MODIFIED METHOD OF CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME DEPENDENT NAVIER-STOKES PROBLEMS

Zhiyong Si,Xinlong Feng,Yunxia Wang

doi:10.3846/13926292.2015.1091394

Abstract

In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent NavierStokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

Highlights

In this paper, we consider the time-dependent Navier-Stokes problems ut − ν u + (u · ∇)u + ∇p = f,∇ · u = 0, u(x, 0) = u0(x), u(x, t) = 0, x ∈ Ω × [0, T ], x ∈ Ω × [0, T ], x ∈ Ω, x ∈ ∂Ω × [0, T ], (1.1)where Ω is a bounded domain in R2 assumed to have a Lipschitz continuous boundary ∂Ω. u = (u1(x, t), u2(x, t))T represents the velocity vector, p(x, t) represents the pressure, f (x, t) is the body force, ν = 1/Re the viscosity number, Re is the Reynolds number.Developing efficient finite element methods for the Stokes and Navier-Stokes equations is a key component in the incompressible flow simulation
We present a modified method of characteristics variational multiscale (VMS) finite element method based on the L2 projection for the time dependent NavierStokes equations
In order to show the efficiency of the modified method of characteristics variational multiscale (MMOCVMS) finite element method, we firstly present some numerical results of analytical solution problems

Summary

Introduction

There are many works devoted to this method, e.g. VMS method for the Navier-Stokes equations [26]; a two-level VMS method for convectiondominated convection-diffusion problems [29]; VMS methods for turbulent flow [9, 30, 31]; large-eddy simulation (LES) [24, 25, 32]; subgrid-scale models for the incompressible flow [22, 44]. The main difference is the definition of the large scales projection (L2-projection in [28] and elliptic projection in [27]). There is another class of VMS method which rely on a three-scale decomposition of the flow field into large, resolved small and unresolved scales [8]

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Results

Conclusion