On the Development of Harmonic Balance Methods for Multiple Fundamental Frequencies

2018 ◽  
Author(s):  
Laura Junge ◽  
Graham Ashcroft ◽  
Hans-Peter Kersken ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Fundamental Frequency ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Cross Coupling ◽  
Nonlinear Frequency ◽  
Fundamental Frequencies ◽  
Frequency Domain Methods ◽  
Sampling Points ◽  
The Common

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.

scholarly journals A New Harmonic Balance Approach Using Multidimensional Time

2021 ◽  
Author(s):  
Laura Junge ◽  
Christian Frey ◽  
Graham Ashcroft ◽  
Edmund Kuegeler
Keyword(s):  
Frequency Domain ◽  
Fundamental Frequency ◽  
Harmonic Balance ◽  
Nonlinear Frequency ◽  
Base Frequency ◽  
Fundamental Frequencies ◽  
Balance Approach ◽  
Frequency Domain Methods ◽  
Sampling Points ◽  
The Common

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.

On the Application of Frequency-Domain Methods to Multistage Turbomachinery

2015 ◽  
Author(s):  
Laura Junge ◽  
Graham Ashcroft ◽  
Peter Jeschke ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Harmonic Balance ◽  
Axial Compressor ◽  
Harmonic Balance Method ◽  
Nonlinear Frequency ◽  
Solution Quality ◽  
Multi Stage ◽  
Frequency Domain Methods ◽  
Blade Row

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.

scholarly journals A New Harmonic Balance Approach Using Multidimensional Time

2020 ◽  
Author(s):  
Laura Junge ◽  
Christian Frey ◽  
Graham Ashcroft ◽  
Edmund Kügeler
Keyword(s):  
Frequency Domain ◽  
Fundamental Frequency ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Cartesian Product ◽  
Discrete Fourier Transforms ◽  
Base Frequency ◽  
Balance Approach ◽  
Sampling Points ◽  
The Common

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.

scholarly journals Comparison of Common Methods in Dynamic Response Predictions of Rotor Systems with Malfunctions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongliang Yao ◽  
Qian Zhao ◽  
Qi Xu ◽  
Bangchun Wen
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Newmark Method ◽  
Incremental Harmonic Balance Method ◽  
Rotor Systems ◽  
Double Frequency ◽  
Frequency Domain Methods ◽  
Incremental Harmonic Balance

The efficiency and accuracy of common time and frequency domain methods that are used to simulate the response of a rotor system with malfunctions are compared and analyzed. The Newmark method and the incremental harmonic balance method are selected as typical representatives of time and frequency domain methods, respectively. To improve the simulation efficiency, the fixed interface component mode synthesis approach is combined with the Newmark method and the receptance approach is combined with the incremental harmonic balance method. Numerical simulations are performed for rotor systems with single and double frequency excitations. The inherent characteristic that determines the efficiency of the two methods is analyzed. The results of the analysis indicated that frequency domain methods are suitable single and double frequency excitation rotor systems, whereas time domain methods are more suitable for multifrequency excitation rotor systems.

Simulations of Unsteady Blade Row Interactions Using Linear and Non-Linear Frequency Domain Methods

2015 ◽  
Author(s):  
Christian Frey ◽  
Graham Ashcroft ◽  
Hans-Peter Kersken
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Measurement Data ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Frequency Domain Methods ◽  
Blade Row ◽  
Non Linear ◽  
Rotor Stator Interaction ◽  
Unsteady Simulation

This paper compares various approaches to simulate unsteady blade row interactions in turbomachinery. Unsteady simulations of turbomachinery flows have gained importance over the last years since increasing computing power allows the user to consider 3D unsteady flows for industrially relevant configurations. Furthermore, for turbomachinery flows, the last two decades have seen considerable efforts in developing adequate CFD methods which exploit the rotational symmetries of blade rows and are therefore up to several orders of magnitude more efficient than the standard unsteady approach for full wheel configurations. This paper focusses on the harmonic balance method which has been developed recently by the authors. The system of equations as well as the iterative solver are formulated in the frequency domain. The aim of this paper is to compare the harmonic balance method with the time-linearized as well as the non-linear unsteady approach. For the latter the unsteady flow fields in a fan stage are compared to reference results obtained with a highly resolved unsteady simulation. Moreover the amplitudes of the acoustic modes which are due to the rotor stator interaction are compared to measurement data available for this fan stage. The harmonic balance results for different sets of harmonics in the blade rows are used to explain the minor discrepancies between the time-linearized and unsteady results published by the authors in previous publications. The results show that the differences are primarily due to the neglection of the two-way coupling in the time-linearized simulations.

On the Extension of a Harmonic Balance Method for the Simulation of Compressor Casing Treatments

2016 ◽  
Author(s):  
Christian Voigt ◽  
Graham Ashcroft
Keyword(s):  
Time Domain ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Nonlinear Frequency ◽  
Transonic Compressor ◽  
Frequency Domain Methods ◽  
Blade Tip ◽  
The Time Domain ◽  
Casing Treatments

In recent years both linear and nonlinear frequency domain methods have become increasingly popular in the simulation and investigation of time-periodic flows in turbomachinery. In this work the extension of an alternating frequency/time domain Harmonic Balance method to support arbitrary inter-domain block interfaces, with possibly different frames of reference, is described in detail. The approach outlined is based on the time-domain, area-based interpolation algorithm originally developed for the investigation of casing treatments. In this paper, it is shown that by solving the domain coupling problem in the time-domain it is possible to accurately and efficiently capture the flow physics of such complex, nonlinear problems as blade tip interaction with casing treatments in transonic compressors. To demonstrate and verify the basic algorithm the advection of a simple entropy disturbance in a subsonic duct flow is first computed. Secondly, unsteady flow due to rotor-stator interaction in a transonic compressor stage is simulated and the data compared with reference numerical methods. Finally, to validate the method a single stage transonic axial compressor with casing treatments is simulated and the results are compared with previously published time-domain simulations as well as experimental data based on particle image velocimetry measurements in the blade tip region.

Comparison of Linear and Nonlinear Frequency Domain Methods for Flutter Analysis

2018 ◽  
Author(s):  
Hans-Peter Kersken ◽  
Graham Ashcroft ◽  
Christian Frey ◽  
Nina Wolfrum ◽  
Oliver Pütz
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance Method ◽  
Frequency Domain Method ◽  
Nonlinear Frequency ◽  
Solution Approach ◽  
Time Solution ◽  
Domain Method ◽  
Frequency Domain Methods ◽  
Flutter Analysis ◽  
Linear And Nonlinear

Both linear and nonlinear frequency domain methods have been applied successfully to the investigation of time-periodic phenomena in turbomachinery. Linear methods allow to perform flutter analysis of turbomachinery blade rows very efficiently. Nonlinear frequency domain method can be applied to flutter analysis as well. If a pseudo-time solution algorithm is employed as a solver the nonlinear frequency domain method takes advantage of the stabilizing effect of the nonlinear coupling of the harmonics. Additionally, it allows studying the influence of nonlinear effects on the flutter stability. A linear GMRes based method and a harmonic balance method using a pseudo-time solution approach are compared with respect to computational efficiency when applied to the flutter analysis of blades of a stationary gas turbine and a low pressure turbine of a jet engine. It is shown that both methods have their merits and limitation depending on the type of problem at hand.

Forced Response Sensitivity Analysis Using an Adjoint Harmonic Balance Solver

2018 ◽  
Author(s):  
Anna Engels-Putzka ◽  
Jan Backhaus ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Unsteady Aerodynamics ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Algorithmic Differentiation ◽  
Design And Optimization ◽  
Initial Application ◽  
Reverse Mode ◽  
Geometric Changes

This paper describes the development and initial application of an adjoint harmonic balance solver. The harmonic balance method is a numerical method formulated in the frequency domain which is particularly suitable for the simulation of periodic unsteady flow phenomena in turbomachinery. Successful applications of this method include unsteady aerodynamics as well as aeroacoustics and aeroelasticity. Here we focus on forced response due to the interaction of neighboring blade rows. In the CFD-based design and optimization of turbomachinery components it is often helpful to be able to compute not only the objective values — e.g. performance data of a component — themselves, but also their sensitivities with respect to variations of the geometry. An efficient way to compute such sensitivities for a large number of geometric changes is the application of the adjoint method. While this is frequently used in the context of steady CFD, it becomes prohibitively expensive for unsteady simulations in the time domain. For unsteady methods in the frequency domain, the use of adjoint solvers is feasible, but still challenging. The present approach employs the reverse mode of algorithmic differentiation (AD) to construct a discrete adjoint of an existing harmonic balance solver in the framework of an industrially applied CFD code. The paper discusses implemen-tational issues as well as the performance of the adjoint solver, in particular regarding memory requirements. The presented method is applied to compute the sensitivities of aeroelastic objectives with respect to geometric changes in a turbine stage.

scholarly journals Dual Time Stepping Algorithms With the High Order Harmonic Balance Method for Contact Interfaces With Fretting-Wear

2011 ◽  
Author(s):  
Loi¨c Salles ◽  
Laurent Blanc ◽  
Fabrice Thouverez ◽  
Alexander M. Gouskov ◽  
Pierrick Jean
Keyword(s):  
Frequency Domain ◽  
Dry Friction ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Contact Forces ◽  
Balance Method ◽  
Fretting Wear ◽  
Rate Equations ◽  
Time Stepping ◽  
Dual Time Stepping

Contact interfaces with dry friction are frequently used in turbomachinery. Dry friction damping produced by the sliding surfaces of these interfaces reduces the amplitude of bladed-disk vibration. The relative displacements at these interfaces lead to fretting-wear which reduces the average life expectancy of the structure. Frequency response functions are calculated numerically by using the multi-Harmonic Balance Method (mHBM). The Dynamic Lagrangian Frequency-Time method is used to calculate contact forces in the frequency domain. A new strategy for solving non-linear systems based on dual time stepping is applied. This method is faster than using Newton solvers. It was used successfully for solving Nonlinear CFD equations in the frequency domain. This new approach allows identifying the steady state of worn systems by integrating wear rate equations a on dual time scale. The dual time equations are integrated by an implicit scheme. Of the different orders tested, the first order scheme provided the best results.

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