Convected Helmholtz Equation Research Articles

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

Construction and analysis of a HDG solution for the total-flux formulation of the convected Helmholtz equation

We introduce a hybridizable discontinuous Galerkin (HDG) method for the convected Helmholtz equation based on the total flux formulation, in which the vector unknown represents both diffusive and convective phenomena. This HDG method is constricted with the same interpolation degree for all the unknowns and a physically informed value for the penalization parameter is computed. A detailed analysis including local and global well-posedness as well as a super-convergence result is carried out. We then provide numerical experiments to illustrate the theoretical results.

The factorization method and Capon’s method for random source identification in experimental aeroacoustics

Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modeled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show that a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon’s method or as the minimum variance method. Numerical examples illustrate our theoretical findings.

Low-order Prandtl-Glauert-Lorentz based Absorbing Boundary Conditions for solving the convected Helmholtz equation with Discontinuous Galerkin methods

We construct Absorbing Boundary Conditions (ABCs) for the convected Helmholtz equation that are easy to implement in a Hybridizable Discontinuous Galerkin (HDG) formulation. The construction is based upon the Prandtl-Glauert-Lorentz map which transforms the convected Helmholtz equation into the regular Helmholtz equation. The new ABCs are thus issued from classical Bayliss-Gunztburger-Turkel ABCs and are valid for carrier flows that vary inside the computational domain but become uniform far away from the source. They lead to accurate numerical results for low and intermediate Mach numbers using a HDG formulation with the acoustic potential and the total flux as unknowns.

Generalization of Amiet’s theory for small reduced-frequency and nearly-critical gusts

Amiet’s theory is an airfoil noise prediction technique based on the resolution of a convected Helmholtz equation for small velocity potential and pressure disturbances. In this approach, the partial differential equation can be classified as hyperbolic or elliptic depending on the sign of the Helmholtz equation parameter. Existing solutions present non-physical results at the transition between these two regimes. In this study, a new approximation aiming at accurately predict back-scattering effects of small reduced-frequency and nearly-critical gusts is proposed. Our approach reduces the necessity of regularization and ad-hoc corrections adopted in previous literature. Moreover, our solution matches published results for gusts of large reduced-frequency. We present analytical derivations of the pressure trace that are applicable to the leading- and the trailing-edge airfoil noise prediction. The proposed formulation is particularly applicable to acoustically compact airfoils. In practical applications, this condition is verified for the low-frequency aerodynamic noise produced by ventilation systems and automotive fans.

Low-Order Prandtl-Glauert-Lorentz Based Absorbing Boundary Conditions for Solving the Convected Helmholtz Equation with Discontinuous Galerkin Methods

Low-Order Prandtl-Glauert-Lorentz Based Absorbing Boundary Conditions for Solving the Convected Helmholtz Equation with Discontinuous Galerkin Methods

Acoustic-roughness receptivity in subsonic boundary-layer flows over aerofoils

The generation of a viscous–inviscid instability through scattering of an acoustic wave by localised and distributed roughness on the upper surface of a NACA 0012 aerofoil is studied with a time-harmonic compressible adjoint linearised Navier–Stokes approach. This extends previous work by the authors dedicated to flat plate geometries. The key advancement lies in the modelling of the inviscid acoustic field external to the aerofoil boundary layer, requiring a numerical solution of the convected Helmholtz equation in a non-uniform inviscid field to determine the unsteady pressure field on the curved aerofoil surface. This externally imposed acoustic pressure field subsequently drives the acoustic boundary layer, which fundamentally determines the amplitudes of acoustic-roughness receptivity. A study of receptivity in the presence of Gaussian-shaped roughness and sinusoidally distributed roughness at Mach number $M_\infty =0.4$ and Strouhal numbers $\mathcal {S} \approx \{46,69,115\}$ shows the effects of various parameters, most notably angle of attack, angle of incidence of the externally imposed plane acoustic wave and geometry of surface roughness; the latter is varied from viewpoint of its placement on the aerofoil surface and its wavelength. The parametric study suggests that non-parallel effects are quite substantial and that considerable differences arise when using parallel flow theory to estimate the optimal width of Gaussian-shaped roughness elements to provoke the greatest response. Furthermore, receptivity amplitudes for distributed roughness are observed to be generally higher for lower angles of attack, i.e. for less adverse pressure gradients. It is also shown that the boundary layer is more receptive to upstream-travelling acoustic waves.

IFAR liner benchmark challenge #1 – DLR impedance eduction of uniform and axially segmented liners and comparison with NASA results

This paper presents the contribution from the German Aerospace Center (DLR) to the first liner benchmark challenge under the framework of the International Forum for Aviation Research (IFAR). Therefore, two sets of acoustically damping wall treatments, called ‘liner samples’, have been produced by additive manufacturing based on the design data provided by NASA coordinating this benchmark. These liner samples have been integrated and acoustically characterized in the liner flow test facility DUCT-R at DLR Berlin as well as in the liner flow test facility GFIT at NASA Langley. Besides the dissipation coefficients and the axial pressure profiles, the liner wall impedance was educed by first determining the axial wave numbers and then applying a straightforward method based on the one-dimensional Convected Helmholtz Equation. Finally, the comparison of the liner impedance values to the NASA results show a fairly good agreement.

Robust Three-Dimensional Acoustic Performance Probabilistic Model for Nacelle Liners

This paper is devoted to the robust optimization of nacelle liners (acoustic treatments). The full computational aeroacoustic model is based on the convected Helmholtz equation in presence of a nonhomogeneous flow velocity field, which is computed by solving the potential Euler equations. Uncertainties are taken into account in order to increase the robustness of the predictions. A reduced-order computational aeroacoustic model is constructed in order to implement the nonparametric probabilistic model of uncertainties induced by the modeling errors in the aeroacoustic model and in the liner model. In addition, the uncertainties on the acoustic excitation induced by the fan are taken into account using the parametric probabilistic approach. The developed methodology is applied to a three-dimensional nacelle intake and allows for computing the confidence regions of the random far-field radiated pressure in terms of random sound pressure level, which are compared with experiments for one flight condition, one frequency, and for two types of liners.

Stable Perfectly Matched Layers with Lorentz transformation for the convected Helmholtz equation

Perfectly Matched Layers (PMLs) appear as a popular alternative to non-reflecting boundary conditions for wave-type problems. The core idea is to extend the computational domain by a fictitious layer with specific absorption properties such that the wave amplitude decays significantly and does not produce back reflections. In the context of convected acoustics, it is well-known that PMLs are exposed to stability issues in the frequency and time domain. It is caused by a mismatch between the phase velocity on which the PML acts, and the group velocity which carries the energy of the wave. The objective of this study is to take advantage of the Lorentz transformation in order to design stable perfectly matched layers for generally shaped convex domains in a uniform mean flow of arbitrary orientation. We aim at presenting a pedagogical approach to tackle the stability issue. The robustness of the approach is also demonstrated through several two-dimensional high-order finite element simulations of increasing complexity.

A modal boundary element method for axisymmetric acoustic problems in an arbitrary uniform mean flow

This paper presents a new numerical analysis approach based on an improved Modal Boundary Element Method (MBEM) formulation for axisymmetric acoustic radiation and propagation problems in a uniform mean flow of arbitrary direction. It is based on the homogeneous Modal Convected Helmholtz Equation (MCHE) and its convected Green’s kernel using a Fourier transform method. In order to simplify the flow terms, a general modal boundary integral solution is formulated explicitly according to two new operators such as the particular and convected kernels. Through the use of modified operators, the improved MBEM approach with flow takes a convective form of the general MBEM approach and has a similar form of the nonflow MBEM formulation. The reference and reduced Helmholtz Integral Equations (HIEs) are implicitly taken into account a new nonreflecting Sommerfeld condition to solve far field axisymmetric regions in a uniform mean flow. For isolating the singular integrations, the modal convected Green’s kernel and its modified normal derivative are performed partly analytically in terms of Laplace coefficients and partly numerically in terms of Fourier coefficients. These coefficients are computed by recursion schemes and Gauss-Legendre quadrature standard formulae. Specifically, standard forms of the free term and its convected angle resulting from the singular integrals can be expressed only in terms of real angles in meridian plane. To demonstrate the application of the improved MBEM formulation, three exterior acoustic case studies are considered. These verification cases are based on new analytic formulations for axisymmetric acoustic sources, such as axisymmetric monopole, axial and radial dipole sources in the presence of an arbitrary uniform mean flow. Directivity plots obtained using the proposed technique are compared with the analytical results.

Uniqueness of an inverse source problem in experimental aeroacoustics

This paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in or . As a main result we prove that an unknown, compactly supported source power function is uniquely determined by idealized, noise-free correlation measurements. Our framework further allows for a precise characterization of state-of-the-art source reconstruction methods and their interrelations.

An analytic solution for gust–cascade interaction noise including effects of realistic aerofoil geometry

This paper presents an analytic solution for the sound generated by rotor–stator interaction for aerofoils with small camber and thickness subject to a background flow with small angle of attack. The interaction is modelled as a convected, unsteady vortical or entropic gust incident on an infinite rectilinear cascade of staggered aerofoils in a background flow that is uniform far away from the cascade. Applying rapid distortion theory (RDT) and transforming to an orthogonal coordinate system reduces the cascade of aerofoils to a cascade of flat plates. By seeking a perturbation expansion in terms of the disturbance of the background flow from uniform flow, leading- and first-order governing equations and boundary conditions are obtained for the acoustic potential. The system is then solved analytically using the Wiener–Hopf method. The resulting expression is inverted to give the acoustic potential function in the entire domain, i.e. a solution to the inhomogeneous convected Helmholtz equation with inhomogeneous boundary conditions in a cascade geometry. The solution significantly extends previous analytical work that is restricted to flat plates or only calculates the far-upstream radiation, and as such can give insight into the role played by blade geometry on the acoustic field upstream, downstream and in the important inter-blade region of the cascade. This new solution is validated against solutions that only account for flat plates at zero angle of attack. Various aeroacoustic results, including the scattered pressure, unsteady lift and sound power output, are discussed for a range of geometries and angles of attack.

ON IMPROVEMENT OF THE IMPEDANCE EDUCATION ACCURACE BY ACCOUNT OF IMPEDANCE VARIABILITY ALONG THE ACOUSTIC LINER

An impedance eduction method considering impedance variability along the acoustic liner is proposed. Impedance eduction is based on minimization of the residual functional, which is the discrepancy between the experimental and computed values of acoustic pressure at the interferometer microphones. The computed values are obtained by solving the Helmholtz equation, convected Helmholtz equation and linearized Navier-Stokes equations for a two-dimensional duct with an impedance wall and flow based on finite-element method. The cases of constant and variable impedance along the liner sample are considered. It is established that when taking into account the variable impedance, the residual functional decreases more than twofold against the case of constant impedance.

Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems

AbstractWe analyse the convergence of finite element discretizations of time-harmonic wave propagation problems. We propose a general methodology to derive stability conditions and error estimates that are explicit with respect to the wavenumber $k$. This methodology is formally based on an expansion of the solution in powers of $k$, which permits to split the solution into a regular, but oscillating part, and another component that is rough, but behaves nicely when the wavenumber increases. The method is developed in its full generality and is illustrated by three particular cases: the elastodynamic system, the convected Helmholtz equation and the acoustic Helmholtz equation in homogeneous and heterogeneous media. Numerical experiments are provided, which confirm that the stability conditions and error estimates are sharp.

Computation of uniform mean flow acoustic scattering by single layer regularized meshless method

The simulation of the acoustic fields propagating in moving flows and the interaction of such fields with the mean flow itself is of importance in aviation acoustic prediction. Volume methods are preferred in this area. Boundary integral method is also concerned because of its own advantages. The transformation space is usually adopted in solving the convective Helmholtz Equation, physical space is also preferred for the convenient boundary condition implementation and combination with other methods. In this paper, the Convective Helmholtz Equation is solved by the Single layer Regularized Meshless Method (SRMM) in the physical space. The SRMM has been proposed by the author for the Laplace equation and Helmholtz Equation. SRMM has similar idea with the Equivalent Source Method (ESM), which represents the fields by the monopoles. But the source points overlap physical points. It overcomes the difficulty of the source points location choice, but the singularity arises. The subtraction and adding-back technique is adopted to get the amplitude of the singularity. The Dual-Surface Method is adopted to avoid irregular frequencies.

A BEM–FMM approach applied to the combined convected Helmholtz integral formulation for the solution of aeroacoustic problems

Exterior and interior acoustic calculations often exploit the advantages of Boundary Element Method (BEM) numerical scheme. Due to the large dimension of dense matrices arising in real-case applications, direct matrix–vector products become prohibitive, since the computational cost of standard methods increases quadratically. In this work, the black-box Fast Multipole Method (bbFMM), based on the Chebyshev interpolation, is extended to the integral solution of the convective Helmholtz equation stabilized using the Combined Helmholtz Integral Equation Formulation (CHIEF). The CHIEF approach relies on the solution of an over-determined linear system achieved by adding a set of points internal to the scattering body. In order to handle the distribution of the CHIEF points within the body volume, a robust stochastic approach has been also developed. Moreover, an Adaptive bbFMM (AbbFMM) is presented for handling the oscillating kernels, such as those arising in the solution of the convective Helmholtz equation. The proposed approach, which has a complexity of O(NlogN), is interesting for industrial applications, especially in the aeroacoustic field.

On the frequency domain formulation of the generalized sound extrapolation method

A frequency domain formulation of the generalized sound extrapolation method is proposed. The Fourier transform is performed to the flow variables to prepare the source terms as the input of the sound extrapolation solver, and the inhomogeneous convected Helmholtz equation is solved using the Green's functions in the frequency domain. The proposed formulations are applied to typical aeroacoustics problems including two dimensional (2D) and three dimensional monopoles in uniform mean flows, flat plate-gust interaction noise and the sound produced by a co-flowing jet. The formulations can yield close agreements with the predictions by the time domain solution. Substantial time saving can be achieved compared with a time domain solver, as the estimation of emission/retarded time, which needs to conduct interpolation of the time sequence of the flow variables, is not required. Also, the computation in the 2D space is much easier since the Green's function is used directly. The results suggest that the frequency formulation can potentially be used for sound projection computation in aeroacoustics.

A LBIE-RBF solution to the convected wave equation for flow acoustics

In this paper, the meshless local boundary integral equation (LBIE) method is implemented for the solution of convected wave equation in frequency domain. Such a problem plays an important role for the noise prediction in flow acoustics. The case of a uniform flow in x-direction is analyzed. For the resultant convected Helmholtz equation, the free-space Green's function has a more complicated form than the Green's function for the classical Helmholtz equation, though it reduces to the latter when a coordinate (Prandtl–Glauert) transformation is applied. Furthermore, the LBIE method requires the determination of the solution of the corresponding Dirichlet's problem in local subdomains (companion solution). Therefore, the relevant companion solution is derived. Radial basis functions are used for the interpolations of the field variable. Numerical examples are shown for cases up to frequency of 5000 Hz and Mach number of 0.5, and the results are compared with FEM software (COMSOL).

Accuracy of a Low Mach Number Model for Time-Harmonic Acoustics

We study the time-harmonic acoustic radiation in a fluid in flow. To go beyond the convected Helmholtz equation adapted only to potential flows, starting from the Goldstein equations, coupling exac...