Stable and accurate pressure approximation for unsteady incompressible viscous flow

Jian-Guo Liu,Robert L Pego,Jie Liu

doi:10.1016/j.jcp.2010.01.010

Abstract

How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier-Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy of computing pressure in two kinds of existing time-discrete projection methods implicit in viscosity only, and (b) devise new higher-order accurate time-discrete projection methods that extend a slip-correction idea behind the well-known finite-difference scheme of Kim and Moin. We test these schemes for stability and accuracy using various combinations of C^0 finite elements. For all three kinds of time discretization, one can obtain third-order accuracy for both pressure and velocity without a time-step stability restriction of diffusive type. Furthermore, two kinds of projection methods are found stable using piecewise-linear elements for both velocity and pressure.

Highlights

IntroductionWe consider the Navier-Stokes equations (NSE) for incompressible fluid flow in a domain Ω in RN (N = 2 or 3) with velocity specified on the boundary Γ = ∂Ω
We consider the Navier-Stokes equations (NSE) for incompressible fluid flow in a domain Ω in RN (N = 2 or 3) with velocity specified on the boundary Γ = ∂Ω.We write the momentum equation and boundary conditions in the form∂tu + ∇p = ν∆u + F in Ω, (1) u=g on Γ. (2)Here u is the fluid velocity, p the pressure, and ν = 1/Re is the kinematic viscosity coefficient, taken to be a fixed positive constant
The results for the (2,2) and (3,2) Pressure approximation (PA) and Slip correction (SC) schemes indicate that the eigenvalues always have magnitude less than 1. We found this result insensitive to spatial resolution, and it holds with various finite-element pairs for spatial approximation that were tested

Summary

Introduction

We consider the Navier-Stokes equations (NSE) for incompressible fluid flow in a domain Ω in RN (N = 2 or 3) with velocity specified on the boundary Γ = ∂Ω. The pressure field p should ensure that the velocity is divergence-free, with ∇ · u = 0 in Ω. This incompressibility condition is the source of many difficulties associated with the numerical approximation of solutions of NSE, especially in the presence of boundaries. Projection methods, deriving from classic work of Chorin and Temam, aim to deal efficiently with incompressibility through strategies that involve the Helmholtz decomposition of an arbitrary vector field into a sum of a gradient plus a divergence-free field. For many years, projection methods were plagued by large and poorly understood numerical boundary-layer errors

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