Time-domain Harmonic Balance Research Articles

Graphic processing unit accelerated time-domain harmonic balance method for multi-row turbomachinery flow simulation

Graphic processing unit accelerated time-domain harmonic balance method for multi-row turbomachinery flow simulation

An Approximate Time Domain Nonlinear Harmonic Method for Analyzing Unsteady Flows With Multiple Fundamental Modes

Abstract In this article, a new hybrid time and frequency domain method called the approximate time domain nonlinear harmonic method is proposed for a robust and efficient analysis of turbomachinery unsteady flow with more than one fundamental mode. The proposed method combines the features of the nonlinear harmonic method and the time domain harmonic balance method. Unsteady flow components are grouped according to their fundamental frequencies and inter blade phase angles, and separate unsteady flow governing equations are constructed for each mode group. These mode groups are connected using the least squares method based Fourier transform. The proposed method is expected to be exact for a linear problem but approximate for a nonlinear problem. Three sets of analyses with increasing levels of difficulties are presented to demonstrate the validity and effectiveness of the proposed method by comparing it with the time domain harmonic balance method based on the almost periodic Fourier transform.

Efficient analysis of unsteady flows within multi-stage turbomachines using the coupled time and passage spectral method

A coupled time and passage spectral method has been proposed very recently for tracking blade wakes penetrating the immediate downstream blade row and reaching far downstream blade rows. To achieve an efficient numerical analysis, the number of time and space modes to be retained has to be limited, as the computational cost of such an analysis is at least proportional to the number of modes. In this study, time and space modes related to downstream propagation of blade wakes reaching beyond their immediate downstream blade row are ranked according to their amplitudes of flow quantities through a time domain harmonic balance analysis using a domain consisting of multiple blade passages for the third row to capture the wakes of the first row of a two-stage fan. Modes with significant amplitudes are identified and they are really sparse. This sparsity of significant modes provides the premise for an efficient analysis using the coupled time and passage spectral method. A guideline for <italic>a priori</italic> selection of time and space modes has been developed by analyzing the frequencies and nodal diameters of those significant modes. The guideline is subsequently verified through four different coupled time and passage spectral analyses with different levels of accuracy by including different time and space modes.

Fully-turbulent adjoint method for the unsteady shape optimization of multi-row turbomachinery

The possibility of taking into account unsteady flow effects if performing turbomachinery shape optimization is attractive to accurately address inherently time dependent design problems. The harmonic balance method is an efficient solution for computational fluid dynamics problems of turbomachinery characterized by quasi-periodic flows. If applied in combination with adjoint methods, it enables the possibility to deal with unsteady fluid-dynamic design in a cost effective manner, opening the way towards multi-disciplinary applications. This paper presents the development of a novel fully-turbulent discrete adjoint based on the time domain harmonic balance method and its application to the constrained fluid dynamic optimization of an axial turbine stage. As opposed to previous works, the proposed method does not require any assumption on the turbulent eddy viscosity and on the set of input frequencies. The results show that the method provides accurate gradients, if compared with second order finite differences, and significant deviation with respect to the sensitivity computed with the constant eddy viscosity approximation. The application of the method to the fluid-dynamic shape optimization of the exemplary stage leads to improve the total-to-static efficiency of 0.8%. The efficiency increase is found to be higher than that obtained by means of a steady state optimization method.

Time-Domain Harmonic Balance Method for Turbomachinery Aeroelasticity

The present paper investigates a time-domain harmonic balance method as an alternative to classical time-marching schemes for stability studies of turbomachineries toward flutter. A weak-coupling approach is applied, which requires computing the fluid response to prescribed harmonic motions of the structure. The harmonic balance method, formulated in the arbitrary Lagrangian/Eulerian framework, is adapted to single-passage reduction using phase-lag boundary conditions expressed purely in the time domain. Validation against experimental data for the 11th standard configuration for aeroelasticity is performed, showing good agreement. Finally, an industrial test case is presented: a fan designed by Safran Snecma. The results show the good accuracy of the proposed harmonic balance method as well as significant reductions in computational time.

Time-domain harmonic balance method for aerodynamic and aeroelastic simulations of turbomachinery flows

A time-domain Harmonic Balance method is applied to simulate the blade row interactions and vibrations of state-of-the-art industrial turbomachinery configurations. The present harmonic balance app...

Time-domain harmonic balance method for aerodynamic and aeroelastic simulations of turbomachinery flows

A time-domain Harmonic Balance method is applied to simulate the blade row interactions and vibrations of state-of-the-art industrial turbomachinery configurations. The present harmonic balance approach is a time-integration scheme that turns a periodic or almost-periodic flow problem into the coupled resolution of several steady computations at different time samples of the period of interest. The coupling is performed by a spectral time-derivative operator that appears as a source term of all the steady problems. These are converged simultaneously making the method parallel in time. In this paper, a non-uniform time sampling is used to improve the robustness and accuracy regardless of the considered frequency set. Blade row interactions are studied within a 3.5-stage high-pressure axial compressor representative of the high-pressure core of modern turbofan engines. Comparisons with reference time-accurate computations show that four frequencies allow a fair match of the compressor performance, with a reduction of the computational time up to a factor 30. Finally, an aeroelastic study is performed for a counter-rotating fan stage, where the rear blade is submitted to a prescribed harmonic vibration along its first torsion mode. The aerodynamic damping is analysed, showing possible flutter.

Non-uniform time sampling for multiple-frequency harmonic balance computations

A time-domain harmonic balance method for the analysis of almost-periodic (multi-harmonics) flows is presented. This method relies on Fourier analysis to derive an efficient alternative to classical time marching schemes for such flows. It has recently received significant attention, especially in the turbomachinery field where the flow spectrum is essentially a combination of the blade passing frequencies. Up to now, harmonic balance methods have used a uniform time sampling of the period of interest, but in the case of several frequencies, non-necessarily multiple of each other, harmonic balance methods can face stability issues due to a bad condition number of the Fourier operator. Two algorithms are derived to find a non-uniform time sampling in order to minimize this condition number. Their behavior is studied on a wide range of frequencies, and a model problem of a 1D flow with pulsating outlet pressure, which enables to prove their efficiency. Finally, the flow in a multi-stage axial compressor is analyzed with different frequency sets. It demonstrates the stability and robustness of the present non-uniform harmonic balance method regardless of the frequency set.

A Time-Domain Harmonic Balance Method for Rotor/Stator Interactions

In the absence of instabilities, the large deterministic scales of turbomachinery flows resulting from the periodic rotation of blades can be considered periodic in time. Such flows are not simulated with enough efficiency when using classical unsteady techniques as a transient regime must be bypassed. New techniques, dedicated to time-periodic flows and based on Fourier analysis, have been developed recently. Among these, harmonic balance methods cast a time-periodic flow computation in several coupled steady flow computations. A time-domain harmonic balance method is derived and adapted to phase lag periodic conditions to allow the simulation of only one blade passage per row regardless of row blade counts. Sophisticated space and time interpolations are involved and detailed. The test case is a single stage subsonic compressor. A convergence study of the present harmonic balance is performed and compared with a reference well-resolved classical unsteady flow simulation. The results show, on one hand, the good behavior of the harmonic balance and its ability to correctly predict global quantities as well as local flow pattern; on the other hand, the simulation time is drastically reduced.

Applications of the Multi-Interval Chebyshev Collocation Method in RF Circuit Simulation

In this paper, we discuss the multi-interval Chebyshev (MIC) method for analog and radio-frequency (RF) circuit simulations. The method has been applied in a wide variety of circuit analyses including periodic steady-state, time-varying small signal, and cyclostationary noise analyses for both driven and autonomous circuits. In contrast to traditional analog/RF circuit simulations using either low-order time-domain integration methods or harmonic balance methods, we show that the MIC method can efficiently achieve high accuracy on strongly nonlinear circuits possessing waveforms with rapid transitions. In addition, it has the same flexibility in simulating frequency-dependent components as the harmonic balance methods but can use similar preconditioning strategies and matrix-implicit Krylov-subspace solvers as time-domain techniques, leading to good convergence on nonlinear problems.

Continuation of Higher-Order Harmonic Balance Solutions for Nonlinear Aeroelastic Systems

The harmonic balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants were applied to nonlinear aeroelastic systems over the last two decades. This paper presents a detailed description of several harmonic balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing subcritical and supercritical Hopf bifurcations as well as a particular type of explosive bifurcation. The methods investigated in this work are applied to a nonlinear aeroelastic model of a generic transport aircraft featuring polynomial or free-play stiffness nonlinearity in the control surface. It is shown that high-order harmonic balance solutions will accurately capture the complete bifurcation behavior of this system for both types of nonlinearity. Low-order solutions can become inaccurate in the presence of numerous folds in the limit-cycle oscillation branch but can still yield practical engineering information at a fraction of the cost of higher-order solutions. Time-domain harmonic balance schemes are shown to be more computationally expensive than the standard harmonic balance approach.