Aeroacoustic Problems Research Articles

Grid-optimized upwind DRP finite difference scheme on curvilinear grids for computational aeroacoustics

Based on the curvilinear grids, grid-optimized upwind dispersion-relation-preserving (GOUPDRP) finite difference schemes are studied in the paper. GOUPDRP finite difference schemes can automatically resolve waves with much shorter wavelengths and need not consider the filter or the explicit dissipation terms, and have much smaller dispersive and dissipative errors. Recently, GOUPDRP schemes on the non-uniform Cartesian meshes have been developed and tested by authors of this paper, others optimized upwind finite schemes just only on the uniform Cartesian grids also have been investigated; nevertheless, practical problems existing in aeroacoustics are unusually restricted to the uniform or non-uniform Cartesian grids, the corresponding computational grids could be curvilinear ones owing to the different complex geometries. In contrast to other optimized upwind schemes on the uniform Cartesian grids, GOUPDRP schemes can be easily extended to the curvilinear meshes. In order to determine the optimization coefficients of GOUPDRP schemes on the curvilinear grids, a general curvilinear transformation is introduced from physical domain to computational domain. Mathematic formulations related to GOUPDRP schemes on the curvilinear meshes are also derived in detail in the paper. To demonstrate the effectiveness of GOUPDRP schemes, benchmark problems are solved using the curvilinear grids; calculated solutions are compared with their analytical counterparts. Numerical results show that GOUPDRP schemes on the curvilinear grids are highly efficient and relatively potential to solve practical problems in aeroacoustics.

Highly stable RK time advancing schemes for Computational Aero Acoustics

In this paper a brief survey of finite difference methods and time discretization schemes for the numerical simulation of problems in Computational Aero Acoustics (CAA), with special emphasis in the contributions of the authors in the last years to the subject, is presented. Due to the specific properties of these problems it is shown by means of some illustrative examples that standard schemes have some drawbacks and new numerical schemes have been derived taking into account not only the usual stability and accuracy requirements but also the dissipation and dispersion properties as well as low storage requirements. Some relevant contributions to the subject are presented comparing the relative merits by means of a Fourier analysis and numerical experiments.

CABARET scheme for the numerical solution of aeroacoustics problems: Generalization to linearized one-dimensional Euler equations

A generalization of the CABARET finite difference scheme is proposed for linearized one-dimensional Euler equations based on the characteristic decomposition into local Riemann invariants. The new method is compared with several central finite difference schemes that are widely used in computational aeroacoustics. Numerical results for the propagation of an acoustic wave in a homogeneous field and the refraction of this wave through a contact discontinuity obtained on a strongly nonuniform grid are presented.

Directivity of acoustic emissions from wave packets to the far field

We investigate the acoustic emission from wave packets to the far field. To this end, we develop a theory for one- and two-dimensional source fields in the shape of wave packets with Gaussian envelopes. This theory is based on an approximation to Lighthill's acoustic analogy for distant observers. It is formulated in the spectral domain in which a Gaussian wave packet is represented again by a Gaussian. This allows us to determine the directivity of the acoustic emission (e.g. superdirectivity and Mach waves) by simple geometric constructions in the spectral domain. It is shown that the character of the acoustic emission is mainly governed by the aspect ratio and the Mach number of the wave packet source. To illustrate the relevance of this theory we use it to study two prominent problems in subsonic jet aeroacoustics.

Acoustic intensity-based method for sound radiations in a uniform flow

An acoustic intensity-based method (AIBM) is extended and verified for predicting sound radiation in a subsonic uniform flow. The method assumes that the acoustic propagation is governed by the modified Helmholtz equation on and outside of a control surface, which encloses all the noise sources and nonlinear effects. With acoustic pressure derivative and its co-located acoustic pressure as input from an open control surface, the unique solution of the modified Helmholtz equation is obtained by solving the least squares problem. The AIBM is coupled with near-field Computational Fluid Dynamics (CFD)/Computational Aeroacoustics (CAA) methods to predict sound radiation of model aeroacoustic problems. The effectiveness of this hybrid approach has been demonstrated by examples of both tonal and broadband noise. Since the AIBM method is stable and accurate based on the input acoustic data from an open surface in a radiated field, it is therefore advantageous for the far-field prediction of aerodynamics noise propagation when an acoustic input from a closed control surface, like the Ffowcs Williams-Hawkings surface, is not available [Philos. Trans. R. Soc. London, Ser. A 264, 321-342 (1969)].

Resolution of computational aeroacoustics problems on unstructured grids with a higher-order finite volume scheme

Computational fluid dynamics (CFD) has become increasingly used in the industry for the simulation of flows. Nevertheless, the complex configurations of real engineering problems make the application of very accurate methods that only work on structured grids difficult. From this point of view, the development of higher-order methods for unstructured grids is desirable. The finite volume method can be used with unstructured grids, but unfortunately it is difficult to achieve an order of accuracy higher than two, and the common approach is a simple extension of the one-dimensional case. The increase of the order of accuracy in finite volume methods on general unstructured grids has been limited due to the difficulty in the evaluation of field derivatives. This problem is overcome with the application of the Moving Least Squares (MLS) technique on a finite volume framework. In this work we present the application of this method (FV-MLS) to the solution of aeroacoustic problems.

Non-stationary problem of active sound control in bounded domains

The present paper deals with the non-stationary problem of active shielding of a domain from undesirable external sources of noise. Active shielding is achieved by constructing additional (secondary) sources in such a way that the total contribution of all sources leads to the desirable effect. The problem is formulated as an inverse source problem with the secondary sources positioned outside the domain to be shielded. Along with the undesirable field (noise) to be shielded the presence of a desirable component (“friendly” sound) is accepted in the analysis. The constructed solution of the problem requires only the knowledge of the total field (noise) on the perimeter of the shielded domain. Some important aspects of the problem are addressed in the paper for the first time, such as the non-stationary formulation of the problem, existence of the resonance regimes and sensitivity of the solution to the input errors. The obtained solution is applicable to aeroacoustics problems.

Acoustic intensity based data analysis and assessment in computational aeroacoustics

A methodology for assessing the quality and analysing the sound field of a numerical solution for an aeroacoustic problem is presented using an in-duct propagation problem as example. Namely an analysis of the acoustic energy conservation provides insight about the solution quality; sources of numerical error as well as physical source of sound are located by an analysis of the acoustic intensity; a mode analysis provides information about the sound field in the duct and a visualisation of the average pathway of sound by the acoustic intensity helps the interpretation of the modal amplitude data. The problem chosen for the presentation deals with the propagation of a single acoustic mode upstream through the tapering inlet duct of a centrifugal compressor. The propagation of tone noise is a typical problem for computational aeroacoustics. The numerical approach employs high-order finite-difference discretisation schemes with structured body-fitted meshes. An overset grid approach allows to overcome the geometrical complexity of the problem. The conservation of the acoustic energy is found to be violated by less than 3% in case of a potential base flow. In this case major sources of error are found in the interpolation between overset and background mesh. In case of a non-potential flow with boundary layers at the wall, the conservation of acoustic energy cannot be claimed. Consequently, it shows an increase of the acoustic energy, for which the accelerated flow in the nozzle is identified as source. It is shown that the filtering scheme affect the solution more than the spatial discretisation, if the grid resolution is relatively high. A strong scattering of energy into higher radial modes is detected. The fifth and sixth radial mode are found to be preferred in the current example. They are excited in the nozzle with a similar level. For these high radial modes, the transport of acoustic energy takes place further away from the duct wall. This explains the relatively large drop of about 20 dB for the average sound pressure level at the wall over the inlet nozzle.

Broadband impedance boundary conditions for the simulation of sound propagation in the time domain

An accurate and practical surface impedance boundary condition in the time domain has been developed for application to broadband-frequency simulation in aeroacoustic problems. To show the capability of this method, two kinds of numerical simulations are performed and compared with the analytical/experimental results: one is acoustic wave reflection by a monopole source over an impedance surface and the other is acoustic wave propagation in a duct with a finite impedance wall. Both single-frequency and broadband-frequency simulations are performed within the framework of linearized Euler equations. A high-order dispersion-relation-preserving finite-difference method and a low-dissipation, low-dispersion Runge-Kutta method are used for spatial discretization and time integration, respectively. The results show excellent agreement with the analytical/experimental results at various frequencies. The method accurately predicts both the amplitude and the phase of acoustic pressure and ensures the well-posedness of the broadband time-domain impedance boundary condition.

Modeled Boltzmann Equation and Its Application to Shock-Capturing Simulation

A modified equilibrium distribution function for the Bhatnagar-Gross-Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity-acoustic and entropy-acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier-Stokes equations is required.

Optimized compact filtering schemes for computational aeroacoustics

AbstractSpurious waves are inevitably introduced in the simulation of computational aeroacoustics problems by finite difference schemes. Filters are proposed to damp the spurious waves as soon as they are generated. A family of filtering schemes, both interior and boundary, are optimized in this paper. The transfer functions of the filters are approximated to modified Gaussian functions and sharp spectral cutoffs are obtained. The advanced sequential quadratic programming method is employed to get the approximation. Although all of the optimized filters are stable and perform well in the computation, we find that the optimized fourth‐order filters are sufficient for different kinds of requirements. The improvements of the accuracy and convenient application of the optimized filters are demonstrated when applied to one‐ and two‐dimensional problems. Copyright © 2008 John Wiley & Sons, Ltd.

A Weak Compressible Flow Solution for Fluid and Air-Borne Acoustics Coupled Problems in a Nonlinear System

This paper presents the implementation and validation of a numerical simulation method that takes into account the air-borne acoustic wave propagation within the turbulent flow solution at low Mach numbers. The method is called weak compressibility, and is used in order to cope with fully coupled aero-acoustics problem in a nonlinear system. The weak compressibility was implemented into an explicit incompressible flow solver using an edgebased finite element method. A single 3D cavity case was used to validate the method in comparison with the experiment. The simulated results showed good agreement with the experimental data with regards to the acoustic pressure and frequency for the dominant peak of the sound pressure level (SPL) spectra over a wide speed range from 15 to 55 meters per second.

Modeled Boltzmann Equation and Its Application to Direct Aeroacoustic Simulation

The Bhatnagar, Gross, and Krook modeled Boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success. However, its application to direct aeroacoustic computation is less successful. One possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound. The present study reports on the development of an improved modeled Boltzmann equation for aeroacoustics simulation. The approach is to modify the Maxwellian distribution normally assumed for the equilibrium particle distribution function. Constraints imposed are the exact recovery of the state equation for a diatomic gas and the Euler equations without invoking the small Mach number assumption. Thus formulated, a distribution function consisting of the Maxwellian distribution plus three other terms that attempt to account for particle-particle collisions is obtained. A velocity lattice method is used to solve the improved modeled Boltzmann equation using an equivalent lattice equilibrium distribution function. The simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the Euler equations. The results of the improved modeled Boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain, thus verifying the viability and correctness of the modified equilibrium distribution function.

Grid-optimized upwind dispersion-relation-preserving scheme on non-uniform Cartesian grids for computational aeroacoustics

Grid-optimized upwind dispersion-relation-preserving (GOUPDRP) finite difference schemes on non-uniform Cartesian grids are proposed in this paper. With the high-order accuracy and the robustness, GOUPDRP schemes cannot only minimize the dispersion error and the dissipation error, but also damp out the spurious short waves automatically without the demand of filter or explicit dissipation terms. So far, some optimized upwind schemes have been implemented only on uniform grids. It is well known that most of aeroacoustic problems are seldom confined to uniform grids. Unlike other optimized upwind schemes, GOUPDRP schemes can be easily applied to the non-uniform meshes, which give finite difference equations locally the same dispersion relation as original partial differential equations on the non-uniform grids. Mathematic formulations of the GOUPDRP schemes are derived on the non-uniform meshes in the paper. In order to investigate the effectiveness of GOUPDRP schemes, benchmark problems are solved by this scheme; numerical solutions are compared with their analytical counterparts. Numerical results suggest that GOUPDRP schemes are highly efficient and relatively potential to solve acoustical problems on non-uniform grids.

Multidimensional optimization of finite difference schemes for Computational Aeroacoustics

Because of the long propagation distances, Computational Aeroacoustics schemes must propagate the waves at the correct wave speeds and lower the isotropy error as much as possible. The spatial differencing schemes are most frequently analyzed and optimized for one-dimensional test cases. Therefore, in multidimensional problems such optimized schemes may not have isotropic behavior. In this work, optimized finite difference schemes for multidimensional Computational Aeroacoustics are derived which are designed to have improved isotropy compared to existing schemes. The derivation is performed based on both Taylor series expansion and Fourier analysis. Various explicit centered finite difference schemes and the associated boundary stencils have been derived and analyzed. The isotropy corrector factor, a parameter of the schemes, can be determined by minimizing the integrated error between the phase or group velocities on different spatial directions. The order of accuracy of the optimized schemes is the same as that of the classical schemes, the advantage being in reducing the isotropy error. The present schemes are restricted to equally-spaced Cartesian grids, so the generalized curvilinear transformation method and Cartesian grid methods are good candidates. The optimized schemes are tested by solving various multidimensional problems of Aeroacoustics.

A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes

This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes. In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials. The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered. The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme. Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems.

A Hermite upwind WENO scheme for solving hyperbolic conservation laws

This paper proposes a new WENO procedure to compute problems containing both discontinuities and a large disparity of characteristic scales. In a one-dimensional context, the WENO procedure is defined on a three-points stencil and designed to be sixth-order in regions of smoothness. We define a finite-volume discretization in which we consider the cell averages of the variable and its first derivative as discrete unknowns. The reconstruction of their point-values is then ensured by a unique sixth-order Hermite polynomial. This polynomial is considered as a symmetric and convex combination, by ideal weights, of three fourth-order polynomials: a central polynomial, defined on the three-points stencil, is combined with two polynomials based on the left and the right two-points stencils. The symmetric nature of such an interpolation has an important consequence: the choice of ideal weights has no influence on the properties of the discretization. This advantage enables to formulate the Hermite interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, nonlinear weights are then defined. To deal with the peculiarities of the Hermite interpolation near discontinuities, we define a new procedure in order for the nonlinear weights to smoothly evolve between the ideal weights, in regions of smoothness, and one-sided weights, otherwise. The resulting scheme is a sixth-order WENO method based on central Hermite interpolation and TVD Runge–Kutta time-integration. We call this scheme the HCWENO6 scheme. Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In these experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems.

Implementation of nonreflecting boundary conditions for the nonlinear Euler equations

Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier–Bessel modes. This leads to an ‘exact’ nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier–Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.

Spectrum of gravitational radiation from primordial turbulence

Energy injection into the early universe can induce turbulent motions of the primordial plasma, which in turn act as a source for gravitational radiation. Earlier work computed the amplitude and characteristic frequency of the relic gravitational wave background, as a function of the total energy injected and the stirring scale of the turbulence. This paper computes the frequency spectrum of relic gravitational radiation from a turbulent source of the stationary Kolmogoroff form which acts for a given duration, making no other approximations. We also show that the limit of long source wavelengths, commonly employed in aeroacoustic problems, is an excellent approximation. The gravitational waves from cosmological turbulence around the electroweak energy scale will be detectable by future space-based laser interferometers for a substantial range of turbulence parameters.

Lattice Boltzman Method Simulation of Aeroacoustics and Nonreflecting Boundary Conditions

A lattice Boltzmann method that can recover the first coefficient of viscosity and the specific heat ratio correctly has been adopted for one-step aeroacoustic simulations because it can recover the speed of sound correctly. Instead of solving the Navier-Stokes equations as in the case of direct numerical simulation, the lattice Boltzmann method only needs to solve one transport equation for the collision function. Other flow properties are obtained by integrating this collision function over the particle velocity space. The lattice Boltzmann method is effective only if appropriate nonreflecting boundary conditions for open computational boundaries are available, just like the direct numerical simulation. Four different nonreflecting boundary conditions are commonly used with direct numerical simulation for one-step aeroacoustic simulations. Among these are the characteristics-based method, the perfectly matched layer method, the C 1 continuous method, and the absorbing layer method. Not all nonreflecting boundary conditions are applicable when used with the lattice Boltzmann method; some might not be appropriate, whereas others could be rather effective. This paper examines some existing nonreflecting boundary conditions plus other new proposals, their appropriateness, and their suitability for the lattice Boltzmann method. The assessment is made against two classic aeroacoustic problems: propagation of a plane pressure pulse and propagation of acoustic, entropy, and vortex pulses in a uniform stream. A reference solution is obtained using direct numerical simulation assuming a relatively large computational domain without any specified nonreflecting boundary conditions. The results, obtained with a computational domain half the size of that used for the direct numerical simulation calculations, show that the absorbing layer method and the extrapolation method with assumed filter perform the best.