Richard Pasquetti

Entropy viscosity method for nonlinear conservation laws

A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. This new approach does not use any flux or slope limiters, applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation. Various benchmark problems are solved with finite elements, spectral elements and Fourier series to illustrate the capability of the proposed method.

High-order large-eddy simulation of flow over the “Ahmed body” car model

The structure of the turbulent flow over a simplified automotive model, the Ahmed body (S. R. Ahmed and G. Ramm, SAE Paper No. 8403001, 1984) with a 25° slanted back face, is investigated using high-order large-eddy simulations (LESs) at Reynolds number Re=768000. The numerical approach is carried out with a multidomain spectral Chebyshev–Fourier solver and the bluff body is modeled with a pseudopenalization method. The LES capability is implemented thanks to a spectral vanishing viscosity (SVV) technique, with particular attention to the near wall region. Such a SVV-LES approach is extended for the first time to an industrial three-dimensional turbulent flow over a complex geometry. In order to better understand the interactions between flow separations and the dynamic behavior of the released vortex wake, a detailed analysis of the flow structures is provided. The topology of the flow is well captured showing a partial separation of the boundary layer over the slanted face and the occurrence of two stro...

BOUNDARY ELEMENT APPROACH FOR INVERSE HEAT CONDUCTION PROBLEMS: APPLICATION TO A BIDIMENSIONAL TRANSIENT NUMERICAL EXPERIMENT

A boundary element approach is used to solve inverse heat conduction problems in multidimensional and nonlinear situations. In stationary and transient cases, discretized boundary integral equations are expressed and, to obtain satisfying results, are associated with regularization procedures over space and time (transient case). The efficiency of the method, which depends on the accuracy of the measurements, is explored through an example of transient bidimensional application. A comparison with the superposition method (linear problems) is presented.

Spectral Vanishing Viscosity Method for Large-Eddy Simulation of Turbulent Flows

An efficient spectral vanishing viscosity method for the large-eddy simulation of incompressible flows is proposed, both for standard spectral and spectral element approximations. The approach is integrated in a collocation spectral Chebyshev-Fourier solver and then used to compute the turbulent wake of a cylinder in a crossflow confined geometry (Reynolds number Re=3900)

Spectral Element Methods on Unstructured Meshes: Comparisons and Recent Advances

Spectral element approximations for triangles are not yet as mature as for quadrilaterals. Here we compare different algorithms and show that using an integration rule based on Gauss-points for simplices is of interest. We point out that this can be handled efficiently and allows to recover the convergence rate theoretically expected, even with curved elements.

Spectral vanishing viscosity method for LES: sensitivity to the SVV control parameters

Where spectral methods are concerned, the spectral vanishing viscosity (SVV) method offers an interesting way of computing high Reynolds number flows since it allows stabilization of the calculations whilst preserving the exponential rate of convergence of the spectral approximation. Here we first show how to implement the SVV method in an existing Navier–Stokes solver and then investigate the sensitivity of the numerical results to its main characteristic parameters, namely the SVV amplitude and the SVV activation mode, by focusing on the computation of a turbulent wake in a cylinder, embedded in a channel-like domain, at Reynolds number Re = 3900.

High-Order Algorithms for Large-Eddy Simulation of Incompressible Flows

Abstract“Defiltering-Transport-Filtering” (DTF) algorithms are proposed for the large eddy simulation of incompressible flows by using high order methods. These new algorithms are based (i) on an approximate deconvolution method for the modeling of the sub-grid scale stress tensor and (ii) on a semi-Lagrangian method to handle the convective term. Such algorithms are implemented in 3D spectral solvers (one homogeneous direction), using differential operators to handle in an approximate way the filtering and defiltering operations. Stability and dissipation properties of the schema are discussed. Preliminary results, obtained with a Chebyshev collocation solver, for the 3D wake of a cylinder with Reynolds number equal to 1000 are presented.

Spectral element methods on unstructured meshes: which interpolation points?

In the field of spectral element approximations, the interpolation points can be chosen on the basis of different criteria, going from the minimization of the Lebesgue constant to the simplicity of the point generation procedure. In the present paper, we summarize some recent nodal distributions for a high order interpolation in the triangle. We then adopt these points as approximation points for the numerical solution of an elliptic partial differential equation on an unstructured simplicial mesh. The L2-norm of the approximation error is then analyzed for a model problem.

Entropy Viscosity Method for High-Order Approximations of Conservation Laws

A stabilization technique for conservation laws is presented. It introduces in the governing equations a nonlinear dissipation function of the residual of the associated entropy equation and bounded from above by a first order viscous term. Different two-dimensional test cases are simulated – a 2D Burgers problem, the “KPP rotating wave” and the Euler system – using high order methods: spectral elements or Fourier expansions. Details on the tuning of the parameters controlling the entropy viscosity are given.

From Suitable Weak Solutions to Entropy Viscosity

This paper focuses on the notion of suitable weak solutions for the three-dimensional incompressible Navier-Stokes equations and discusses the relevance of this notion to Computational Fluid Dynamics. The purpose of the paper is twofold (i) to recall basic mathematical properties of the three-dimensional incompressible Navier-Stokes equations and to show how they might relate to LES (ii) to introduce an entropy viscosity technique based on the notion of suitable weak solution and to illustrate numerically this concept.