[MUSIC] Well, we have obtained a model of the flow-induced damping in a pipe using a single mode approximation, but we could not predict this famous fluid-conveying pipe instability or garden hose instability. The next step is to improve the model by enriching the description of the dynamics of the pipe. Let us use a two modes approximation. [MUSIC] To keep the equation simple and without changing the physics, we are going to work with the fluid-conveying bi-pendulum. Without fluid the dynamics of the single pendulum was governed by theta double dot + theta = 0. Let us consider a bi-pendulum by combining two simple pendulums. Its deformation is now defined by two angles, theta and phi. Without fluid, and using the same dimension parameters, the dynamics is governed by two equations that couple the evolutions in time of theta and phi. You can obtain these two equations by any standard methods of rigid body dynamics. In these two equations, I have two degrees of freedom, theta and phi. But their evolutions are coupled. This is not exactly a two modes approximation as I've used before. Let us build this. I can look for the two modes of this coupled system, here's what I get. The first mode has a frequency of square root of 2- 1. The motion is quite pendular, with the two masses moving in the same direction. The second mode has a higher frequency and the mode shape shows the two masses moving in opposite direction. First mode in phase, second mode out of phase. I can write the state of my bi-pendulum theta phi using a combination of these two modes with the modal variables q1 and q2 as I have done often before. Now, q1 and q2 satisfy simple uncoupled modal oscillator equations. Now, that we have the model for the dynamics of the solid, let us include the interaction with the fluid. For the fluid conveying single pendulum, we could handle the case of an arbitrary reduced velocity. It is a bit heavy to handle for the bi-pendulum, although it can be done too. But what we're looking for is the simplest model of the instability that occurs when we increase the flow velocity. You remember that for other systems, such as an air foil, the approximation of quasistatic aeroelasticiy allowed to predict dynamic instabilities using a two modes approximation. Let us try this. [MUSIC] Here is the fluid-conveying bi-pendulum in a deformation state frozen in time. What is the fluid loading on the solid in this geometry with a reduced velocity UR? This is a classical problem of fluid mechanics. Flow in a pipe with an elbow. Using the fluid momentum balance, the magnitude of the force exerted by the flow on an elbow is rho SU squared (theta- phi) in dimensional form, for small angles. I can define the Cauchy number based on the stiffness of the springs and my dimensionless force reads simply Cauchy times theta minus phi. What is the effect of this force on the dynamics of the pipe? All I have to do is to project the same force on my two modes. Because f depends on theta and phi, it depends on q1 and q2. I end up with my equations in q1 and q2, including new terms, depending on the Cauchy number. Here they are. On the left hand side, I have the modal oscillator terms for q1 and q2. On the right hand side, all the terms are proportional to the Cauchy number as expected. And importantly, there are terms in q1 and q2 in both equations. I can put on the left hand side the stiffness terms of each equations, and I get something very similar to cases we have seen before. On mode one, I have a stiffness that's going to increase with the Cauchy number. Conversely, on mode two the stiffness is going to decrease, and I have a coupling between the two modes that is not symmetric at all. This is going to give me an instability with a merging of the two frequencies and non symmetry coupling. Let us see that. I can compute the modes of the system as a function of the Cauchy number. At zero flow velocity, two modes with frequencies, omega one and omega two as before. As the flow velocity is increased the frequencies come closer and closer. When they meet the non symmetric couplings give two modes, one stable and one unstable. Here is what the unstable mode looks like. It is a combination of the two initial modes. It has a same look as the experimental motion after instability. Why is it unstable? It is unstable because over one cycle, the force exerted by the flow on the elbow brings a positive work as we can see here. Although the configuration is quite different this is exactly the same mechanism as in coupled mode flutter of an air foil. Actually in many other instabilities of this type. The dynamics combining two modes allows the fluid force to bring a positive work in a cycle. [MUSIC] Much more can be done on the fluid conveying pipe problem and has been done. Solving for all reduced velocities using beam modes instead of mass spring pendulum, changing the boundary conditions and many other things. This serves as a toy model for a very large set of problems of fluid solid interactions. There are other flow induced instabilities that are somehow related to the fluid conveying pipe instability. For instance flag flapping in some regimes, the outer flow is along the flexible system as in the pipe and the unstable mode is similar. Some of the early models are based almost on the same equations. Snoring is also a case of flutter with flow along a flexible system. To model it you have to take into account the flexibility of the palate and the flow on both sides. Let us summarize the key findings here. First by using a single mode approximation we could obtain models for various ranges of reduced velocity, small, high, and intermediate. And because the system is so simple, we could have the solution for all reduced velocities. This showed that the various effects of the couplings somehow added up. But that only accounted for part of the dynamics observed in fluid conveying pipes. Increase of damping, but not the instability. Second, by enriching the dynamic of the solid, you could predict the instability. This is in fact the basis of building models. Using the simplest approximation for the dynamics and for the interaction, articulating the resulting models, and if necessary, using more elaborate models. Are there some fluid interactions problems that cannot at all be treated by what we done up to now? Yes, there are and many. Next we shall address one of them of great particular importance, vortex induced vibrations. [MUSIC]
