Almost Periodic Fourier Transformations for the Analysis of Periodic Phenomena in Turbomachinery

Abstract

In the modern multidisciplinary design of turbomachinery components unsteady simulations are essential to understand such topics as flutter, noise generation or blade-row interaction effects. Although (nonlinear) frequency-based methods can often be used for such applications, these methods typically require a detailed knowledge of the frequency content of the unsteady flow field. However, as blade-row interactions in multistage turbomachinery components may give rise to a plethora of solution harmonics, a full specification of these frequencies is not always feasible. In this context time-domain simulations are attractive since an a priori specification of the solution harmonics is not required and, within the limits of the simulation duration and time step size, all unsteady effects can be directly captured. It nevertheless remains a challenge in such time-domain simulations to efficiently extract the frequency content. This is particularly so when incommensurable frequencies are present in a simulation. To address this issue we investigate in this work the application of Almost Periodic Fourier Transforms (APFTs) to compute the harmonic content of time-domain simulations online. In contrast to a conventional Discrete Fourier Analysis (DFT), APFTs do not require the definition and use of a beat frequency, and therefore often require significantly fewer sampling points, allowing the discrete spectra of unsteady simulations to be determined more quickly with reduced numerical effort. In this paper we present a brief introduction to the APFT methodology and describe its implementation in the CFD code TRACE. To investigate and demonstrate the application of the method we consider two academic test cases: the advection of an entropy disturbance in an empty duct, and the interaction of an artificial vorticity disturbance with an oscillating compressor blade corresponding to the well-known aeroelasticity test case Standard Configuration 10.