[MUSIC] So, we are exploring the framework of the approximation of pseudo-static aeroelasticity. We showed that taking into account the velocity of the solid at the interface led to flow induced damping forces. This damping could be negative and instability could result. We have illustrated this on the case of stall flutter of an airfoil, but the framework is actually quite general. Let us now see what would happen for a body with arbitrary shape moving in a rather slow flow. I will consider here the simple cases of a rigid body moving in the direction of the flow or across the flow. Let us start by moving in the direction of the flow, and to make things even simpler, let us have a cylinder of diameter L moving with the velocity x dot. If we have a flow velocity that satisfies our condition for the pseudo static aeroelastic approximation the velocity, X dot maybe considered as frozen which means time independent. So, we have a cylinder moving with the velocity x dot in the same direction as a flow velocity U. This is equivalent to considering a fixed cylinder with a flow velocity V equals to U minus X dot. This is now quite simple What is the force resulting from a flow of magnitude V on a fixed cylinder? It reads F(V) = one half of rhoV2 LCd (Re) where Cd is the drag coefficient that depends on a Reynolds number which is here, rho VL / mu. Here is a typical plot of the evolution of the drag coefficient on a cylinder with the Reynolds number. It first decreases and then reaches a kind of plateau, except in a small region before Reynolds of 10 to the 6. So, as we have done before, we may expand the force F considering that the velocity of the cylinder is small versus that of the flow. We have a first term corresponding to the drag when the cylinder is not moving, but we are interested in the next term which takes into the account the velocity of the cylinder. It reads, X dot dF over dX dot and this is the damping force that we expecting. What is the sign of the damping coefficient? We have to go back to the model of the drag force where the velocity, V, is actually dependent here on X dot. This is a straightforward derivation, which gives finally the coefficient dF over dX dot, proportional to the velocity U, and to a factor which depends on the drag coefficient only. [MUSIC] What is the sign of this damping coefficient? It only depends on the sign of the term related to the drag coefficient. If this quantity is positive, then the force will have a damping effect on any motion of the cylinder. Conversely, if it is negative, then any motion would be amplified, and we shall have a dynamic instability called drag galloping. In practice can we have a negative damping coefficient? We know that the drag coefficient is positive but it decreases with Reynolds number so that the second term in the bracket is negative. How is the sum of the two? Well, it is negative but only in a very small Reynolds number near one million where there is a sudden drop of the drag coefficient. This drop corresponds to what is called drag crisis and in that region the sum of the two terms is negative. We have then what is called a drag instability of the cylinder which is going to oscillate in the direction of the flow. The mechanisms of the drag crisis instability is easy to understand. When the cylinder moves in the downstream direction, the apparent flow velocity decreases. In the drag crisis region it happens that the drag then increases. Conversely, when going in the upstream direction the apparent velocity increases causing the drag crisis, and the drag drops. As a result, the trend of drag is always in the direction of the velocity and the power of the fluid force over the cycle will be positive. This will cause a flow induced dynamic instability. Actually, this instability may occur on any bluff body oscillating in the direction of a flow provided there is a drag crisis. What is drag crisis? It is a sudden change in the flow pattern, particularly in the organization of the boundary layer near the solid wall. This only happen if the body has smooth walls, not with corners. Here's the form of the evolution of drag coefficient of several shapes. The rounded shapes in red may be unstable because there is a sudden decrease in the drag coefficient. The shapes with sharp edges in green, here the square, and the triangle, will be stable. [MUSIC] Well, using this same idea, let us see what happens if a body of arbitrary shape moves transversely to the flow. Let theta naught be the angle that defines its angular opposition relatively to the flow. If again the velocity y dot is frozen in time we have only to consider a fixed body with a relative velocity of V which is a combination of U and Y dot. This is equivalent to considering a change in the incidence of the flow which becomes theta nought plus alpha where alpha depends on the ratio between y dot and U. The lift force acting in that configuration is as for the airfoil, defined by the lift coefficient CL, which depends on the angle of incidence of the flow. As this angle is dependant on the velocity y dot again, we obtain a flow induced damping. And the coefficient depends on the variation of the lift coefficient with the angle of incidence. At this stage we are exactly at the same point than when we developed the force for an airfoil in plunge. We found a condition of stall flutter directly dependent on the slope of lift coefficient dCL/dtheta. [MUSIC] But the difference here is that we do not have a streamlined body as for the airfoil. This means that the evolution of the lift coefficient with respect to the angle of the incidence can take any form. The only thing we may say is that it is periodic in theta naught and if it is periodic in theta naught , there is certainly going to be a region where the slope is positive, and a region where the slope is negative. So there will always be a range of orientation of the body with respect to the flow, such that motion in the lit direction will be negatively damped. This is much more general than the conditional of stall. We shall call this lift galloping, which occurs as soon as the lift coefficient has a decreasing slope with the angle-of-attack. For instance, here is the case of a square section. Because of the symmetries, we only need to plot the lift coefficient between 0 and pi over 2. By just looking at the shape of the curve, we see that we may expect lift galloping for small angles and near the diamond position. Actually, as a general feature it is observed that only when bodies are elongated in the cross-flow direction, the lift coefficient has a negative slope first. Schematically, for a rectangle aligned with the flow, the lift motion is damped. For a rectangle transversed to the flow the lift motion is unstable. Look at the movie; what you see here is kind of poster suspended in a shop. It is placed closed to the shop entrance where you have a continuous air current. Because the pendulum motion is very slightly damped, the flow induced forces result in an instability with a very large amplitude of motion. To summarize, we have found that the dynamic instability may occur when a bluff body is free to move in a flow. When flow induced forces are considered to depend on the velocity of the body, we have a flow induced damping force that destabilize the motion. This is often referred to as galloping, although this term describes more the motion than the cause. Our two cases illustrated here, drag galloping of a cylinder and lift galloping of a rectangle are the simplest cases. More generally, you can imagine that the bluff body that moves in a direction that is inclined to the flow, will combine these effects. Moreover, one may take into account the simultaneous changes of lift and drag forces that occur when the solid moves. There are rather elaborate models for this, but the elementary mechanisms are those we just saw. In practice, you will find these issues of galloping in many systems. For instance, in transmission lines ice may aggregate and transform the cylindrical section into a shape elongated across the flow. Also in offshore engineering, the current may come across what is called riser towers which are arrangements of connected pipes. These systems may have non-zero lift under some incident angles, and this galloping may occur. Galloping also occurs in systems moving in much more complex motions such as gates in dams. Tainter gates were found to vibrate under flow and this can be analyzed by using our models. Finally, bridges. There are a lot of galloping issues in bridges and some of them require the use of models more complex than what we studied here. In particular, the famous Tacoma Narrows Bridge failure is not an easy case. Let us just say here that it falls in the category of coupling with the flow, slow flow, where you have to take into account the local velocity of the bridge deck. More generally, models exist that allow to take into account not only the displacements but also the rotation of sections in the flow. Let us conclude our chapter on coupling with a slow flow. By slow flow we meant that its velocity is not so large that the solid velocity can be neglected. In that case we could define an assumption of pseudo-static aeroelascity, which predicts that a flow induced damping force is going to apply on the solid, and that the damping force maybe stabilizing or destabilizing. When it is destabilizing we have a dynamic instability that builds up on a single mode. No coupling of modes needed. We have illustrated this on stall flutters of airfoils drag galloping and lift galloping. [MUSIC] At this stage of the course, we have models for interaction with fast flows, very high reduce velocities, and for slow flows, intermediate reduce velocities. We also have models for very slow flows when the fluid velocity is negligible. Can we cover the whole axis of reduced velocities and connect all these models? We shall try and do this in the next chapter and show that we can give a consistent picture. [MUSIC]