Harmonic Balance Analysis of Blade Row Interactions in a Transonic Compressor

In this paper we apply the harmonic balance technique to analyze an inlet guide vane and rotor interaction problem, and compare the computed flow solutions to existing experimental data. The computed results, which compare well with the experimental data, demonstrate that the technique can accurately and efficiently model strongly nonlinear periodic flows, including shock/vane interaction and unsteady shock motion. Using the harmonic balance approach, each blade row is modeled using a computational grid spanning just a single blade passage regardless of the actual blade counts. For each blade row, several sublime level solutions that span a single time period are stored. These sublime level solutions are related to each other through the time derivative term in the Euler (or Navier-Stokes) equations, which is approximated by a pseudo-spectral operator, by complex periodicity conditions along the periodic boundary of each blade row's computational domain, and by the interface boundary conditions between the vane and rotor. Casting the governing equations in harmonic balance form removes the explicit dependence on time. Mathematically, the equations to be solved are similar in form to the steady Euler (or Navier-Stokes) equations with an additional source term proportional to the fundamental frequency of the unsteadiness. Thus, conventional steady-state computational fluid dynamics techniques, including local time stepping and multigrid acceleration, are used to accelerate convergence, resulting in a very efficient unsteady flow solver.