Unsteady Turbomachinery Simulations Using Harmonic Balance on a Discontinuous Galerkin Discretization

The Harmonic Balance method is well suited for analyzing unsteadiness in turbomachinery flows comprised of a few dominant frequencies. A harmonic condition is imposed on the temporal derivatives through a Fourier transform operation. The solution is then reinterpreted as a time-domain problem, where several instances of time (lying within the largest period) are solved for simultaneously with the enforcement of the time-harmonic condition providing coupling between time instances. A discontinuous Galerkin discretization is used together with overset grids to provide higher-order spatial accuracy and flexibility in representing complex geometry. In this work, the discontinuous Galerkin infrastructure is extended for unsteady problems with a Harmonic Balance method and a Diagonally Implicit Runge-Kutta time-integrator. Verification results are presented for both time integration approaches in addition to results for a turbine blade with unsteadiness driven by a prescribed unsteady inlet boundary condition. Comparisons of results from the Harmonic Balance and Diagonally Implicit Runge-Kutta approaches are very close, with some small discrepancies that require further investigation. Significantly, rapid convergence from the Newton solver is obtained for the Harmonic Balance approach applied to the Euler equations for the turbine blade problem. Solutions converged by 8–10 orders of magnitude are obtained in between 5 and 16 Newton steps.