Solution of thermo‐fluid problems by collocation with local pressure correction

Purpose – The purpose of this paper is to explore the application of the mesh‐free local radial basis function collocation method (RBFCM) in solution of coupled heat transfer and fluid‐flow problems.Design/methodology/approach – The involved temperature, velocity and pressure fields are represented on overlapping five nodded sub‐domains through collocation by using multiquadrics radial basis functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time stepping.Findings – The performance of the method is assessed on the classical two dimensional de Vahl Davis steady natural convection benchmark for Rayleigh numbers from 103 to 108 and Prandtl number 0.71. The results show good agreement with other methods at a given range.Originality/value – The pressure‐velocity coupling is calculated iteratively, with pressure correction, predicted from the local mass continuity equat...

[1]  Božidar Šarler.  Towards a mesh-free computation of transport phenomena , 2002 .

[2]  G. D. Davis.  Natural convection of air in a square cavity: A bench mark numerical solution , 1983 .

[3]  W. Chen,et al.  New RBF collocation methods and kernel RBF with applications , 2001, ArXiv.

[4]  S. Atluri,et al.  The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods , 2002 .

[5]  B. Šarler,et al.  Primitive variable dual reciprocity boundary element method solution of incompressible Navier-Stokes equations , 1999 .

[6]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[7]  Satya N. Atluri,et al.  MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD FOR CONVECTION DIFFUSION PROBLEMS , 2000 .

[8]  B. Šarler,et al.  Meshless local radial basis function collocation method for convective‐diffusive solid‐liquid phase change problems , 2006 .

[9]  S. Atluri.  The meshless method (MLPG) for domain & BIE discretizations , 2004 .

[10]  H. Sadat,et al.  PERFORMANCE AND ACCURACY OF A MESHLESS METHOD FOR LAMINAR NATURAL CONVECTION , 2000 .

[11]  Ching-Shyang Chen,et al.  Radial basis function collocation method solution of natural convection in porous media , 2004 .

[12]  Alain J. Kassab,et al.  An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer , 2007 .

[13]  Božidar Šarler,et al.  From Global to Local Radial Basis Function Collocation Method for Transport Phenomena , 2007 .

[14]  Mehrdad T. Manzari,et al.  An explicit finite element algorithm for convection heat transfer problems , 1999 .

[15]  M. Hortmann,et al.  Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions , 1990 .

[16]  Andrew D. Back,et al.  Radial Basis Functions , 2001 .

[17]  B. Šarler,et al.  Solution of transient direct-chill aluminium billet casting problem with simultaneous material and interphase moving boundaries by a meshless method , 2006 .

[18]  O. C. Zienkiewicz,et al.  Characteristic‐based‐split (CBS) algorithm for incompressible flow problems with heat transfer , 1998 .

[19]  Decheng Wan,et al.  A NEW BENCHMARK QUALITY SOLUTION FOR THE BUOYANCY-DRIVEN CAVITY BY DISCRETE SINGULAR CONVOLUTION , 2001 .

[20]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[21]  B. Šarler.  A Radial Basis Function Collocation Approach in Computational Fluid Dynamics , 2005 .

[22]  Robert Vertnik,et al.  Meshfree explicit local radial basis function collocation method for diffusion problems , 2006, Comput. Math. Appl..