A stable second‐order mass‐weighted upwind scheme for unstructured meshes

In this paper, an original second-order upwind scheme for convection terms is described and implemented in the context of a Control-Volume Finite-Element Method (CVFEM). The proposed scheme is a second-order extension of the first-order MAss-Weighted upwind (MAW) scheme proposed by Saabas and Baliga (Numer. Heat Transfer 1994; 26B:381–407). The proposed second-order scheme inherits the well-known stability characteristics of the MAW scheme, but exhibits less artificial viscosity and ensures much higher accuracy. Consequently, and in contrast with nearly all second-order upwind schemes available in the literature, the proposed second-order MAW scheme does not need limiters. Some test cases including two pure convection problems, the driven cavity and steady and unsteady flows over a circular cylinder, have been undertaken successfully to validate the new scheme. The verification tests show that the proposed scheme exhibits a low level of artificial viscosity in the pure convection problems; exhibits second-order accuracy for the driven cavity; gives accurate reattachment lengths for low-Reynolds steady flow over a circular cylinder; and gives constant-amplitude vortex shedding for the case of high-Reynolds unsteady flow over a circular cylinder. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  S. B. Nasrallah,et al.  Effect of the blockage ratio on the flow in a channel with a built-in square cylinder , 2003 .

[2]  M. Braza,et al.  Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder , 1986, Journal of Fluid Mechanics.

[3]  M. Hainke,et al.  Comparison of linear and quadratic shape functions for a hybrid control‐volume finite element method , 2002 .

[4]  J. Li,et al.  Numerical study of Laminar flow past one and two circular cylinders , 1991 .

[5]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[6]  C. Hwang,et al.  Locally implicit total-variation-diminishing schemes on unstructuredtriangular meshes , 1991 .

[7]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[8]  C. Prakash,et al.  AN IMPROVED CONTROL VOLUME FINITE-ELEMENT METHOD IFOR HEAT AND MASS TRANSFER, AND FOR FLUID FLOW USING EQUAL-ORDER VELOCITY-PRESSURE INTERPOLATION , 1986 .

[9]  B. P. Leonard,et al.  Why You Should Not Use 'Hybrid', 'Power-Law' or Related Exponential Schemes for Convective Modelling—There Are Much Better Alternatives , 1995 .

[10]  L. J. Lee,et al.  Simulation of Liquid Composite Molding Based on Control-Volume Finite Element Method , 2002 .

[11]  Sassi Ben Nasrallah,et al.  Two-dimensional laminar fluid flow and heat transfer in a channel with a built-in heated square cylinder , 2003 .

[12]  Christian Masson,et al.  Development of a fully coupled control‐volume finite element method for the incompressible Navier–Stokes equations , 2004 .

[13]  Christian Masson,et al.  Co‐located equal‐order control‐volume finite element method for two‐dimensional axisymmetric incompressible fluid flow , 1994 .

[14]  S. G. Rubin,et al.  A diagonally dominant second-order accurate implicit scheme , 1974 .

[15]  Suhas V. Patankar,et al.  A CONTROL VOLUME-BASED FINITE-ELEMENT METHOD FOR SOLVING THE NAVIER-STOKES EQUATIONS USING EQUAL-ORDER VELOCITY-PRESSURE INTERPOLATION , 1984 .

[16]  Gerry E. Schneider,et al.  A SKEWED, POSITIVE INFLUENCE COEFFICIENT UPWINDING PROCEDURE FOR CONTROL-VOLUME-BASED FINITE-ELEMENT CONVECTION-DIFFUSION COMPUTATION , 1986 .

[17]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

[18]  Bantwal R. Baliga,et al.  CO-LOCATED EQUAL-ORDER CONTROL-VOLUME FINITE-ELEMENT METHOD FOR MULTIDIMENSIONAL, INCOMPRESSIBLE, FLUID FLOW—PART II: VERIFICATION , 1994 .

[19]  B. R. Baliga,et al.  A NEW FINITE-ELEMENT FORMULATION FOR CONVECTION-DIFFUSION PROBLEMS , 1980 .

[20]  Subhash C. Mishra,et al.  TRANSIENT CONDUCTION AND RADIATION HEAT TRANSFER WITH HEAT GENERATION IN A PARTICIPATING MEDIUM USING THE COLLAPSED DIMENSION METHOD , 2001 .