Jérémie Gressier

Shock wave instability and the carbuncle phenomenon : same intrinsic origin?

The theoretical linear stability of a shock wave moving in an unlimited homogeneous environment has been widely studied during the last fty years. Important results have been obtained by D yakov (1954), Landau & Lifchitz (1959) and then by Swan & Fowles (1975) where the fluctuating quantities are written as normal modes. More recently, numerical studies on upwind nite dierence schemes have shown some instabilities in the case of the motion of an inviscid perfect gas in a rectangular channel. The purpose of this paper is rst to specify a mathematical formulation for the eigenmodes and to exhibit a new mode which was not found by the previous stability analysis of shock waves. Then, this mode is conrmed by numerical simulations which may lead to a new understanding of the so-called carbuncle phenomenon.

Robustness versus accuracy in shock-wave computations

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd – even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk’s analysis of Roe’s scheme, general criteria are derived to predict the odd – even decoupling. This analysis is applied to Roe’s scheme, the Equilibrium Flux Method, the Equilibrium Interface Method and the AUSM scheme. Strict stability is shown to be desirable to avoid most of these flaws. Finally, the link between marginal stability and accuracy on shear waves is established.

ON THE PATHOLOGICAL BEHAVIOR OF UPWIND SCHEMES

Despite constant progress in the developement of upwind schemes, some failings still remain. Quirk recently reported that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of them is the odd-even decoupling that occurs along planar shocks aligned with the mesh. Quirk proposed to test this shortcoming with the propagation of a planar shock in a duct. First, we give a few results on some failings. Then, following Quirks analysis of Roes scheme, general criteria are derived to predict the odd-even decoupling. This analysis is applied to Roes scheme, EFM Pullins scheme, EIM Macrossans scheme and AUSM Lious scheme. Strict stability is shown to be desirable to avoid must of these flaws.

A Cure for the Sonic Point Glitch

Among the various numerical schemes developed since the ’80s for the computation of the compressible Euler equations, the vast majority produce in certain cases spurious pressure glitches at sonic points. This flaw is particularly visible in the computation of transonic expansions and leads lo unphysical “expansion shocks” when the flow undergoes rapid change of direction. The analysis of this flaw is presented, based on a series of numerical experiments. For Flux-Vector Splitting methods, it is suggested that it is not the order of differentiability of the numerical flux which is crucial but the way the pressure at an interface is calculated. A new way of evaluating the pressure at the interface is proposed, based upon kinetic theory, and is applied to most current available algorithms including Flux-Vector Splitting and Flux-Difference Splitting methods as well as recent hybrid schemes (AUSM, HUS).

A High-Order Compact Limiter Based on Spatially Weighted Projections for the Spectral Volume and the Spectral Differences Method

This paper exposes the theoretical developments needed to design a class of spatially weighted polynomial projections used in the definition of a compact limiter dedicated to high-order methods. The spectral volume framework and its integral representation of the solution is used to introduce the degree reduction of the polynomial interpolation. The degree reduction is conducted through a linear projection onto a smaller polynomial space. A particular care is taken regarding the conservativity property and results in a parametric framework where projections can be monitored with spatial weights. These projections are used to define a simple and compact high-order limiting procedure, the SWeP limiter. Then, numerical evaluations are performed using the spectral differences method for the mono-dimensional Euler equations and demonstrate the high-order behavior of the SWeP limiter.

Revisiting Froude’s Theory for Hovering Shrouded Rotor

This paper extends Froude’s momentum theory for free propellers to the analysis of shrouded rotors. A one-dimensional analytical approach is provided, and a homokinetic normal inlet surface model is proposed. Formulations of thrusts and power for each system component are derived, leading to the definition of optimum design criteria and providing insight into the global aerodynamics of shrouded rotors. In the context of micro-air vehicles applications, assessment of the model is conducted with respect to numerical data. Overall, comparison between numerical and analytical results shows good agreement and highlights the sensitivity of the model to viscous effects.

Assessment of the Spectral Volume Method on Inviscid and Viscous Flows

The compact high-order ‘Spectral Volume Method’ (SVM, Wang, J. Comput. Phys. 178(1):210–251) designed for conservation laws on unstructured grids is presented. Its spectral reconstruction is exposed briefly and its applications to the Euler equations are presented through several test cases to assess its accuracy and stability. Comparisons with classical methods such as MUSCL show the superiority of SVM. The SVM method arises as a high-order accurate scheme, geometrically flexible and computationally efficient.

On the Positivity of FVS Schemes

Over the last ten years, robustness of schemes has raised an increasing interest among the CFD community. One mathematical aspect of scheme robustness is the positivity preserving property. At high Mach numbers, solving the conservative Euler equations can lead to negative densities or internal energy. Some schemes such as the flux vector splitting (FVS) schemes are known to avoid this drawback. In this study, a general method is detailed to analyze the positivity of FVS schemes. As an application, three classical FVS schemes (Van Leer’s, Hanel’s variant and Steger and Warming’s) are proved to be positively conservative under a CFL-like condition. Finally, it is proved that for any FVS scheme, there is an intrinsic incompatibility between the desirable property of positivity and the exact resolution of contact discontinuities.