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

TLDR

Many unsteady flows in turbomachinery are periodic in time. A harmonic balance technique for modeling unsteady nonlinear flows in turbomachinery is presented. The method represents unsteady flow variables as Fourier series in time with spatially varying coefficients, yielding a harmonic‑balance form of the Euler or Navier–Stokes equations that can be solved efficiently as a steady problem using conventional CFD techniques such as pseudotime marching, local time stepping, and multigrid acceleration. The technique is 10–100× faster than conventional nonlinear time‑domain CFD, accurately models strongly nonlinear transonic viscous flows in a high‑pressure compressor rotor with few Fourier terms, and shows that fluid nonlinearities can significantly affect even small blade vibrations.

Abstract

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