An Upwind Nodal Integral Method for Incompressible Fluid Flow

An upwind nodal solution method is developed for the steady, two-dimensional flow of an incompressible fluid. The formulation is based on the nodal integral method, which uses transverse integrations, analytical solutions of the one-dimensional averaged equations, and node-averaged uniqueness constraints to derive the discretized nodal equations. The derivation introduces an exponential upwind bias by retaining the streamwise convection term in the homogeneous part of the transverse-integrated convection-diffusion equation. The method is adapted to the stream function-vorticity form of the Navier-Stokes equations, which are solved over a nonstaggered nodal mesh. A special nodal scheme is used for the Poisson stream function equation to properly account for the exponentially varying vorticity source. Rigorous expressions for the velocity components and the no-slip vorticity boundary condition are derived from the stream function formulation. The method is validated with several benchmark problems. An idealized purely convective flow of a scalar step function indicates that the nodal approximation errors are primarily dispersive, not dissipative, in nature. Results for idealized and actual recirculating driven-cavity flows reveal a significant reduction in false diffusion compared with conventional finite difference techniques.

[1]  T. Shih,et al.  Effects of grid staggering on numerical schemes , 1989 .

[2]  Yassin A. Hassan,et al.  a Stable Mass-Flow Two-Dimensional Skew Upwind Scheme , 1983 .

[3]  A. Ougouag,et al.  ILLICO-HO: A Self-Consistent Higher Order Coarse-Mesh Nodal Method , 1988 .

[4]  I. Christie.  Upwind compact finite difference schemes , 1985 .

[5]  D. N. De G. Allen,et al.  RELAXATION METHODS APPLIED TO DETERMINE THE MOTION, IN TWO DIMENSIONS, OF A VISCOUS FLUID PAST A FIXED CYLINDER , 1955 .

[6]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[7]  Michael A. Leschziner,et al.  Discretization of nonlinear convection processes: A broad-range comparison of four schemes , 1985 .

[8]  Ching-Jen Chen,et al.  Finite-analytic numerical solution of heat transfer in two-dimensional cavity flow , 1981 .

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

[10]  O. C. Zienkiewicz,et al.  An ‘upwind’ finite element scheme for two‐dimensional convective transport equation , 1977 .

[11]  W. C Horak,et al.  A nodal coarse-mesh method for the efficient numerical solution of laminar flow problems , 1981 .

[12]  G. D. Raithby,et al.  Skew upstream differencing schemes for problems involving fluid flow , 1976 .

[13]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

[14]  B. P. Leonard,et al.  Simple high-accuracy resolution program for convective modelling of discontinuities , 1988 .

[15]  R. D. Lawrence.  PROGRESS IN NODAL METHODS FOR THE SOLUTION OF THE NEUTRON DIFFUSION AND TRANSPORT EQUATIONS , 1986 .

[16]  M. Cross,et al.  A critical evaluation of seven discretization schemes for convection–diffusion equations , 1985 .

[17]  D. Spalding.  A novel finite difference formulation for differential expressions involving both first and second derivatives , 1972 .

[18]  Gerry E. Schneider,et al.  A MODIFIED STRONGLY IMPLICIT PROCEDURE FOR THE NUMERICAL SOLUTION OF FIELD PROBLEMS , 1981 .

[19]  C. Prakash,et al.  APPLICATION OF THE LOCALLY ANALYTIC DIFFERENCING SCHEME TO SOME TEST PROBLEMS FOR THE CONVECTION-DIFFUSION EQUATION , 1984 .

[20]  George D. Raithby,et al.  IMPROVED FINITE-DIFFERENCE METHODS BASED ON A CRITICAL EVALUATION OF THE APPROXIMATION ERRORS , 1979 .

[21]  Ching-Jen Chen,et al.  Finite analytic numerical method for unsteady two-dimensional Navier-Stokes equations , 1984 .

[22]  H. B. Keller,et al.  Driven cavity flows by efficient numerical techniques , 1983 .