AIAA 2001–0152 Acceleration of Convergence to a Periodic Steady State in Turbomachinery Flows

This paper presents a technique used to accelerate the convergence of unsteady calculations of time-periodic flows to a periodic steady state. The basis of the procedure is the use of the discrete Fourier transform in time, and is similar to the harmonic balance procedure that has been pursued by Hall et. al. The technique is amenable to parallel processing, and convergence acceleration techniques such as multi-grid and implicit residual averaging. The computational efficiency of this method is compared with dual time stepping algorithms. Sample calculations are provided, and a comparison between solutions with varying temporal resolution is presented. The results show that the computational efficiency of the harmonic balance technique is largely a function of the temporal resolution. Initial experiments confirm the promise of the harmonic balance method to achieve significant reductions in computational cost. Introduction The calculation of periodic unsteady flows in tur-bomachinery continues to present a severe challenge to Computational Fluid Dynamics (CFD). The un-steadiness stems mainly from the relative motion of the rotating blade fields, and has a fundamental period which depends on the rate of rotation and the number of blade passages. Currently, a popular approach to the computation of this problem is to introduce a fully implicit A-stable time discretization of the flow equations requiring the solution of a set of coupled nonlinear equations at each physical time step. These equations are typically solved with an explicit inner iteration by introducing a pseudo-time variable and marching to a steady state. Converged solutions attained at the end of the inner iterations represent a solution of the implicit equations at the end of the physical time step. 1–4 A variety of techniques such as variable local pseudo-time steps, implicit residual averaging and multigrid are used to accelerate the convergence of the inner iterations. The efficiency of this dual time stepping method depends on the effectiveness of these acceleration techniques in restricting the number of inner iterations. If the physical time step can be increased in comparison to an explicit scheme by a factor larger than the number of inner iterations, the implicit scheme will be more efficient than the explicit scheme. In practice it has been found that for simple geometries and flows, on the order of 24-36 implicit time steps are sufficient to resolve the features of one period in the oscillation of the flow. Typically one restricts the number of inner iterations to the order …