Non-uniform Mean Flow Research Articles

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.

Analysis of Combustion Noise Sources Using Doak's Momentum Potential Theory

Noise emissions of modern lean combustors are related to different sources. Direct combustion noise is generated by heat release fluctuations, while indirect noise sources include the acceleration of entropy and vorticity inhomogeneities through the nozzle-guide-vane or at the combustor exit. The latter noise source is characterized by the coupling of fluctuations convected in non-uniform mean flow, which can e.g. cause vortical or entropic fluctuations to be partly transferred into acoustics. Due to the complexity of the sources, a clear and quantitative separation of the different phenomena in terms of primitive variables presents a significant challenge. This study therefore proposes an alternative framework for the description of combustion noise based on Doak's Momentum Potential Theory (MPT). The MPT defines a Generalized Acoustic Field (GAF) and describes the sound production in terms of mean energy fluxes carried by the respective acoustic, thermal and turbulent fluctuating momentum components. To confirm the ability to identify the different combustion noise sources, the method was applied to Large-Eddy Simulation data of a non-reacting swirl-combustor simulator. Finally, the coherent character and spectral behavior of the GAF were investigated using a Spectral Proper Orthogonal Decomposition (SPOD) analysis and correlated to the source distributions.

A high-order stabilized finite-element model for the Linearized Navier-Stokes equations

A numerical approach based on high-order stabilized finite elements is proposed to solve thermo-acoustic and visco-acoustic problems accounting for non-uniform mean flow effects. The approach is based on the Linearized Navier-Stokes equations written in conservative form in the frequency domain. An adaptive polynomial Finite-Element Method (FEM) using hierarchic shape functions is applied for accuracy and ease-of-use. A new enrichment strategy, inspired by the extended Finite Element Method (X-FEM), is developed to resolve the finer scales near the walls at a reasonable computational cost. It relies on a re-orthogonalization procedure proposed to preserve both the continuity of the solution and the conditioning properties of the discrete model. The performance of the method is first evaluated by performing two-dimensional simulations of acoustic waves affected by visco-thermal wall losses and mean flow effects while propagating in a duct. Numerical results are in good agreement with the analytical solution. The applicability of this new methodology is then demonstrated by computing the sound absorption of an acoustic liner installed in an impedance tube in the presence of a grazing flow in 2D. Numerical predictions of the sound pressure levels in the tube are compared to experimental data from the literature.

Novel Wentzel–Kramers–Brillouin Solutions to the Nonisentropic Helmholtz Equation in a Nonuniform Duct With Mean Temperature Gradient and Mean Flow

Abstract We derive the generalized Helmholtz equation (GHE) governing nonisentropic acoustic fluctuations in a quasi 1D duct with nonuniform cross section, mean temperature gradient, and nonuniform mean flow. Nonisentropic effects are included via heat conduction terms in the mean and fluctuating energy equations. To derive the Helmholtz equation exclusively in terms of the fluctuating pressure field, a relationship between density and pressure fluctuations is needed, which is shown to be a second-order differential equation for nonisentropic motions. Novel analytical solutions that are accurate for both low/high frequencies and small/large mean gradients are presented for the GHE based on the Wentzel–Kramers–Brillouin (WKB) method. WKB solutions are developed using the ansatz that the pressure fluctuation field has a travelling wave form, p^(x)=exp[∫0x(a+ib)dx], where x is the axial coordinate. Substituting this form into the Helmholtz equation yields coupled, nonlinear ordinary differential equations (ODEs) for a and b. Analytical solutions to the ODEs are obtained using the approximations of high frequency and slowly varying mean properties. This simplification allows us to obtain the lower order solutions b0 and a0. We then enhance solution accuracy by using a0 to solve for b1 without any approximations. Finally, b1 is employed to get a1, giving us the higher order solution. The p^ calculated from (a1, b1) is in good to excellent agreement with numerical solution of the GHE for both low and high frequencies and for a range of mean Mach numbers, including M¯≳1.

Application of Reciprocity to Calculating the Scattering Matrix of a Complex Muffler Without and With Nonuniform Meanflow

Application of Reciprocity to Calculating the Scattering Matrix of a Complex Muffler Without and With Nonuniform Meanflow

Construction and Numerical Assessment of Local Absorbing Boundary Conditions for Heterogeneous Time-Harmonic Acoustic Problems

This article is devoted to the derivation and assessment of local absorbing boundary conditions (ABCs) for numerically solving heterogeneous time-harmonic acoustic problems. To this end, we develop a strategy inspired by the work of Engquist and Majda to build local approximations of the Dirichlet-to-Neumann operator for heterogeneous media, which is still an open problem. We focus on three simplified but characteristic examples of increasing complexity to highlight the strengths and weaknesses of the proposed ABCs: the propagation in a duct with a longitudinal variation of the speed of sound, the propagation in a nonuniform mean flow using a convected wave operator, and the propagation in a duct with a transverse variation of the speed of sound and density. For each case, we follow the same systematic approach to construct a family of local ABCs and explain their implementation in a high-order finite element context. Numerical simulations allow us to validate the accuracy of the ABCs and to give recommendations for the tuning of their parameters.

On the convective wave equation for the investigation of combustor stability using FEM-methods

Solving the Helmholtz equation with spatially resolved finite element method (FEM) approaches is a well-established and cost-efficient methodology to numerically predict thermoacoustic instabilities. With the implied zero Mach number assumption all interaction mechanisms between acoustics and the mean flow velocity including the advection of acoustic waves are neglected. Incorporating these mechanisms requires higher-order approaches that come at massively increased computational cost. A tradeoff might be the convective wave equation in frequency domain, which covers the advection of waves and comes at equivalent cost as the Helmholtz equation. However, with this equation only being valid for uniform mean flow velocities it is normally not applicable to combustion processes. The present paper strives for analyzing the introduced errors when applying the convective wave equation to thermoacoustic stability analyses. Therefore, an acoustically consistent, inhomogeneous convective wave equation is derived first. Similar to Lighthill’s analogy, terms describing the interaction between acoustics and non-uniform mean flows are considered as sources. For the use with FEM approaches, a complete framework of the equation in weak formulation is provided. This includes suitable impedance boundary conditions and a transfer matrix coupling procedure. In a modal stability analysis of an industrial gas turbine combustion chamber, the homogeneous wave equation in frequency domain is subsequently compared to the Helmholtz equation and the consistent Acoustic Perturbation Equations (APE). The impact of selected source terms on the solution is investigated. Finally, a methodology using the convective wave equation in frequency domain with vanishing source terms in arbitrary mean flow fields is presented.

On the generation of entropy noise in a shock containing nozzle of high-performance aircraft at afterburner

High performance aircraft operate at high jet temperature and velocity. Measured noise data indicate that at afterburner condition their noise contain components not found in high temperature supersonic laboratory model jets. The combustion process inside an afterburner is highly unsteady and incomplete. Residual combustion takes place in the jet plume. This suggests that the flow from the afterburner into the jet nozzle would contain significant amount of hot and cold temperature blobs, generally referred to as entropy waves. It is known that when entropy waves move through a non-uniform mean flow, indirect combustion noise (also called entropy noise) would be generated. Military styled nozzles have a rather abrupt area change at and near the throat. This induces the formation of internal shocks inside the nozzle. This paper investigates the generation of entropy noise when strong entropy blobs or waves are convected through the internal shocks. It is found that the result of entropy blobs-shock interaction could lead to the radiation of intense entropy noise out of the nozzle exit into the jet plume as fast waves with a speed equal to sound speed plus jet flow speed. These fast waves create highly supersonic flow disturbances in the jet flow leading to strong Mach wave radiation. It is proposed that this Mach wave radiation is the source of a new noise component observed in the spectra of the F-22A aircraft. Detailed analysis of the F-22A noise data provides support for the proposal.

Analytical solutions of acoustic field in annular combustion chambers with non-uniform cross-sectional surface area and mean flow

Low-order acoustic network models, treating the complex combustor geometry as a network of simple geometry elements, are typically used to analyse circumferential combustion instabilities in annular combustion chambers. These elements are typically assumed to have uniform cross-sectional surface area and average radius so that the analytical solutions of the acoustic field within them can be directly obtained. However, this may lead to errors in the combustion instability analyses. The present work derives the analytical solutions of the acoustic field in annular combustion chambers with both varying cross-sectional surface area and average radius sustaining a mean flow. A wave equation for the pressure perturbation is firstly derived based on very few assumptions. Analytical solutions of the acoustic field are then derived based on a modified WKB approximation. These solutions are then validated by comparing them to results by numerically resolving the linearised Euler equations. Results show that accurate predictions can always be obtained for a smooth change of the cross-sectional surface area and small-to-moderate subsonic axial Mach numbers as long as the frequency is larger than a certain value.

A global linearized framework for modelling shear dispersion and turbulent diffusion of passive scalar fluctuations

Abstract

Study of Velocity-Potential Integral Formulations for Sound Scattered by Moving Bodies

This paper deals with the theoretical–numerical study of integral formulations for the prediction of noise scattered by moving bodies, through the velocity-potential description of flow disturbance. Specifically, three potential-flow acoustic scattering integral formulations are examined: a linearized boundary-field integral formulation, recently introduced by the authors, is compared with two widely applied boundary integral approaches based on Taylor and Taylor–Lorentz transformations. The objective of this work is to provide a detailed analysis of the theoretical differences among these formulations and corresponding limits of applicability, followed by the assessment of their capability of capturing the influence of nonuniform mean flow around moving sound scatterers. This is accomplished by a numerical investigation that concerns a rigid, rectangular, nonlifting wing in uniform rectilinear translation impinged by sound waves emitted by a comoving pulsating point source.

Development of 3D boundary element method for the simulation of acoustic metamaterials/metasurfaces in mean flow for aerospace applications

Low-cost airlines have significantly increased air transport, thus an increase in aviation noise. Therefore, predicting aircraft noise is an important component for designing an aircraft to reduce its impact on environmental noise along with the cost of testing and certification. The aim of this work is to develop a three-dimensional Boundary Element Method (BEM), which can predict the sound propagation and scattering over metamaterials and metasurfaces in mean flow. A methodology for the implementation of metamaterials and metasurfaces in BEM as an impedance patch is presented here. A three-dimensional BEM named as BEM3D has been developed to solve the aero-acoustics problems, which incorporates the Fast Multipole Method to solve large scale acoustics problems, Taylor’s transformation to account for uniform and non-uniform mean flow, impedance and non-local boundary conditions for the implementation of metamaterials. To validate BEM3D, the predictions have been benchmarked against the Finite Element Method (FEM) simulations and experimental data. It has been concluded that for no flow case BEM3D gives identical acoustics potential values against benchmarked FEM (COMSOL) predictions. For Mach number of 0.1, the BEM3D and FEM (COMSOL) predictions show small differences. The difference between BEM3D and FEM (COMSOL) predictions increases further for higher Mach number of 0.2 and 0.3. The increase in difference with Mach number is because Taylor’s Transformation gives an approximate solution for the boundary integral equation. Nevertheless, it has been concluded that Taylor’s transformation gives reasonable predictions for low Mach number of up to 0.3. BEM3D predictions have been validated against experimental data on a flat plate and a duct. Very good agreement has been found between the measured data and BEM3D predictions for sound propagation without and with the mean flow at low Mach number.

Numerical and Asymptotic Solutions of the Pridmore-Brown Equation

A study is made of acoustic duct modes in two-dimensional and axisymmetric three-dimensional lined ducts with an isentropic inviscid transversely nonuniform mean flow and sound speed. These modes are described by a one-dimensional eigenvalue problem consisting of a Pridmore-Brown equation complemented by hard-wall or impedance-wall boundary conditions. A numerical solution, based on a Galerkin projection and an efficient method for the resulting nonlinear eigenvalue problem, is compared with analytical approximations for low and high frequencies. A collection of results is presented and discussed. Modal wave numbers are traced in the complex plane for varying impedance, showing the usual regular modes and surface waves. A study of a vanishing boundary layer (the Ingard limit) showed that, in contrast to the smoothly converging acoustic modes and downstream running acoustic surface wave, the convergence of the other surface waves is numerically more difficult. Effects of (transverse) turning points and exponential decay are discussed. Especially the occurrence of modes insensitive to the wall impedance is pointed out. Cut-on wave numbers of hard-wall modes are presented as a function of frequency. A strongly nonuniform mean flow gives rise to considerable differences between the modal behavior for low and for high frequencies.

An analytical correction to Amiet’s solution of airfoil leading-edge noise in non-uniform mean flows

Gust/turbulence–leading edge interaction is a significant source of airfoil broadband noise. An approach often used to predict the sound is based on Amiet’s flat-plate solution. Analytical studies have been conducted to investigate the influences of airfoil geometries, non-uniform mean flows and turbulence statistics, which, however, were often too convoluted. In this work, the problem is revisited by proposing simple corrections to the standard flat-plate solution to account for the effect of non-uniform mean flows of real airfoils. A key step in the method is to use a new space–time transformation that is analogous to the Prandtl–Glauert transformation to simplify the sound governing equation with spatially varying coefficients to a classical wave equation, which is then solved using the Schwarzschild technique as in Amiet’s solution. The impacts of Mach number, wavenumber and airfoil geometry on the prediction accuracy are investigated for both single-frequency and broadband cases, and the results are compared against high-fidelity simulations. It predicts the sound reduction by the airfoil thickness, and reveals that the reduction is caused by the non-uniform streamwise velocity. The limitations of the model are discussed and the approximation errors are estimated. In general, the prediction error increases with the airfoil thickness, the sound frequency and the flow Mach number. Nevertheless, in all cases studied in this work, the proposed correction can effectively improve the prediction accuracy of the flat-plate solution much more efficiently compared to numerical solutions of the Euler equations using computational aeroacoustics.

Entropy noise from shock containing nozzles of military jets at afterburner

This paper considers the noise of military jets operating at afterburner. Boosting by the afterburner, the jet velocity and temperature are much higher than those of laboratory jets. Under these conditions, do we anticipate the noise components of these jets the same as those of laboratory jets, i.e., consisting mainly turbulent mixing noise? Do we expect new noise sources in these jets, since they operate at much higher velocities and temperatures and that the combustion processes in the afterburner are highly unsteady. In this paper, we explore one plausible new noise mechanism; the generation of entropy noise inside the shock containing nozzles of military jets. It is well-known that hot and cold spots (entropy waves) from unsteady combustion when convecting through a non-uniform mean flow would create fluctuating pressure and hence acoustic radiation. We investigate this possibility by performing numerical simulation of the flow and noise inside a rectangular nozzle typical of that of the F-22 Raptor. These nozzles are imperfectly expanded. There are shocks inside. We find that because shocks have very steep velocity gradients, their presence leads to the generation and emission of strong entropy noise when entropy waves from the afterburner pass through them.

An efficient computation of cascade-gust interaction noise based on a hybrid analytical and boundary element method

This paper presents a hybrid analytical and boundary element method to compute the airfoil cascade-gust interaction noise. We assume that the acoustic response of an airfoil to the oncoming gust depends on the locally non-uniform mean flow such that the analytical solution based on a generalised Prandtl-Glauert transformation is utilised to compute the sound generation. The radiation of sound from an airfoil is scattered by other airfoils, which is resolved by a high-efficiency boundary element method. An averaging approach is proposed to efficiently compute the periodic Green's function due to the cascade. The predicted results are validated against numerical simulations for flat plate cascades, and the results match reasonably well. The cut-off properties due to the geometry restriction are discussed, and the critical gust wavelengths for the different orders of acoustic mode are derived. The low-frequency components are trapped in the space between the blades if the stagger angle is zero, while the plane waves are likely to propagating otherwise. For cascades with real airfoil geometry, the background mean flows are computed based on the potential theory. The omission of the refraction effect and approximation errors in the analytical solution of the acoustic response can cause errors for real airfoils, but the hybrid method can capture the sound reduction due to the finite thickness, and yield closer agreement with the numerical solution than the flat plate.

Pressure-field permeable-surface integral formulations for sound scattered by moving bodies

This paper presents two novel integral formulations for the prediction of sound scattered by moving bodies, derived from the Lighthill and the Ffowcs Williams and Hawkings equations for the pressure field. These are expressed in the frequency-domain, over a permeable (fictitious) surface surrounding the scatterer(s), and are numerically evaluated through application of a boundary element technique. The aims of the paper are the assessment of the influence of the nonlinear terms of Lighthill and Ffowcs Williams and Hawkings equations on sound scattering prediction in the presence of nonuniform mean-flow due to scatterer motion, the assessment of the corresponding limits of applicability of the widely-used linear formulations for solid-wall boundaries, and the development of integral formulations capable to predict accurately and efficiently the noise scattered in the far field by moving bodies. The numerical investigation concerns a non-lifting wing in uniform translation impinged by an acoustic disturbance generated by a co-moving source, and includes the comparison of the results obtained through the proposed scattering formulations with those provided by a boundary-field velocity-potential approach recently validated for moving-body problems. Its main outcomes reveal that, when the proposed pressure-field formulations include the nonlinear terms through application of a suitable permeable-surface, their predictions match those provided by the boundary-field velocity-potential solver, whereas the fully linear versions of both pressure-field approaches yield underestimated scattered noise predictions, significantly less accurate than those given by the linear version of the velocity-potential formulation.

Extensions to the acoustic scattering analysis for cloaks in non-uniform mean flows

Most of the acoustic cloak designs are based on the stationary medium, which, however, may be inaccurate in many practical applications with non-uniform flows. The optimization of the acoustic cloaking performance was described as an active noise control problem, and the theoretical model by Huang, Zhong, and Stalnov [J. Acoust. Soc. Am. 135(5), 2571-2580 (2014b)] was employed for the quick evaluation of the sound scattering. In this work, extensions are made to address the unsolved but essential issues in the model. First, the impact of the discontinuities at the interface between the cloak and surrounding fluids is investigated. Second, the high-order Born's approximation is employed to solve the sound governing equation, which can quickly improve the prediction accuracy. Finally, the optimized cloaking strategy is applied to airfoils in turbulent flows to demonstrate the capability of the proposed modelling for cases of practical importance. Also, it is found that the performance of the optimized cloak is insensitive to the frequency of the incident wave. The promising results suggest that an optimized cloaking design can effectively suppress the sound scattering, providing the confidence with the mathematical framework for the potential aeronautics and marine applications.

Geometry effect on the airfoil-gust interaction noise in transonic flows

Numerical simulations are conducted to investigate the impacts of airfoil thickness, the angle of attack and camber on the airfoil-gust interaction noise in transonic flows where locally supersonic regimes and terminating shocks are present. The conclusions about the geometry effects based on the extensively studied subsonic cases are revisited. With the increase of airfoil thickness, the sound generation is reduced in the downstream direction as in subsonic flows. More sound is produced in the upstream direction for thicker airfoils due to the non-uniform mean flow and shocks in the near field. The compensative effect makes the overall sound reduction by the airfoil thickness less than the subsonic cases despite the significant difference in the radiation patterns. The acoustic responses to the single frequency gusts are sensitive to the airfoil angle of attack in transonic flows. However, the overall differences are reduced when multiple wavenumber components are superposed in isotropic turbulence, and the sound pressure levels are therefore close as in subsonic flows. Similarly, the significant variations in single frequency acoustic responses by airfoil camber are averaged by the superposition of various wavenumber components. However, apparent variations are still found in the upstream direction, especially for the turbulences with small integral length scales.

On the effect of streamwise disturbance on the airfoil-turbulence interaction noise.

Usually the prediction of airfoil-gust interaction noise is mainly based on the flat plate solution, and only the transverse velocity disturbance that is perpendicular to the flat plate surface is considered. However, the solution might underestimate the sound generation in non-uniform mean flows for finite thickness airfoils, in which the velocity disturbance in the streamwise direction could probably produce sound as well. The effect is studied in this work by using both numerical simulations and theoretical analysis. To highlight the effect of streamwise disturbance, the anisotropic turbulences with constant transverse velocity components while varying streamwise components are employed, which can be realised by adjusting the integral length scales in the Gaussian energy spectra. The simulation results explicitly show that more sound is produced by the streamwise components for airfoils at higher Mach numbers when the flow non-uniformities are more significant. An analytical model based on acoustic analogy to account for the sound generation by the coupling between the gust and non-uniform mean flows is proposed, which explains the observations in the numerical simulation well. Also, a correction to Amiet's flat plate solution using the proposed model yields better agreement with the numerical results, especially for anisotropic turbulences with larger integral length scale in the streamwise direction.