OPTIMIZED DUAL-TIME STEPPING TECHNIQUE FOR TIME-ACCURATE NAVIER-STOKES CALCULATIONS

Mikhail Nyukhtikov

Moscow Institute of Physics and Technology Moscow, Russia

Natalia V. Smelova, Brian E. Mitchell, D. Graham Holmes

General Electric Global Research Center Niskayuna, NY USA

Abstract This paper presents an optimized discretization of the time derivative term for the dual-time stepping method. The proposed discretization is second order ac­curate and has a lower level of dissipation and dispersion errors than the conven­tional non-optimized second order discretization. Sample calculations demon­strate that the optimized scheme requires approximately 45-50% less time steps per unsteady cycle compared to the standard non-optimized scheme to resolve an unsteady fbw within a certain margin of amplitude error. The number of time steps per cycle can be reduced by 10-15% to keep the phase error less than a cer­tain level when the optimized scheme is used. Since time-accurate calculations are expensive, the proposed approach leads to significant savings of computa­tional time and resources.

1. Introduction

Flows in the turbomachinery environment are inherently unsteady. Physi­cal phenomena like vortex shedding, wake/blade row interaction, tip leakage etc. can be modeled correctly only by time-accurate non-linear methods. A number of such methods were developed over the past few years. One of them, the dual-time stepping method [Jameson, 1991], is widely used and is fa­vored for its convergence properties and its ease of implementation. It employs well-known convergence acceleration techniques like multigridding, false time marching, and residual smoothing. It also has been shown [Melson et al., 1993] that this method allows an implicit treatment of the real time derivative and this removes the upper stability limit from the size of the time step. However, the

449

K. C. Hall et al. (eds.),

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 449-459. © 2006 Springer. Printed in the Netherlands.

consideration of the stability limit is not sufficient because the dissipation and dispersion errors also limit the size of the time step.

This paper focuses on the development of a new optimized time derivative discretization, which has a better dissipation and dispersion properties as com­pared to conventional non-optimized discretization. This new discretization allows larger time steps (hence, reduced computational time) for a given level of numerical error. The technique for constructing a low-dissipation and low – dispersion scheme [Tam and Webb, 1993, Hu et al., 1994] is based on intro­ducing more than the minimum required number of points to the discretization stencil. The coefficients for these extra points are determined by minimizing the dissipation and dispersion errors. Note that the introduction of extra points increases the memory requirements.

In this paper, a second order accurate discretization of the time derivative is considered. The optimized scheme with low dissipation and dispersion errors is constructed by introducing only one additional point. A detailed description of the optimization process is provided. The paper also presents test problems, which demonstrate the overall improvements and speedup.