[MUSIC] Vortex induced vibrations, as we have just seen it, seems a very simple phenomenon. The fluid dynamics is oscillating and this causes oscillating forces on the solid which oscillates a lot and there is a resonance. But something happens in practice which does not fit with such a simple picture. You remember that the frequency of oscillation of the wake of a fixed cylinder satisfies, as I've said, the Strouhal law. In other terms, the frequency of vortex shedding is proportional to the reduced velocity UR. Its dimensionless value, f vortex, is equal to SUR, where S is the Strouhal number. But when the cylinder is let free to move under the force caused by the wake, the frequency of vortex shedding deviates from this and very significantly. What happens is shown here schematically. If we increase our flow velocity above the point where there is resonance, the frequency of shedding is said to lock on the frequency of oscillation of the cylinder. It very strongly deviates from the Strouhal law. We say it is locked on the motion of the cylinder. Eventually at higher reduced velocities, it jumps back to the Strouhal law. This lock-in phenomenon results in something like an extended resonance. The wake continues to excite the cylinder at its own frequency, even at higher reduced velocities. Instead of having VIV only when the frequencies match, we have them over a wide range. The amplitude of motion schematically shown here remains high even after the pure resonance condition is satisfied. Do we have a model for this? The basic model of VIV is based on a lift force that oscillates at the frequency of the Strouhal law. Then this force is applied on the oscillating cylinder, and if resonance occurs, the amplitude of motion is large. But here, by construction, the Strouhal law is respected and they cannot be lock in. How can the model be improved? First of all, we have to keep in mind that the cause of oscillation of the lift is an oscillatory dynamics that sets on in the flow. Actually it is a result of what is called hydrodynamic instability. The uniform flow solution is not stable anymore and an oscillating solution emerges. It seems reasonable to take into account somehow that the lift force is actually the consequence of something that oscillates. Something that satisfies an oscillator equation. Instead of my formula for F, I can write, F double dot plus S UR squared F equals zero. But doing this just gives me the same result as before. F(t) is applied on Y and I have a resonance. What I'm looking for is a way to take into account the feedback from the solid to the wake. This means that I want to take into account the displacement Y in the equations that governs the lift F. How can I do that? How does the wake respond to a motion of the cylinder that causes the wake? People have done plenty of experiments on wakes behind a moving cylinder, and what they found shows that the wakes are sensitive to the acceleration of the cylinder. The forcing term on the wake equation is actually proportional to Y double dot. So to summarize, instead of having a given wake force shaking the cylinder, we have a coupled system made of an oscillating force and an oscillating cylinder. This is completely different. But how does this coupled system behave? First of all, this set of equation is linear. So there is no way that this is going to give me an amplitude of motion or an amplitude of the force. But I built this model to try to understand lockin which is just a question of frequencies. So what are the frequencies involved in this coupled system? To know them, I just have to compute the modes. Let us look for solutions in the forms of e to the i omega t. Because I have two degrees of freedom, I have two modes, and, therefore, two frequencies. And for each value of UR, I will have a different set of frequencies. For such a simple system I can do all the computations by hand. But here's schematically the result. The horizontal axis is the reduced velocity, the vertical axis gives the frequency, the value of 1 being the cylinder free frequency. The dotted line, is the Strouhal law. At SUr equals 1, the frequency of shedding is expected to be equal to that of the cylinder. But what does the coupled model give? At zero reduce velocity, I have two frequencies. One is that of the solid and the other zero because the frequency of shedding is zero with no flow. With a non-zero flow velocity, the frequency of the wake is close to the Strouhal law. But as the reduced velocity is increased further, I see the two frequencies coming closer and closer. Then they merge and there is only one frequency, and then they split again. The wake frequency goes up and back to the Strouhal law. What does happen in the yellow zone here where I have only one common frequency? Here is the growth rate of the modes. Opposite to the imaginary part of the frequencies. It tells me if a mode is damped or negative, neutral or unstable when positive. For loaw and high reduced velocity they are equal to zero. But in the range I've identified before, one of the mode has a positive growth rate, and one has a negative. This tells me that I have an unstable coupled mode. What does the unstable mode look like? [MUSIC] To visualize it, I have to show the evolution of the two variables. The displacement Y and the lift force F. Here is the mode. As you can see, the cylinder oscillates and the force has a phase shift with it. And thanks to this phase shift, there is a positive work of the force over a cycle. Let us summarize. To understand and predict lock-in, this mixup of the two frequencies, I have to incorporate an effect of the cylinder motion on the wake dynamics. The instability we obtained here is very similar to coupled mode flutter as we've seen it for air flois or for fluid conveying pipes, but what is very special is that instead of having two solid modes being coupled by the flow, we have one solid mode coupled to one fluid mode. We already had coupling between solid modes and fluid modes, in the case of sloshing of a free surface. It was the dynamic Froude number that governed it, but that coupling was conservative, it did not bring any instability. We also had coupling between solid modes, and at time-dependent process, viscous diffusion in the fluid. That brought damping, and the Stokes number was the key parameter. Here, we have coupling between a solid mode and a wake mode. And these coupling bring some instability in a limited range of reduced velocity. Let us conclude. To address the various kinds of interactions between fluids and solids, you know that the key thing to look for is the comparison of all the time scales in the fluid and in the solid. The dimensionless parameters are built just for this. The Stokes number, the dynamic Froude number, the reduced velocity, the Cauchy number. All the models that exist and that you can use in all fields of engineering have in the background some assumptions on these time scales. [MUSIC]
