×

zbMATH — the first resource for mathematics

Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. (English) Zbl 1222.76072
Summary: The local multi-quadric differential quadrature (LMQDQ) method is applied on three-dimensional incompressible flow problems. The LMQDQ method is among the newly proposed mesh-free methods. Unlike the traditional differential quadrature (DQ) method, the weighting coefficients of LMQDQ method are determined by using the radial basis functions as trial functions instead of high-order polynomials. The main concern of this paper is to discuss the effectiveness of using LMQDQ method to solve 3D incompressible Navier-Stokes equations in the primitive-variable form. Three-dimensional lid-driven cavity flow problem with Reynolds numbers of 100, 400 and 1000 was chosen as a test case to validate the LMQDQ method. The computed velocity profiles along the vertical and horizontal centre lines are given and compared with available data in the literature.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lucy, L.B., A numerical approach to the testing of the fission hypothesis, Astron. J., 8, 1013-1024, (1977)
[2] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[3] Liu, W.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[4] Babuska, I.; Melenk, J., The partition of unity method, Int. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[5] C.A. Duarte, J.T. Oden, Hp clouds-a meshless method to solve boundary-value problems, TICAM Report 95-05.
[6] Oñate, E.; Idelsohn, S.; Zienkiewicz, O.C.; Taylor, R.L., A finite point method in computational mechanics: application to convective transport and fluid flow, Int. J. numer. methods engrg., 39, 3839-3866, (1996) · Zbl 0884.76068
[7] Atluri, S.N.; Zhu, T., New meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. mech., 22, 2, 117-127, (1998) · Zbl 0932.76067
[8] Doblare, M.; Cueto, E.; Calvo, B.; Martinez, M.A.; Garcia, J.M.; Cegonino, J., On the employ of meshless methods in biomechanics, Comput. methods appl. mech. engrg., 194, 801-821, (2005) · Zbl 1112.74563
[9] Q.W. Ma, Meshless local Petrov-Galerkin method for two-dimensional nolinear water wave problems, J Comput. Phys., in press. · Zbl 1087.76531
[10] Noguchi, H.; Kawashima, T., Meshfree analyses of cable-reinforced membrane structures by ALE-EFG method, Engrg. anal. bound. elem., 28, 5, 443-451, (2004) · Zbl 1130.74491
[11] Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I. surface approximations and partial derivative estimates, Comput. math. appl., 19, 6-8, 127-145, (1990) · Zbl 0692.76003
[12] Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II. solutions to parabolic, hyperbolic, and elliptic partial differential equations, Comput. math. appl., 19, 6-8, 147-161, (1990) · Zbl 0850.76048
[13] G.E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, in: A.L. Mehaute, C. Rabut, L.L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, 1997, pp. 131-138. · Zbl 0938.65140
[14] Jumarhon, B.; Amini, S.; Chen, K., The Hermite collocation method using radial basis functions, Engrg. anal. bound. elem., 24, 607-611, (2000) · Zbl 0967.65107
[15] Franke, C.; Schaback, R., Convergence order estimates of meshless collocation methods using radial basis functions, Adv. comput. math., 8, 4, 381-399, (1998) · Zbl 0909.65088
[16] Driscoll, T.A.; Forberg, B., Interpolation in the limit of increasingly flat radial basis functions, Comput. math. appl., 43, 413-422, (2002) · Zbl 1006.65013
[17] Hon, Y.C.; Wu, Z.M., A quasi-interpolation method for solving stiff ordinary differential equations, Int. J. numer. methods engrg., 48, 1187-1197, (2000) · Zbl 0962.65060
[18] Chen, W.; Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. math. appl., 43, 379-391, (2002) · Zbl 0999.65142
[19] Shu, C.; Ding, H.; Yeo, K.S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 192, 941-954, (2003) · Zbl 1025.76036
[20] H. Ding, C. Shu, D.B. Tang, Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments, Int. J. Numer. Methods Engrg., in press. · Zbl 1089.65119
[21] Shu, C.; Ding, H.; Chen, H.Q.; Wang, T.G., An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput. methods appl. mech. engrg., 194, 2001-2017, (2005) · Zbl 1093.76052
[22] Bellman, R.; Kashef, B.G.; Casti, J., Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations, J. comput. phys., 10, 40-52, (1972) · Zbl 0247.65061
[23] Micchelli, C.A., Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. approx., 2, 11-22, (1986) · Zbl 0625.41005
[24] Franke, R., Scattered data interpolation: tests of some methods, Math. comput., 38, 181-199, (1982) · Zbl 0476.65005
[25] Shu, C., Differential quadrature and its application in engineering, (2000), Springer-Verlag London Limited · Zbl 0944.65107
[26] Chorin, A.J., Numerical solution of the Navier-Stokes equations, Math. comput., 22, 745-762, (1968) · Zbl 0198.50103
[27] J.R. Meiling, J. Dalheim, Numerical prediction of the response of a vortex-excited cylinder at low Reynolds numbers, in: Proceedings 7th International Offshore and Polar Engineering Conference, 1997, Honolulu, USA.
[28] Ding, H.; Shu, C.; Yeo, K.S.; Lu, Z.L., Simulation of natural convection in eccentric annuli between a square outer cylinder and a circular inner cylinder using local MQ-DQ method, Numer. heat transfer, part A—applications, 47, 291-313, (2005)
[29] Kim, J.; Moin, P., Application of fractional-step method to incompressible Navier-Stokes equations, J. comput. phys., 59, 308-323, (1985) · Zbl 0582.76038
[30] Ku, H.C.; Hirsh, R.S.; Taylor, T.D., A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations, J. comput. phys., 70, 439-462, (1987) · Zbl 0658.76027
[31] Jiang, B.N.; Lin, T.L.; Povinelli, L.A., Large-scale computation of incompressible viscous flow by least-squares finite element method, Comput. methods appl. mech. engrg., 114, 213-231, (1994)
[32] Fujima, S.; Tabata, M.; Fukasawa, Y., Extension to three-dimensional problems of the upwind finite element scheme based on the choice of up- and down-wind points, Comput. methods appl. mech. engrg., 112, 109-131, (1994) · Zbl 0854.76051
[33] Guj, G.; Stella, F., A vorticity-velocity method for the numerical solution of 3D incompressible flows, J. comput. phys., 106, 286-298, (1993) · Zbl 0770.76045
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.