[MUSIC] What is sloshing? It is the oscillating motion of a fluid that you can see in swimming pools, in a cup of coffee, or in France, in a glass of French wine, of course, and in the tanks of liquid natural gas ships or almost everywhere. In all these motions, there might be some coupling with a solid around it for two reasons. First, this sloshing motion may be caused by the moving solid. This is the case when you move your cup of coffee or your glass of wine. That may cause coffee or wine to spill. Second, the mass of moving fluid may be large, and that may result in a large feedback force on the solid. This is a well-know effect in tank trucks for instance The motion of the fluid inside te truck may cause a truck to overturn. How can we model this? First, let us try to model sloshing itself. To understand sloshing, let us consider a fluid domain with a free surface. By free surface, I mean that the fluid is in contact with a gas at a constant pressure, say, the atmospheric pressure. The other boundaries are fixed. I'm going to consider small motions and to neglect the effect of viscosity, which means that the Stokes number is large. Here are the equations satisfied in the fluid domain. They are satisfied by small u and small p. So, although the solid is fixed for the moment, I imagine that there might be motions in the fluid. Now, I have to be a bit more specific about what happens at the free surface. There, as I just have stated, the fluid is in contact with air at atmospheric pressure, but the free surface deforms. I should write that the contact is made at the point X + Ksi where Ksi is the displacement of the fluid particle. Let us write this total pressure at the point X + Ksi . As I did before, I expand total P and get P naught as the zero order term. The first order term reads small p (X) + grad of P naught Ksi . And because P total is a constant equal to P atmospheric, the first other term must be zero. So I had a condition that p plus grad of P naught Ksi = 0. Now, because P naught is the hydrostatic pressure, grad of P naught is vertical. All this reduces to condition between the local pressure p, and the local vertical displacement of the interface. To summarize, the free surface conditions reads that any vertical motion of a surface is opposed by an increasing pressure. It is just like a spring. So, having gravity and a free surface is equivalent to a mattress of springs on the free surface opposing motion. The effect of gravity, which is active everywhere in the fluid domain, actually materializes only at the free surface. Now, what is the relation to sloshing? Well, at the free surface, we have stiffness and we have inertia everywhere. So there is certainly going to be some oscillatory dynamics. Let us see that. [MUSIC] We are not going to explore the general models of sloshing, because what we are looking for is, eventually, the coupling between sloshing and the solid dynamics. Here is a very simple fluid domain in 2D. It is rectangular and the free surface is there. It is like a swimming pool if you want. Here are the equations for the fluid domain. On the walls and at the free surface. I can eliminate the velocity and get a set of equations on pressure only. This is a Laplace problem, but with a rather special condition at the free surface. Here are the equations with the boundary conditions. What is the first mode of free oscillation of this system? Well, I'll look for P in the form of e to the i omega t times a function of space, Phi p. Because of the simple geometry I can look for Phi p as a function of x times the function of z. Because delta p equals 0, the function of x , F and the function of z, G, both satisfy a very simple differential equation. Because F prime must be equal to 0 on both sides, F is going to be a cosine. And then G is going to be a hyperbolic cosine. By combining these functions, F and G, I get the modal shape of pressure Phi p. Here are the isovalues of Phi p. This is not very easy to understand, so let us look at the associated velocity field. Phi u is obtained out of Phi p by taking its gradient. Here is what the velocity modal shape Phi u looks like. Now, you see clearly the sloshing motion of the fluid. So, if I consider a free oscillating motion along this mode, here is what it looks like. On the free surface, the deformation is a cosine, as you would expect. Finally, using the condition of a free surface, we have the frequency of sloshing omega. It depends on the Froude number, as you would expect, because this is where you have the effect of gravity. [MUSIC] To summarize, I have computed the modal shapes in pressure and velocity of the first sloshing mode. I also have its frequency. This is just like any vibration mode, except it is for the fluid domain. I would like now to use this mode for a single mode approximation of the dynamics of the fluid. For this, I would like to write the modal oscillator equation, and I need the fluid modal mass and fluid modal stiffness. How do I get them? Well, this is similar to what is generally done in modal dynamics, except that the system is fluid. To get the modal mass and the stiffness of this mode, here is the direct way. I take the oscillating form of p. I then insert it in the free surface condition considered not locally but in a projected form, where I take the sum over whole free surface times Phi p. I get a minus omega square times a constant plus another constant equals 0. This defines my fluid sloshing modal mass, Mf, the coefficient of Omega square, and modal stiffness, Kf. [MUSIC] We have everything. Modal shapes, frequency, modal mass, and modal stiffness. Now, we can use this mode into a single mode approximation of the solution dynamics of our fluid. This means writing u equals Q dot Phi u. And p equals capital Q double dot phi p. And the variable capital Q that governs the dynamics of my fluid satisfies the oscillator equation of the mode. Mf Q double dot + Kf Q = 0 for free motions. And of course, if the sloshing motion is forced, then I will have a projected forced F on the right-hand side. With all that you can have resonance like for any oscillator. This is what happens here on the pool of a cruise ship. It is a forced oscillator response following essentially a single mode. What we did for simple rectangular domain can be done on any shape. You just have to solve a different Laplace problem to get Phi p, and then Phi u, and the frequency, and the oscillator equation. As you can see, a single mode approximation and the associated oscillator equation is a simple way to do some fluid sloshing dynamics. We have used it before for the solid only, but it can be done for the fluid too. Now that we know how to enter the sloshing of the fluid domain, let us couple that with the dynamics of the solid. [MUSIC]
