[MUSIC] So, for slow flows under the approximation of pseudo-static aeroelasticity, we have explored the effect of taking into account the velocity of the solid at the interface. And we predicted that a dynamic instability of the solid may appear. Let us see how this would apply to an airfoil. We consider the same airfoil than before. Because our new dynamic instability mechanism only requires one mode in the motion, let us assume that the rotation theta is blocked at a constant value, say theta nought. So we only have the plunge displacement y of t, the chord L, the mass M, and the stiffness K. In the absence of flow the equation of motion is simply that of the oscillator. We shall keep using the dimensional form of the equation here because they are so simple. If we now add a flow velocity U what will happen? It all depends on the range of flow velocity we are considering. Let us define the reduced velocity as U / L times the frequency of the mode and D the displacement number with y nought the amplitude of motion. Imagine first that the reduced velocity is very large and that we are under the quasi-static aeroelasticity assumption. This means that the reduced velocity is much larger than D. In that case, the fluid force that applies on the airfoil is assumed to depend only on the position of the airfoil, defined here by the dimensionless plunge y. But the lift force on an airfoil, depends on the angle of attack, here theta nought, but not on the position y. So we already know that the flow induced stiffness, which gives the dependence of the force on the position y, is going to be 0. In other terms, there is going to be no net force acting on the right hand side of the oscillator equation. So, there is absolutely no effect of the flow on the dynamics of the airfoil, which is the same as before. Of course, the lift force is going to change the mean position of the airfoil but the dynamics is unchanged. The free oscillations have the same frequency and no damping. Let us consider now the case where the flow velocity is not so large and where we cannot neglect the airfoil velocity. We are in the range of pseudo-static aeroelasticity. U R squared much larger than D. We have schematically an airfoil with an angle of attack, theta nought that moves at a constant velocity y dot in a freestream velocity U. Can we say what the force on the airfoil is going to be? Well, in terms of fluid mechanics, we may consider the frame of reference of the airfoil where the airfoil velocity is zero. There the flow velocity relative to the airfoil, V, is the combination of the original velocity U and of minus y dot. This new velocity differs from U essentially by its inclination alpha, such that tangent of alpha of course equals minus y dot over U. Now, if I turn all these of an angle alpha, this is the same problem as a horizontal flow of velocity V acting on a fixed airfoil with an inclination that becomes theta nought plus alpha. In that case, we know what the lift force is going to be. It will be proportional to the lift coefficient for this angle of attack theta nought plus alpha. And we may expand the lift coefficient in terms of the angle considering that alpha is small. We have then the mean lift as expected and the next term in alpha. We know that alpha is at the first order, the ratio between y dot and U. With that, the fluctuating part of the force appears to be proportional to y dot. And this is the expected flow-induced damping term. So, with that, we can predict the behavior of an airfoil oscillating in plunge of flow. Under our approximation, the effect of the flow is to modify the oscillator equation for plunge by adding a damping term. The damping coefficient varies linearly with the flow velocity, as expected. More important, its sign is going to depend essentially on the slope of the lift coefficient curve, dCL over d theta. What is the slope? [MUSIC] Here is a typical lift coefficient curve. Up to now we only considered very small angles of attack. In fact, for a wide range of angle of attck, the slope remains positive. Lift increases if you increase the angle of attack. This corresponds to a well attached flow pattern on the upper side of the airfoil. But above an angle of attack of about 10 to 15 degrees, the lift coefficient starts to decrease. This is related to a change in the flow pattern which is no more attached on the upper surface. This is the region of stall. Now in terms of instability, here is what we can say now. All we have to look for is the sign of dCL over d theta. Clearly, for angles lower than the angle of stall, the slope is positive, and the flow is going to bring, therefore, a positive damping. There the flow tends to damp any vibration of the wing. Conversely, for higher angle of attacks, the slope is negative, and so is the flow-induced damping. We then expect a dynamic instability, which is called stall flutter. Let us give a simple explanation of why we have an instability here. Consider an airfoil with a small angle of attack, with an upward velocity of y dot. Because of the upward motion, the apparent angle of attack on the airfoil is going to be even smaller, which will reduce the lift. So the variation of the lift is opposed to the vertical upward velocity. A force opposed to the velocity of a solid is a damping force. As a consequence, we have stability there. Conversely, for angles of attack above stall, here's what happens. An upward velocity would result in a smaller angle of attack too. But this will increase the lift by de-stalling. So, the variations of the force then goes in the same direction as the velocity causing the instability. You can easily compute the force and the work of the force of a cycle of the oscillator. You will see that in the region of negative slopes, above stall, the fluid force transfers energy to the solid. Stall flutter of airfoils is a very important phenomenon in practice. Why? Because a wing is made to bring lift. And to have more lift, you may want to go in the region near the beginning of stall. And if you enter the region of negative slope, the lift does not decrease much. But all of a sudden, you destabilize the vibrations of your wing. This may have disastrous consequences. What happens when stall flutter starts? Actually it may not be the end of the story. In fact, the strong motion of the airfoil has a very strong influence on the flow pattern near the airfoil. As a result, the original lift coefficient curve that we used does not really apply. And some limit cycle may be found where in the average of a cycle there is no energy transfer. [MUSIC] So with all these models we have found several ways for an airfoil to be unstable on the flow. The first one was the static instability, buckling in torsion. The second one was the dynamic instability in combinde torsion and plunge also called coupled mode flutter. And now we found that an airfoil could also be unstable dynamically in pure plunge in what is called stall flutter. Actually, these instabilities which would all result in a large flow-induced motion of the airfoil, correspond to very different mechanisms of coupling between the flow and the oscillating solid. The first two may be found for fast flows in the framework of quasi-static aeroelasticity when the velocity of the airfoil can be neglected. The last one, requires that the velocity of the airfoil is not totally negligible and this is the framework of pseudo-static aeroelasticity. Airfoils are not just found in planes. You will find them in wind mills, helicopter blades, turbines, hydrofoils and many other devices where you want to have lift. All these are prone to all these kinds of instabilities. Of course, the geometries, the kinematics, and the flows are different. But in the design of such systems, you have to take into account, all these ways by which the solid maybe coupled to the flow. Let us go back to what we learned about pseudo-static aeroelasticity. We knew as a general result that by taking into account the velocity of the solid in the fluid dynamics, we might have some flow inducing stabilities by negative damping. For airfoils, this is exactly what happens in what is called stall stutter. Next, we shall try and use this approach to understand the motion of other bodies in rather slow flow. [MUSIC]