Nonlinear Harmonic Method Research Articles

The nonlinear harmonic method: from single stage to multi-row effects

The nonlinear harmonic method (NLH) and its extension to the modelling of the clocking effects can simulate the unsteady flows in multistage machines at much lower CPU cost. The bladerow interactions that can be reproduced are those between adjacent rotors and stators and also between successive rotors or successive stators, including the contra-rotating open rotors (CROR). However, the simulation of the unsteady effects of the engine pylon on the flow in the rear rotor requires the modelling of the time-varying disturbance from a stator of periodicity 1 (the pylon) into a rotor that is not adjacent, requiring an extension of the current NLH method. In addition, it has been observed that in some stator1/rotor/stator2 configurations, the amplitude of the perturbation associated with a blade passing frequency (BPF) in a row is not only created by the adjacent row but is also modified in space by the interaction of the periodic disturbances coming from the other two rows. For instance, for a struts/rotor/stator machine, the spatial frequencies of the periodic disturbance in the stator could be provoked not only by the interaction with the adjacent rotor but also by an additional interaction with the periodic flow coming from the struts. The present paper introduces a method that enables the identification of these spatial frequencies for a general row1/row2/row3 configuration. Besides, any row can be a stator or a rotor and the time-varying disturbances are not only produced by the interaction with the adjacent row but also with the other rows. A more general NLH method is therefore obtained. The proposed method was validated against a three-cylinder test case, a CROR test case, a struts/rotor/stator case and an impeller/bladed diffuser/volute case.

Differential evolution based soft optimization to attenuate vane–rotor shock interaction in high-pressure turbines

This article presents a soft computing methodology to design turbomachinery components experiencing strong shock interactions. The study targets a reduction of unsteady phenomena using evolutionary optimization with robust, high fidelity, and low computational cost evaluations. A differential evolution (DE) algorithm is applied to optimize the transonic vane of a high-pressure turbine. The vane design candidates are examined by a cost-effective Reynolds-averaged Navier–Stokes (RANS) solver, computing the downstream pressure distortion and aerodynamic efficiency. A reduction up to 55% of the strength of the shock waves propagating downstream of the stand-alone vane was obtained. Subsequently to the vane optimization, unsteady computations of the vane–rotor interaction were performed using a non-linear harmonic (NLH) method. Attenuation above 60% of the unsteady forcing on the rotor (downstream of the optimal vane) was observed, with no stage-efficiency abatement. These results show the effectiveness of the proposed soft optimization to improve unsteady performance in modern turbomachinery exposed to strong shock interactions.

3D Finite Element Analysis of the Stray Loss in Power Transformer Structure Parts

In order to analyze the leakage magnetic field and stray loss in power transformer, leakage magnetic field and stray loss in structure parts of a power transformer are calculated by three-dimensional (3-D) non-linear time harmonic finite element method (FEM). The results show that stray loss and loss density in structure parts are large and which may lead to local overheating and affect performance of the transformer. The magnetic shields are used to reduce the stray loss and loss density of power transformer. Effects of these shields on stray loss and loss density of structure parts are discussed. The results show that stray loss and local overheating can be reduced and eliminated effectively by adding magnetic shields. It provides some references for the analysis of stray loss and optimization design in transformer.

Detailed Unsteady Simulation of a Counterrotating Aspirated Compressor with a Focus on the Aspiration Slot and Plenum

An unsteady analysis of the MIT counterrotating aspirated compressor (CRAC) has been conducted using the Numeca FINE/Turbo 3D viscous turbulent flow solver with the Nonlinear Harmonic (NLH) method. All three blade rows plus the aspiration slot and plenum were included in the computational domain. Both adiabatic and isothermal solid wall boundary conditions were applied and simulations with and without aspiration were completed. The aspirated isothermal boundary condition solutions provide the most accurate representation of the trends produced by the experiment, particularly at the endwalls. These simulations provide significant insight into the flow physics of the aspiration flow path. Time histories and spanwise distributions of flow properties in the aspiration slot and plenum present a flow field with significant temporal and spatial variations. In addition, the results provide an understanding of the aspiration flow path choking mechanism that was previously not well understood and is consistent with experimental results. The slot and plenum had been designed to aspirate 1% of the flow path mass flow, whereas the experiment and simulations show that it chokes at about 0.5% mass flow.

Prediction of Near-And Far-Field Noise Generated by Contra-Rotating Open Rotors

An innovative aeroacoustic prediction method for Contra-Rotating Open Rotors (CROR) based on the nonlinear harmonic method (NLH) for the CFD computations and on the Ffowcs Williams and Hawkings (FW-H) equations for the far-field acoustic propagation is described. This method is tested on a generic 8 × 8 puller CROR at typical take-off conditions. The outstanding efficiency of the prediction method in terms of CPU cost is demonstrated in this study. An analysis of the aerodynamic interactions between the two rotors as well as of the aeroacoustic field radiated in the far-field is presented, allowing for an improved understanding of noise generation mechanisms.

Unsteady simulation of a counter–rotating research compressor using the nonlinear harmonic method

In this article, the nonlinear harmonic method implemented in Fine/Turbo code has been evaluated for unsteady rotor–rotor interaction analysis. Strengths and weaknesses of the nonlinear harmonic method are highlighted through direct comparison with experimental data and time-domain unsteady simulation for the Northwestern Polytechnical University counter-rotating research compressor at the design and off-design conditions. The rotor–rotor interactions were studied in detail to advance the understanding of the flow physics involved. Based on the unsteady simulation, deterministic stresses were computed and compared to the nonlinear harmonic method to analyze distinct sources of unsteadiness and their effect on the time-averaged flow field. For the present case, the nonlinear harmonic method efficiently computes unsteady flows and shows major improvements regarding the computational time (to a factor up to 10).

Toward a High Order Throughflow––Investigation of the Nonlinear Harmonic Method Coupled With an Immersed Boundary Method for the Modeling of the Circumferential Stresses

Capturing a level of modeling of the flow inside a multistage turbomachine, such as unsteadiness for example, can be done at different levels of detail, either by capturing all deterministic features of the flow with a pure unsteady method or by settling for an approximated solution at a lower computational cost. The harmonic methods stand in this second category. Among them, the “nonlinear harmonic method” (NLHM) from He and Ning [1998, “Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines,” AIAA J., 36, pp. 2005–2012] revealed the most efficient. This method consists of solving the fully nonlinear 3D steady problem and a linearized perturbation system in the frequency domain. As it has been shown by the authors that the circumferential variations exhibit a harmonic behavior, it is proposed here to adapt the NLHM to the throughflow model, where the main nonlinear system would be the common throughflow equations and the auxiliary system would give access to the circumferential stresses. As the numerical local explicit impermeability conditions are unsupported by Fourier series, the adaptation of this technique to the throughflow model relies on a reformulation of the blade effect by a smooth force field as in the “immersed boundary method” from Peskin [2002, “The Immersed Boundary Method,” Acta Numerica, 11, pp. 1–39]. A simple example of an inviscid flow around a cylinder will illustrate the preceding developments, bringing back the mean effect of the circumferential nonuniformities into the meridional flow.

Frequency domain unsteady transonic aerodynamics for flutter and limit cycle oscillation prediction

The purpose of this study was to develop new approaches for predicting transonic flutter and limit cycle oscillations (LCO) using computational methods. The TSD equation is separated into the in-phase and out-of-phase components through a nonlinear harmonic averaging method. It is then solved in the frequency domain to obtain the aerodynamic forcing function which is needed in the flutter and LCO analyses. To predict flutter, equations are developed using the concept of generalized coordinates. The flutter speed is determined by examining the frequency-domain matrix equation eigenvalues. Flutter characteristics of the AGARD I-445.6 wing are analyzed. Flutter speed and frequency are well predicted in subsonic speed, but are overestimated in supersonic flow. To predict limit cycle oscillations, the frequency-domain aerodynamic coefficients are used to obtain a nonlinear time-domain expression for the aerodynamic force. Limit cycle oscillation characteristics of the DAST ARW-2 wing are analyzed. The results show LCO for Mach numbers ranging from 0.915 to 0.940.

Concurrent Blade Aerodynamic-Aero-elastic Design Optimization Using Adjoint Method

Increasing aerothermal and aero-elastic performance requirements and constraints are closely linked in modern blading designs. There is thus a need for more concurrent interaction between the disciplines at earlier stages of a design process. Presented in this paper are the development, validation, and demonstration of the adjoint approach to concurrent blading aerodynamic and aero-elastic design optimizations. A nonlinear harmonic phase solution method is adopted to solve the unsteady Reynolds-averaged Navier–Stokes equations. The flow field response in terms of both the mean aerothermal performance and aero-elastic stability to a geometrical perturbation can be obtained by three “steadylike” flow solutions at three distinctive temporal phases. This unsteady flow solution method is computationally very efficient and provides a convenient and consistent base for formulating the corresponding adjoint equations. The adjoint system for the unsteady flow solver is solved effectively by a relatively simple extension of the method and techniques previously developed for a steady flow adjoint solver. As a result, the sensitivities of both the steady (time-mean) flow loss and the aerodynamic damping/forcing to detailed blade geometry changes can be very efficiently obtained by solving equivalently three steadylike adjoint equations. Several case studies are presented to illustrate the validity and effectiveness of this new concurrent approach.

Application of a non-linear damped harmonic analysis method to the normal modes of the Earth

The search for weak oscillations in noisy records requires the development of analysis tools more sophisticated than the Fourier spectrum analysis. We present in this paper a new non-linear damped harmonic analysis (NLDHA) method that enables us to detect any damped harmonic signal hidden in noise and to simultaneously estimate the amplitude, frequency, phase and damping factor of the harmonic signal. We also compare this newly developed NLDHA method with a conventional auto-regressive method called ESPRIT. The application of the NLDHA method to the seismic normal modes 0S2 and 0S3 excited after the 2004 Sumatra-Andaman earthquake highlights its ability to precisely recover the frequencies and quality factors of the singlets that constitute these two seismic modes. The application of the NLDHA in the sub-seismic frequency band reveals the presence of some unmodelled waves but no detection of the translational oscillation of the inner core has been claimed.

On decoupled and fully-coupled methods for blade forced response prediction

Two highly efficient fully-coupled methods of predicting the resonant forced response of turbomachinery blades have been developed with the intention of increased computational efficiency over a decoupled method. The flow and structural equations are solved simultaneously, based on the frequency-domain nonlinear harmonic method and the modal reduction technique. By combining the aerodynamic forcing and damping calculations into a single analysis, the coupled solution at a single excitation frequency is approximately half that of the decoupled method. Significant flow–structure coupling effects were discovered, leading to a study into the impact of frequency shift on the fully-coupled solution. A case study on the NASA Rotor 67 transonic aero fan rotor shows a significant reduction in vibration amplitude for the fully-coupled solution due to the resonant frequency shift, caused by the aerodynamic added mass effect. Prompting the development of a novel resonance-tracking algorithm to solve the additional degree-of-freedom in resonant frequency, the increase in computational efficiency in the fully-coupled method is lost due to the need for multiple solutions. A study into the added mass effect and the implications on the coupled solution is undertaken and an evaluation is made between the use of decoupled and fully-coupled forced response systems. It is shown that the decoupled method can accurately predict the resonant vibration level from a single calculation at the natural frequency and is insensitive to frequency shift for lightly damped cases.

Nonlinear Frequency-Modulation Distortion Analysis for VCO Based on Harmonic Balance Method

The analysis methods for the steady-state responses of the voltage tuning negative resistance oscillator (voltage-controlling oscillator, VCO) by the microwave nonlinear autonomous circuit harmonic balance method in millimeter-wave bands are studied in the paper. Firstly, the quasi-periodic characteristic of the steady-state response of the VCO modulated by a periodic signal is proved. Then, on the bases of the harmonic balance analysis and the inter-modulation balance analysis, a novel method for obtaining the steady-state tuning performance and the nonlinear frequency-modulation distortion characteristic of the VCO is presented. The total analysis process is aimed to a kind of NRD-guide Gunn diode VCO. The large-signal lumped equivalent circuit model of the millimeter-wave P+N-junction varactor is also given for explaining the algorithm and the principle of the NRD-guide VCO.

Electrical measurements of the quality factor of microresonators and its dependence on the pressure

The pressure dependence of the quality factors of several lateral microelectromechanical resonators is measured, by exploiting an electrical, non-linear second harmonic method that eliminates the effect of capacitive parasitics without requiring additional driving and/or frequency conversions. Analytical models for the possible causes of energy loss due to the residual gas surrounding the resonators, namely slide-film and squeeze-film losses, are compared with the experimental data.

Steady-State Responses of NRD-Guide LSE/LSM Mode Oscillators in MM-Wave Bands

The analysis methods for the steady-state responses of the mm-wave band NRD-guide negative impedance oscillators based on nonlinear microwave autonomous circuits harmonic balanced method are presented in the paper in details. Firstly, the large-signal nonlinear lumped equivalent circuits of the Gunn diode are studied in mm-wave bands. Then, the performances of two kinds of NRD-guide autonomous circuits, the probe-exciting LSM11o-mode oscillator and the LSE11o-mode oscillator, are analyzed by the way of extracting the large-signal dynamic harmonic admittance (conductance) of the Gunn diode or directly configuring the harmonic balance equations for the oscillator. The input impedance of the exciting probe in the oscillator and the performances of the load-pulling and the local stability of the NRD-guide oscillator are also involved.

Computation of Unsteady Flows Around Oscillating Blades Using Linear and Nonlinear Harmonic Euler Methods

A quasi-three-dimensional time-linearized Euler method has been developed to compute unsteady flows around oscillating blades. In the baseline method, unsteady flow is decomposed into a steady flow plus a linear harmonically varying unsteady flow. Both the steady flow equations and the unsteady perturbation equations are solved using a pseudo-time-marching method. Based upon this method, a novel nonlinear harmonic Euler method has been developed. Due to the nonlinearity of the aerodynamic governing equations, time-averaging generates extra “unsteady stress” terms. These nonlinear effects are included by a strongly coupled approach between the perturbation equations and the time-averaged equations. Numerical results demonstrate that nonlinear effects are very effectively modeled by the nonlinear harmonic method.