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A New Harmonic Balance Approach Using Multidimensional Time

Junge, Laura and Frey, Christian and Ashcroft, Graham and Kügeler, Edmund (2021) A New Harmonic Balance Approach Using Multidimensional Time. In: ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition, GT 2020. ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition, 21.-25. Sep 2020, Virtual Conference, Online. doi: 10.1115/GT2020-16224. ISBN 978-079188419-5.

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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 as they are capable of fully exploiting the given spatial and temporal periodicities, as well as modelling 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 method requires extension. One possible approach is to use almost periodic Fourier transforms with equidistant or non-equidistant 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 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 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, that 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.

Item URL in elib:https://elib.dlr.de/138554/
Document Type:Conference or Workshop Item (Speech)
Title:A New Harmonic Balance Approach Using Multidimensional Time
Authors:
AuthorsInstitution or Email of AuthorsAuthor's ORCID iD
Junge, Lauralaura.junge (at) dlr.deUNSPECIFIED
Frey, ChristianChristian.Frey (at) dlr.deUNSPECIFIED
Ashcroft, GrahamGraham.Ashcroft (at) dlr.deUNSPECIFIED
Kügeler, EdmundEdmund.Kuegeler (at) dlr.dehttps://orcid.org/0000-0002-9719-626X
Date:11 January 2021
Journal or Publication Title:ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition, GT 2020
Refereed publication:Yes
Open Access:Yes
Gold Open Access:No
In SCOPUS:Yes
In ISI Web of Science:No
DOI :10.1115/GT2020-16224
ISBN:978-079188419-5
Status:Published
Keywords:Computer simulation, Fourier transforms, Turbomachinery, Unsteady flow
Event Title:ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition
Event Location:Virtual Conference, Online
Event Type:international Conference
Event Dates:21.-25. Sep 2020
Organizer:The American Society of Mechanical Engineers
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Aeronautics
HGF - Program Themes:propulsion systems
DLR - Research area:Aeronautics
DLR - Program:L ER - Engine Research
DLR - Research theme (Project):L - Virtual Engine and Validation methods (old)
Location: Köln-Porz
Institutes and Institutions:Institute of Propulsion Technology > Numerical Methodes
Deposited By: Junge, Laura
Deposited On:25 Jan 2021 08:31
Last Modified:08 Feb 2021 10:37

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