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

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

Large Eddy Simulation of flows in industrial compressors: a path from 2015 to 2035

A better understanding of turbulent unsteady flows is a necessary step towards a breakthrough in the design of modern compressors. Due to high Reynolds numbers and very complex geometry, the flow that develops in such industrial machines is extremely hard to predict. At this time, the most popular method to simulate these flows is still based on a Reynolds Averaged Navier-Stokes (RANS) approach. However there is some evidence that this formalism is not accurate for these components, especially when a description of time-dependent turbulent flows is desired. With the increase in computing power, Large Eddy Simulation (LES) emerges as a promising technique to improve both knowledge of complex physics and reliability of flow solver predictions. The objective of the paper is thus to give an overview of the current status of LES for industrial compressor flows as well as to propose future research axes regarding the use of LES for compressor design. While the use of wall-resolved LES for industrial multistage compressors at realistic Reynolds number should not be ready before 2035, some possibilities exist to reduce the cost of LES, such as wall-modelling and the adaptation of the phase lag condition. This paper also points out the necessity to combine LES to techniques able to tackle complex geometries. Indeed LES alone, i.e. without prior knowledge of such flows for grid construction or the prohibitive yet ideal use of fully homogeneous meshes to predict compressor flows, is quite limited today

Multi-frequential harmonic balance approach for the computation of unsteadiness in multi-stage turbomachines

In turbomachines, the relative motion of fixed and rotating blades gives rise to deterministic unsteady interactions at frequencies termed BPFs (Blade Passing Frequencies). In a multi-stage turbomachine, a row sandwiched between two other rows is submitted to (at least) two BPFs, hence the need for multiple frequency methods. Initially developed for single frequency problems, harmonic methods have been extended to account for multiple frequencies. All the variations of the Harmonic Balance (HB) technique proposed in the literature rely on a uniform time sampling of the longest period of interest (though the number of samples can differ). This can compromise the efficiency of the method, as too many time samples are computed. Besides, as demonstrated in the present contribution, uniform time sampling can also raise stability issues. To overcome these computational limitations, a new approach using non-uniform time sampling is proposed in the present contribution. This paper will be organized as follows: first, the multi-frequency HB methods is presented, and the impact of time sampling on numerical stability is discussed. Then, algorithms for an automatic choice of the time samples are presented and compared. The proposed non-uniform sampling is assessed for a model problem (i.e. a pulsating channel). Finally, a section is dedicated to the application to a turbomachinery configuration, with emphasis on the choice of frequencie

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

Convergence of Fourier-based time methods for turbomachinery wake passing problems

The convergence of Fourier-based time methods applied to turbomachinery flows is assessed. The focus is on the harmonic balance method, which is a time-domain Fourier-based approach standing as an efficient alternative to classical time marching schemes for periodic flows. In the literature, no consensus exists concerning the number of harmonics needed to achieve convergence for turbomachinery stage configurations. In this paper it is shown that the convergence of Fourier-based methods is closely related to the impulsive nature of the flow solution, which in turbomachines is essentially governed by the characteristics of the passing wakes between adjacent rows. As a result of the proposed analysis, a priori estimates are provided for the minimum number of harmonics required to accurately compute a given turbomachinery configuration. Their application to several contra-rotating open-rotor configurations is assessed, demonstrating the practical interest of the proposed methodology

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

Contrasting the Harmonic Balance and Linearized Methods for Oscillating-Flap Simulations

In the framework of unsteady aerodynamics, forced-harmonic-motion simulations can be used to compute unsteady loads. In this context, the present paper assesses two alternatives to the unsteady Reynolds-averaged Navier–Stokes approach, the linearized unsteady Reynolds-averaged Navier–Stokes equations method, and the harmonic balance approach. The test case is a NACA 64A006 airfoil with an oscillating ␣ap mounted at 75% of the chord. Emphasis is put on examining the performances of the methods in terms of accuracy and computational cost over a range of physical conditions. It is found that, for a subsonic ␣ow, the linearized unsteady Reynolds-averaged Navier–Stokes method is the most ef␣cient one. In the transonic regime, the linearized unsteady Reynolds-averaged Navier–Stokes method remains the fastest approach, but with limited accuracy around shocks, whereas a one- harmonic harmonic balance solution is in closer agreement with the unsteady Reynolds-averaged Navier–Stokes solution. In the case of separation in the transonic regime, the linearized unsteady Reynolds-averaged Navier–Stokes method fails to converge, whereas the harmonic balance remains robust and accurate

Numerical flutter analysis of turbomachinery bladings based on time-linearized, time-spectral and time-accurate simulations

Within the European research project FUTURE a transonic compressor rotor has been evaluated with respect to its aeroelastic stability. Based on the identical computational mesh different numerical approaches of the DLR flow solver TRACE and the ONERA code elsA are compared in this paper: whereas a time-linearized method based on small time-harmonic perturbations of the steady flow field is chosen in TRACE, elsA uses a non-linear time-domain harmonic balance method in order to determine local aerodynamic work entries and the corresponding global damping values for all inter-blade phase angles (IBPA). Furthermore, nonlinear reference solutions for the critical IBPA range will be provided by both codes using their classical time-marching schemes. In this context the difference in terms of computational effort between the fast linearized or harmonic balance methods and the classical nonlinear techniques shall be highlighted as well