Due to the relative motion between adjacent blade rows the aerodynamic flow fields within turbomachinery are normally dominated by deterministic, periodic phenomena. In the numerical simulation of such unsteady flows (nonlinear) frequency-domain methods are therefore attractive as they are capable of fully exploiting the given spatial and temporal periodicity, as well as capturing or modelling flow nonlinearity. Central to the efficiency and accuracy of such frequency-domain methods is the selection of the frequencies and the circumferential modes to be resolved in simulations. Whilst trivial in the context of the simulation of a single compressor- or turbine-stage, the choice of solution modes becomes substantially more involved in multi-stage configurations. In this work the importance of mode scattering, in the context of the unsteady aerodynamic field, is investigated and quantified. It is shown that scattered modes can substantially impact the unsteady flow field and are essential for the accurate modelling of wake propagation within multistage configurations. Furthermore, an iterative approach is outlined, based on the spectral analysis of the circumferential modes at the interfaces between blade rows, to identify the dominant solution modes that should be resolved in the adjacent blade row. To demonstrate the importance of mode scattering and validate the approach for their identification the unsteady blade row interaction within a 4.5 stage axial compressor is computed using both the harmonic balance method and, based on a full annulus midspan simulation, a time-domain method. Through the inclusion of scattered modes it is shown that the solution quality of the harmonic balance results is comparable to that of the nonlinear time-domain simulation.

Due to the relative motion between adjacent blade rows the aerodynamic flow fields within turbomachinery are usually dominated by deterministic, periodic phenomena. In the numerical simulation of such unsteady flows, (nonlinear) frequency-domain methods are therefore attractive as they are capable of fully exploiting the given spatial and temporal periodicity, as well as modelling flow nonlinearities. A nontrivial issue in the application of frequency-domain methods to turbomachinery flows is to simultaneously capture disturbances with multiple fundamental frequencies in one relative system. In case of harmonically related frequencies, the interval spanned by the sampling points typically resolves the common fundamental frequency. To avoid signal aliasing the highest harmonic of the common frequency should be sampled with an appropriate number of sampling points. However, when the common fundamental frequency is very low in relation to the frequencies of primary interest, equidistant time sampling leads to a high number of sampling points, hence frequency-domain methods can become computationally inefficient. Furthermore, 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, the standard discrete Fourier transform is no longer suitable and the basic harmonic balance method requires extension. In this article two nonlinear frequency-domain approaches for dealing with the accounted issues are demonstrated and compared. The first approach is a generalized harmonic balance method based on almost periodic Fourier transforms with non-equidistant time sampling. Then the so-called harmonic set approach, developed by the authors, is evaluated. Based on the neglection of the nonlinear, quadratic cross-coupling terms between higher harmonics of different fundamental frequencies, the harmonic set approach allows the superposition of periodic disturbances with different fundamental frequencies as well as the separated, equidistant sampling of the highest harmonic of each base frequency. The aim of this paper is to compare the computational efficiency and accuracy of the two methods and assess the impact of neglecting the quadratic cross-coupling terms.

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. 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.To overcome the issues regarding multi-frequency problems described above, 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. In this way the proposed method is capable of integrating the nonlinear coupling effects between higher harmonics of different fundamental frequencies.

The harmonic balance method has emerged as an efficient and accurate approach for computing periodic, as well as almost periodic, solutions to nonlinear ordinary differential equations. The accuracy of the harmonic balance method can however be negatively impacted by aliasing. Aliasing occurs because Fourier coefficients of nonlinear terms in the governing equations are approximated by a discrete Fourier transform (DFT). Understanding how aliasing occurs when the DFT is applied is therefore essential in improving the accuracy of the harmonic balance method. In this work, a new operator that describe the fold-back, i.e. aliasing, of unresolved frequencies onto the resolved ones is developed. The norm of this operator is then used as a metric for investigating how the time sampling should be performed to minimize aliasing. It is found that a time sampling which minimizes the condition number of the DFT matrix is the best choice in this regard, both for single and multiple frequency problems. These findings are also verified for the Duffing oscillator. Finally, a strategy for oversampling multiple frequency harmonic balance computations is developed and tested.

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