Nonlinear Frequency Domain Method Research Articles

An eigenvalue-shift and matching-signal based FOE method for discrete spectrum modulated NFDM system

We propose an improved frequency offset estimation (FOE) algorithm based on eigenvalue-shift and matching-signal (ESMS) for discrete spectrum modulated nonlinear frequency division multiplexing (NFDM) system, which can realize high estimation accuracy without the constraint of the Baud-rate. Meanwhile, its computational complexity is lower in contrast with nonlinear frequency domain (NFD) method. Therefore, the proposed method is more powerful for eigenvalue NFDM transmission systems with large FO variation.

Nonlinear wave energy dissipator with wave attenuation and energy harvesting at low frequencies

It is important to determine how to reduce the wave load and utilize the ocean energy through the hybrid wave attenuation and energy harvesting structure (WAAEHS) as it is the basic scientific problem in the field of ocean engineering. The existing linear hybrid devices have poor performance due to the mismatch between the natural frequency and the excitation frequency in the low frequency region. The method of increasing the mass is unreliable and low economic, and the introduction of nonlinear mechanism to reduce the stiffness has complex fluid-structure coupling problem between waves and structures. In this paper, a new type of negative stiffness mechanism with a simple structure and good stability has been proposed and applied to a hybrid WAAEHS, and a nonlinear frequency domain method combining eigenfunction expansion matching method(EEMM)and multi-harmonic balance method(MHBM) has been proposed to solve the nonlinear wave–structure interaction dynamic model. The influence of the key parameters of the mechanism on the wave attenuation and energy harvesting performance is examined, and the “phase control” mechanism of the negative stiffness mechanism is revealed to improve the low-frequency wave attenuation and energy harvesting performance.

A New Harmonic Balance Approach Using Multidimensional Time

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 modeling 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 (HB) method requires extension. One possible approach is to use almost periodic Fourier transforms with equidistant or nonequidistant 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 multifrequency problems described earlier, 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, which 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.

Development of a fully-coupled harmonic balance method and a refined energy method for the computation of flutter-induced Limit Cycle Oscillations of bladed disks with nonlinear friction contacts

Flutter stability is a dominant design constraint of modern gas and steam turbines. To further increase the feasible design space, flutter-tolerant designs are currently explored, which may undergo Limit Cycle Oscillations (LCOs) of acceptable, yet not vanishing, level. Bounded self-excited oscillations are a priori a nonlinear phenomenon, and can thus only be explained by nonlinear interactions such as dry stick–slip friction in mechanical joints. The currently available simulation methods for blade flutter account for nonlinear interactions, at most, in only one domain, the structure or the fluid, and assume the behavior in the other domain as linear. In this work, we develop a fully-coupled nonlinear frequency domain method which is capable of resolving nonlinear flow and structural effects. We demonstrate the computational performance of this method for a state-of-the-art aeroelastic model of a shrouded turbine blade row. Besides simulating limit cycles, we predict, for the first time, the phenomenon of nonlinear instability, i.e., a situation where the equilibrium point is locally stable, but for sufficiently strong perturbation (caused e.g.by an impact), the dry frictional dissipation cannot bound the flutter vibrations. This implies that linearized theory does not necessary lead to a conservative design of turbine blades. We show that this phenomenon is due to the nonlinear contact interactions at the tip shrouds, which cause a change of the vibrational deflection shape and frequency, which in turn leads to a loss of aeroelastic stability. Finally, we extend the well-known energy method to capture these effects, and conclude that it provides a good approximation and is useful for initializing the fully-coupled solver.

Frequency offset estimation for nonlinear frequency division multiplexing with discrete spectrum modulation.

Although fruitful studies have been conducted on carrier frequency offset (CFO) estimations in linear coherent optical fiber communication systems, there are few studies on CFO estimations and recoveries in the systems based on the nonlinear Fourier transform (NFT). Although the CFO is originated from the linear frequency domain, it definitely has effects on nonlinear spectra, including the shift of the nonlinear frequency and the phase rotations of the scattering data, which are similar to its effects on linear spectra. This work indicates that it is feasible to estimate frequency offset (FO) by capturing symbol variations in the nonlinear frequency domain (NFD) rather than in the linear frequency domain; the latter was usually exploited in the literature. Based on a thorough investigation of the FO induced behavior that appears in a nonlinear frequency division multiplexing (NFDM) system, we proposed a nonlinear frequency domain estimation method aided by training symbols (TS) using an angle search algorithm after NFT operations at the receiver. The discussions in this paper prove that the proposed method is generally applicable to the NFDM systems regardless of whether using single or multiple eigenvalues. A performance comparison between the NFD method and the conventional method in the linear frequency domain is performed with different modulation formats for both single and multiple eigenvalue NFDM transmission systems. The analysis results show that the proposed method holds the better stability and estimation accuracy in contrast with the linear domain estimation method. The TS overhead can also be deduced dramatically, which implies better transmission efficiency. Therefore, the NFD method is more powerful for eigenvalue NFDM transmission systems, especially for the scenarios where high order modulation formats and multiple eigenvalues are utilized.

Rotor–Stator Interactions in a 2.5-Stage Axial Compressor—Part I: Experimental Analysis of Tyler–Sofrin Modes

Abstract This two-part paper investigates the influence of rotor–stator interactions on the blade vibrational stresses of the first rotor, excited by the downstream stator. To this end, aeroacoustic and aeroelastic measurements and numerical setup studies for the solver TRACE are conducted in order to improve the predictive accuracy of blade vibrational stresses. Part I compares tip timing data for resonance crossings of three blisk modes to numerical predictions. Due to the single-row analysis within the linearized version of the flow solver TRACE, unsteady rotor–stator interactions are excluded by default. The findings show that leaving out these interactions in the numerical setup can lead to 97% lower vibrational stress predictions with respect to the absolute value measured. To validate the prediction of rotor–stator interactions by the nonlinear frequency domain method of TRACE, unsteady pressure measurements were conducted at the casing in the inter-row section of the first stage. The results were analyzed using an optimized measuring grid and applying a compressed sensing-based azimuthal mode analysis. Predicted azimuthal mode numbers are in accordance with the experiment, whereas amplitudes deviate from the measurements in part. Part II focuses on the prediction of blade vibrational stresses. To this end, a detailed grid study is performed and comparisons to steady and unsteady measurement data are made. In summary, this two-part paper confirms the importance of rotor–stator interactions for blade vibrational stresses excited by downstream vanes at a state-of-the-art high-pressure compressor.

On the Geometric Conservation Law for the Non Linear Frequency Domain and Time-Spectral methods

The aim of this paper is to present and validate two new procedures to enforce the Geometric Conservation Law (GCL) on a moving grid for an Arbitrary Lagrangian Eulerian (ALE) formulation for the Euler equations discretized in time for either the Non Linear Frequency Domain (NLFD) or Time-Spectral (TS) methods. The equations are spatially discretized by a structured finite-volume scheme on a hexahedral mesh. The derived methodologies follow a general approach where the positions and the velocities of the grid points are known at each time step. The integrated face mesh velocities are derived either from the Approximation of the Exact Volumetric Increments (AEVI) relative to the undeformed mesh or exactly computed based on a Trilinear Mapping (TRI-MAP) between the physical space and the computational domain. The accuracy of the AEVI method highly depends on the computation of the volumetric increments and limits the temporal-order of accuracy of the deduced integrated face mesh velocities to between one and two. Thus defeating the purpose of the NLFD method which possesses spectral rate of convergence. However, the TRI-MAP method has proven to be more computationally efficient, ensuring the satisfaction of the GCL once the convergence of the temporal derivative of the cell volume is reached in Fourier space. The methods are validated numerically by verifying the conservation of uniform flow and by comparing the integrated face mesh velocities to the exact values derived from the mapping. Their performances for aerodynamic simulation are evaluated for a pitching NACA0012 airfoil.

A Nonlinear Frequency-Domain Approach for Numerical Simulation of Wave Energy Converters

Nonlinear, analytical wave-energy converter (WEC) models are generally simulated through time-domain (TD) numerical integration. However, the relatively high computational requirements of TD integration are not compatible with applications where a large number of simulations are needed. Spectral domain (SD) linearization has also been proposed to take into account some nonlinear effects, while being much faster than TD integration. However, as explained in this paper, such SD models have limited accuracy, and cannot extend to the static nonlinear forces. In this paper, a nonlinear frequency-domain (NLFD) method is investigated for WEC simulation, using a projection of the dynamical equations onto a basis of trigonometric functions. A comparison with TD integration (second-order Runge$-$Kutta-RK2) and SD methods is provided, through theoretical considerations, and by means of numerical simulations based on two case studies. In the cases considered, the proposed NLFD method allows for significant computational savings compared to RK2 integration, without requiring any approximation for the radiation forces. NLFD thus shows promising potential for applications involving extensive WEC simulation, such as power production assessment, while preserving the WEC model accuracy. Although slower than SD, NLFD has significant benefits in terms of accuracy and range of applicability.

Three-Dimensional Aeroelastic Solutions via the Nonlinear Frequency-Domain Method

An aeroelastic solver is developed using a nonlinear frequency domain flow solver coupled to a plate-bending finite element linear structural solver. A methodology for determining the flow conditions leading to flutter and limit-cycle oscillations is proposed, based on a root-finding Newton–Raphson iterative method. The novelty of the approach lies in the constant size of the Newton–Raphson system of equations, regardless of the number of degrees of freedom of the structural model. To serve this purpose, a new method of computing mesh velocities for nonlinear frequency-domain flow solvers is developed, and a technique for solving the geometric conservation law within the nonlinear frequency-domain framework is presented accordingly. The new approach for computing mesh velocities is validated against existing experimental data on the Lockheed, Air Force, NASA and Netherlands wing (run 73, CT5), whereas the aeroelastic solver is validated via experimental results of the AGARD I.-wing weakened model 3 and solid model 2 in air and R-12, respectively. The proposed framework is expected to perform limit-cycle oscillation computations an order of magnitude faster than a typical aeroelastic time-marching approach.

Numerical analysis of synthetic jet flows using a diagonally implicit harmonic balance method with preconditioning

Abstract Synthetic jet flow is one of the active flow control mechanisms used to suppress flow instabilities including separation and wake vortices. It induces turbulent flows by ejecting or suctioning periodic jet streams into freestream air. A single piezoelectric actuator is often used to generate a periodic synthetic jet flow into quiescent air. The main purpose of the current study is to develop an efficient diagonally implicit harmonic balance method based on the nonlinear frequency domain method. This method is implemented in a compressible RANS flow solver with a preconditioning treatment to improve the performance of the numerical analysis. The transpiration wall boundary condition for periodic velocity of jet flows is used to avoid mesh deformation associated with the movement of the jet actuator diaphragm, where the periodic velocity profile is modeled from existing experimental data. With varying numbers of harmonics of up to fifteen the results are compared with the experimental data as well as those of the time-accurate computations. Reasonable agreement with the experimental data is obtained at reduced computational costs.

Comparison of Frequency Domain and Time-Domain Methods for Aeromechanical Analysis

Unsteady flow around an oscillating plate cascade and that through a single compressor rotor subject to vibration have been computationally studied, aimed at examining the predictive ability of two low fidelity frequency methods compared with a high fidelity time-domain solution method for aeroelasticity. The computational solutions demonstrate the capabilities of the frequency domain methods compared with the nonlinear time-domain solution method in capturing small perturbations in the unsteady flow. They also show the great advantage of significant CPU time saving by the frequency methods over the nonlinear time method. Comparisons of two different frequency methods, nonlinear harmonic and phase solution method, show that these methods can produce different results due to the differences in numeric and physical conditioning. The results obtained using phase solutions method are in better agreement with the nonlinear time-domain solution. This is because the same numeric and physical conditioning are used in both the nonlinear time-domain method and phase solution frequency domain method.

Modeling of thermoacoustic systems using the nonlinear frequency domain method.

When modeling thermoacoustic (TA) devices at high amplitude, nonlinear effects such as time-average mass flows, and the generation of higher harmonics can no longer be neglected. Thus far, modeling these effects in TA devices required a generally computationally costly time integration of the nonlinear governing equations. In this paper, a fast one-dimensional nonlinear model for TA devices is presented, which omits this costly time integration by directly solving the periodic steady state. The model is defined in the frequency domain, which eases the implementation of phase delays due to viscous resistance and thermoacoustic heat exchange. As a demonstration, the model is used to solve an experimental standing wave thermoacoustic engine. The obtained results agree with experimental results, as well as with results from a nonlinear time domain model from the literature. The low computational cost of this model opens the possibility to do optimization studies using a nonlinear TA model.

An implicit and adaptive nonlinear frequency domain approach for periodic viscous flows

An implicit nonlinear Lower–Upper symmetric Gauss–Seidel (LU-SGS) solver has been extended to the adaptive Nonlinear Frequency Domain method (adaptive NLFD) for periodic viscous flows. The discretized equations are linearized in both spatial and temporal directions, yielding an innovative segregate approach, where the effects of the neighboring cells are transferred to the right-hand-side and are updated iteratively. This property of the solver is aligned with the adaptive NLFD concept, in which different cells have different number of modes; hence, should be treated individually. The segregate analysis of the modal equations prevents assembling and inversion of a large left-hand-side matrix, when high number of modes are involved. This is an important characteristic for a selected flow solver of the adaptive NLFD method, where a high modal content may be required in highly unsteady parts of the flow field. The implicit nonlinear LU-SGS solver has demonstrated to be both robust and computationally efficient as the number of modes is increased. The developed solver is thoroughly validated for the laminar vortex shedding behind a stationary cylinder, high angle of attack NACA0012 airfoil, and a plunging NACA0012 airfoil. An order of magnitude improvement in the computational time is observed through the developed implicit approach over the classical modified 5-stage Runge–Kutta method.

An adaptive Non-Linear Frequency Domain method for viscous flows

An adaptive Nonlinear Frequency Domain method (adaptive NLFD) has been developed and validated for the Navier–Stokes equations. Although the compute time for periodic flows is reduced by using the NLFD approach over classical time marching schemes, the adaptive NLFD approach leads to further reductions in the overall computational effort. The adaptive NLFD method is validated for the laminar vortex shedding behind a stationary two-dimensional cylinder. A novel adaptation strategy that allows both mode augmentation and reduction, enables the adaptive approach to gain a factor of two reduction in memory and computational cost at an equivalent solution accuracy to the non-adaptive approach.

An efficient approach for calculation of pitching moment in nonlinear reduced frequency method at low Mach number transonic flows

AbstractIn this research, an efficient methodology for calculation of pitching moment coefficient at low Mach number transonic flows by using the perturbed nonlinear reduced frequency approach is presented. The proposed approach uses the perturbation technique in the nonlinear frequency domain (NLFD) method to estimate the solution at high harmonics. In this approach, the density and velocity fields at high harmonics are perturbed about those at low harmonics. Perturbing the density and velocity fields, the semi‐linear form of the governing equations is obtained. The resulting solution vector and spatial operator are then approximated by discrete form of Fourier transformation and governing equations are solved by using the pseudo‐spectral approach. Numerical results show that the proposed approach predicts good pitching moment coefficient at low Mach number transonic flows with up to 50% savings in computational time. Copyright © 2011 John Wiley & Sons, Ltd.

Analysis of pressure field in time domain using nonlinear reduced frequency approach in unsteady transonic flows

PurposeThe purpose of this paper is to discuss the capability of nonlinear frequency domain (NLFD) method in predicting surface pressure coefficient presented in the time domain in unsteady transonic flows.Design/methodology/approachIn this research, the solution and spatial operator are approximated by discrete form of Fourier transformation and resulting nonlinear equations are solved by use of pseudo‐spectral approach. Considered transonic flows involve different flow pattern on the airfoil surfaces. One of the test cases involves moving shocks on both lower and upper airfoil surfaces and in the two other test cases a moving shock occurs only on the upper surface.FindingsPressure distributions presented in the time domain using NLFD are compared with three test cases. Results show that NLFD predicts reasonable pressure distributions in time domain except in vicinity of shock positions. Although this method may predict unfair results near shock positions, however gives good estimates for global properties such as lift coefficient.Originality/valueIn the previous works on NLFD method, the flow field results have been limited to representing the pressure in the frequency domain or global coefficients such as lift coefficients. No details of pressure distributions in the time domain have been provided in such investigations. In this research, by presenting the pressure in the time domain, the conditions on which good pressure distributions are obtained are demonstrated.

Aeroelastic Solutions Using the Nonlinear Frequency-Domain Method

In this work, both fully implicit time-domain and nonlinear frequency-domain methods are used to compute aeroelastic solutions in transonic flows. Specifically, flutter boundaries for a two-dimensional NACA64A010 airfoil are calculated and compared with preexisting numerical results. The second-order backward-difference time-accurate scheme will serve as our numerical benchmark when determining the computational efficiency of the nonlinear frequency-domain method. Comparable results between the time-domain and nonlinear frequency-domain approaches will further justify the nonlinear frequency-domain method as an efficient alternative to the study of unsteady periodic problems. A temporal resolution study will establish the required number of modes or time steps per period required for unsteady transonic flows.

Multi-objective aerodynamic shape optimization for unsteady viscous flows

This paper presents an adjoint method for the multi-objective aerodynamic shape optimization of unsteady viscous flows. The goal is to introduce a Mach number variation into the Non-Linear Frequency Domain (NLFD) method and implement a novel approach to present a time-varying cost function through a multi-objective adjoint boundary condition. The paper presents the complete formulation of the time dependent optimal design problem. The approach is firstly demonstrated for the redesign of a helicopter rotor blade in two-dimensional flow and in three-dimensional viscous flow, the technique is employed to validate and redesign the NASA Rectangular Supercritical Wing (RSW).

Optimum Shape Design for Unsteady Three-Dimensional Viscous Flows Using a Nonlinear Frequency-Domain Method

Thispaperpresentsanadjointmethodfortheoptimumshapedesignofunsteadythree-dimensionalviscous flows. The goal is to develop a set of discrete unsteady adjoint equations and the corresponding boundary condition for the nonlinear frequency-domain method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the nonlinear frequency-domain adjoint equations for threedimensional viscous transonic flows. Third, we present results that demonstrate the application of the theory to a three-dimensional wing.

Convergence studies of the time accurate and non-linear frequency domain methods for optimum shape design

This paper presents convergence studies of the time accurate and non-linear frequency domain (NLFD) methods for optimum shape design. The NLFD method provides an attractive alternative for aerodynamic shape optimisation for unsteady flows. It dramatically reduces the number of time steps required to resolve the periodic flow field. In this work, the convergence of the gradients are investigated as the number of time steps are increased to determine the required number of modes that is adequate for shape optimisation for unsteady flows. The formulation of the adjoint equations for both the time accurate and NLFD techniques are presented. The techniques are employed for the redesign of a pitching airfoil to reduce the time-averaged drag coefficient while maintaining the lift coefficient.