A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations. (English) Zbl 1383.76366
Summary: The two-dimensional Navier-Stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions (MAPS) in its global formulation. The fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a Stokes non-homogeneous system of equations with multiquadric (MQ) radial basis function as the source term. The nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field, and the resulting linear system of equations is solved by a simple direct iterative scheme (Picard iteration), with the velocity guess given by the solution at the previous iteration. Although the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation. The proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the Kovasznay flow problem at different Reynolds numbers, \(\mathrm{Re}\). From this analysis, it is observed that the MAPS results are stable and accurate for a wide range of shape parameter values. In addition, lid-driven cavity flow problems in rectangular and triangular domains up to \(\mathrm{Re}=3200\) and \(\mathrm{Re}=1000\), respectively, and the backward-facing step at \(\mathrm{Re}=800\) are solved, and the results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement.
MSC:
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
References:
| [1] | Patankar, S. V.; Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transfer, 15, 1787-1806 (1972) · Zbl 0246.76080 |
| [2] | Issa, R. I., Solution of the implicitly discretised fluid flow equations by operator-splitting, J. Comput. Phys., 62, 40-65 (1986) · Zbl 0619.76024 |
| [3] | Chorin, A., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 12-26 (1967) · Zbl 0149.44802 |
| [4] | Suh, J.-C.; Kim, K.-S., A vorticity-velocity formulation for solving the two-dimensional Navier-Stokes equations, Fluid Dyn. Res., 25, 195-216 (1999) |
| [5] | Davies, C.; Carpenter, P. W., A novel velocity-vorticity formulation of the Navier-Stokes equations with applications to boundary layer disturbance evolution, J. Comput. Phys., 172, 119-165 (2001) · Zbl 1065.76573 |
| [6] | Hribersek, M.; Skerget, L., Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries, Comput. Methods Appl. Mech. Engrg., 194, 4196-4220 (2005) · Zbl 1151.76537 |
| [7] | Lo, D. C.; Young, D. L.; Tsai, C., High resolution of 2D natural convection in a cavity by the DQ method, J. Comput. Appl. Math., 203, 219-236 (2007) · Zbl 1172.76372 |
| [8] | Dworkin, S. B.; Bennett, B. A.V.; Smooke, M. D., A mass-conserving vorticity-velocity formulation with application to nonreacting and reacting flows, J. Comput. Phys., 215, 430-447 (2006) · Zbl 1173.76373 |
| [9] | Bourantas, G. C.; Skouras, E. D.; Loukopoulos, V. C.; Nikiforidis, G. C., Numerical solution of non-isothermal fluid flows using local radial basis functions (LRBF) interpolation and a velocity-correction method, J. Comput. Methods Eng. Sci. Mech., 64, 187-212 (2010) · Zbl 1231.76070 |
| [10] | Demirkaya, G.; Soh, C. W.; Ilegbusi, O., Direct solution of Navier-Stokes equations by radial basis functions, Eng. Anal. Bound. Elem., 32, 1848-1858 (2008) · Zbl 1145.76346 |
| [11] | Divo, E.; Kassab, A. J., An efficient localized radial basis function meshless method for fluid flow and conjugate heat transfer, J. Heat Transfer, 129, 124-136 (2007) |
| [12] | Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—II solution to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19, 127-145 (1990) · Zbl 0692.76003 |
| [13] | Chantasiriwan, S., Cartesian grid methods using radial basis functions for solving Poisson, Helmholtz, and diffusion convection equations, Eng. Anal. Bound. Elem., 28, 1417-1425 (2004) · Zbl 1098.76585 |
| [14] | Mai-Duy, N.; Tran-Cong, T., Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations, Eng. Anal. Bound. Elem., 26, 133-156 (2002) · Zbl 0996.65131 |
| [15] | Boztosuna, I.; Charafi, A., An analysis of the linear advection-diffusion equation using mesh-free and mesh-dependent methods, Eng. Anal. Bound. Elem., 26, 889-895 (2002) · Zbl 1038.76030 |
| [16] | Chinchapatnam, P. P.; Djidjeli, K.; Nair, P. B., Radial basis function meshless method for the steady incompressible Navier-Stokes equations, Int. J. Comput. Math., 84, 1509-1526 (2007) · Zbl 1123.76048 |
| [17] | Madych, W. R.; Nelson, S., Multivariate interpolation and conditionally positive definite functions, Math. Comp., 54, 211-230 (1990) · Zbl 0859.41004 |
| [18] | Schaback, R., Multivariate interpolation and approximation by translates of basis functions, (Chui, C. K.; Schumaker, L. L., Approximation Theory VIII: Wavelets and Multilevel Approximation (1995), World Scientific Pub. Co. Inc.: World Scientific Pub. Co. Inc. College Station), 1-8 |
| [19] | Brown, D., On approximate cardinal preconditioning methods for solving PDEs with radial basis functions, Eng. Anal. Bound. Elem., 29, 343-353 (2005) · Zbl 1182.65174 |
| [20] | Ling, L.; Opfer, R.; Schaback, R., Results on meshless collocation techniques, Eng. Anal. Bound. Elem., 30, 247-253 (2006) · Zbl 1195.65177 |
| [21] | Sarler, B., A radial basis function collocation approach in computational fluid dynamics, CMES Comput. Model. Eng. Sci., 7, 770-790 (2005) · Zbl 1189.76380 |
| [22] | Mai-Duy, T. T.-C. N., Approximation of function and its derivatives using radial basis function networks, Appl. Math. Model., 27, 197-220 (2003) · Zbl 1024.65012 |
| [23] | Mai-Duy, N.; Tran-Cong, T., Numerical solution of differential equations using multiquadric radial basis function networks, Neural Netw., 14, 185-199 (2001) |
| [24] | Ding, H.; Shu, C.; Yeo, K.; Xu, D., Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method, Comput. Methods Appl. Mech. Engrg., 195, 516-533 (2006) · Zbl 1222.76072 |
| [25] | Shu, C.; Ding, H.; Yeo, K., 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 |
| [26] | Sanyasiraju, Y.; Chandhini, G., Local radial basis function based gridfree scheme for unsteady incompressible viscous flows, J. Comput. Phys., 227, 8922-8948 (2008) · Zbl 1146.76045 |
| [27] | Chinchapatnam, P. P.; Djidjeli, B. N.K.; Tan, M., A compact RBF-FD based meshless method for the incompressible Navier-Stokes equations, Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ., 223, 275-290 (2009) |
| [28] | Chen, C.; Fan, C.; Wen, P., The method of approximated particular solutions for solving certain partial differential equations, Numer. Methods Partial Differential Equations, 28, 506-522 (2012) · Zbl 1242.65267 |
| [29] | Golberg, M.; Cheng, C., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (Goldberg, M. A., Boundary Integral Methods: Numerical and Mathematical Aspects (1998), Comput. Mech. Publ.: Comput. Mech. Publ. Southampton), 103-176 · Zbl 0945.65130 |
| [30] | Florez, W. F.; Power, H., DRM multidomain mass conservative interpolation approach for the bem solution of the two-dimensional Navier-Stokes equations, Comput. Math. Appl., 43, 457-472 (2002) · Zbl 1034.76038 |
| [31] | Happel, J.; Brenner, H., Low Reynolds Number Hydrodynamics (1983), Martinus Nijhoff Publishers: Martinus Nijhoff Publishers The Hague |
| [32] | Power, H.; Wrobel, L., Boundary Integral Methods in Fluid Mechanics (1995), Comput. Mech. Publ.: Comput. Mech. Publ. Southampton · Zbl 0815.76001 |
| [33] | Samaan, M. F.; Rashed, Y. F., Free vibration multiquadric boundary elements applied to plane elasticity, Appl. Math. Model., 33, 2421-2432 (2009) · Zbl 1185.74023 |
| [34] | Erturk, E., Numerical solutions of 2-D steady incompressible flow over a backward-facing step, part I: high Reynolds number solutions, Comput. Fluids, 37, 633-655 (2008) · Zbl 1237.76102 |
| [35] | Kovasznay, L., Laminar flow behind a two-dimensional grid, Proc. Cambridge Philos. Soc., 44, 58-62 (1948) · Zbl 0030.22902 |
| [36] | Durmus, A.; Boztosum, I.; Yasuk, F., Comparative study of the multiquadric and thin-plate spline radial basis functions for the transient-convective diffusion problems, Internat. J. Modern Phys. C, 17, 1151-1169 (2006) · Zbl 1137.76501 |
| [37] | Huang, C.; Lee, C.; Cheng, A. H.D., Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method, Eng. Anal. Bound. Elem., 31, 614-623 (2007) · Zbl 1195.65176 |
| [38] | Bayona, V.; Moscoso, M.; Kindelan, M., Optimal constant shape parameter for multiquadric based RBF-FD method, J. Comput. Phys., 230, 7384-7399 (2011) · Zbl 1343.65128 |
| [39] | Stevens, D.; Power, H.; Cliffe, K., A solution to linear elasticity using locally supported RBF collocation in a generalised finite-difference mode, Eng. Anal. Bound. Elem., 37, 32-41 (2013) · Zbl 1351.74160 |
| [40] | Chantasiriwan, S., An alternative approach for numerical solutions of the Navier-Stokes equations, Internat. J. Numer. Methods Engrg., 69, 1331-1344 (2007) · Zbl 1194.76198 |
| [41] | Ghia, U.; Ghia, K. N.; Shin, C., High-resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48, 387-411 (1982) · Zbl 0511.76031 |
| [42] | Botella, O.; Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids, 27, 421-433 (1998) · Zbl 0964.76066 |
| [43] | Kobayashi, M. H.; Pereira, J. M.C.; Pereiraz, J. C.F., A conservative finite-volume second-order-accurate projection method on hybrid unstructured grids, J. Comput. Phys., 150, 40-75 (1999) · Zbl 0934.76049 |
| [44] | Perron, S.; Boivin, S.; Hérard, J. M., A finite volume method to solve the 3D Navier-Stokes equations on unstructured collocated meshes, Comput. Fluids, 33, 1305-1333 (2004) · Zbl 1075.76044 |
| [45] | Gao, Wei; Duan, Ya-li; Liu, Ru-xun, The finite volume projection method with hybrid unstructured triangular collocated grids for incompressible flows, J. Hydrodyn. Ser. B, 21, 201-211 (2009) |
| [46] | Liu, C. H.; Leung, D. Y.C., Development of a finite element solution for the unsteady Navier-Stokes equations using projection method and fractional \(\theta \)-scheme, Comput. Methods Appl. Mech. Eng., 190, 4301-4317 (2001) · Zbl 1015.76041 |
| [47] | Erturk, E.; Gokcol, O., Fine grid numerical solutions of triangular cavity flow, Eur. Phys. J. Appl. Phys., 38, 97-105 (2007) |
| [48] | Rani, H. P.; Sheu, T. W.H., Nonlinear dynamics in a backward-facing step flow, Phys. Fluids, 18, 84-101 (2006) · Zbl 1185.76504 |
| [49] | Ravnik, J.; Skerget, L.; Hribersek, M.; Zunic, Z., Numerical simulation of dilute particle laden flows by wavelet BEM-FEM, Comput. Method Appl. M., 197, 789-805 (2008) · Zbl 1169.76403 |
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