A Study of the Short Wave Components in Computational Acoustics

Abstract

It is shown by using a Dispersion-Relation-Preserving [Formula: see text] finite difference scheme that it is feasible to perform direct numerical simulation of acoustic wave propagation problems. The finite difference equations of the [Formula: see text] scheme have essentially the same Fourier-Laplace transforms and hence dispersion relations as the original linearized Euler equations over a broad range of wavenumbers (here referred to as long waves). Thus it is guaranteed that the acoustic waves, the entropy and the vorticity waves computed by the [Formula: see text] scheme are good approximations of those of the exact solutions of Euler equations as long as the wavenumbers are in the long wave range. Computed waves with higher wavenumber, or the short waves, generally have totally different propagation characteristics. There are no counterparts of such waves in the exact solutions. The short waves of a computation scheme are, therefore, contaminants of the numerical solutions. The characteristics of these short waves are analyzed here by group velocity consideration and standard dispersive wave theory. Numerical results of direct simulations of these waves are reported. These waves can be generated by discontinuous initial conditions. To purge the short waves so as to improve the quality of the numerical solution, it is suggested that artificial selective damping terms be added to the finite difference scheme. It is shown how the coefficients of such damping terms may be chosen so that damping is confined primarily to the high wavenumber range. This is important for then only the short waves are damped leaving the long waves basically unaffected. The effectiveness of the artificial selective damping terms is demonstrated by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.

A Numerical and Experimental Investigation of the Dissipation Mechanisms of Resonant Acoustic Liners

“…E!ectively, the multiple-size-mesh multiple-time-step DRP scheme is used as the computation algorithm. In addition, to eliminate spurious short numerical waves and to promote numerical stability, arti"cial selective damping, see reference [11], is added to the computation algorithm. The same distribution of the arti"cial mesh Reynolds number used in reference [8] is employed in the present work.…”

Time-domain implementation of an impedance boundary condition with boundary layer correction

“…This implies that any effective wavenumber κ∆x is found for two different values of k∆x. The point k∆x = α 0 corresponds to the transition from dissipative to parasitic damping referred to by Tam et al [14]. For the DRP scheme used here [11], α 0 ≈ 0.634π ≈ 1.993, or approximately 3 points per wavelength.…”

“…Requiring low dissipation means that instabilities are often found at under-resolved scales, of the order of half the Nyquist frequency (i.e. four points per wavelength spatially), necessitating selective filtering [13][14][15]. In time domain simulations, this selective filtering often takes the form of a weak low-pass spatial filter applied at every point at every time step.…”

Straightforward high-order numerical dissipation via the viscous term for direct and large eddy simulation

“…The most popular method to control spurious oscillations is to use upwind schemes to compute the convective terms in order to reinforce artificially the dissipation near the mesh cutoff wavenumber. A similar effect can be obtained using a specific artificial damping term [16,33] or a filtering procedure [9,5,4]. Upwinding, damping or filtering techniques are essentially non-conservative methods that introduce more or less explicitly some numerical dissipation.…”