Document Type: Research Paper

Author

Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, 53751-71379, Iran

Abstract

In this paper, the laminar incompressible flow equations are solved by an upwind least-squares meshless method. Due to the difficulties in generating quality meshes, particularly in complex geometries, a meshless method is increasingly used as a new numerical tool. The meshless methods only use clouds of nodes to influence the domain of every node. Thus, they do not require the nodes to be connected to form a mesh and decrease the difficulty of meshing, particularly around complex geometries. In the literature, it has been shown that the generation of points in a domain by the advancing front technique is an order of magnitude faster than the unstructured mesh for a 3D configuration. The Navier–Stokes solver is based on the artificial compressibility approach and the numerical methodology is based on the higher-order characteristic-based (CB) discretization. The main objective of this research is to use the CB scheme in order to prevent instabilities. Using this inherent upwind technique for estimating convection variables at the mid-point, no artificial viscosity is required at high Reynolds number. The Taylor least-squares method was used for the calculation of spatial derivatives with normalized Gaussian weight functions. An explicit four-stage Runge-Kutta scheme with modified coefficients was used for the discretized equations. To accelerate convergence, local time stepping was used in any explicit iteration for steady state test cases and the residual smoothing techniques were used to converge acceleration. The capabilities of the developed 2D incompressible Navier-Stokes code with the proposed meshless method were demonstrated by flow computations in a lid-driven cavity at four Reynolds numbers. The obtained results using the new proposed scheme indicated a good agreement with the standard benchmark solutions in the literature. It was found that using the third order accuracy for the proposed method could be more efficient than its second order accuracy discretization in terms of computational time.

Keywords

Main Subjects

[1] Y. Morinishi, T. S. Lund, O. V. Vasilyev and P. Moin, “Fully conservative higher order finite difference schemes for incompressible flow”, Journal of Computational Physics, Vol. 143, pp. 90-124, (1998).

[2] T. Ikeno and T. Kajishima, “Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations”, Journal of Computational Physics, Vol. 226, No. 2, pp. 1485-1508, (2007).

[3] S. E. Razavi, K. Zamzamian and A. Farzadi, “Genuinely multidimensional characteristic-based scheme for incompressible flows”, International Journal  for Numerical Methods in Fluids, Vol. 57, No. 8, pp. 929-949, (2008).

[4] M. Y. Hashemi and A. Jahangirian, “Simulation of high-speed flows by an unstructured grid implicit method including real gas effects”, International Journal  for Numerical Methods in Fluids, Vol. 56, No. 8, pp. 1281-1287, (2007).

[5] X. Wang and X. Li,  “Numerical simulation of three dimensional non Newtonian free surface flows in injection molding using ALE finite element method”, Finite Elements in Analysis and Design, Vol. 46, No. 7, pp. 551-562, (2010).

[6] R. W. Lewis, K. Ravindran and A. S. Usmani, “Finite element solution of incompressible flows using an explicit segregated approach”, Archives of Computational Methods in Engineering, Vol. 2, No. 4, pp. 69-93, (1999).

[7] J. H. Kent, “Air Conditioning Modelling by Computational Fluid Dynamics”,  Architectural Science Review, Vol. 37, No. 3, pp. 103-113, (1994).

[8] S. Wang and D. Zhub, “Application of CFD in retrofitting air-conditioning systems in industrial buildings”, Energy and Buildings, Vol. 35, No. 9, pp. 893-902, (2003).

[9] S. W. Hwang, D. H. Kim, J. K. Min and J. H. Jeong, “CFD analysis of fin tube heat exchanger with a pair of delta winglet vortex generators”, Journal of Mechanical Science and Technology, Vol. 26, No. 9, pp. 2949-2958, (2012).

[10] M. Yataghene and J. Legrand, “A 3D-CFD model thermal analysis within as craped surface heat exchanger”, Computers & Fluids, Vol. 71, pp. 380-399, (2013).

[11] S. Aradag, U. Olgun, F. Aktrk and B. Basibyk, “CFD analysis of cooling of electronic equipment as an undergraduate design project”, International Journal of Hydrogen Energy, Vol. 20, No. 1, pp. 103-113, (2012).

[12] H. Sadat-Hosseini H, Pablo Carrica, F. Stern, N. Umeda, H. Hashimoto, S. Yamamura and A. Mastuda,“CFD, system-based and EFD study of ship dynamic instability events: Surf-riding, periodic motion, and broaching”, Ocean Engineering, Vol. 38, No. 1, pp. 88-110, (2011).

[13] J. F. Thompson, F. C. Thomas and C. W. Mastin, “Automated numerical generation of body fitted curvilinear co-ordinate system for field containing any number of arbitrary 2D bodies”, Journal of Computational Physics, Vol. 15, pp. 299-319, (1974).

[14] N. P. Weatherill, “A method for generating irregular computational grids in multiply connected planar domains”, International Journal for Numerical Methods in Fluids, Vol. 8, pp. 181-197, (1998).

[15] R. Lohner and E. Onate, “An advancing front point generation technique”, Communications in Numerical Methods in Engineering, Vol. 14, No. 12, pp. 1097-1108, (1998).

[16] R. Lohner and E. Onate, “A general advancing front technique for filling space with arbitrary objects”, International Journal for Numerical Methods in Engineering, Vol. 61, No. 12, pp. 1977-1991, (2004).

[17] P. W. Randles and L. D. Libersky, “Smoothed particle hydrodynamics: some recent improvements and applications”, Computer Methods in Applied Mechanics and  Engineering, Vol. 139, No. 1, pp. 375- 408, (1996).

[18] C. S. Chew CS, 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, (2006).

[19] H. Ding, C. Shu, K. S. Yeo and D. Xu, “Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavity”, Computers & Fluids, Vol. 33, No. 1, pp. 137-154, (2004).

[20] S. N. Atluri and T. Zhu, “A new meshless local Petrov-Galerkin (MLPG) approach”, Computational Mechanics, Vol. 22, No. 2, pp. 117-127, (1998).

[21] J. S. Chen, C. T. Wux, S. Yoon and Y. You, “A stabilized conforming nodal integration for Galerkin mesh-free methods”, International Journal for Numerical Methods in Engineering, Vol. 50, No. 2, pp. 435-466, (2001).

[22] M. Polner, L. Pesch and J. J. W. van der Vegt, “Construction of stabilization operators for Galerkin least-squares discretizations of compressible and incompressible flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 2431-2448, (2007).

[23] C. Shu, and B. E. Richards, “Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations”, International Journal  for Numerical Methods in Fluids, Vol. 15, No. 7, pp. 791-798, (1992).

[24] H. Q. Chen and C. Shu, “An efficient implicit mesh-free method to solve two dimensional compressible Euler equations”, International Journal of Modern Physics, Vol. 16, No.3, pp. 439-454, (2005).

[25] W. K. Liu, S. Jun and Y. F. Zhang, “Reproducing kernel particle methods”, International Journal  for Numerical Methods in Fluids, Vol. 20, No. 8-9, pp. 1081-1106, (1995).

[26] M. Najafi, A. Arefmanesh, and V. Enjilela, “Meshless local Petrov-Galerkin method-higher Reynolds numbers fluid flow applications”, Engineering Analysis with Boundary Elements, Vol. 36, No. 11, pp. 1671-1685, (2012).

[27] Y. L. Wu, G. R. Liu and Y. T. Gu, “Application of meshless local Petrov-Galerkin (MLPG) approach to simulation of incompressible flow”, Numerical Heat Transfer, Vol. 48, No. 5, pp. 459-475, (2005).

[28] R. Lohner, C. Sacco, E. Onate and S. Idelsohn, “A finite point method for compressible flow”, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 1765-1779, (2002).

[29] E. Ortega, E. Onate, S. Idelsohn and R. Flores, “A meshless finite point method for three-dimensional analysis of compressible flow problems involving moving boundaries and adaptivity”, International Journal  for Numerical Methods in Fluids, Vol. 73, No. 4, pp. 323-343, (2013).

[30] Z. Ma, H. Chen and C. Zhou, “A study of point moving adaptivity in gridless method”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 926-1937, (2008).

[31] Y.  Hashemi,  and A.  Jahangirian, “Implicit fully mesh-less method for compressible viscous flow calculations”, Journal of Computational and Applied Mathematics, Vol. 235, No. 16, pp. 4687-4700, (2011).

[32] M. Y. Hashemi, and A. Jahangirian,“An efficient implicit mesh-less method for compressible flow calculations”, International Journal  for Numerical Methods in Fluids, Vol. 67, No. 6, pp. 754-770, (2011).

[33] X. K. Zhang, K. C. Kwon and S. K. Youn, “The least-squares meshfree method for the steady incompressible viscous flow”, Journal of Computational Physics, Vol. 206, pp. 182-207, (2005).

[34] X. Su, S.Yamamoto and K. Nakahashi, “Analysis of a meshless solver for high Reynolds number flow”, International Journal  for Numerical Methods in Fluids, Vol. 72, No. 5, pp. 505-527, (2013).

[35] A. J. Chorin, “A numerical method for solving incompressible viscous flow problems”, Journal of Computational Physics, Vol. 2, No. 1, pp. 12-26, (1967).

[36] J. Farmer, L. Martinelli and A. Jameson, “Fast multigrid method for solving incompressible hydrodynamic problems with free surface”, American Institute of Aeronautics and Astronautics Journal, Vol. 32, pp. 1175-1182, (1994).

[37] V. Esfahanian and P. Akbarzadeh, “The Jameson’s numerical method for solving the incompressible viscous and in viscid flows by means of artificial compressibility and preconditioning method”, Applied Mathematics and Computation, Vol. 206, pp. 651-661, (2008).

[38] C. Liu, X. Zheng and C. H. Sung, “Preconditioned multigrid methods for unsteady incompressible flows”, Journal of Computational Physics, Vol. 139, No. 1, pp. 35-57, (1998).

[39] Y. Kallinderis and H. T. Ahn, “Incompressible Navier-Stokes method with general hybrid meshes”, Journal of Computational Physics, Vol. 210, No. 1, pp. 75-108, (2005).

[40] D. Drikakis, P. A. Govatsos and D. E. Papantonis, “A characteristic based method for incompressible flows”, International Journal  for Numerical Methods in Fluids, Vol. 19, No. 8, pp. 667-685, (1994).

[41] D. Drikakis, O. P. Iliev and D. P. Vassileva, “A nonlinear multigrid method for the three-dimensional incompressible Navier-Stokes equations”, Journal of Computational Physics, Vol. 146, No. 1, pp. 301-321, (1998).

[42] K. Siong and C. Y. Zhao, “Numerical study of steady/unsteady flow and heat transfer in porous media using a characteristics-based matrix-free implicit FV method on unstructured grids”, International Journal of Heat and Fluid Flow, Vol. 25, pp. 1015-1033, (2004).

[43] E. Shapiro and D. Drikakis, “Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part I. Derivation of different formulations and constant density limit”, Journal of Computational Physics, Vol. 210, No. 2, pp. 584-607, (2005).

[44] E. Shapiro and D. Drikakis, “Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part II. Multi grid implementation and numerical tests”, Journal of Computational Physics, Vol. 210, pp. 608-631, (2005).

[45] M. Y. Hashemi and K. Zamzamian,“A multidimensional characteristic-based method for making incompressible flow calculations on unstructured grids”, Journal of Computational and Applied Mathematics, Vol. 259(B), pp.795-805, (2014).

[46] S. Sridar and N. Balakrishnan , “An upwind finite difference scheme for meshless solvers”, Journal of Computational Physics, Vol. 189, No. 1, pp.1-29, (2003).

[47] C. Praveen C. and S. M. Deshpande, “Kinetic meshless method for compressible flows”, International Journal  for Numerical Methods in Fluids, Vol. 55, No. 11, pp.1059-1089, (2007).

[48] H. Luo, J. D. Baum and R. Lohner, “Hybrid building-block and gridless method for compressible flows”, International Journal  for Numerical Methods in Fluids, Vol. 59, No. 4, pp. 459-474, (2009).

[49] A. Katz, “Meshless methods for computational fluid dynamics”, .2009, PhD Thesis, Stanford University, (2009).

[50] A. Katz and A. Jameson, “Meshless scheme based on alignment constraints”, American Institute of Aeronautics and Astronautics Journal, Vol. 48, No. 11, pp. 2501-2511, (2010).

[51] G. May and Jameson, “Unstructured algorithms for inviscid and viscousflows embedded in a unified solver architecture Flo3xx”, AIAA 43rd Aerospace Sciences Meeting 2005; 0318, Reno, Nevada, (2005).

[52] C. Hirsch, Numerical Computation of Internaland External Flows (Volume 1:Fundamentals of Computational FluidDynamics), 2ndedition, Elsevier, Burlington, pp. 357-360, (2007).

[53] K.   Zamzamian  and  S.  E.   Razavi, “Multidimensional up winding for incompressible flows based on characteristics”,  Journal of Computational Physics, Vol. 227, No. 19, pp. 8699-8713, (2008).

[54] Y. Zhao and B. Zhang, “A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grid”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 5, pp. 733-756, (2000).

[55] C. H. Tai and Y. Zhao, “Parallel unsteady incompressible viscous flow computations using an unstructured multigrid method”, Journal of Computational Physics, Vol. 192, No. 1, pp. 277-311, (2003).

[56] R. Lohner, C. Sacco, E. Onate, S. A. Idelsohn,“Finite point method for compressible flow”, International Journal for Numerical Methods in Engineering, Vol. 53, pp.1765–1779, (2002).

[57] D. J. Mavriplis, A. Jameson, and L. Martinelli,“Multigrid solution of the Navier-Stokes equations on the triangular meshes”, AIAA paper, 27th Sciences Meeting, Reno, Nevada, USA, January 9-12, (1989).

[58] U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method”, Journal of Computational Physics, Vol. 48, pp.387-411, (1982).

CAPTCHA Image