[MUSIC] The computation of added stiffness is easy because it only depends on some integrals over the interface. To compute the added mass we have to go a bit further. The added mass m A was defined as the inertia coefficient, when we obtained the force acting on the solid. It is made of the mass number, M, and of the sum of an interface of quantities that we know. Normal n, mode shape of the solid motion phi, and a quantity that we have to compute, the pressure shape, phi p. What is phi p? It was related to the velocity shape Phi u through phi u equals minus grad of phi p, and phi u was the solution of div of phi u equals zero, and phi u n equals phi n, at the interface. By taking the divergence of this momentum equation, we have div of grad of phi_p equals zero. So, we just have to find the field phi p, that satisfies Laplace equation, delta of phi p. equals zero, with a boundary condition at interface. Once we have phi p, we will obtain the added mass ma by a simple sum of the interface. [MUSIC] So, to summarize, for a given problem of fluid solid interaction, all we have to do is to solve a Laplace equation over the fluid domain, with a classical boundary condition at the interface to match with the shape of the motion of the solid. Solving a Laplace equation is very simple, using numerical methods, for instance. It does not require advanced computing techniques. And remember, we just have to solve it once. This is much, much simpler than CFD. But the Laplace equation is so simple and so common in physics that there are many exact solutions. Let us use some of these solutions to see what happens. Imagine the case of a cylinder, oscillating horizontally in an infinite fluid. On the interface, Phi is the unit horizontal vector, and so Phi n equals cos theta. I'm using here the polar coordinates r and theta. The solution of the Laplace equation with this condition is easy. And we have Phi p=1/r(cos theta). Here is what the pressure field looks like. The corresponding velocity field, minus grad of Phi p, is associated with a unit displacement of the cylinder as shown here. As you can see, the fluid turns around the cylinder as it moves. Here is an animated motion based on this solution. It is much amplified of course, so that we can see what happens, but remember, that our theory is only valid for small oscillations. Now, remember that we need all these to have the added mass. All we have to do now is integrate over the interface, the quantity phi p, n phi. Because the dimensionless radius of the cylinder is one, we obtain mA, the added mass equals M, the mass number, times pi. What does it mean? Let us go back to the dimensional quantities. The masses have been scaled by the mass of the solid. So, the dimensionless mass, mA, is the ratio of the true added mass large MA, over the mass of the solid, M solid. We are talking here of a 2D problem, so of course, these are masses per unique length. On the other hand, the mass number M is the ratio of Rho R square over the mass of the solid. Here R is the radius of the cylinder. By combining these, we get the dimensional added mass. large MA equal to Rho pi r square. Now, this is very interesting, because we have an added mass here, that is equal to the mass of the displaced water, just as in Archimedes' principle. This result has a very important particular application in the field of ocean engineering. If you're interested in the vibrations of a cable or pipe underwater, You know now that the coupling with the water is going to bring an adding mass. And you know that the adding mass is going to be something like the mass of displaced water. Very often, underwater systems are designed to be neutrally buoyant to avoid forces upward or downward. To be naturally buoyant, the mass per unit length of the pipe or cable must be similar to the mass of displaced water, so that gravity is compensated by Archimedes force. So both M solid and M A are equal to the mass of displaced water. This means that the apparent mass of the pipe, the sum of the two is twice the mass of the pipe itself. This is the basis for rule of thumb used in ocean engineering that says that the interaction with water typically doubles all the masses. No need for CFD or even to solve the Laplace equation again. All fluid solid interaction reduces here to doubling the masses. So, we found theh the added mass of a cylinder in an infinite fluid, was that of the displaced fluid. Is that a general rule? No, not at all, unfortunately. For instance, imagine a cylinder in a confined, circular, fluid domain. Let delta be the dimensionless radius of the fluid domain. We can solve here the Laplace problem to have phi p with now the condition that radial velocity is zero at the radius of delta. We can get phi p in close form, and eventually by integrating as before, we have added mass here. The result is a bit surprising. First, the added mass is not the displaced mass. Second, it is higher than the displaced fluid mass, although there is less fluid. And the more confined the fluid is, the larger the added mass. You can see that when the gap closes, delta approaches one, and then the added mass goes to infinity. For instance, here are two geometries with increased confinement. Delta equals 2 and then delta equals 1.1. And here are the added masses using the formula of both. This is very strange. the less fluid there is the higher the fluid added mass is. Let us try and explain this. [MUSIC] To try and understand, we should have a look at the kinetic energy of the fluid as the cylinder moves. The dimensionless kinetic energy, K C, is the sum of 1/2 of M u square over the fluid domain. Because the velocity u has the particular form of q dot times phi u, we can take the q dot out of integral, because it is independent of space. And because phi u is related to phi p everywhere, and because phi p is related to phi at interface, we can work on integrals, and it can be shown that the sum over the whole domain of phi u square, is equal to the sum over the interface of phi, p, n, phi. And this is exactly our added mass. So, the key result is that, the added mass, is the mass that the allows expressing the kinetic energy of the fluid, using the velocity of the solid. Now, we understand why the added mass for a confined cylinder is larger than for a non-confined one. Look at the fluid motion in the confined case. The fluid velocities are much higher. And if the fluid velocities are much higher, even on a small domain, the total fluid kinetic energy is higher, and so, the added mass is higher. So it is difficult to move a cylinder in a confined field space, not because of viscosity, but because it is difficult to accelerate the fluid around the cylinder. This is fluid inertia. Fluid added mass effect. [MUSIC] The same idea can help you understand that added mass is a directional mass. The solid can have a different added mass, depending on the direction it is moving. For instance, here. An ellipse with a ratio epsilon between the two axis. The added mass can be derived exactly. In the motion, transverse to the main axis, it is the same as a cylinder M pi. In a motion along the axis, it is M pi time epsilon square. Much smaller. The added mass there is smaller, because this motion does not move much fluid. So the fluid kinetic energy is small, and so is the added mass. The added mass in the vertical motion, is much larger than in the horizontal direction. Added mass is a very important concept in engineering. It is the simplest model of the effects of a fluid on a moving solid. And of course, the higher the fluid density, the higher the added mass. So in designing submerged systems or a floating systems in interaction with water, you have to take this into account, as it is the main effect of the fluid on the dynamics of the solid. But added mass may also be important in air. Why? Consider a lock of hair. Its weight may be of a few grams. And the added mass, of a few grams also. This means that in computer graphics applied to video games or animated films, you have to take into account the added mass of air, on hairs. Let us summarize. The added mass effect is a major effect in fluid solid interactions. It is remarkable, that the result of all the complex flow caused by the motion of the solid is so simple. Of course, this is because we have made some very strong assumptions. Low reduced velocity, small amplitude, negligible effects of viscosity. In the next step, you will learn how to take into account the effects of viscosity. [MUSIC]
