An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier–Stokes equations

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Abstract

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier–Stokes equations. This proposed method solves a smaller system of nonlinear Navier–Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier–Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection–diffusion–reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier–Stokes solutions.

Introduction

Numerical simulation of incompressible viscous fluid flows remains an area of continuous challenges due to the approximation of advective terms in the multi-dimensional domain. In addition to the notorious convective instability problem, approximation of these direction relevant terms can introduce false diffusion error [1]. Therefore, a well-suited multi-dimensional upwinding scheme should effectively dispense with the crosswind diffusion error without at the sacrifice of scheme destabilization. Splitting of the equation has been known to be a proper way to solve the multi-dimensional equation by obtaining the solutions more efficiently and accurately using the analytical one-dimensional model [2]. We propose in this paper, however, a truly two-dimensional flux discretization scheme to avoid slow convergence in the operator sweeping. To reduce the afore-mentioned false diffusion error, the general solution of the investigated two-dimensional model transport equation will be taken into account in the approximation of flux terms for velocities.

When simulating the steady incompressible Navier–Stokes equations in co-located (or non-staggered) grids, node-to-node oscillatory pressure solutions arising from the decoupling of velocity and pressure fields have been frequently reported [1]. This motivated us to discretize the currently investigated elliptic-type partial differential equations within the non-staggered grid context to prevent the oscillatory pressure solutions. When solving the incompressible Navier–Stokes in non-staggered grids, central approximation of the pressure gradient terms may lead to pressure odd–even decoupling. In order to eliminate this problem, an adequate amount of artificial dampings can be added to the scheme implicitly or explicitly for the sake of stability enhancement [3], [4], [5], [6].

Linearization of the nonlinear terms in the incompressible Navier–Stokes equations plays another essential role in the assessment of computational efficiency. Improper linearization of convection terms in the flow equations may slow down convergence or can even cause the divergent solution to occur. Amongst the methods reported in the literature for the linearization of nonlinear Navier–Stokes equations, the multi-level method has gained an increasing acceptance in the past few years. As the name of this class of methods indicates, the multi-level method [7], [8] involves calculating the solutions at different levels of the grid system. Take the two-level method as an example, the differential equation is solved firstly at nodes in the coarse grid system, at which the solutions can be computed less expensively. This is followed by a computationally more intensive calculation of the same differential equation on the fine mesh. Note that the convergent solutions must be calculated in the coarse mesh. As a result, the linearization method chosen for the convective terms shown in the momentum equation plays also an essential role. For this reason, the computationally more efficient Oseen-type linearization method will be employed to render the linearized equation cast in the convection–diffusion–reaction differential form.

The reminder of this paper is organized as follows. In Section 2, the governing equations cast in the primitive variable form are solved along with the prescribed pressure boundary value. This is followed by presenting the proposed prolongation operator for effectively mapping the convergent solutions obtained at the coarse mesh to those obtained at the fine mesh. In Section 4, the underlying five-point convection–diffusion–reaction (CDR) scheme will be presented to accurately solve the linearized momentum transport equations. Two theoretically derived discrete pressure gradient operators are also presented in Section 4 in order to save the CPU time without suffering from even–odd oscillations in the non-staggered grids. In Section 5, the two-level Oseen model implemented with the proposed implicit and explicit compact pressure gradient approximation schemes is analytically validated by solving the problem which is amenable to the exact solution. Finally, some conclusions are drawn in Section 6.

Section snippets

Governing equations

The incompressible viscous flow motion, which is governed by the following continuity and momentum equations, will be dealt with in this paper:·u=0,(u·)u=-p+1Re2u+f.The chosen primitive variables (u,p) will be sought subject solely to the specified boundary condition for u [9]. All lengths have been normalized by L, the velocity components by u, the time by L/u, and the pressure by ρu2, where ρ denotes the fluid density. The resulting Reynolds number Re(ρuL/μ) represents the measure of

Two-level Navier–Stokes solver

In this section, the linearization method for the convective term (u̲·)u̲ will be presented in both coarse and fine meshes. Within the Newton linearization framework, expansion of ST with respect to the two arbitrary variables S and T at the iteration level k leads to the following updated expression for ST, namely, Sk+1Tk+1=Sk+1Tk+SkTk+1-SkTk++H.O.T. By virtue of this expansion equation, (u2)xk+1 and (uv)yk+1 shown in the x- and y-momentum equations can be approximated to render the

Five-point convection–diffusion–reaction (CDR) scheme

The original idea of the analytical CDR scheme presented in [2] will be extended to the analysis of two-dimensional equation. In view of Eqs. (4), (5), the following model equation for ϕ is considered for the sake of description of the CDR scheme:aϕx+bϕy-k2ϕ+cϕ=f.To eliminate the convective instability problem and to retain the prediction accuracy, the following general solution of the above model equation is employed:ϕ(x,y)=A1eλ1x+A2eλ2x+A3eλ3y+A4eλ4y+fc,where λ1,2=a±a2+4ck2k and λ3,4=b±b2+4ck

Analytic Navier–Stokes problem

We will verify the proposed two-level Navier–Stokes solver by solving the problem, defined in 0x,y1, amenable to the following exact solutions:u=-2(1+y)(1+x)2+(1+y)2,v=2(1+x)(1+x)2+(1+y)2,p=-2(1+x)2+(1+y)2.We plot the values of log(err1err2) against log(h1h2), where the L2 error norms err1 and err2 are obtained at two consecutively refined mesh sizes h1 and h2, to calculate the scheme’s rate of convergence. In Fig. 4, the predicted quadratic spatial rates of convergence for u and p, computed

Concluding remarks

The main feature of the two-level method proposed for effectively solving the incompressible Navier–Stokes solutions in non-staggered grids is the derived prolongation operator aimed to accurately communicate the nodal velocities obtained at the grid points in two mesh levels. Another distinct feature of the present scheme development is the transformation of the convection–diffusion differential equation into the convection–diffusion–reaction equation so as to be able to apply the rigorously

Acknowledgment

This work was supported by the National Science Council of the Republic of China under Grants NSC94-2611-E-002-021 and NSC94-2745-P-002-002.

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