Mathematical and numerical analysis of an alternative well-posed two-layer turbulence model

In this article, we wish to investigate the behavior of a two-layer k - ε turbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations. First, we explain the difficulties inherent in the model. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspects of the model are preserved by the new formulation, and in particular, we show how the physicists can help us to prove the existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulence model and we show that the alternative model may be preferred to the original one, because of its good properties (existence of a solution of the coupled problems)

Non-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Computations

A time-domain harmonic balance method for the analysis of almost-periodic (multi-harmonics) flows is presented. This method relies on Fourier analysis to derive an efficient alternative to classical time marching schemes for such flows. It has recently received significant attention, especially in the turbomachinery field where the flow spectrum is essentially a combination of the blade passing frequencies. Up to now, harmonic balance methods have used a uniform time sampling of the period of interest, but in the case of several frequencies, non-necessarily multiple of each other, harmonic balance methods can face stability issues due to a bad condition number of the Fourier operator. Two algorithms are derived to find a non-uniform time sampling in order to minimize this condition number. Their behavior is studied on a wide range of frequencies, and a model problem of a 1D flow with pulsating outlet pressure, which enables to prove their efficiency. Finally, the flow in a multi-stage axial compressor is analyzed with different frequency sets. It demonstrates the stability and robustness of the present non-uniform harmonic balance method regardless of the frequency set

Recursive regularization step for high-order lattice Boltzmann methods

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of non-equilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second (and higher) order non-hydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from $10^4$ to $10^6$, and where a thorough analysis of the case at $Re=3\times 10^4$ is conducted. In the latter, results obtained using both regularization steps are compared against the BGK-LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.Comment: Accepted for publication as a Regular Article in Physical Review

Shock-cell noise of supersonic underexpanded jets

International audienceShock-cell noise is a particular noise that appears in imperfectly expanded jets. Under these expansion conditions a series of expansions and compressions appear following a shock-cell type structure. The interaction between the vortices developed at the lip of the nozzle and the shock-cells generates what is known as shock-cell noise. This noise has the particularity to be propagated upstream with a higher intensity. This publication will focus on the shock-cell noise generated by an axisymmetric under-expanded 10 to the power of 6 Reynolds single jet. The LES computations are carried out using the elsA code developed by ONERA and extended by CERFACS with high-order compact schemes. They are validated against experimental results. The LES simulation is initialized with a RANS solution where the nozzle exit conditions are imposed. Even though no inflow forcing is applied, good agreement is obtained in terms of flow structures and broadband shock-cell noise that is propagated to the farfield by means of the Ffowcs-Williams & Hawkings analogy

Hydrodynamic - acoustic filtering of a supersonic-underexpanded jet

The main noise that is perceived inside the cabin of an airplane originates in the turbofan engines and their associated exhaust jets. Due to flight conditions, a pressure difference appears at the exit of the secondary flow which engenders a series of expansion and compression waves known as shock-cells

Wall-Laws for High Speed Flows over Adiabatic and Isothermal Walls

Projet M3NWe present the extension of our wall-laws developed for low-speed flows to super and hypersonic configurations. In particular, we are interested in flows over isothermal walls and account for heat transfer. We recall the main steps of the development: - Obtention of generalized wall functions for low-speed fluids, valid for all $y^+$, - Taking into account transversal effects. - Accounting for the compressible feature of the flow on adiabatic walls without using informations on the local boundary layer structure but only those available at the fictitious wall. - Extension to isothermal walls. A posteriori evaluation of the heat flux at the real wall using informations at the fictitious one. - Only use informations available on unstructured meshes and avoid those coming from a cartesian hypothesis for the mesh in near-wall regions. These ingredients are validated on hypersonic configurations on adiabatic and isothermal walls for expansion and compression ramps as well as for reentry geometries

A coupled implicit-explicit time integration method for compressible unsteady flows

This paper addresses how two time integration schemes, the Heun's scheme for explicit time integration and the second-order Crank-Nicolson scheme for implicit time integration, can be coupled spatially. This coupling is the prerequisite to perform a coupled Large Eddy Simulation / Reynolds Averaged Navier-Stokes computation in an industrial context, using the implicit time procedure for the boundary layer (RANS) and the explicit time integration procedure in the LES region. The coupling procedure is designed in order to switch from explicit to implicit time integrations as fast as possible, while maintaining stability. After introducing the different schemes, the paper presents the initial coupling procedure adapted from a published reference and shows that it can amplify some numerical waves. An alternative procedure, studied in a coupled time/space framework, is shown to be stable and with spectral properties in agreement with the requirements of industrial applications. The coupling technique is validated with standard test cases, ranging from one-dimensional to three-dimensional flows

Revisiting the spectral analysis for high-order spectral discontinuous methods

The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we wish to revisit such analysis, focusing attention on two key points. The first point concerns the effects of time integration on the spectral analysis. It is shown with standard high-order Finite Difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength. The situation depends on the choice of the coupled schemes (one for time integration, one for space derivative and one for the filter) and here, the compact scheme with its eighth-order filter seems to have a better spectral accuracy than the considered dispersion-relation preserving scheme with its associated filter, especially in terms of dissipation. Secondly, such a coupled space–time approach is applied to the new class of high-order spectral discontinuous approaches, focusing especially on the Spectral Difference method. A new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in [0,π][0,π], following the Matrix Power method. For wavenumbers above π, an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing depends on the polynomial degree and on the number of time steps. A new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than π. Since the new criteria recover the previous results for wavenumbers below π, the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors. At last, since the standard Finite Difference schemes can serve as reference solution for their capability in aeroacoustics, it is shown that the Spectral Difference method is as accurate as (or even more accurate) than the considered Finite Difference schemes

Modal structure of a supersonic under-expanded jet

Le bruit de choc est un bruit particulier qui intervient lorsque qu'un jet n'est pas parfaitement détendu. On observe alors des cellules de choc en aval de la tuyère, composées d'onde de compression et de détente. Les interactions entre la turbulence et ces cellules de choc sont responsable de la génération du bruit de choc. Ce bruit se caractérise par une directivité marqué vers l'amont de l'écoulement ainsi qu'une forte intensité. Dans cette étude, nous nous intéressons à l'analyse modale de la structure d'un jet supersonique sous-détendu caractérisé par un nombre de Reynolds Re=UjDj/vj = 10(6), calculé par simulation aux grandes échelles (SGE) à l'aide du code elsA développée par l'ONERA avec l'intégration de schémas d'ordre élevé du CERFACS