Takéo Takahashi

Collisions in Three-Dimensional Fluid Structure Interaction Problems

This paper deals with a system composed of a rigid ball moving into a viscous incompressible fluid over a fixed horizontal plane. The equations of motion for the fluid are the Navier–Stokes equations, and the equations for the motion of the rigid ball are obtained by applying Newtons laws. We show that for any weak solution of the corresponding system satisfying the energy inequality, the rigid ball never touches the plane. This result is the extension of that obtained in [M. Hillairet, Comm. Partial Differential Equations, 32 (2007), pp. 1345–1371] in the two-dimensional setting.

Solving Inverse Source Problems Using Observability. Applications to the Euler-Bernoulli Plate Equation

The aim of this paper is to provide a general framework for solving a class of inverse source problems by using exact observability of infinite dimensional systems. More precisely, we show that if a system is exactly observable, then a source term in this system can be identified by knowing its intensity and appropriate observations which often correspond to measurements of some boundary traces. This abstract theory is then applied to obtain new identifiability results for a system governed by the Euler-Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation.

A control theoretic approach to the swimming of microscopic organisms

In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that, there exists a large class of finite dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body in any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers. Our aim is to give a control theoretic interpretation of the swimming mechanism of micro-organisms (like ciliata) which is one of the fascinating problems in fluid mechanics.

Convergence of the Lagrange--Galerkin Method for the Equations Modelling the Motion of a Fluid-Rigid System

In this paper, we consider a Lagrange--Galerkin scheme to approximate a two-dimensional fluid-rigid body problem. The equations of the system are the Navier--Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme.

Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid

In this paper we study a three-dimensional fluid-structure interaction problem. The motion of the fluid is modeled by the Navier-Stokes equations and we consider for the elastic structure a finite dimensional approximation of the equation of linear elasticity. The time variation of the fluid domain is not known a priori, so we deal with a free boundary value problem. Our main result yields the local in time existence and uniqueness of strong solutions for this system.

Liquid jet generation and break-up

We consider the dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapour phase. As the basic mathematical model we use the Euler equations for a sharp interface approach and local and global versions of the NavierStokes-Korteweg equations for the diffuse interface approach. The mathematical models are discussed and we introduce discretization methods for both approaches. Finally numerical simulations in one and two space dimensions are presented.This work is motivated by the numerical simulation of the generation and break-up of droplets after the impact of a rigid body on a tank filled with a compressible fluid. This paper splits into two very different parts. The first part deals with the modeling and the numerical resolution of a spray of liquid droplets in a compressible medium like air. Phenomena taken into account are the breakup effects due to the velocity and pressure waves in the compressible ambient fluid. The second part is concerned with the transport of a rigid body in a compressible liquid, involving reciprocal effects between the two components. A new one-dimensional algorithm working on a fixed Eulerian mesh is proposed. The GENJET (GENeration and breakup of liquid JETs) project has been proposed by the Centre d’Études de Gramat (CEG) of the Délégation Générale de l’Armement (DGA). It concerns the general study of the consequences of a violent impact of a rigid body against a reservoir of fluid. Experiments show that once the solid has pierced the shell of the reservoir, it provokes a dramatic increase of the pressure inside the reservoir, whose effect is the ejection of some fluid through the pierced hole. The generated liquid jet then expands into the ambient air, where it can interact with some air pressure waves, leading to a fragmentation of the jet into small droplets. These experiments show that after having pierced the shell, the projectile behaves as a rigid body. They also show that the liquid inside the reservoir behaves as a compressible fluid (indeed, the projectile velocity, around 1000 m.s, is in the same order of magnitude than the sound speed in the liquid). The modeling of such a complex flow requires to take into account very different regimes, from the pure compressible and/or incompressible flow condition to a droplet regime (such a regime sharing some similarities with kinetic modeling of Liquid jet generation and breakup 3 particles). Moreover many scales are needed to correctly describe the complete experiments, from the large hydrodynamic scale to the small droplet scale. The study done during CEMRACS 2004 focused on the fluid regime and on the droplet regime, since some important difficulties are still there for both regimes separately. • Concerning the breakup of droplets in the air, we have focused on physical and numerical modeling issues. • Concerning the fluid regime, an important difficulty at the numerical level is that we want to get an accurate numerical description of the transport of a rigid body inside a compressible fluid. Even if the rigid body is of course not a fluid, the situation shares at the numerical level a lot of similarities with the coupling an incompressible fluid with a compressible one. Thus this part of the study concerns more numerical algorithms than the modeling. The present paper follows this cutout of the study. Section 1 presents the modeling of the breakup of droplets, whereas section 2 treats the coupling of the rigid body and the fluid. In both sections, numerical results are reported. In view of the main goal of the GENJET project, a natural perspective of the work described below would be the coupling of the models, algorithms and numerical methods. 1. A kinetic modeling of a breaking up spray with high Weber numbers In this section, we aim to model a spray of droplets which evolve in an ambient fluid (typically the air). That kind of problem was first studied by Williams for combustion issues [32]. The works of O’Rourke [20] helped to set the modeling of such situations and their numerical simulation through an industrial code, KIVA [1]. The main phenomenon that occuring in the spray is the breakup of the droplets. Any other phenomena, such as collisions or coalescence, will be neglected in this work, but they are reviewed in [3] for example. Instead of using the TAB model (see [2]), which is more accurate for droplets with low Weber numbers, we choose the so-called Reitz wave model [27], [21], [4]. Then this breakup model is taken into account in a kinetic model [14], [2]. The question of the spray behavior with respect to the breaking up has arised in the context of the French military industry. One aims to model with an accurate precision the evolution of a spray of liquid droplets inside the air. In that situation, the droplets of the spray are assumed to remain incompressible (the mass density ρd is a constant of the problem) and spherical. We also assume that the forces on the spray are negligible with respect to the drag force, at least at the beginning of the computations. After a few seconds, the gravitation may become preponderant. Note that the aspects of energy transfer will not be tackled in this report.In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N +2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction.In this paper, we introduce a new PIC method based on an adaptive multi-resolution scheme for solving the one dimensional Vlasov–Poisson equation. Our approach is based on a description of the solution by particles of unit weight and on a reconstruction of the density at each time step of the numerical scheme by an adaptive wavelet technique: the density is firstly estimated in a proper wavelet basis as a distribution function from the current empirical data and then “de-noised” by a thresholding procedure. The so-called Landau damping problem is considered for validating our method. The numerical results agree with those obtained by the classical PIC scheme, suggesting that this multi-resolution procedure could be extended with success to plasma dynamics in higher dimensions.

Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L∞ data is also given.

Small solids in an inviscid fluid

We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.

On the motion of a rigid body with a cavity filled with a viscous liquid

We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier-Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Katos scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.

On the identifiability of a rigid body moving in a stationary viscous fluid

This paper is devoted to a geometrical inverse problem associated with a fluid–structure system. More precisely, we consider the interaction between a moving rigid body and a viscous and incompressible fluid. Assuming a low Reynolds regime, the inertial forces can be neglected and, therefore, the fluid motion is modelled by the Stokes system. We first prove the well posedness of the corresponding system. Then we show an identifiability result: with one measure of the Cauchy forces of the fluid on one given part of the boundary and at some positive time, the shape of a convex body and its initial position are identified.