Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

Glass, Olivier; Sueur, Franck; Takahashi, Takéo

doi:10.24033/asens.2159

Cited by 26 publications

(44 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Similarly, in the fluid, the velocity U = (u, 0, w) T has no transverse component, and the horizontal discharge takes the form Q = (q, 0). The water waves equations (19) in (ζ, Q) variables therefore simplify into a system of two scalar equations on (ζ, q), in which the operators R(h, Q) and a NH (h, Q) defined in (14)- (15) are therefore denoted R(h, q) and a NH (h, q) for the sake of clarity. • For the interior domain.…”

Section: Coupling With the Solid Dynamicsmentioning

confidence: 99%

“…In order to compute the elementary potentials that appear in the expression for the added mass, one needs to solve a d-dimensional elliptic problem in the (bounded) interior region I(t). This has to be compared with the (d + 1)dimensional elliptic equation one has to solve in the (unbounded) fluid region Ω(t) in order to compute the Kirchoff potential that appear classically in the expression for the added mass (see for inctance [19,18]). this is a classical computation in solid mechanics that we reproduce for the sake of completeness.…”

Section: Coupling With the Solid Dynamicsmentioning

confidence: 99%

“…This is because a rigid body has to accelerate not only itself but also the fluid around it. Exploiting this effect is necessary for a sharp mathematical analysis of the equations [19,18] and plays also a crucial role for the stability of numerical simulations in many fluid-structure interaction problems [9]. This added-mass effect can be quite complex however, since it depends strongly on the location of the solid with respect to the boundaries of the fluid domain [18]; in the case a floating body considered here, the analysis is complicated by the fact that the boundary of the fluid domain is a free surface, which moreover intersects the surface of the body.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On the Dynamics of Floating Structures

Lannes

2017

Ann. PDE

View full text Add to dashboard Cite

Abstract. This paper addresses the floating body problem which consists in studying the interaction of surface water waves with a floating body. We propose a new formulation of the water waves problem that can easily be generalized in order to take into account the presence of a floating body. The resulting equations have a compressible-incompressible structure in which the interior pressure exerted by the fluid on the floating body is a Lagrange multiplier that can be determined through the resolution of a d-dimensional elliptic equation, where d is the horizontal dimension. In the case where the object is freely floating, we decompose the hydrodynamic force and torque exerted by the fluid on the solid in order to exhibit an added mass effect; in the one dimensional case d = 1, the computations can be carried out explicitly. We also show that this approach in which the interior pressure appears as a Lagrange multiplier can be implemented on reduced asymptotic models such as the nonlinear shallow water equations and the Boussinesq equations; we also show that it can be transposed to the discrete version of these reduced models and propose simple numerical schemes in the one dimensional case. We finally present several numerical computations based on these numerical schemes; in order to validate these computations we exhibit explicit solutions in some particular configurations such as the return to equilibrium problem in which an object is dropped from a non-equilibrium position in a fluid which is initially at rest.

show abstract

Section: Coupling With the Solid Dynamicsmentioning

confidence: 99%

Section: Coupling With the Solid Dynamicsmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

On the Dynamics of Floating Structures

Lannes

2017

Ann. PDE

View full text Add to dashboard Cite

show abstract

“…Remark 2 Let us observe that this class is slightly larger than the log-convex class used in Ref. [30], in which the weights satisfy the following equivalent properties:…”

Section: Remark 1 Using the Leibniz Differentiation Rules Log-superlmentioning

confidence: 98%

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

Besse¹,

Frisch²

2017

Commun. Math. Phys.

View full text Add to dashboard Cite

The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries.For a good understanding, it is crucial to carry out, besides mathematical studies, highaccuracy and well-resolved numerical exploration. Such studies can be very demanding in computational resources, but recently it has been shown that very substantial gains can be achieved first, by using Cauchy's Lagrangian formulation of the Euler equations and second, by taking advantages of analyticity results of the Lagrangian trajectories for flows whose initial vorticity is Hölder-continuous. The latter has been known for about twenty years (Serfati, J. Math. Pures Appl. 74: 95-104, 1995), but the combination of the two, which makes use of recursion relations among time-Taylor coefficients to obtain constructively the time-Taylor series of the Lagrangian map, has been achieved only recently (Frisch and Zheligovsky, Commun. Math. Phys. 326: 499-505, 2014; Podvigina et al., J. Comput. Phys. 306: 320-342, 2016 and references therein).Here we extend this methodology to incompressible Euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity. Non-constructive regularity results for these cases have already been obtained by Glass et al. (Ann. Scient.Éc. Norm. Sup. 4 e série 45: 1-51, 2012). Using the invariance of the boundary under the Lagrangian flow, we establish novel recursion relations that include contributions from the boundary. This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy-Lagrangian method.

show abstract

“…Let us give a few references about the Cauchy problem concerning this system. In the context of regular solutions (say at least C 1 ) with finite energy, the problem has been considered by Ortega, Rosier and Takahashi [25] in the full plane, by Rosier and Rosier [26] in the full space and by Houot, San Martin and Tucsnak [16] in a bounded domain (see also G., Sueur and Takahashi [14]). …”

Section: Iii-3 Andmentioning

confidence: 99%

Dynamics of a small rigid body in a perfect incompressible fluid

Glass¹

2014

Journées Équations Aux Dérivées Partielles

Self Cite

View full text Add to dashboard Cite

We consider a solid in a perfect incompressible fluid in dimension two. The fluid is driven by the classical Euler equation, and the solid evolves under the influence of the pressure on its surface. We consider the limit of the system as the solid shrinks to a point. We obtain several different models in the limit, according to the asymptotics for the mass and the moment of inertia, and according to the geometrical situation that we consider. Among the models that we get in the limit, we find Marchioro and Pulvirenti's vortexwave system and a variant of this system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force. These results are obtained in collaboration with Christophe Lacave (Paris-Diderot), Alexandre Munnier (Nancy) and Franck Sueur (Bordeaux).

show abstract

Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

Cited by 26 publications

References 14 publications

On the Dynamics of Floating Structures

On the Dynamics of Floating Structures

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

Dynamics of a small rigid body in a perfect incompressible fluid

Contact Info

Product

Resources

About