[MUSIC] Hello, we already have a lot of models that can be used to estimate the effect of the presence of a fluid on the dynamics of a solid. Some models are applicable when the fluid is almost at rest and some when the fluid is moving fast. These limit cases correspond respectively to very low and very high reduced velocities. We obtained general results on the form of the force applied on the solid as a consequence of its interactions with the fluid. For fast flow or more precisely for high reduced velocity, the force acting on the solid is a stiffness force, and this stiffness depends on the flow velocity. We have seen that this may cause static instabilities or buckling and even dynamic instabilities such as coupled mode flutter. But to obtain these results, we had to make a very strong assumption on the flow velocity. It had to be so high that the velocity of the fluid-solid interface could be comparatively neglected. This was the assumption of quasi-static aeroelasticity. But there are cases where the velocity does not satisfy such a condition. We shall refer that to as "coupling with a slow flow". In that case, can we say anything? Do we have simple models to predict what is going to happen? Let us explore this. So in terms of reduced velocity, we want to explore cases where UR is neither very small nor very large. In general that would mean that the two time scales, T solid and T fluid maybe of the same order of magnitude. In that domain, there's actually a range of reduced velocities where something simple can be built. Using the same ideas than in the last chapter, let us compare the evolution in time of two quantities, one pertaining to the dynamics of the solid and one to the dynamics of the fluid. If T solid is very large in comparison with T fluid, we are in the case of a fast flow or quasi static velocity. If T solid is larger than T fluid but not that large, then we may be in the situation illustrated here. During a time interval of T fluid, the quantities in the solid do change, but at a constant rate. Let us see in the equations how this intuition would materialize. [MUSIC] We may reuse exactly the same equations for the fluid domain and the solid than in the last chapter, and I shall not write them again, but let us have a closer look at the boundary conditions on the fluid domain. We have already estimated the order of magnitude of the two types of boundary conditions on the fluid. At the interface, the dimensionless velocity is of the order of D over UR. On the other boundaries of the fluid, it is of the order of 1. But now neither of them can be neglected in comparison with the other. Now let us estimate the magnitude of the variation of the interface conditions. I call this Gamma tilde, the acceleration, which is the variation in time of the velocity. What is the order of magnitude of Gamma tilde? To estimate this variation, we can rely on the same type of arguments than before. We know that the solid displacement scale as Ksi nought, and that it involves in a timescale T solid. This means that the dimensionless displacements scale as Ksi nought over L, which is exactly the displacement number D. This displacement evolves in a dimensionless time scale of T solid over T fluid, which is precisely the reduced velocity UR. So to summarize, Ksi bar is of order D and it varies over a time scale of UR. Then we can say that the order of d squared ksi bar over d t bar squared is d over UR squared. So, the order of the variation of the fluid velocity at the interface is also D over UR squared. Let us summarize, the dynamics of the fluid is governed on one hand by a condition of order one, and on the other hand by a condition of order D over UR at the interface. This condition at the interface does vary in time. During a unit time interval in the fluid, delta T tilde = 1, its variation is of order D over UR squared. Now, certainly if UR squared is much larger than D, the variation can be set to 0 without changing much the results. This corresponds exactly to what we meant by neglecting the variation of the solid dynamics. The solid moves so slowly that its velocity can be considered to be constant in time in terms of the fluid dynamics. This is a bit more complicated than the very strong approximation we made in quasi-static aeroelasticity. Here, we do not ask for the solid to be fixed. We ask for the solid to have a fixed velocity. [MUSIC] Let us summarize. In the general case, we have an interface within the fluid and the solid which has a deformation and a velocity. In the quasi-static aeroelasticity approximation, it has a deformation but no velocity. We say then that the deformation is frozen in time in comparison with fluid dynamics. Now, we have defined what is called the pseudo-static aeroelasticity approximation where the interface has a deformation and a velocity, but the velocity is frozen in time. Note that on the axis of the reduced velocity, a condition that UR squared is larger than D is less restrictive than UR larger than D. Remember that D is the displacement member. It is of the order of magnitude of the displacements in the solid divided by the size of the solid. In practice, the orders of interest are rather small. Particularly, if we are concerned with vibrations. So you may imagine that D is say of the order of 0.01 or 0.1 at the most. As a consequence, when we say that we need UR squared to be larger than D, it may typically only require that UR is larger than 10, which is not much. Let us compare these various approximations in terms of their respective time scales involved. In the general case, the fluid dynamics and the solid dynamics are coupled by their kinematic and dynamic condition at interface and they evolve simultaneously. In the quasi-static aeroelasticity approximation, we had two dynamics. One slow, and one fast. The solid dynamics gives the position of the interface through a kinematic condition that is considered as time independent for the fluid dynamics. Then the solution of the fluid dynamics gives a load at the interface. And now, in the pseudo-static aeroelasticity approximation, we still have two dynamics, one slow and one fast. The solid dynamics give the position and the velocity of the interface through a kinematic condition that is considered as time independent for the free dynamics. This means that in the fluid, we are back again to classical problem of fluid mechanics with time independent boundary conditions, then the solution of the fluid dynamics gives the load at interface. With that, we can compute the solid dynamics and so on. So as in the static aeroelasticity approximation, the two dynamics are coupled. But again, because we have a separation of time scales, we do not have to solve them simultaneously. So to summarize, we have found that for intermediate reduced velocity, we can build a model where the velocity of the solid at interface can be considered as constant in time while we solve the fluid dynamics. This allows you using again all classical results we have in fluid mechanics for problems which time independent boundary conditions. Next, we shall see how this new approximation can give us interesting models of the fluid induced forces [MUSIC]
