Iterative projection method for unsteady Navier-Stokes equations with high Reynolds numbers

TL;DR

This paper introduces a novel iterative projection method for solving unsteady Navier-Stokes equations at high Reynolds numbers, achieving weak divergence-free velocities with enhanced stability and efficiency over traditional methods.

Contribution

The paper presents a new iterative projection technique incorporating BDF2 discretization, skew-symmetric convection, and iterative projections, improving accuracy and convergence for high Reynolds number flows.

Findings

Effectively handles high Reynolds numbers with few iterations

Achieves weak divergence-free velocity in practice

Outperforms traditional explicit convection methods

Abstract

A new iterative projection method is proposed to solve the unsteady Navier-Stokes equations with high Reynolds numbers. The convectional projection method attempts to project the intermediate velocity to the divergence free space only once per time step. However, such a velocity is not genuinely divergence free in general practice, which can yield large errors when the Reynolds number is high. The new method has several important features: the BDF2 time discretization, the skew-symmetric convection in a semi-implicit form, two modulating parameters, and the iterative projections in each time step. A major difficulty in the proof of iteration convergence is the nonlinear convection. We solve this problem by first analyzing the non-convective scheme with a focus on the spectral properties of the iterative matrix, and then employing a delicate perturbation analysis for the convective scheme. The work achieves the weakly divergence free velocity (strongly divergence free for divergence free finite element spaces), and the rigorous stability and error analysis when the iterations converge. The three dimensional numerical tests confirm that this new method can effectively treat high Reynolds numbers with only a few iterations per time step, where the convectional projection method and the iterative projection method with the explicit convection would fail.