[MUSIC] We found many possible interactions between a fluid and a solid. We could see that the same fundamental models could be used to understand and predict couplings in very different systems. For instance, the couple mode flutter of an airfoil, and the garden hose instability had a common structure, a frequency merging with non-symmetric coupling between the two modes. In fact, there are many possible applications of these models we have built together and most of the engineering practice in fluid interactions is based on them. But that does not cover all observed phenomena. For instance, here is an experiment on a flexible system, a chain, that oscillates under flow. Some dye in the flow shows that this is simultaneous with the shedding of vortices from the solid boundary into the flow. Here is a computation of the same phenomenon, with a flexible cylinder that moves also in relation with vortices. This is called vortex-induced vibrations, or VIV, and has nothing to do apparently with what we have been talking about up to now. Why is it so different? First, it happens in a limited range of reduced velocity, say, 5 to 10, and then disappears at higher flow velocities. This does not look at all like the instabilities we have found. When an airfoil was unstable, increasing the flow velocity made things worse. Second, it happens even with uniform flow on a cylinder flexible just in the lift direction. And we have seen that a cylinder cannot experience lift galloping. [MUSIC] What we see is different. It seems related to some internal dynamics of the fluid, here the shedding of vortices. But so far we have considered only one time scale in the fluid domain. That was T Fluid, the time corresponding to the convection of a particle across the length L, so that T Fluid = L/U. In fact, if you look at the flow behind the fixed cylinder when the Reynolds number is high enough, you'll see oscillations. So there is another time scale. Let us call it T vortex because these oscillations correspond to the shedding of vortices behind a cylinder. If there is another time scale for the fluid, then all our models which are based on the comparison between T Fluid and T Solid, using the reduced velocity, are just not relevant. What do we know about this time scale of vortex shedding? Well, T vortex can be estimated from the oscillation of a quantity such as pressure in the wake. On an instantaneous image of a flow, we can also estimate it from the wavelengths of the pattern of vortices that by the flow. We can do that for several flow velocities, several cylinder diameters. And what is remarkable is that this new time scale, T vortex, is actually proportional to the one we have been using, T Fluid. This means that when we increase the velocity, the frequency of shedding increases. This is call the Strouhal law, and the coefficient S between the two time scales is the Strouhal number. The Strouhal number for a cylinder would be around 0.2 if the Reynolds number is high enough. Remember that the reduced velocity was built to compare the dynamics of the fluid and of the solid. U R = T Solid/T Fluid. When U R = 1, the time of convection and the time of oscillation of a solid were the same. Now using the Strouhal law, we can make a comparison between T Solid and T Vortex, because T Vortex = (1/Strouhal) T Fluid. The ratio T Solid/T Vortex is not U R but now S times U R. What does it mean? It means that when SU R = 1, the shedding of vortices and the motion of the solid happen at the same time scale. [MUSIC] When we observe vortex-induced vibrations, we had U R of about 5 and now S equals 0.2, so SU R equals about 1. Something is happening here between the vortices and the motion of the solid. It is easy to imagine what happens then. We have oscillations in the flow at a period T Vortex. Certainly this causes oscillation of the lift on the cylinder at that same period. If that period of forcing is equal to the period of free oscillation of the cylinder, we might have a resonance. Here is a simple model of that effect. Let F vortex be that oscillating lift caused by vortex shedding on the cylinder per unit time. We can write F vortex = one-half of rho U squared L times C l, the lift coefficient, times sinus(2 pi t/T vortex). This force oscillates at the frequency of vortex shedding. Its magnitude is defined by C l called the fluctuating lift coefficient, which is about 0.3 for a cylinder. Now if the cylinder is a mass-spring system that is allowed to move in the lift direction, we have a forced oscillator equation. On the left-hand side, the dynamics of the cylinder. On the right-hand side, the forcing by the wake. But we know how an oscillator responds to a sinusoidal forcing. Here is the response curve in dimensionless form. At the reduced velocity of 1/S we have a resonance. Of course, if we have damping, the resonance is limited in amplitude. But there's also something special with this resonance curve. It does not decrease after the resonance. It stays on a plateau. Why? Because as you increase the velocity, you increase the frequency of the forcing but also the magnitude of the forcing. This is the basis of modeling of vortex-induced vibrations. The solid is forced to oscillate because the fluid dynamics oscillates itself. Vortex-induced vibrations, or VIV, occur not only in cylinders, but on all bluff bodies where there is an oscillating wake, as soon as the Reynolds number is high enough. How can you avoid vortex-induced vibrations? Now that you know the cause, you can imagine many ways. First, you can think of avoiding the resonance. Because VIV occur in limited range of the used velocity, you just have to check that the main frequency of free oscillation of your system is away from the frequency of vortex shedding. For instance, you compute the frequency of a free oscillation of an antenna. Then for a given wind velocity, you compute the frequency of vortex shedding and you compare them. But this might not be enough. Damping can also help because it reduces the amplitude at resonance. There are many damping systems that can be added to vibrating structures of all sizes. And finally, you can try to suppress the cause, which is the oscillation of the wake. This is done usually by adding devices such as splitter plates, helical strikes, bumps, and so on. [MUSIC] Let us summarize. We have here a new dynamical system, the wake. It has its own timescale of oscillation. And when the timescale matches that of a solid, we have a resonance. This is a pure forcing of the solid by the fluid, vortex-induced vibration. But what about the feedback? You can imagine that when a cylinder moves a lot, it influences the wake. What happens then? We have found plenty of useful models when considering the feedback coupling between the fluid and the solid. Let us do that next. [MUSIC]
