Application of an Iterative High Order Difference Scheme Along With an Explicit System Solver for Solution of Stream Function-Vorticity Form of Navier–Stokes Equations

Abstract

This paper describes the general convection-diffusion equation in 2D domain based on a particular fourth order finite difference method. The current fourth-order compact formulation is implemented for the first time, which offers a semi-explicit method of solution for the resulting equations. A nine point finite difference scheme with uniform grid spacing is also put into action for discretization purpose. The proposed numerical model is based on the Navier–Stokes equations in a stream function-vorticity formulation. The fast convergence characteristic can be mentioned as an advantage of this scheme. It combines the enhanced Fournié's fourth order scheme and the expanded fourth order boundary conditions, while offering a semi-explicit formulation. To accomplish this, some coefficients which do not influence the solutions are also omitted from Fournié's formulation. Consequently, very accurate results can be acquired with a relatively coarse mesh in a short time. The robustness and accuracy of the proposed scheme is proved using the benchmark problems of flow in a driven square cavity at medium and relatively high Reynolds numbers, flow over a backward-facing step, and flow in an L-shaped cavity.

A computational methodology for two‐dimensional fluid flows

“…Analysis of the simulated data shows that the average CPU times in the wall-clock unit, for CFL numbers 0:25.t D 10 3 / and 2:5.t D 10 2 /, are about 12 days and 1 day, respectively, for the same dimensionless integration time. The pattern of the velocity profile with increasing Re is similar to what was presented by Perrin and Hu [28], Ghia et al [67] and Ghadimi et al [66]. Most classical CFD codes would use a CFL<1 because the advection term is typically treated with an explicit scheme.…”

“…E and Liu [2] discussed the occurrence of artificial numerical boundary layer if a classical fractional step method is employed to solve the incompressible Navier-Stokes equation (see also [7]). Although the present numerical method and the set of equations are different than those used by Ghia et al [67] and Ghadimi et al [66], these reference results are useful feedback for assessing a simulation of shear-driven flow using the proposed collocation method. This is an incompressible flow in a square cavity OE0; 1 OE0; 1 with no slip conditions u D 0 D v on boundaries, x D 0; x D 1 and y D 0.…”

“…A shear-driven flow is a circulation in a confined box [28,66,67], where an imposed shear stress drives the fluid, and is a classical test problem for the assessment of CFD codes. The simulation of a shear-driven flow using the incompressible Navier-Stokes equation is a challenging endeavor in the field of CFD.…”

“…The simulation of a shear-driven flow using the incompressible Navier-Stokes equation is a challenging endeavor in the field of CFD. Ghia et al [67] and Ghadimi et al [66] examined a similar shear-driven flow, using the steady-state vorticity equation (see Equations (1) and (2) of [66]). In this article, we do not have enough room to address these unresolved challenges with this fractional step projection method; however, we aim to demonstrate the potential of the present discretization technique to the field of CFD, using a simulation of the classical shear-driven flow.…”

Numerical Simulations of Flow around Square Cylinder Using an Iterative High Order Difference Scheme

“…Similarly, Lubcke et al (2001) tested both RANS an LES method in simulation of flow around rectangular cylinder at Reynolds number Re=22,000. Ghadimi et al (2013) presented a new forth order finite difference scheme to solve the steady Navier-Stokes equation in the form of vorticity-streamfunction. They extended Fourni'e's (2006) formulation and applied it to solve Navier-Stokes equation in the benchmark problem of flow in the driven cavity up to .…”