Efficient Solution of Unsteady Nonlinear Flows Using a Multiple Zone Harmonic Balance Technique

Abstract

An efficient two-dimensional multiple zone harmonic balance solver for calculating unsteady nonlinear flows is presented. The solver adapts Roe's flux difference splitting algorithm such that it can be used to discretize the harmonic balance equations. It is demonstrated that the numerical solutions produced by this solver are in good agreement with known results for a variety of unsteady flows, including cascade flows. The solver incorporates a multiple zone technique in which the number of harmonics is allowed to vary in different zones of a flow domain. In the present study, the multiple zone technique is optimized for unsteady nonlinear transonic flows. It is shown that the multiple zone technique reduces computational time by 50-60% in comparison to single zone solutions. It is additionally shown that these computational savings come with no change in the accuracy of the solution. An analysis of the temporal and spatial behavior of the waves associated with harmonic balance discretization schemes is also presented. In the temporal analysis, the numerical stability limits of several discretization schemes are worked out in detail, and a numerical instability associated with the first-order upwind discretization is removed. The numerical stability limits are verified through experimentation. In the spatial analysis, spatial wave amplification factors are derived for the same set of discretization schemes. A novel upwind approximation of the harmonic source term is introduced, and it is demonstrated that, for one-dimensional flows, this approximation eliminates the spatial wave dissipation associated with previously used cell-centered discretizations of the source term. However, it is found that the difference between the dissipation associated with each approximation of the source term is less pronounced in two-dimensions.