Time Integration Schemes for the Unsteady Navier-Stokes Equations

The efficiency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the efficiency of higher-order Runge-Kutta schemes in comparison with the popular Backward Differencing Formulations. For this comparison an unsteady two-dimensional laminar flow problem is chosen, i.e. flow around a circular cylinder at Re=1200. It is concluded that for realistic error tolerances (smaller than 10,1) fourth- and fifth-order Runge Kutta schemes are the most efficient. For reasons of robustness and computer storage, the fourth-order Runge-Kutta method is recommended. The efficiency of the fourth-order Runge-Kutta scheme exceeds that of second-order Backward Difference Formula (BDF2) by a factor of 2.5 at engineering error tolerance levels (10^-1-10^-2). Efficiency gains are more dramatic at smaller tolerances

A new implicit algorithm with multigrid for unsteady incompressible flow calculations

We present a new fully-implicit algorithm for unsteady incompressible flow calculations for both the Euler and Navier-Stokes equations. The new method couples the artificial compressibility approach with an implicit A-stable discretization of the unsteady terms in order to advance the solution in a time-accurate manner with no stability limitations on the time step. A pseudotransient steady state problem is solved at each time step to provide a direct coupling between the velocity and pressure fields, and to satisfy the divergence-free constraint. The present algorithm solves the pseudotransient problem by using the highly efficient multigrid time stepping technique, originally developed by Jameson [I] for compressible flow calculations. Both viscous and inviscid test problems are presented. An inviscid two-dimensional flow over an oscillating cylinder is used to validate the method by comparison with analytic results. The mean quantities of the unsteady viscous flow over a circular cylinder for Re 5 200 are computed and found to be in good agreement with the computational and cxperimental data obtained by other authors. Results for the unsteady viscous flow over a NACA0012 airfoil at 20 " angle of attack are also presented, and domain truncation and time resolution effects are discussed