Nicolas de Cacqueray

A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations

A shock-capturing methodology is developed for non-linear computations using low-dissipation schemes and centered finite differences. It consists in applying an adaptative second-order filtering to handle discontinuities in combination with a background selective filtering to remove grid-to-grid oscillations. The shock-capturing filtering is written in its conservative form, and its magnitude is determined dynamically from the flow solutions. A shock-detection procedure based on a Jameson-like shock sensor is derived so as to apply the shock-capturing filtering only around shocks. A second-order filter with reduced errors in the Fourier space with respect to the standard second-order filter is also designed. Linear and non-linear 1D and 2D problems are solved to show that the methodology is capable of capturing shocks without providing dissipation outside shocks. The shock detection allows in particular to distinguish shocks from linear waves, and from vortices when it is performed from dilatation rather than from pressure. Finally the methodology is simple to implement and reasonable in terms of computational cost.

Finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations

In this paper, the errors generated by the computation of derivatives in the azimuthal direction ? when flow equations are solved in cylindrical coordinates using finite differences are investigated. They might be large for coarse discretizations even using high-order schemes, which led us to design explicit finite differences specially for 8, 16, 32 and 64 points per circle. These schemes are shown to improve accuracy with respect to standard finite differences, and to provide solutions for a two-dimensional propagation problem similar to those obtained using Fourier spectral methods in the direction ?. A method is then presented to alleviate the time-step limitation resulting from explicit time integration near cylindrical origin, when finite differences are used. It consists in calculating azimuthal derivatives at coarser resolutions than permitted by the grid, in the same way as usually done using spectral methods. In practice, a series of doublings of the effective discretization in ? is implemented. Thus simulations can for instance be performed on a grid containing n?=256 points with a time step 32 times larger, with an accuracy comparable to that achieved in corresponding simulations involving Fourier spectral methods in the direction ?.

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

a n = relative contribution of azimuthal modes n c = local sound speed c1 = ambient field sound speed f = frequency fc = cutoff frequency fshock = central frequency of the broadband shockassociated noise fup = frequency of the upstream-propagating shockassociated noise LVS = first shock length from vortex sheet, 2 M j 1 rj= 1, 1 2:40483 L1 = first shock length measured on the jet axis Ma = acoustic Mach number Mc = convection Mach number Me = exit Mach number Mj = equivalent fully expanded exit Mach number, uj=c1 n = azimuthal mode pe = exit static pressure pj = equivalent fully expanded exit static pressure p0 = fluctuating static pressure hpi = mean static pressure Re = Reynolds number based on exit conditions re = nozzle radius rj = equivalent fully expanded nozzle radius Stc = cutoff Strouhal number Ste = Strouhal number based on exit conditions Stup = Strouhal number of the upstream-propagating shock-associated noise Te = exit temperature uaxis = centerline mean axial velocity uc = convection velocity ue = exit velocity uj = equivalent fully expanded exit velocity huzi = mean axial velocity hu02 z i = mean-square axial velocity fluctuations hu02 z in = mean-square axial velocity fluctuations due to mode n juz j Ste; n = two-dimensional power spectral densities z1 = 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

Noise of an overexpanded Mach 3.3 jet: non-linear propagation effects and correlations with flow

The noise emitted by an overexpanded round jet at a Mach number of 3.3 and a Reynolds number of 105, computed in a previous study using large-eddy simulation (LES), is investigated. In a first step, the non-linear sound propagation effects are quantified by performing two far-field wave extrapolations from the LES near-field data. The extrapolations are carried out by solving the linearized Euler equations in one case and the full Euler equations in the other, without atmospheric absorption, up to a distance of 240 radii from the jet nozzle exit. The non-linear effects are shown to be quite significant, resulting in a series of N-shaped waves in the pressure signals, and in weaker mid-frequency components and stronger high-frequency components in the spectra. Close to the peak directivity radiation angle, for instance, they lead to about a 8 dB loss and a 6 dB gain at the Strouhal numbers of 0.2 and 1, respectively. In a second step, noise generation mechanisms are discussed by calculating correlations between far-field pressure fluctuations and turbulent quantities in the jet. High levels of correlation are found with the centerline flow fluctuations at the end of the potential core, with the shear-layer flow fluctuations over a large axial distance, and with the centerline density fluctuations between the 3rd and the 5th shock cells. They are attributed to the intermittent intrusion of low-speed vortical structures in the potential core, to the supersonic convection of turbulent structures, and to the shock motions at the screech tone frequency.

A computational study of shear stress in smooth rectangular channels

The assumption of uniform flow permits the computation of the bed mean shear stress in open channels from the knowledge of the bed slope, hydraulic radius and fluid density alone. It is however important to distinguish between bank and bed shear values, in particular for sediment transport applications and research on this issue has been ongoing since the 1930s. This line of work was revived recently by Yang and Lim (1997) and Guo and Julien (2005) who have proposed division methods and formulations for the calculations of the shear stress values. This paper examines their work and the formulation of Guo and Julien in particular, using detailed Computational Fluid Dynamics simulation results. Numerical results for the various components of shear are presented and the type of division lines computed here compared with the existing propositions, in particular those of Leighly (1932), Keulegan (1938) and the recent findings ofYang and Lim, and Guo and Julien.

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.

Direct noise computation of a shocked and heated jet at a Mach number of 3.30

An overexpanded axisymmetric jet at an exit Mach number of 3.30 and a Reynolds number of 10 5 is computed by compressible large-eddy simulation (LES) to determine directly its radiated sound field, using low-dissipation schemes in combination with an adaptative shock-capturing method. At the jet nozzle exit, static pressure and temperature are respectively equal to 0.5×10 5 Pa and 360 K, and a laminar flow profile is imposed. To assess the validity of the simulation, the mean aerodynamic field and the near-field pressure levels, obtained directly by LES, are compared to available literature data. The axial velocity fluctuations in the shear-layer are also characterized using azimuthal decomposition, and exhibit properties close to those observed in screeching jets. The acoustic field is dominated by axisymmetric and first azimuthal modes, and noise sources are investigated from the acoustic near field: Mach waves, shock-associated noise and turbulent mixing noise occuring around the end of the potential core are identified.

Self-adjusting shock-capturing spatial filtering for high-order non-linear computations

A shock-capturing methodology is developed for non-linear computations using lowdissipation schemes and centered finite differences. It consists in applying an adaptative second-order filtering to handle discontinuities in combination with a background selective filtering to remove grid-to-grid oscillations. The shock-capturing filtering is written in its conservative form, and its magnitude is determined dynamically from the flow solutions. A shock-detection procedure based on a Jameson-like shock sensor is derived. A secondorder filter with reduced errors in the Fourier space with respect to the standard secondorder filter is also designed. Linear and non-linear problems are solved to show that the methodology is capable of capturing shocks without providing dissipation outside shocks. The shock detection allows in particular to distinguish shocks from linear waves, and from vortices when it is performed from dilatation rather than from pressure.

Investigation of high supersonic jet noise: non-linear propagation effects and flow-acoustics correlations

In this paper, the data obtained by LES for an initially laminar and overexpanded jet at Mach number 3.3 and Reynolds number 10 are reexamined in order to investigate the nonlinear effects on the propagation of the acoustic waves, and the normalized flow/acoustics cross-correlations. To study the non-linear propagation effects at the direction φ = 60◦ and up to 240 radii from the nozzle exit, the LES near field is propagated in far-field by solving either the isentropic linearized Euler equations or the full Euler equations. The comparisons of the acoustic data obtained from the two methods clearly show that the non-linear effects are strong up to about 240 radii from the nozzle exit. Using the far-field acoustic results from the non-linear propagation, the normalized cross-correlations between the turbulent flow quantities and the acoustic pressure signals at the direction φ = 60◦ are then evaluated to give some information on sound generation. A sound source which may be similar to that one observed in subsonic jets is first found on the jet axis in the vicinity of the end of the potential core. Other sound sources attributed to the supersonic convection of turbulent vortices are noticed. Finally, the normalized cross-correlations between the fluctuating density along the jet axis and the acoustic pressure display correlation bands between the 3rd and the 5th shock cells. These bands might be linked with the screech generation mechanism.

A Dynamic Spatial Filtering Procedure for Shock Capturing in High-Order Computations

A shock-capturing method is developed for high-order non-linear computations. It consists in applying an adaptative second-order conservative filtering to handle discontinuities, in combination with a background selective filtering to remove grid-to-grid oscillations. The magnitude of the shock-capturing filtering is determined dynamically from the flow solutions using a procedure based on a Jameson-like shock detector. Results obtained for a shock-propagation problem are shown to assess the validity of the method.