On the spectral properties of shock-capturing schemes

Pirozzoli, Sergio

doi:10.1016/j.jcp.2006.07.009

Cited by 241 publications

(194 citation statements)

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“…An appropriate shock-capturing method is also required specifically in supersonic flows to remove Gibbs oscillations near shocks [43,44]. In this case, particular attention must be paid to the possible spurious dissipation of the turbulent structures and the acoustic waves by the shock-capturing procedure [45].…”

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Investigation of a High-Mach-Number Overexpanded Jet Using Large-Eddy Simulation

2011

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An initially laminar overexpanded round jet at an exit Mach number of 3.30 and a Reynolds number of 10 5 is calculated by compressible large-eddy simulation. The near field obtained by large-eddy simulation is also propagated to the acoustic far field by solving the full Euler equations to take into account the nonlinear propagation effects. Both computations are performed using low-dissipation finite differences in combination with an adaptive shock-capturing method. The jet originates from a straight pipe nozzle of radius r e , including lips of thickness 0:05r e . At the pipe exit, Blasius mean flow profiles are imposed, and static pressure and temperature are equal to 0:5 10 5 Pa and 360 K, resulting in fully adapted and acoustic Mach numbers, respectively, of 2.83 and 3.47. The jet flowfields, as well as the acoustic near and far fields, are described in detail and compared with data of the literature. The turbulent mechanisms developing in the jet are investigated, using spectral and azimuthal decompositions of the velocity fluctuations along the shear layers. Same analyses are applied to the acoustic fields, in order to discuss the properties of jet noise components. In this way, Mach waves, shock-associated noise, screech tones, and turbulent mixing noise are identified. = first shock location on the jet axis = specific heat ratio r = mesh size in the radial direction z = mesh size in the axial direction = boundary-layer thickness in the pipe nozzle 0:5 = jet half-width e = exit molecular viscosity e = exit density = radiation angle measured according to the flow direction

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Investigation of a High-Mach-Number Overexpanded Jet Using Large-Eddy Simulation

2011

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AbstractFollowing the study of Pirozzoli [1], the objective of the present work is to provide a detailed theoretical analysis of the spectral properties and the conservation properties of nonlinear finite difference discretizations. First, a Nonlinear Spectral Analysis (NSA) is proposed in order to study the statistical behaviour of the modified wavenumber of a nonlinear finite difference operator, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase.

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On the spectral and conservation properties of nonlinear discretization operators

Fauconnier

Dick

2011

Journal of Computational Physics

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Following the study of Pirozzoli [1], the objective of the present work is to provide a detailed theoretical analysis of the spectral properties and the conservation properties of nonlinear finite difference discretizations. First, a Nonlinear Spectral Analysis (NSA) is proposed in order to study the statistical behaviour of the modified wavenumber of a nonlinear finite difference operator, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. Second, the necessary conditions for local and global conservation of momentum and kinetic energy are derived and verified for nonlinear discretizations. Because the nonlinear mechanisms result in a violation of the energy conservation conditions, the NSA is used to quantify the energy imbalance. Third, the effect of aliasing errors due to the nonlinearity is analyzed. Finally, the theoretical observations are verified for two simple, thought relevant, numerical simulations.

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“…For a nonlinear scheme, no analytical formula of the spectral relations can be obtained. However, using the technique suggested by Pirozzoli [23], the approximate dispersion relation (ADR) of the nonlinear scheme still can be obtained. The two parameters  m and  s can be determined numerically.…”

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Nonlinear spectral-like schemes for hybrid schemes

He¹,

Liang

2014

Sci. China Phys. Mech. Astron.

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In spectral-like resolution-WENO hybrid schemes, if the switch function takes more grid points as discontinuity points, the WENO scheme is often turned on, and the numerical solutions may be too dissipative. Conversely, if the switch function takes less grid points as discontinuity points, the hybrid schemes usually are found to produce oscillatory solutions or just to be unstable. Even if the switch function takes less grid points as discontinuity points, the final hybrid scheme is inclined to be more stable, provided the spectral-like resolution scheme in the hybrid scheme has moderate shock-capturing capability. Following this idea, we propose nonlinear spectral-like schemes named weighted group velocity control (WGVC) schemes. These schemes show not only high-resolution for short waves but also moderate shock capturing capability. Then a new class of hybrid schemes is designed in which the WGVC scheme is used in smooth regions and the WENO scheme is used to capture discontinuities. These hybrid schemes show good resolution for small-scales structures and fine shock-capturing capabilities while the switch function takes less grid points as discontinuity points. The seven-order WGVC-WENO scheme has also been applied successfully to the direct numerical simulation of oblique shock wave-turbulent boundary layer interaction. The multi-scales phenomena in such problems requires the numerical method to be high order and high resolution. To be reliable, a DNS of such flows must resolve these various scales, particularly smaller ones with accuracy in both amplitude and phase [2]. Concerning high-order and highresolution schemes, there have been upwind compact schemes [3,4], dispersion-relation preserving schemes [5] and wavenumber-extended finite difference schemes [6]. These schemes are all designed for high resolution of short waves with respect to the computational grid. Another highresolution schemes are explicit/compact central schemes [7] which have no dissipation resulted in providing spurious solutions and leading to inevitable stability problems [8,9].However, these methods above are limited to compute flows without discontinuities. To capture shocks, treating these methods nonlinearly is required [4]. For flows with discontinuities, WENO [10] schemes which have been demonstrated very promising shock-capturing capabilities and high order accuracy are widely used. However, numerical tests also indicate that classical WENO schemes are usually not optimal for computing turbulent flows or aeroacoustic fields because they are too diffusive for short waves [11].

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On the spectral properties of shock-capturing schemes

Cited by 241 publications

References 15 publications

Investigation of a High-Mach-Number Overexpanded Jet Using Large-Eddy Simulation

Investigation of a High-Mach-Number Overexpanded Jet Using Large-Eddy Simulation

On the spectral and conservation properties of nonlinear discretization operators

Nonlinear spectral-like schemes for hybrid schemes

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