AN ALIASED DISCRETE FOURIER TRANSFORM FOR ARBITRARY-FREQUENCY HARMONIC BALANCE

Kurzfassung

n recent years, the harmonic balance method has become an important tool for evaluating periodic flow phenomena in the design of turbomachinery components. Usually, this method works in both the time and frequency domains. In the time domain, nonlinearities of the investigated flow equations can be easily described, whereas in the frequency domain, approximate solutions for the periodic flow problem can be found efficiently. Switching between these two representations in the context of a Discrete Fourier Transform (DFT) is therefore of fundamental importance for the quality of the simulation method. DFTs that require a large number of sampling points increase the simulation time considerably, whereas inaccurate DFTs such as the almost periodic Fourier transform can significantly impair the quality of the results obtained due to poor mapping properties. In this paper, we present an aliased discrete Fourier transform. In contrast to the classical DFT, it is able to represent arbitrary frequency combinations with a small number of sampling points by mapping the relevant spectrum to an equivalent spectrum that is as compact as possible. At the same time, the new transform is unitary, i.e., the change between frequency and time domains is exact. Additionally, the time domain representation is exact w.r.t. the modes’ phase positions. Furthermore, a novel targeted treatment of aliasing makes it possible to exclude interference with unresolved, but energy-carrying, modes at low additional cost. The new method is compared with existing methods from the literature. Its advantages are demonstrated using a selection of turbomachinery test cases, showing accuracy comparable to reference simulations at a reduced computational cost.