Analysis and Simulation of Fluid Dynamics

Table of Contents

1. Frontmatter

2. Some Recent Asymptotic Results in Fluid Mechanics

   Thomas Alazard

   Abstract

   The general equations of fluid mechanics are the law of mass conservation, the Navier-Stokes equation, the law of energy conservation and the laws of thermodynamics. These equations are merely written in this generality. Instead, one often prefers simplified forms. To obtain reduced systems, the easiest route is to introduce dimensionless numbers which quantify the importance of various physical processes. Many recent works are devoted to the study of the classical solutions when such a dimensionless number goes to zero. A few results in this field are here reviewed.

3. Recent Mathematical Results and Open Problems about Shallow Water Equations

   Didier Bresch, Benoît Desjardins, Guy Métivier

   Abstract

   The purpose of this work is to present recent mathematical results about the shallow water model. We will also mention related open problems of high mathematical interest.

4. Direct Numerical Simulation and Analysis of 2D Turbulent Flows

   Charles-Henri Bruneau

   Abstract

   Efficient methods are used to approximate incompressible Navier-Stokes equations. 2D turbulent flows are simulated in the cavity and behind arrays of cylinders in a channel. They confirm on one hand the presence of an attractor and on the other hand the coexistence of both direct enstrophy and inverse energy cascades. The use of a threshold directly on the vorticity intensity or on the wavelets packets coefficients separate the flow into two parts, each part corresponding to one cascade.

5. Numerical Capture of Shock Solutions of Nonconservative Hyperbolic Systems via Kinetic Functions

   Christophe Chalons, Frédéric Coquel

   Abstract

   This paper reviews recent contributions to the numerical approximation of solutions of nonconservative hyperbolic systems with singular viscous perturbations. Various PDE models for complex compressible materials enter the proposed framework. Due to lack of a conservative form in the limit systems, associated weak solutions are known to heavily depend on the underlying viscous regularization. This small scales sensitiveness drives the classical approximate Riemann solvers to grossly fail in the capture of shock solutions. Here, small scales sensitiveness is encoded thanks to the notion of kinetic functions so as to consider a set of generalized jump conditions. To enforce for validity these jump conditions at the discrete level, we describe a systematic and effective correction procedure. Numerical experiments assess the relevance of the proposed method.

6. Domain Decomposition Algorithms for the Compressible Euler Equations

   V. Dolean, F. Nataf

   Abstract

   In this work we present an overview of some classical and new domain decomposition methods for the resolution of the Euler equations. The classical Schwarz methods are formulated and analyzed in the framework of first order hyperbolic systems and the differences with respect to the scalar problems are presented. This kind of algorithms behave quite well for bigger Mach numbers but we can further improve their performances in the case of lower Mach numbers. There are two possible ways to achieve this goal. The first one implies the use of the optimized interface conditions depending on a few parameters that generalize the classical ones. The second is inspired from the Robin-Robin preconditioner for the convection-diffusion equation by using the equivalence via the Smith factorization with a third order scalar equation.

7. The Two-Jacobian Scheme for Systems of Conservation Laws

   Rosa Donat, Pep Mulet

8. Do Navier-Stokes Equations Enable to Predict Contact Between Immersed Solid Particles?

   M. Hillairet

   Abstract

   We present here a short overview of recent results on a paradox appearing in the area of fluid-solid interactions. This paradox states that, in two space dimensions, strong solutions to viscous models describing fluid-solid interactions do not permit rigid solids inside the fluid to collide.

9. The Reduced Basis Element Method for Fluid Flows

   Alf Emil Løvgren, Yvon Maday, Einar M. Rønquist

   Abstract

   The reduced basis element approximation is a discretization method for solving partial differential equations that has inherited features from the domain decomposition method and the reduced basis approximation paradigm in a similar way as the spectral element method has inherited features from domain decomposition methods and spectral approximations. We present here a review of the method directed to the application of fluid flow simulations in hierarchical geometries. We present the rational and the basics of the method together with details on the implementation. We illustrate also the rapid convergence with numerical results.

10. Asymptotic Stability of Steady-states for Saint-Venant Equations with Real Viscosity

    Corrado Mascia, Frederic Rousset

11. Numerical Simulations of the Inviscid Primitive Equations in a Limited Domain

    A. Rousseau, R. Temam, J. Tribbia

    Abstract

    This work is dedicated to the numerical computations of the primitive equations (PEs) of the ocean without viscosity with the nonlocal (mode by mode) boundary conditions introduced in [RTT05b]. We consider the 2D nonlinear PEs, and firstly compute the solutions in a “large” rectangular domain Ω₀ with periodic boundary conditions in the horizontal direction. Then we consider a subdomain Ω₁, in which we compute a second numerical solution with transparent boundary conditions. Two objectives are achieved. On the one hand the absence of blow-up in these computations indicates that the PEs without viscosity are well posed when supplemented with the boundary conditions introduced in [RTT05b]. On the other hand they show a very good coincidence on the subdomain Ω₁ of the two solutions, thus showing also the computational relevance of these new boundary conditions. We end this study with some numerical simulations of the linearized primitive equations, which correspond to the theoretical results established in [RTT05b], and evidence the transparent properties of the boundary conditions.

12. Some Recent Results about the Sixth Problem of Hilbert

    Laure Saint-Raymond

    Abstract

    The sixth problem proposed by Hilbert, in the occasion of the International Congress of Mathematicians held in Paris in 1900, asks for a global understanding of the gas dynamics. For a perfect gas, the kinetic equation of Boltzmann provides a suitable model of evolution for the statistical distribution of particles. Hydrodynamic models are obtained as first approximations when collisions are frequent. In incompressible regime, rigorous convergence results are now established by describing precisely the corrections to the hydrodynamic approximation, namely physical phenomena such as relaxation or oscillations on small spatio-temporal scales, and checking that they do not disturb the mean motion.

13. On Compressible and Incompressible Vortex Sheets

    Paolo Secchi

    Abstract

    We introduce the main known results of the theory of incompressible and compressible vortex sheets. Moreover, we present recent results obtained by the author with J.F. Coulombel about compressible vortex sheets in two space dimensions, under a supersonic condition that precludes violent instabilities. The problem is a nonlinear free boundary hyperbolic problem with two difficulties: the free boundary is characteristic and the Lopatinski condition holds only in a weak sense, yielding losses of derivatives. In [18, 20] we prove the existence of such piecewise smooth solutions to the Euler equations close enough to stationary vortex sheets. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme.

14. Existence and Stability of Compressible and Incompressible Current-Vortex Sheets

    Yuri Trakhinin

    Abstract

    Recent author’s results in the investigation of current-vortex sheets (MHD tangential discontinuities) are surveyed. A sufficient condition for the neutral stability of planar compressible current-vortex sheets is first found for a general case of the unperturbed flow. In astrophysical applications, this condition can be considered as the sufficient condition for the stability of the heliopause, which is modelled by an ideal compressible current-vortex sheet and caused by the interaction of the supersonic solar wind plasma with the local interstellar medium (in some sense, the heliopause is the boundary of the solar system). The linear variable coefficients problem for nonplanar compressible current-vortex sheets is studied as well. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces. The a priori estimate deduced for this problem is a necessary step to prove the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of ideal compressible MHD. Analogous results are obtained for incompressible current-vortex sheets. In the incompressibility limit the sufficient stability condition found for compressible current-vortex sheets describes exactly the half of the whole parameter domain of linear stability of planar discontinuities in ideal incompressible MHD.