A New Harmonic Balance Approach Using Multidimensional Time

Abstract

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.

Simulation of Indexing and Clocking with a New Multidimensional Time Harmonic Balance Approach

“…Furthermore, the non-zero frequency disturbances are represented in the 2π-periodic hyperphase domain τ = ω b t; cf. [8]. By computing the Cartesian product of the time instants with the sampling phases, the domain is extended to a multidimensional 2π-periodic time-phase domain, indicated by the grey background.…”

“…TRACE is a parallel Navier-Stokes flow solver developed at the Institute of Propulsion Technology of the German Aerospace Center (DLR) to model and investigate complex turbomachinery flows [13]. Details on the implementation and validation of the MDFT approach in TRACE can be found in [8].…”

“…However, it still remains a significant challenge to capture nonlinear interaction effects within the context of configurations with multiple fundamental frequencies. To overcome this issue, a new harmonic balance approach based on multidimensional discrete Fourier transforms (MDFTs) in time has recently been introduced by the authors [8]. The crucial idea of the multidimensional Fourier approach is to extend the one-dimensional time to multiple time dimensions so that each periodic flow phenomenon is described by a combination of sampling points in the multidimensional time.…”

Analyzing Unsteady Turbomachinery Flow Simulations With Mixing Entropy

“…Note that this is in stark contrast to configurations of two blade rows, where the flows in the different passages are identical up to a certain phase shift. Over the last ten years, several authors (see, e.g., [8][9][10][11][12]) have adapted the original harmonic balance approach to tackle the above problems.…”

Minimizing Aliasing in Multiple Frequency Harmonic Balance Computations

“…The harmonic set approach does however neglect nonlinear coupling between different harmonic sets, except through common frequencies such as the zeroth frequency. These coupling terms may be accounted for by introducing several time variables, one for each fundamental frequency in the basis [14,22]. This approach once again relies on the standard discrete Fourier transform with uniform time samples, but requires a substantially larger computational cost than the harmonic set approach.…”