Over the past years, nonlinear frequency-domain methods have become a state-of-the-art technique for the numerical simulation of unsteady flow fields within multistage turbomachinery as they are capable of fully exploiting the given spatial and temporal periodicities, as well as modeling flow nonlinearities in a computationally efficient manner. Despite this success, it still remains a significant challenge to capture nonlinear interaction effects within the context of configurations with multiple fundamental frequencies. If all frequencies are integer multiples of a common fundamental frequency, the interval spanned by the sampling points typically resolves the period of the common base frequency. For configurations in which the common frequency is very low in relation to the frequencies of primary interest, many sampling points are required to resolve the highest harmonic of the common fundamental frequency and the method becomes inefficient. In addition, when a problem can no longer be described by harmonic perturbations that are integer multiples of one fundamental frequency, as it may occur in two-shaft configurations or when simulating the nonlinear interaction in the context of forced response or flutter, then the standard discrete Fourier transform is no longer suitable, and the basic harmonic balance (HB) method requires extension. One possible approach is to use almost periodic Fourier transforms with equidistant or nonequidistant time sampling. However, the definition of suitable sampling points that lead to well-conditioned Fourier transform matrices and small aliasing errors is an intricate issue and far from straightforward. To overcome the issues regarding multifrequency problems described earlier, a new harmonic balance approach based on multidimensional Fourier transforms in time is presented. The basic idea of the approach is that instead of defining common sampling points in a common time period, separate time domains, one for each base frequency, are spanned, and the sampling points are computed equidistantly within each base frequency's period. Since the sampling domain is now extended to a multidimensional time-domain, all time instant combinations covering the whole multidimensional domain are computed as the Cartesian product of the sampling points on the axes. In a similar fashion, the frequency-domain is extended to a multidimensional frequency-domain by the Cartesian product of the harmonics of each base frequency, so that every point defined by the Cartesian product is an integer linear combination of the occurring frequencies. In this way, the proposed method is capable of fully integrating the nonlinear coupling effects between higher harmonics of different fundamental frequencies by using multidimensional discrete Fourier transforms within the harmonic balance solution procedure. The aim of this paper is to introduce the multidimensional harmonic balance method in detail and demonstrate the capability of the approach to simultaneously capture unsteady disturbances with arbitrary excitation frequencies. Therefore, the well-established aeroelasticity testcase standard configuration 10 in the presence of an artificial inflow disturbance, which mimics an upstream blade wake, is investigated. The crucial aspect of the proposed testcase is that a small ratio of the frequency of the inflow disturbance and the blades vibration frequency is chosen. To demonstrate the advantages of the newly proposed multidimensional harmonic balance approach, the results are compared to unsteady simulations in the time-domain and to state-of-the-art frequency-domain methods based on one-dimensional discrete Fourier transforms.

The aim of this study was to assess the capabilities of different simulation approaches to predict the flutter stability of a steam turbine rotor. The focus here was on linear and nonlinear frequency domain solvers in combination with the energy method, which is widely used for the prediction of flutter onset. Whereas a GMRES solver was used for the linear problem, the nonlinear methods employed a time-marching procedure. The solvers were applied to the flutter analysis of the first rotor bending mode of the open Durham Steam Turbine test case. This test case is representative of the last stage of modern industrial steam turbines. We compared our results to those published by other researchers in terms of aerodynamic damping and local work per cycle coefficients. Time-domain, harmonic balance, and time-linearised methods were compared to each other in terms of CPU efficiency and accuracy.

The aim of this paper is to demonstrate how the harmonic balance method can be used to predict rotor–rotor and stator–stator interactions in turbomachinery. These interactions occur in the form of clocking and indexing. Whereas clocking refers to the dependency of the performance on the relative circumferential positioning of the rotors or stators, the term indexing is used when different blade (or vane) counts lead to an aperiodic time-averaged flow. The approach developed here is closely related to the one presented by He, Chen, Wells, Li, and Ning, who generalised the Nonlinear Harmonic method to zero-frequency disturbances. In particular, configurations with only one passage per blade row are used for the simulations. We validate the methods by means of the simulation of a fan stage configuration with rotationally asymmetric inlet conditions. It is demonstrated that the harmonic balance solver is able to accurately predict the inhomogeneity of the time-averaged flow field in the stator row. Moreover, the results show that the approach offers a considerable gain in computational efficiency.