**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30517

##### A Hybrid Mesh Free Local RBF- Cartesian FD Scheme for Incompressible Flow around Solid Bodies

**Authors:**
A. Javed,
K. Djidjeli,
J. T. Xing,
S. J. Cox

**Abstract:**

A method for simulating flow around the solid bodies has been presented using hybrid meshfree and mesh-based schemes. The presented scheme optimizes the computational efficiency by combining the advantages of both meshfree and mesh-based methods. In this approach, a cloud of meshfree nodes has been used in the domain around the solid body. These meshfree nodes have the ability to efficiently adapt to complex geometrical shapes. In the rest of the domain, conventional Cartesian grid has been used beyond the meshfree cloud. Complex geometrical shapes can therefore be dealt efficiently by using meshfree nodal cloud and computational efficiency is maintained through the use of conventional mesh-based scheme on Cartesian grid in the larger part of the domain. Spatial discretization of meshfree nodes has been achieved through local radial basis functions in finite difference mode (RBF-FD). Conventional finite difference scheme has been used in the Cartesian ‘meshed’ domain. Accuracy tests of the hybrid scheme have been conducted to establish the order of accuracy. Numerical tests have been performed by simulating two dimensional steady and unsteady incompressible flows around cylindrical object. Steady flow cases have been run at Reynolds numbers of 10, 20 and 40 and unsteady flow problems have been studied at Reynolds numbers of 100 and 200. Flow Parameters including lift, drag, vortex shedding, and vorticity contours are calculated. Numerical results have been found to be in good agreement with computational and experimental results available in the literature.

**Keywords:**
CFD,
Meshfree particle methods,
Hybrid grid,
Incompressible Navier Strokes equations,
RBF-FD

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1088114

**References:**

[1] S. Dennis, and G.-Z. Chang, “Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100,” J. Fluid Mech, vol. 42, no. 3, pp. 471-489, 1970.

[2] M. Braza, P. Chassaing, and H. H. Minh, “Numerical Study and Physical Analysis of the Pressure and Velocity-Fields in the near Wake of a Circular-Cylinder,” Journal of Fluid Mechanics, vol. 165, pp. 79- 130, Apr, 1986.

[3] H. Takami, and H. B. Keller, “Steady Two‐Dimensional Viscous Flow of an Incompressible Fluid past a Circular Cylinder,” Physics of Fluids, vol. 12, no. 12, pp. II-51-II-56, 1969.

[4] L. B. Lucy, “A numerical approach to the testing of fission hypothesis,” Astronomical Journal, vol. 8, pp. 1013-1024, 1977.

[5] B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: Diffuse approximation and diffuse elements,” Computational Mechanics, vol. 10, no. 5, pp. 307-318, 1992.

[6] T. Belytschko, Y. Y. Lu, and L. Gu, “ELEMENT-FREE GALERKIN METHODS,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229-256, 1994.

[7] W. K. Liu, S. Jun, S. F. Li, J. Adee, and T. Belytschko, “Reproducing Kernel Particle Methods for Structural Dynamics,” International Journal for Numerical Methods in Engineering, vol. 38, no. 10, pp. 1655-1679, May 30, 1995.

[8] J. M. Melenk, and I. Babuska, “The partition of unity finite element method: Basic theory and applications,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1-4, pp. 289-314, Dec 15, 1996.

[9] E. Onate, S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor, and C. Sacco, “A stabilized finite point method for analysis of fluid mechanics problems,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1-4, pp. 315-346, Dec 15, 1996.

[10] C. Liu, X. Zheng, and C. Sung, “Preconditioned multigrid methods for unsteady incompressible flows,” Journal of Computational Physics, vol. 139, no. 1, pp. 35-57, 1998.

[11] H. Ding, C. Shu, K. S. Yeo, and D. Xu, “Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 9-11, pp. 727-744, 2004.

[12] Y. Sanyasiraju, and G. Chandhini, “Local radial basis function based gridfree scheme for unsteady incompressible viscous flows,” Journal of Computational Physics, vol. 227, no. 20, pp. 8922-8948, Oct, 2008.

[13] J. P. Morris, P. J. Fox, and Y. Zhu, “Modeling low Reynolds number incompressible flows using SPH,” Journal of Computational Physics, vol. 136, no. 1, pp. 214-226, Sep 1, 1997.

[14] J. J. Monaghan, “Simulating Free-Surface Flows with Sph,” Journal of Computational Physics, vol. 110, no. 2, pp. 399-406, Feb, 1994.

[15] M. B. Liu, G. R. Liu, and K. Y. Lam, “Constructing smoothing functions in smoothed particle hydrodynamics with applications,” Journal of Computational and Applied Mathematics, vol. 155, no. 2, pp. 263-284, Jun 15, 2003.

[16] M. Liu, G. Liu, Z. Zong, and K. Lam, "Numerical simulation of incompressible flows by SPH."

[17] C. Shu, H. Ding., and K. S. Yeo., “Computation of Incompressible Navier-Stokes Equations by Local RBF-based Differential Quadrature Method,” Computer Modeling in Engineering and Sciences, vol. 7, no. 2, pp. 195-206, 2005.

[18] C. Shu, H. Ding, and K. S. Yeo, “Local radial basis function-based differential quadrature method and its application to solve twodimensional incompressible Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 7-8, pp. 941-954, 2003.

[19] C. G. Koh, M. Gao, and C. Luo, “A new particle method for simulation of incompressible free surface flow problems,” International Journal for Numerical Methods in Engineering, vol. 89, no. 12, pp. 1582-1604, Mar 23, 2012.

[20] S.-y. Tuann, and M. D. Olson, “Numerical studies of the flow around a circular cylinder by a finite element method,” Computers & Fluids, vol. 6, no. 4, pp. 219-240, 12//, 1978.

[21] C. S. Chew, K. S. Yeo, and C. Shu, “A generalized finite-difference (GFD) ALE scheme for incompressible flows around moving solid bodies on hybrid meshfree–Cartesian grids,” Journal of Computational Physics, vol. 218, no. 2, pp. 510-548, 11/1/, 2006.

[22] A. Belov, L. Martinelli, and A. Jameson, “A new implicit algorithm with multigrid for unsteady incompressible flow calculations,” AIAA paper, vol. 95, pp. 0049, 1995.

[23] X. He, and G. Doolen, “Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder,” Journal of Computational Physics, vol. 134, no. 2, pp. 306-315, 1997.

[24] R. Mei, and W. Shyy, “On the finite difference-based lattice Boltzmann method in curvilinear coordinates,” Journal of Computational Physics, vol. 143, no. 2, pp. 426-448, 1998.

[25] E. J. Kansa, “Multiquadrics - a Scattered Data Approximation Scheme with Applications to Computational Fluid-Dynamics .2. Solutions to Parabolic, Hyperbolic and Elliptic Partial-Differential Equations,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 147- 161, 1990.

[26] J. G. Wang, and G. R. Liu, “On the optimal shape parameters of radial basis functions used for 2-D meshless methods,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 23-24, pp. 2611- 2630, 2002.

[27] P. P. Chinchapatnam, K. Djidjeli, and P. B. Nair, “Radial basis function meshless method for the steady incompressible Navier–Stokes equations,” International Journal of Computer Mathematics, vol. 84, no. 10, pp. 1509-1521, 2007.

[28] Z. Guo, B. Shi, and N. Wang, “Lattice BGK model for incompressible Navier–Stokes equation,” Journal of Computational Physics, vol. 165, no. 1, pp. 288-306, 2000.

[29] A. I. Tolstykh, and D. A. Shirobokov, “On using radial basis functions in a "finite difference mode" with applications to elasticity problems,” Computational Mechanics, vol. 33, no. 1, pp. 68-79, Dec, 2003.

[30] G. B. Wright, and B. Fornberg, “Scattered node compact finite difference-type formulas generated from radial basis functions,” Journal of Computational Physics, vol. 212, no. 1, pp. 99-123, Feb 10, 2006.

[31] A. R. Firoozjaee, and M. H. Afshar, “Steady-state solution of incompressible Navier–Stokes equations using discrete least-squares meshless method,” International Journal for Numerical Methods in Fluids, vol. 67, no. 3, pp. 369-382, 2011.

[32] P. P. Chinchapatnam, K. Djidjeli, P. B. Nair, and M. Tan, “A compact RBF-FD based meshless method for the incompressible Navier-Stokes equations,” Proceedings of the Institution of Mechanical Engineers Part M-Journal of Engineering for the Maritime Environment, vol. 223, no. M3, pp. 275-290, Aug, 2009.

[33] C. Perng, and R. Street, “A coupled multigrid‐domain‐splitting technique for simulating incompressible flows in geometrically complex domains,” International journal for numerical methods in fluids, vol. 13, no. 3, pp. 269-286, 1991.

[34] M. Hinatsu, and J. Ferziger, “Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique,” International Journal for Numerical Methods in Fluids, vol. 13, no. 8, pp. 971-997, 1991.

[35] P. Chow, and C. Addison, “Putting domain decomposition at the heart of a mesh‐based simulation process,” International journal for numerical methods in fluids, vol. 40, no. 12, pp. 1471-1484, 2002.

[36] D. Calhoun, “A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions,” Journal of Computational Physics, vol. 176, no. 2, pp. 231-275, 2002.

[37] R. B. Pember, J. B. Bell, P. Colella, W. Y. Curtchfield, and M. L. Welcome, “An adaptive Cartesian grid method for unsteady compressible flow in irregular regions,” Journal of computational Physics, vol. 120, no. 2, pp. 278-304, 1995.

[38] J. Falcovitz, G. Alfandary, and G. Hanoch, “A two-dimensional conservation laws scheme for compressible flows with moving boundaries,” Journal of Computational Physics, vol. 138, no. 1, pp. 83- 102, 1997.

[39] A. Gilmanov, F. Sotiropoulos, and E. Balaras, “A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids,” Journal of Computational Physics, vol. 191, no. 2, pp. 660-669, 2003.

[40] R. Glowinski, T.-W. Pan, and J. Periaux, “A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 112, no. 1, pp. 133-148, 1994.

[41] A. Javed, K. Djidjeli, and J. Xing, Tang, “Shape adaptive RBF-FD Implicit Scheme for Incompressible Viscous Navier-Strokes Equations,” Journal of Computer and Fluid, vol. submitted for Publication, 2013.

[42] J. Kim, and P. Moin, “Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations,” Journal of Computational Physics, vol. 59, no. 2, pp. 308-323, June, 1985, 1985.

[43] B. Fornberg, “A Numerical Study of Steady Viscous-Flow Past a Circular-Cylinder,” Journal of Fluid Mechanics, vol. 98, no. Jun, pp. 819-855, 1980.

[44] D. Kim, and H. Choi, “A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids,” Journal of Computational Physics, vol. 162, no. 2, pp. 411-428, 2000.

[45] Y. Zang, R. L. Street, and J. R. Koseff, “A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates,” Journal of Computational Physics, vol. 114, no. 1, pp. 18-33, 1994.