breuer, M., Momentum and heat transfer from cylinders in laminar crossflow at 10^−4⩽re⩽200, Int. J. heat mass transf., 41, 3409-3430, (1998) · Zbl 0918.76012
[29] Leel, S.H.; Leal, L.G., Low Reynolds number flow past cylindrical bodies of arbitrary cross sectional shape, J. fluid mech., 164, 401-427, (1986) · Zbl 0587.76049
[30] Leonard, A., Computing three-dimensional incompressible flows with vortex elements, Annu. rev. fluid mech., 17, 523, (1985) · Zbl 0596.76026
[31] Martin, E.D., A generalized capacity matrix technique for computing aerodynamic flows, Comput. fluids, 2, 79, (1974) · Zbl 0328.76004
[32] Mas-Gallic, S., Deterministic particle method: diffusion and boundary conditions, Lect. appl. math. ser., 28, 433-465, (1991) · Zbl 0751.76055
[33] Napolitano, M.; Pascazio, G.; Quartapelle, L., A review of vorticity conditions in the numerical solution of the equations, Comp. fluids, 28, 139-185, (1999) · Zbl 0964.76075
[34] Ploumhans, P.; Winckelmans, G.S., Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry, J. comput. phys., 165, 354-406, (2000) · Zbl 1006.76068
[35] Ploumhans, P.; Winckelmans, G.S.; Salmon, J.K.; Leonard, A.; Warren, M.S., Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at re=300, 500 and 1000, J. comput. phys., 178, 427-463, (2002) · Zbl 1045.76030
[36] Proudman, I.; Pearson, J.R., Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder, J. fluid mech., 2, 237-262, (1957) · Zbl 0077.39103
[37] J. Salvi, Contribution théorique et numérique des méthodes intégrales de frontière à la résolution des équations de Navier Stokes bidimensionnelles en formulation vitesse tourbillon, Ph.D. Thesis, Ecole polytechnique, 1999.
[38] Tritton, D.J., Experiments on the flow past a circular cylinder at low Reynolds numbers, J. fluid mech., 6, 547-567, (1959) · Zbl 0092.19502
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.