Broadband Time-domain Impedance Boundary Research Articles

Dynamic large-eddy simulation of hypersonic transition delay over broadband wall impedance

Three-dimensional numerical simulations of hypersonic boundary layer transition delay due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) were carried out on a 7 $^{\circ {}}$ half-angle cone for unit Reynolds numbers $Re_m=2.43 \times 10^6$ – $6.40\times 10^6\ \text {m}^{-1}$ , at the free-stream Mach number $M_\infty =7.4$ , for both sharp and 2.5 mm nose tip radii. A broadband time-domain impedance boundary condition was used to model the acoustic effects of the porous surface on the flow field. A quasi-spectral sub-filter-scale dynamic closure was adopted to stabilize the computations upon turbulent breakdown under extreme cooling conditions, with wall-to-adiabatic temperature ratio of $T_{w}/ T_{ad} \simeq 0.08 $ , while accurately recovering the growth rates of the unstable modes present in the early transition stages. Good agreement is observed with the reference experimental data, both in terms of the predicted extent of the transition delay and the measured second-mode frequency spectrum. The latter is strongly modulated by the formation of near-wall low-temperature three-dimensional streaks. Pressure disturbances concentrate in corridors of locally thickened boundary layer, with frequencies lower than what predicted by linear theory. Here, trapped wavetrains are formed, which can persist long into the turbulent region. Finally, it is shown that the presence of a porous wall simply shifts the onset of turbulence downstream, without affecting its structure.

Linear stability analysis of second-mode attenuation via porous carbon-matrix ceramics

Effects of porous carbon-fiber-reinforced carbon-matrix ceramics (C/C) on the stability of second-mode waves on a 7°-half-angle cone were investigated for Reynolds numbers Rem=2.43×106–6.40×106 m−1 at the freestream Mach number of M∞=7.4, for both sharp and 2.5-mm-round nose tips. A broadband time-domain impedance boundary condition was used to model the effects of the C/C porosity on the flow dynamics leveraging direct ultrasonic benchtop experiments and homogenous absorber theory. A spectral linear stability solver based on orthogonal Laguerre functions, naturally vanishing in the free stream, was used to predict linear spatial growth rates, which are in agreement with independent pulsed axisymmetric direct-numerical simulations. The latter were carried out with the quasi-spectral viscosity closure—a dynamic quasi-spectral procedure capable of deactivating the sub-filter scale stresses in the absence of turbulent break down—verifying its suitability to carry out transitional calculations without affecting ultrasonic wave dynamics. The effectiveness of a porous C/C surface is shown to decrease drastically with static pressure and its presence is shown to decrease the second-mode growth rates in regions where it is unstable as well as increasing the attenuation rates in regions where it is stable.

Time-domain impedance boundary condition modeling with the discontinuous Galerkin method for room acoustics simulations.

The time-domain nodal discontinuous Galerkin (TD-DG) method is emerging as a potential wave-based method for three-dimensional (3D) room acoustics modeling, where high-order accuracy in the low frequency range, geometrical flexibility, and accurate modeling of boundary conditions are of critical importance. This paper presents a formulation of broadband time-domain impedance boundary conditions (TDIBCs) of locally-reacting surfaces in the framework of the TD-DG method. The formulation is based on the approximation of the plane-wave reflection coefficient at normal incidence in the frequency domain using a sum of template rational functions, which can be directly transformed to the time-domain. The coupling of the TDIBCs with the discontinuous Galerkin discretization is achieved through the characteristic waves of the upwind flux along the boundary, where a series of first-order auxiliary differential equations is time-integrated in a high-order way. To verify the performance of the formulation, various numerical tests of single reflection scenarios are shown to demonstrate the cost efficiency and memory-efficiency of high-order basis functions, among which a 3D application to an impedance boundary of rigidly backed glass-wool baffle for room acoustic purposes is presented.

High-fidelity simulation of a standing-wave thermoacoustic–piezoelectric engine

We have carried out wall-resolved unstructured fully compressible Navier–Stokes simulations of a complete standing-wave thermoacoustic–piezoelectric engine model inspired by the experimental work of Smoker et al. (J. Appl. Phys., vol. 111 (10), 2012, 104901). The model is axisymmetric and comprises a 51 cm long resonator divided into two sections: a small-diameter section enclosing a thermoacoustic stack and a larger-diameter section capped by a piezoelectric diaphragm tuned to the thermoacoustically amplified mode (388 Hz). The diaphragm is modelled with multi-oscillator broadband time-domain impedance boundary conditions (TDIBCs), providing higher fidelity over single-oscillator approximations. Simulations are first carried out to the limit cycle without energy extraction. The observed growth rates are shown to be grid convergent and are verified against a numerical dynamical model based on Rott’s theory. The latter is based on a staggered grid approach and allows jump conditions in the derivatives of pressure and velocity in sections of abrupt area change and the inclusion of linearized minor losses. The stack geometry maximizing the growth rate is also found. At the limit cycle, thermoacoustic heat leakage and frequency shifts are observed, consistent with experiments. Upon activation of the piezoelectric diaphragm, steady acoustic energy extraction and a reduced pressure amplitude limit cycle are obtained. A heuristic closure of the limit cycle acoustic energy budget is presented, supported by the linear dynamical model and the nonlinear simulations. The developed high-fidelity simulation framework provides accurate predictions of thermal-to-acoustic and acoustic-to-mechanical energy conversion (via TDIBCs), enabling a new paradigm for the design and optimization of advanced thermoacoustic engines.

Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models.

A methodology to design broadband time-domain impedance boundary conditions (TDIBCs) from the analysis of acoustical models is presented. The derived TDIBCs are recast exclusively as first-order differential equations, well-suited for high-order numerical simulations. Broadband approximations are yielded from an elementary linear least squares optimization that is, for most models, independent of the absorbing material geometry. This methodology relies on a mathematical technique referred to as the oscillatory-diffusive (or poles and cuts) representation, and is applied to a wide range of acoustical models, drawn from duct acoustics and outdoor sound propagation, which covers perforates, semi-infinite ground layers, as well as cavities filled with a porous medium. It is shown that each of these impedance models leads to a different TDIBC. Comparison with existing numerical models, such as multi-pole or extended Helmholtz resonator, provides insights into their suitability. Additionally, the broadly-applicable fractional polynomial impedance models are analyzed using fractional calculus.

Progress and prospective of computational aeroacoustics

The origin and research category of Computational Aeroacoustics (CAA) are introduced firstly. The early progress and applications are summarized, focusing on the two key elements of CAA, high accuracy spatial and time discretization schemes and non-reflecting boundary conditions. Furthermore, the research advances of CAA in the past five years are highlighted, including nonlinear non-reflecting boundary condition, non-uniform time integration method, space discretization method applicable for complex geometry and broadband time domain impedance boundary condition. Finally, the future prospective of CAA is presented briefly.

Improved Multipole Broadband Time-Domain Impedance Boundary Condition

ACOUSTIC liners are widely used in industry for noise reduction. They are indispensable noise-suppression devices for jet engines. Acoustic liners are effective over a wide band of frequencies. To account for such characteristics, broadband impedance models are used for time-domain computation. The acoustic impedance of a surface, Z ! , is originally defined in the frequency domain. Problems involving noise propagation over impedance surfaces may be solved in the time domain. Time-domain methods have obvious advantages over frequency-domain methods for broadband and nonlinear problems. However, because an impedance boundary condition is set up for use in the frequency domain, an equivalent time-domain impedance boundary condition is needed. Mathematically, the time-domain equivalent may be derived by taking its inverse Fourier transform. For the equivalent time-domain impedance condition to be physical, the impedance function must be causal, real, and passive [1]. In general, the inverse Fourier transform of a frequency-domain impedance boundary condition would lead to a convolution integral. Computation of the convolution integral is time consuming. However, if the impedance can be represented by a sum of certain special functions, the evaluation of the convolution may be carried out in a simple systematic manner. Tam and Auriault [2] introduced a three-parameter time-domain impedancemodel. Thismodel can provide accurate representation of liner impedance for single discrete frequency problems. But the three-parameter model cannot offer a general representation of liner impedance over awide band of frequencies. Rienstra [1] proposed an extended Helmholtz resonator model. The free parameters of the model are the geometrical dimensions of the resonator. The impedance of the model is derived by the response of the Helmholtz resonator to periodic acoustic input. A comparison study by Richter et al. [3] indicated that a numerical instability could occur in some cases. A filtering technique has to be applied for avoiding this problem. Recently, Reymen et al. [4] developed a new and efficient time-domain impedance boundary condition based on the use of recursive convolution. The key to their method is the representation of the impedance by a series of simple poles in the complex plane. For the method to work efficiently, the value of the function in the convolution integral within a time step must be approximated by a linear or quadratic function. This impedance model has been proven to be physical, causal, and real. The primary objective of this paper is to develop an improved multipole broadband time-domain impedance boundary condition. A secondary objective is to derive an analytical solution for problems involving damping of broadband acoustic input by liners with broadband impedance characteristics. These solutions may serve for benchmarking numerical computations involving broadband sound. This paper is organized as follows. Section II discusses the improved multipole broadband impedance model. The implementation of the time-domain impedance boundary condition is presented in Sec. III. Section IV presents analytical solutions of twodimensional problems with broadband incident acoustic waves. These solutions are then used to validate numerical results with a liner having broadband impedance properties. Section IV concludes this work.

Validation of a Zonal Approach Computing the Sound Radiation from Lined Ducts

A zonal approach for the prediction of the far-field radiation from flows in lined ducts is investigated. The approach combines a high-order computational aeroacoustics scheme with the recently developed acoustic intensitybased method for the calculation of far-field radiations. The advantage over the application of an acoustic analogy is the usage of an open control surface for the acoustic input. The capability of the current hybrid approach was validated on the basis of two benchmark cases. One considers the sound radiation from a two-dimensional semiinfinite duct in which a small fraction of the walls close to outlet nozzle is lined. The second problem concerns the sound radiation from a bypass enginelike annular duct with lined and hardwalled configurations of the infinite centerbody and sheared flow conditions. A broadband time-domain impedance boundary condition based on the extended Helmholtz resonator model and the Ingard–Myers boundary condition is implemented in the computational aeroacoustic code to model the acoustic linings. Although mesh refinement may improve the solutions, the good agreements of analytical solutions and numerical results, in particular for the radiation characteristics, verify this approach as an accurate and efficient prediction tool.

Numerical Investigation of Broadband Slat Noise Attenuation with Acoustic Liner Treatment

The acoustic field produced by broadband noise sources around the slat of a high-lift wing model is predicted using a hybrid method based on a large eddy simulation of the compressible flow for the noise generation and acoustic perturbation equation calculations for the acoustic propagation. The potential for noise attenuation using acoustic liners within the slat gap region of the high-lift wing is investigated by solving the acoustic perturbation equations with a broadband time-domain impedance boundary condition, and the acoustic liner conditions are applied on the slat cove surface and the main element. Results show that attenuation of the broadband noise is obtained with the acoustic liner treatment in the slat gap region.

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.