Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique

A harmonic balance technique for modeling unsteady nonlinear  ows in turbomachinery is presented. The analysis exploits the fact that many unsteady  ows of interest in turbomachinery are periodic in time. Thus, the unsteady  ow conservation variables may be represented by a Fourier series in time with spatially varying coefŽ cients. This assumption leads to a harmonic balance form of the Euler or Navier–Stokes equations, which, in turn, can be solved efŽ ciently as a steady problem using conventional computational  uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efŽ cient, at least one to two orders of magnitude faster than conventional nonlinear time-domainCFD simulations. Computationalresults for unsteady, transonic, viscous  ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear  ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the  ow. Furthermore, in some cases,  uid nonlinearities are found to be important for surprisingly small blade vibrations.