Dans le cadre des probl`emes ´evoqu´es dans le chapitre 1, on s’int´eresse en premier lieu au calcul des chargements hydrodynamiques engendr´es par l’impact sur l’eau des structures consid´er´ees. Le calcul fin du comportement du fluide (jet, s´eparation de l’´ecoulement) n’a donc que peu d’int´erˆet dans le sens ou ces ph´enom`enes sont rarement dimensionnants. L’utilisation de m´ethode en th´eorie potentielle semble donc suffisante et pr´esente de plus l’avantage d’engendrer des coˆuts de calcul moindres par rapport aux m´ethodes CFD qui r´esolvent les ´equations d’Euler et prennent en compte plus finement le comportement du fluide. Au sein des m´ethodes en th´eorie potentielle, la m´ethode de Wagner g´en´eralis´ee pr´esente un bon compromis entre la m´ethode de Wagner lin´earis´ee et une m´ethode compl`etement non-lin´eaire de type BEM. Par rapport `a un mod`ele de Wagner lin´earis´e, l’utilisation d’une m´ethode g´en´eralis´ee permet de s’affranchir en particulier de la limitation portant sur l’angle mort, dont les valeurs possibles ne sont plus born´ees. Cet aspect est particuli`erement int´eressant lorsqu’on souhaite ´etudier l’impact de sructures ´elastiques sur l’eau. En effet, les d´eformations peuvent conduire `a ce que l’angle mort d´epasse les valeurs limites impos´ees par la th´eorie de Wagner lin´earis´ee. Par rapport `a une m´ethode de type BEM, la m´ethode de Wagner g´en´eralis´ee permet de pousser plus loin les d´eveloppements analytiques.
Dans cette th`ese, on se propose donc d’´etudier le mod`ele de Wagner g´en´eralis´e, d’abord dans le cadre de l’impact de corps rigides, puis pour des structures d´eformables.
Chapitre 3
A generalized Wagner model, theory and applications
3.1 Introduction
The two-dimensional water impact of an arbitrary rigid section is considered here. Attention is focused on asymmetric sections, as illustrated in figure 3.1. In the Cartesian coordinates system (Oxy), the liquid is initially at rest and occupies the lower half plane y <0.
y= (x,t)η y=f(x)−H(t) V(t)
H(t)
stagnation point
y
x
stream lines
Fig. 3.1 – Configuration of the two-dimensional hydrodynamic impact problem. In the asym-metric case, a stagnation point is observed and does not coincide with the initial contact point The liquid is assumed to be ideal and incompressible. Both external mass forces and surface tension are neglected. The problem is formulated within the potential flow theory. The flow is described with the velocity potential Φ(x, y, t). This potential satisfies Laplace equation in the flow region which varies in time. Position of the entering section is given by the equation y = f(x)−H(t), where the function f(x) describes the section shape and H(t) is the pene-tration depth. V(t) denotes the water entry velocity of the considered section. Here t(0) = 0, f(x) > 0 for x 6= 0, H(0) = 0 and ˙H(t) = V(t), V(t) >0. The function y = η(x, t) describes the free surface elevation during the penetration. The function η(x, t) can be multi-valued in general. At the initial time instant, t = 0, the body touches the liquid free surface y= 0 at a
single point, x = 0, which is taken as the origin of the coordinate system and penetrates the liquid thereafter at the velocity V(t). Position of the free surface is unknown in advance and must be determined as part of the solution. Kinematic and dynamic boundary conditions are prescribed at the unknown position of the free surface. The former implies that the free surface is material. The dynamic boundary condition implies that the pressure at the free surface is atmospheric at t > 0. This is, the air flow is not taken into account. On the surface of the impermeable body, the kinematic condition prescribes the continuity of the normal velocity.
When the body freely penetrates the liquid with a time varying velocity V(t), the free fall problem is closed with Newton law.
Since the pioneering works by [von Karman, 1929] and [Wagner, 1932], several methods of solution have been developed for this problem. A brief review on Wagner models of impact is given below in order to fix ideas.
linearized free surface
c (t)1 c (t)2 c (t)1 c (t)2 c (t)1 c (t)2
η1 η2
1 2
(a) (b) (c)
η η
Fig. 3.2 – Different linearizations of the impact problem from Karman’s approach (a) and linearized Wagner model (b) to GWM (c). Sketch above : actual modelled flow. Sketch below : full or partial linearization of the flow domain. In the first two cases, the wetted surface is projected onto a horizontal plane whereas in the last one, its exact position is taken into account
In Karman’s approach, the wetted surface is simply determined by taking the intersection of the body surface with the free surface at rest (see figure 3.2 (a)). Since the water surface elevation is neglected, the added mass and the impact loads are underestimated by this model.
As a consequence, a significant error up to a factor two in the prediction of the loads is done.
Linearized Wagner model (see figure 3.2 (b)) is based on the so called “flat disk approxima-tion of the wetted surface”, under the assumpapproxima-tion that the deadrise angle is small (usually less than 20 degrees). Wetting correction is introduced accounting for the so called “piled-up effect”. The boundary conditions on both wetted and free surfaces are linearized in both Karman and Wagner models. They reduce to an homogeneous Dirichlet condition φ = 0 on the free surface and a Neumann condition on the wetted surface φ,y = −V(t). The result-ing solutions in terms of velocity and pressure are sresult-ingular at the contact points. Corrections to the original Wagner model were proposed, using matching expansion technique, such as [Cointe, 1991], [Howisonet al., 1991]. Another remarkable modification has been proposed by [Logvinovich, 1969]. He proposed to add extra terms into the velocity potential expression in order to make its gradient,i.e. the velocity of the flow finite at the contact point. The so-called Modified Logvinovich Method (MLM) [Korobkin, 2004] is of particular interest because it
al-lows us to compute hydrodynamic loads with reasonable accuracy for various two-dimensional shapes.
There are many situations where the linearization of the body boundary condition and wetted surface is not valid any longer. In spite of a fully nonlinear original problem, there are still semi-analytical approaches available to solve the problem for simple shapes such as wedge with arbitrary deadrise angle (symmetric [Dobrovol’skaya, 1969], [Fraenkel and Keady, 2004] or not [Chekin, 2004], [Semenov and Iafrati, 2006]). However, these approaches are not suitable to treat arbitrary shapes. In order to circumvent this limitation, [Zhao and Faltinsen, 1992] pro-posed to solve the fully nonlinear problem of body entry through a boundary integral equation solved in the time domain. In [Iafrati, 2000], this method of solution is extended to asymmetric sections. This method implies numerics and may require significant computational resources.
Hence less expensive models, appropriate for initial design stage, must be developed. In this spirit, [Zhao et al., 1996] built a “simplified” model which is known as Generalized Wagner Model. In this model, free surface is approximated with the horizontal lines emanating from the contact points c1 and c2 on both sides of the section (see figure 3.3). The jets that occur during the impact are not taken into account and the dynamic boundary condition is linearized in the same way as in the original Wagner model. The keypoint is that the boundary condition on the wetted surface is not linearized and prescribed on its exact position. Hence there is no theorical limitation on the deadrise angle. The dynamic boundary condition reduces to an homogeneous Dirichlet condition Φ = 0 while the linearized kinematic boundary condition reads
η,t(x, t) =
Φ,y(x, η(c1(t), t), t), x <0,
Φ,y(x, η(c2(t), t), t), x >0. (3.1) On the wetted surface, the impermeability condition reads Φ,n = V ·n, wherenis the normal on the section pointing in the fluid domain (see figure 3.3) andV the velocity of the impacting section.
Fig. 3.3 – Asymmetric section impacting liquid. Left : the fluid domain is bounded by the linearized free surface. Right : conformal mappings are used to turn the fluid domain into the lower half plane
[Mei et al., 1999] proposed a method of solution of the Generalized Wagner Model based on conformal mapping of the flow domain. Note that the flow domain varies in time.
Appli-cations were given for two-dimensional symmetric sections. This method has been used by [Yettouet al., 2007], for variable velocity of body penetration. In this approach, the fully non-linear pressure is computeda posteriori after having obtained the history of both the wetting correction and the entry velocity by using the linear part of the pressure only. An extension to the three-dimensional case has been proposed by [Faltinsen and Chezhian, 2005]. On the basis of the works by [Meiet al., 1999], we generalize here their method to asymmetric sections.
In the present study, we aim at developing robust algorithm to handle arbitrary shapes. To this end, conformal mappings are used to turn the computational domain into the lower half plane as shown in figure 3.3. The free surface is represented by two lines emanating from the contact points on both sides of the impacting body. This linearization makes the problem artificial since the two lines are not at the same level and fluid is artificially added in the problem. The image of the physical body contour is a flat plate in the transformed plane (figure 3.3 right). The way to proceed will be detailed step by step in section 3.2. In section 3.3, the algorithm to solve the resulting problem is presented. Method to compute the flow around the section at a given time instant is developed. To compute the whole time evolution of the system, complementary developments are required such as computation of the wetting corrections, pressure distribu-tion, hydrodynamic force. This is the aim of section 3.4. It is also shown that, in spite of the artificial aspect of the problem, mass conservation law is verified. Results are shown in section 3.5. Comparisons are made with asymmetric and symmetric sections penetrating liquid with constant velocity. Discussion about the problem of the jet is carried out. Then, the problem of free fall impact is considered for both symmetric and asymmetric sections.