Linearized Euler Equations Research Articles

Impact of a two-dimensional steep hill on wind turbine noise propagation

Abstract. Wind turbine noise propagation in a hilly terrain is studied through numerical simulation in different scenarios. Linearized Euler equations are solved in a moving frame that follows the wavefront, and wind turbine noise is modeled with an extended moving source. We employ large-eddy simulations to simulate the flow around the hill and the wind turbine. The sound pressure levels (SPLs) obtained for a wind turbine in front of a 2D hill and a wind turbine on a hilltop are compared to a baseline flat case. First, the source height and wind speed strongly affect sound propagation downwind. We find that topography influences the wake shape, inducing changes in the sound propagation that drastically modify the SPL downwind. Placing the turbine on the hilltop increases the average sound pressure level and amplitude modulation downwind. For the wind turbine placed upstream of a hill, a strong shielding effect is observed. But, because of the refraction by the wind gradient, levels are comparable with the baseline flat case just after the hill. Thus, considering how terrain topography alters the flow and wind turbine wake is essential to accurately predict wind turbine noise propagation.

Flux reconstruction approach for long-time calculations of the linearized Euler equation

The Linearized Euler Equations (LEE) are one of the fundamental governing equations in acoustics research. Due to the relatively small magnitude of acoustic disturbances compared to the physical quantities of the background flow, ensuring highly precise solutions without dissipation and dispersion over time is essential. The Flux Reconstruction (FR) method, recognized as a high-order numerical method with flexible construction capabilities, has been extensively utilized in recent years for solving partial differential equations numerically. This paper introduces an offsetting numerical flux for one-dimensional LEE. Subsequently, it develops a flux reconstruction scheme with energy-conserving characteristics, referred to as the ECFR scheme, aimed at achieving high-precision numerical solutions for prolonged computations. The energy-conserving feature of the scheme is substantiated through theoretical derivation and further corroborated by numerical validation. Moreover, the impact of Mach numbers and the parameter of the offsetting flux on the scheme's performance is discussed. The stability of the scheme is also analyzed, with the limit of the CFL number being provided. The novel ECFR scheme facilitates obtaining high-precision acoustic numerical solutions, thereby providing strong support for the study of complex engineering problems such as aircraft, submarine, and automobile noise.

Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method

Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

A very fast high-order flux reconstruction for Finite Volume schemes for Computational Aeroacoustics

AbstractGiven the small wavelengths and wide range of frequencies of the acoustic waves involved in Aeroacoustics problems, the use of very accurate, low-dissipative numerical schemes is the only valid option to accurately capture these phenomena. However, as the order of the scheme increases, the computational time also increases. In this work, we propose a new high-order flux reconstruction in the framework of finite volume (FV) schemes for linear problems. In particular, it is applied to solve the Linearized Euler Equations, which are widely used in the field of Computational Aeroacoustics. This new reconstruction is very efficient and well suited in the context of very high-order FV schemes, where the computation of high-order flux integrals are needed at cell edges/faces. Different benchmark test cases are carried out to analyze the accuracy and the efficiency of the proposed flux reconstruction. The proposed methodology preserves the accuracy while the computational time relatively reduces drastically as the order increases.

Heat effects on supersonic jet screech: a linear stability analysis based on parabolized stability equations

In this paper, we use the parabolized stability equations (PSEs) to investigate heat effects on supersonic jet screech. The PSE is derived from the linear Euler equations, and the computations are performed on an empirical mean flow profile. Employing PSE, we can examine several important characteristics of the shear-layer instability waves, including the convection velocity, the growth rate, and the location where the instability waves attain the maximal total amplification. Equipped with these knowledge, we further explore their impacts on the jet screech. Specifically, using a newly proposed iteration scheme, we first identify a more suitable convective Mach number to predict the screech frequency in both heated and cold jets. Second, we find that the transition of the screech mode from the axisymmetric to the helical mode may be explained by the shift in the most amplified instability modes. Using this criterion, we can predict a transition Mach number for the mode staging. Third, by assuming that the effective source location is the position where the instability wave attains its maximal amplitude and using the newly obtained convective Mach number, we can predict the screech mode staging in cold and heated jets using Gao's model. The predictions agree well with the experimental data. Finally, we investigate the heat effects on screech amplitude. It is found that compared to cold jets, the difference between the frequency of the most amplified instability wave and the frequency of the jet screech is much bigger in heated jets, which may lead to a lower screech amplitude.

Gaussian Pulses over Random Topographies for the Linear Euler Equations

This study investigates numerically the interaction between a Gaussian pulse and variable topography using the linear Euler equations. The impact of topography variation on the amplitude and behavior of the wave pulse is examined through numerical simulations and statistical analysis. On one hand, we show that for slowly varying topographies, the incoming pulse almost retains its shape, and little energy is transferred to the small reflected waves. On the other hand, we demonstrate that for rapidly varyingtopographies, the shape of the pulse is destroyed, which is different from previous studies.

Theoretical analysis and numerical validation of the mechanisms controlling composition noise

A new definition of entropy fluctuation is provided in this paper, that allows properly separating entropy and mixture composition fluctuations. This decomposition into linearly independent variables prevents from overestimating compositional noise in indirect noise prediction. When considering quasi one-dimensional flow in nozzles, a new resulting system of linearized Euler equations is obtained. Two analytical solutions of this system of equations are investigated in this study, one dedicated to low-frequency perturbations and another one for all frequency perturbations. This new theory is then validated by comparing the model predictions with direct numerical simulation of nozzle flows in which composition fluctuations are pulsed. To do so, new Navier–Stokes characteristics boundary conditions to account for the new properly defined entropy wave are provided. Furthermore, non-reflecting inlet boundary conditions have been newly derived, relaxing on the axial mass-flow rate, static temperature and species mass fractions. Finally, a parametric study based on an air-kerosene (equivalent C10H20) mixture and an ideal framework shows that composition noise can reach a maximum of 10% of entropy noise for lean combustion and choked nozzle while for rich combustion, composition noise and entropy noise show comparable levels.

Localizing aeroacoustic sources in complex geometries: A hybrid method coupling 3D microphone array and time-reversal

Flow-induced noise is widely encountered in engineering domains, and noise source localization is the first step to understand the generation mechanisms. A large part of aeroacoustic noise measurements is performed in wind tunnels by using microphone arrays. A widespread method to identify the noise sources based on these measurements is the beamforming technique. This relies on a Green function, which ideally accounts for source type, flow, and reflecting boundaries. This function is generally not known analytically. An alternative approach is the time-reversal technique considered here. This method consists in propagating the sound field from the microphones back to focus points interpreted as sources. The back propagation can be done numerically with an acoustic propagation solver, which replaces the Green function needed in the beamforming. To date time reversal has been used to localize sound sources with flow in two-dimensions, without considering any reflecting/diffracting boundary. The objective of this work is to consider a three-dimensional situation with flow, and to include the presence of such boundaries to see if this can improve source localization. A synthetic sound source enclosed in a streamlined body is placed in a wind tunnel, and its sound field is diffracted by a nearby airfoil. A three-dimensional antenna with 768 microphones is used to measure the sound field, and time-reversal is performed by solving the three-dimensional linearized Euler equations with a discontinuous Galerkin method. The interest of taking into account the airfoil geometry for source localization is discussed, and some comparisons with the beamforming method are performed.

On the linear stability of the Lamb–Chaplygin dipole

The Lamb–Chaplygin dipole (Lamb, Hydrodynamics, 2nd edn, 1895, Cambridge University Press; Lamb, Hydrodynamics, 3rd edn, 1906, Cambridge University Press; Chaplygin, Trudy Otd. Fiz. Nauk Imper. Mosk. Obshch. Lyub. Estest., vol. 11, 1903, pp. 11–14) is one of the few closed-form relative equilibrium solutions of the two-dimensional (2-D) Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2-D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time integration of the 2-D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2-D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.

DynAMO: Multi-agent reinforcement learning for dynamic anticipatory mesh optimization with applications to hyperbolic conservation laws

We introduce DynAMO, a reinforcement learning paradigm for Dynamic Anticipatory Mesh Optimization. Adaptive mesh refinement is an effective tool for optimizing computational cost and solution accuracy in numerical methods for partial differential equations. However, traditional adaptive mesh refinement approaches for time-dependent problems typically rely only on instantaneous error indicators to guide adaptivity. As a result, standard strategies often require frequent remeshing to maintain accuracy. In the DynAMO approach, multi-agent reinforcement learning is used to discover new local refinement policies that can anticipate and respond to future solution states by producing meshes that deliver more accurate solutions for longer time intervals. By applying DynAMO to discontinuous Galerkin methods for the linear advection and compressible Euler equations in two dimensions, we demonstrate that this new mesh refinement paradigm can outperform conventional threshold-based strategies while also generalizing to different mesh sizes, remeshing and simulation times, and initial conditions.

Research on the Impact of a Fluid Field on an Acoustic Field in Herschel–Quincke Tube

A study concerning the influence of flow on the Herschel–Quincke duct is presented here, which includes the numerical model, the acoustic source and the absorption condition called the Perfectly Matched Layer. For the excitation of a sound field, a normal mode wave is placed at the inlet of the tube. The function of PML is to simulate the infinite tube and avoid the reflection of acoustic wave. To investigate the influence of flow field on sound field, a coupled calculation method combining the finite element method and computational fluid dynamics is used to solve the linearized Euler equation, named the Galbrun equation. Firstly, the influence of the cross-section of the tube on the acoustic field is considered. Secondly, the effects of flow on the acoustic field is also investigated. Lastly, a comparative analysis of the simulation results reveals the influence of flow and other parameters of the tube on sound propagation. Both the Mach number and the cross-section ratio have an influence on the acoustic resonance, and the resonance frequency decreases with the increase in the cross-section ratio.

Numerical study of the scattering of acoustic waves by an elliptic vortex.

The scattering of the acoustic waves generated by a monopolar source propagating through a two-dimensional elliptic vortex, fixed or convected by a uniform flow, is studied by solving the Linearized Euler Equations in Cartesian coordinates using the Discontinuous Galerkin Method. For a fixed vortex position, the number, amplitudes, and angular spreads of the acoustic interference beams resulting from the sound scattering are found to significantly depend on the orientation of the vortex major axis with respect to the direction of the incident waves and on the vortex maximum tangential velocity. In particular, additional interference beams are obtained at large observation angles for a more elliptical vortex. For a convected elliptic vortex, the interference beams are curved as the angle between the incident acoustic wave and the vortex major axis varies when the vortex travels in the downstream direction. As expected, the scattering of the acoustic waves leads to spectral broadening in this case. Moreover, the widths and the frequencies of the lateral lobes obtained in the spectra on both sides of the peak at the source frequency are different for elliptic and round vortices.

Modeling Symmetric Coherent Mode Interactions in a Supersonic Rectangular Jet

This work presents the use of a reduced-order model (ROM) to predict coherent structure modes and their interaction with one another in a supersonic rectangular jet. The ROM formulation is unique in that it utilizes an integral technique and models the energy exchange between harmonically related modes, which can give insight on how to excite the jet in an experiment or computation. This ROM is used in conjunction with a linearized Euler equation solver to obtain shape functions for various disturbance frequencies. Work is limited here to symmetric modes of the coherent structure, and hence the disturbance in the upper and lower shear layers is in phase. Nonlinear solutions are first obtained without mode–mode interactions, giving a most amplified Strouhal number of 0.15 based on the height of the jet, which is assumed to be the dominant noise source in the jet, referred to as the fundamental frequency. The fundamental’s interaction with either the subharmonic or harmonic is assessed based on their ability to reduce the fundamental. Adding the harmonic is found to be more effective at reducing the peak of the fundamental, and thus this is concluded to be possibly an effective mechanism for noise reduction.

A dynamical approach to the study of instability near Couette flow

AbstractIn this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity , when the perturbations are in the critical spaces . More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size with any small , which implies that is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo‐spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.

Stabilisation of Hydrodynamic Instabilities by Critical Layers in Acoustic Lining Boundary Layers

Acoustics are considered in a straight cylindrical lined duct with an axial mean flow that is uniform apart from a boundary layer near the wall. Within the boundary layer, which may or may not be thin, the flow profile is quadratic and satisfies no-slip at the wall. Time-harmonic modal solutions to the linearized Euler equations are found by solving the Pridmore-Brown equation using Frobenius series. The Briggs–Bers criterion is used to ascertain the spatial stability of the modes, without considering absolute instabilities. The modes usually identified as hydrodynamic instabilities are found to interact with the critical layer branch cut, also known as the continuous spectrum. By varying the boundary-layer thickness, flow speed, frequency, and wall impedance, it is found that these spatial instabilities can be stabilized behind the critical layer branch cut. In particular, spatial instabilities are only found for a boundary layer thinner than a critical boundary-layer thickness . The behaviors observed for the uniform-quadratic sheared flow considered here are further compared to a uniform-linear sheared flow, and a uniform slipping flow under the Ingard–Myers boundary condition, where this process of stabilization is not observed. It is therefore argued that modeling a sufficiently smooth mean flow boundary layer is necessary to predict the correct stability of flow over a lined wall.

Rayleigh–Taylor instability of 3D inhomogeneous incompressible Euler equations with damping in a horizontal slab

In this paper, we consider the Rayleigh–Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. We show that if the steady density profile is non-monotonous along the height, then the Euler system with damping is nonlinearly unstable around the given steady state. In this article, we develop a new variational structure to construct the growing mode solution, and overcome the difficulty in proving the sharp exponential growth rate by exploiting the structures in linearized Euler equations. Then combined with error estimates and a standard bootstrapping argument, we finish the nonlinear instability.

Computational aeroacoustics and phase-conjugation technique for localizing flow-induced sources

This paper presents a framework comprising computational aeroacoustics (CAA) solvers and a numerical phase-conjugation technique to localize flow-induced noise sources generated by bodies immersed in low Mach number flows. Two test-cases were considered, namely, flow over a two-dimensional (a) circular cylinder at Reynolds number Re =150 and Mach number M = 0.2 and (b) NACA-0012 airfoil at Re = 5000 and angle of attack (AoA) α = 5 deg. The CAA simulations were carried out in OpenFOAM wherein the two-dimensional unsteady, incompressible Navier–Stokes equations was solved using either the pressure-implicit splitting of operators (PISO) algorithm or large eddy simulation (LES). The near-field aerodynamic sources were extracted in terms of the unsteady Lighthill stress tensor terms, following which the far-field acoustic data were computed on boundary nodes using different methods which include the numerical solution of the Lighthill equation, Curle’s analogy, and Ffowcs Williams and Hawkings (FWH) analogy as well as linearized Euler equations (LEE). Next, the frequency-domain phase-conjugation (PC) method was implemented in the finite-element (FE) based COMSOL software to reconstruct the radiated acoustic pressure field, whereby the dipole source location was readily identified by pair of focal spots at the cylinder or at the airfoil trailing-edge (TE).

Wind turbine sound propagation: Comparison of a linearized Euler equations model with parabolic equation methods.

Noise generated by wind turbines is significantly impacted by its propagation in the atmosphere. Hence, for annoyance issues, an accurate prediction of sound propagation is critical to determine noise levels around wind turbines. This study presents a method to predict wind turbine sound propagation based on linearized Euler equations. We compare this approach to the parabolic equation method, which is widely used since it captures the influence of atmospheric refraction, ground reflection, and sound scattering at a low computational cost. Using the linearized Euler equations is more computationally demanding but can reproduce more physical effects as fewer assumptions are made. An additional benefit of the linearized Euler equations is that they provide a time-domain solution. To compare both approaches, we simulate sound propagation in two distinct scenarios. In the first scenario, a wind turbine is situated on flat terrain; in the second, a turbine is situated on a hilltop. The results show that both methods provide similar noise predictions in the two scenarios. We find that while some differences in the propagation results are observed in the second case, the final predictions for a broadband extended source are similar between the two methods.

Finite Element Modelling of a Flow-Acoustic Coupling in Unbounded Domains

One of the main issues of design process of HVAC systems and ventilation ducts in particular is correct modelling of coupling of the flow field and acoustic field of the air flowing in such systems. Such a coupling can be modelled in many ways, one of them is using linearised Euler equations (LEE). In this paper, the method of solving these equations using finite element method and open source tools is decribed. Equations were transformed into functional and solved using Python language and FEniCS software. The non-reflective boundary condition called buffer layer was also implemented into equations, which allowed modelling of unbounded domains. The issue, influence of flow on wave propagation, could be adressed using LEE equations, as they take non-uniform mean flow into account. The developed tool was verified and results of simulations were compared with analytical solutions, both in one- and two-dimensional cases. The obtained numerical results are very consistent with analytical ones. Furthermore, this paper describes the use of the developed tool for analysing a more complex model. Acoustic wave propagation for the backward-facing step in the presence of flow calculated using Navier-Stokes equations was studied.

Computation and prediction of intrinsic thermoacoustic oscillations associated with autoignition fronts

A sequential combustor architecture consists of two longitudinally staged combustion zones featuring a propagation-stabilized flame in the first stage and an autoignition-stabilized flame in the second (reheat) stage. This layout allows for increased fuel flexibility compared to single-stage systems and enables clean and efficient combustion of carbon-free fuels such as hydrogen. Interactions between the autoignition-stabilized flame and the acoustic field created by the unsteady ignition front and the surrounding boundaries can result in self-sustained (thermoacoustic) flame oscillations in the reheat stage of the sequential burner, causing performance losses and structural damage. In this paper, an intrinsic flame-acoustic feedback mechanism causing self-sustained thermoacoustic oscillations in a reheat combustor is studied. The intrinsic thermoacoustic feedback is established when an acoustic wave generated by the unsteady ignition front travels upstream and introduces local temperature, pressure and velocity perturbations in the unburnt mixture. These modulate the ignition chemistry and the front kinematics and thus, in turn, act as a new source of perturbation for the ignition front that generates heat release rate oscillations and upstream-traveling acoustic disturbances, closing the feedback loop. Compressible reactive Euler equation computations of autoignition fronts in a simple one-dimensional reheat combustor configuration with fully non-reflecting boundaries are first performed to demonstrate and characterize the intrinsic thermoacoustic oscillations associated with autoignition fronts. These computations show that the autoignition front exhibits distinct harmonic thermoacoustic oscillations which tend to become unstable at some conditions. Next, frequencies and growth rates of the intrinsic thermoacoustic oscillations in the linear regime are predicted using a hybrid approach with flame response transfer functions taken from a forced Euler calculation and using a linearized Euler equation framework to describe the combustor acoustic field. The predictions of the intrinsic thermoacoustic eigenvalue from the linearized Euler equation framework show good agreement with the linear behaviour of these oscillations observed in the Euler computations. The findings of the present work are useful both from a fundamental standpoint for prediction of intrinsic thermoacoustic instabilities and from an industrial perspective for prevention of thermoacoustic instabilities in staged combustion systems.