Julien Berland

High-order, low dispersive and low dissipative explicit schemes for multiple-scale and boundary problems

Explicit high-order numerical schemes are proposed for the accurate computation of multiple-scale problems and for the implementation of boundary conditions. Specific high-order node-centered finite differences and selective filters removing grid-to-grid oscillations are first designed for the discretization of the buffer region between a Δx-grid domain and 2Δx-grid domain. The coefficients of these matching schemes are chosen so that the maximum order of accuracy is reached. Non-centered finite differences and selective filters are then developed with the aim of accurately computing boundary conditions. They are constructed by minimizing the dispersion and the dissipation errors in the wave number space for waves down to four points per wavelength. The dispersion and dissipation properties of the matching and the boundary schemes are described in detail, and their accuracy limits are determined, to show that these schemes calculate accurately waves with at least five points per wavelength. Test problems, including linear convection, wall reflection and acoustic scattering around a cylinder, are finally solved to illustrate the accuracy of the schemes.

Numerical study of screech generation in a planar supersonic jet

The generation of screech tones in an underexpanded jet is investigated by means of compressible large eddy simulation (LES). A three-dimensional planar geometry is considered with the aim of studying screech radiation in a simple jet configuration, whose physics nevertheless remains similar to that of large-aspect-ratio rectangular jets encountered in experimental surveys. The jet operates at fully expanded Mach number Mj=1.55, with Reynolds number Reh=6×104. The LES strategy is based on explicit selective filtering with spectral-like resolution, and low-dispersion and low-dissipation numerical algorithms are implemented to allow the direct noise computation of the phenomenon. The numerical results are first set against experimental data to establish the consistency of the simulation. It is shown that the flow development and the shock-cell structure are in agreement with experiments of the literature. Furthermore, the upstream acoustic field exhibit harmonic tones that compare correctly to screech tones...

Large Eddy Simulation of Screech Tone Generation in a Planar Underexpanded Jet

The screech tones generated by a three-dimensional planar underexpanded jet are computed directly using compressible large eddy simulation (LES). The jet operates at fully expanded Mach number Mj = 1.55, with Reynolds number Reh = 6 × 10 . The LES strategy is based on explicit selective filtering with spectral-like resolution, and low dispersion and low dissipation numerical algorithms are implemented to allow direct noise computation of the phenomenon. The investigation of the numerical results shows that the flow development, the shock cell structure and the upstream acoustic field are well reproduced by the computation. Flow visualization of shock/vortex interactions within the third shock-cell provides evidences that screech sound sources can be interpreted using the shock-leakage theory.

Optimized explicit schemes: matching and boundary schemes and 4th-order Runge-Kutta algorithm.

Some numerical tools for local features of the computational domain are built up by minimizing their dispersion and dissipation properties in the Fourier space for low wavenumbers. First, matching schemes are designed. They allow acoustic waves to travel between a uniform ¢x-mesh and a 2¢x-mesh. Secondly, non-centered finite difference schemes and selective filters for wall boundary conditions are computed. Time integration is also investigated. An optimized fourth-order six-step Runge-Kutta scheme which remains fourth-order with nonlinear operators is developed. Problems of linear and nonlinear propagation and of wall reflection are finally resolved to illustrate accuracy of the numerical methods designed here.

Direct Computation of Turbulent Boundary Layer Noise

Aerodynamic noise from a turbulent boundary layer, also known as flow noise is a fundamental topic in flow-induced noise insofar as it determines the structural response of a surface submitted to this turbulent buffeting. For example, in the automotive applications, sound waves transmitted inside the interior of the vehicle come from two contributions: the direct transfer of the acoustic radiation, and the indirect transfer by the incompressible pressure loading which can excite the vibrational modes of the structure. The first one has a low efficiency but a transfer function close to one, whereas the second is more energetic but the transfer function becomes very weak at high wavenumbers. For the flow around a A-pillar, Alam et al. have shown that the mechanism associated with structural responses to turbulent wall flow is preponderant, but no study has quantified precisely the contributions of the direct and indirect transfers, and the question whether the acoustic part should be considered remains an open question.

A study of differentiation errors in large-eddy simulations based on the EDQNM theory

This paper is concerned with the investigation of numerical errors in large-eddy simulations by means of two-point turbulence modeling. Based on the eddy-damped quasi-normal Markovian (EDQNM) theory, a stochastic model is developed in order to predict the time evolution of the kinetic energy spectrum obtained by a large-eddy simulation (LES), including the effects of the numerics. Using this framework, the influence of the accuracy of the approximate space differencing schemes on LES quality is studied, for decaying homogeneous isotropic incompressible turbulence, with Reynolds numbers Rek based on the transverse Taylor scale equal to 780, 2500 and 8000. The results show that the discretization of the filtered Navier–Stokes equations leads to differentiation and aliasing errors. Error spectra are also presented, and indicate that the numerical errors are mainly originating from the approximate differentiation. In addition, increasing the order of accuracy of the differencing schemes or using algorithms optimized in the Fourier space is found to widen the range of well-resolved scales. Unfortunately, for all the schemes, the smaller scales with wavenumbers close to the grid cut-off wavenumber, are badly calculated and generate differentiation errors over the whole energy spectrum. The eventual use of explicit filtering to remove spurious motions with short wavelength is finally shown to significantly improve LES accuracy.

Numerical Simulation of Supersonic Jet Noise

Jets with complex shock-cell structures appear in numerous technological applications. Most supersonic jets used in aeronautics will be imperfectly expanded in flight, even those from carefully designed convergent-divergent nozzles. The adaption to the ambient pressure takes place in a sequence of oblique shocks which interact with the free shear layers and produce noise. The shock/shear-layer interaction emanates a broadband noise component. This may trigger the young shear layer at the nozzle, forming a feedback loop which results in a discrete noise component called screech . Both components are undesirable from structural and environmental (cabin noise) points of view. Screech tones are known to produce sound pressure levels of 160 dB and beyond.

Investigation using statistical closure theory of the influence of the filter shape on scale separation in large-eddy simulation

The influence of the filter shape on scale separation in large-eddy simulation (LES) is investigated using the eddy-damped quasi-normal Markovian (EDQNM) modelling approach, for discrete filters of order 2–12. The LES subgrid-scale (SGS) stress tensor is split into a represented SGS tensor based on the scales resolved by the mesh, and a non-represented SGS tensor involving the unresolved scales. The investigation of the kinetic energy spectra associated with these quantities shows that the features of the SGS tensor strongly depend on the filter shape. In particular, for the second-order filter, the SGS stress tensor dynamics is dominated by the interactions between the resolved scales. The effective LES cut-off wavenumber is finally evaluated and recast in term of efficiency rates in order to estimate the computational cost required to achieve a given spectral resolution. Sharp cut-off filters appear to be well appropriate to perform an efficient scale separation in LES.

Analysis of Numerical Error Reduction in Explicitly Filtered LES Using Two-Point Turbulence Closure

Numerical errors in large-eddy simulation (LES) are investigated using the eddy-damped quasi-normal Markovian (EDQNM) modeling approach, for finite differences of order 2 to 14, and for optimized differentiation schemes. An EDQNM-LES model is derived to evaluate numerical errors, namely the aliasing and the differentiation errors. The results show that the aliasing errors are negligible whereas the interactions between wavenumbers close to the mesh cut-off wavenumber are responsible for a major part of the differentiation errors. In addition, the accuracy of a LES calculation is seen to be improved when explicit filtering of the higher part of the turbulence spectrum is introduced.

Forces Exerted on a Cylinder in Near-Axial Flow

This study investigates the flow around a cylinder in a near-axial flow at a Reynolds number of 27,000. Both computational fluid dynamics (CFD) calculations and experiments are performed. Time-mean values of lift force coefficient are investigated against the inclination of the cylinder in the domain of low inclinations (<15 deg). A pressure distribution and flow profiles are also measured and extracted from the CFD calculation results for a characteristic inclination α = 5 deg. Numerical results for force and pressure show fair agreement with experiments for inclination below 5 deg and reveal that at low angles, the lift force is proportional to the angle. In the framework of a quasi-static approach, the instantaneous damping force exerted on a cylinder oscillating in axial flow is equivalent to the normal force exerted on a cylinder placed in an oblique flow.