 The last topic that we're going to take a look at is the one that enables us to scale between experimental models and what we call prototypes or engineering prototypes. And this follows under the topic of flow similarity. So what we're going to be using is some terminology here that I'll go over. And it pertains to when conducting experiments on a model, perhaps a model that you would put in a wind tunnel or in a towing tank or something like that. And you compare that to a prototype. So quite often this is an engineering prototype. It would be a full scale or a much larger scale system. And so we've seen through our dimensional analysis, Buckingham pie method that we need to match dimensionless groups. Since we came up with these different dimensionless groups that we've covered, but we must also ensure that flow similarity is met. And so what I'm going to do now is go through what flow similarity implies. So matching the Reynolds number is one of the conditions we need. But we need to do a little bit more between the model and the prototype. And some of these are pretty intuitive, but we will go through them. The first one is the model and the prototype have to be the same sort of shape and dimensions. They have to scale with respect to one another. So we call that geometric similarity. Okay. So there is the definition of geometric similarity. And what it is telling us is that the body shapes need to be similar and they need to scale in all three coordinate dimensions, directions. So we have to have a linear scale ratio. So they have to be scaled with respect to one another. All angles need to be preserved and all flow directions need to be preserved as well. So let's take a look at an example here where we'll put the prototype on the left and then the model on the right. And I'm going to draw an airfoil just because it's one that's fairly easy to draw. So there we have our big airfoil and there we have a smaller airfoil that should look the same. So this is our prototype. Okay. So what we have on the right is a prototype that might be a full scale system. And then we have a model that we might put within a wind tunnel. And you can see all the spatial dimensions scale. So the chord length, one meter to point one, so a scale of 10, the thickness point two down to point zero two. And the angles are preserved as well. So we have the angle between the chord line from the leading edge to the trailing edge and the free stream velocity is 10 degrees in both cases. So this would be an example where we have satisfied geometric similarity. So that would be the first condition of flow similarity. The second condition of flow similarity is that of kinematic similarity. So let's take a look at that. Okay. So the second similarity that we have is that of kinematic similarity. And what it is saying is that the velocities in the flow field at corresponding points need to scale. So again, we're going to use an example of an airfoil. I'll draw a slightly different one with a little bit of camber. So we have on the left hand side our prototype. And on the right hand side, we have our model. And if we were to look at a couple of different points within the flow field, let's say we look there and we look there. Here we would have V1 for our prototype. And then here you might have V2 for the prototype. Going to corresponding points in the model flow field, what we would have to have in order for this to work would be beta V1 model. And then beta V1 prototype, sorry, model as well. And those would need to scale. So our velocity fields between the model and the prototype, I goofed up, I apologize, that should be P, P. So prototype, prototype. So what we find is that the velocity field in the prototype can then scale based on some scaling factor beta. But it would be a common scaling factor at any point within the flow field. And so these velocity vectors, obviously, would probably be less if we're dealing with the much smaller model. And you would scale it with beta. And that would give us the condition of kinematic similarity. So it implies that through a constant, the velocity scale between model and prototype. And the last one is we looked at geometric. We've looked at kinematic, which is a scaling of the velocity fields. The last one is the dynamic similarity, which moves us into the forces acting on the body. So for dynamic similarity, again, what I'll do, I'll draw out the airfoil. We have our prototype and then our model. And let's denote this point here. Let's say that's the quarter chord point. And if we say here we have lift on the prototype and some drag force on the prototype. And this is occurring A away from the quarter chord. Then with our model, we would have vectors at the same location or forces, but they would scale. So you would have zeta times the drag force on the prototype and zeta multiplied by the lift force on the prototype. And that would give you what was on the drag or sorry on the model. And then here this would be how far away or where the force was acting. And there'd be some other scaling factor multiplied by A. So if we have this scenario where our forces and the location that they act is scale, then that would be the case where we have dynamic similarity where the forces on the bodies are scaling. So those are the three types of similarity that give us flow similarity. So beyond just matching the Reynolds number or for example the Mach number if we're looking at a compressible flow, we need to have a condition for flow similarity where we have geometric similarity. That's where the models scale geometrically by a constants in terms of the three spatial dimensions and angles and flow directions. We looked at kinematic similarity. This is the one where the velocity fields need to scale going from the prototype to the model by some scaling constant. And then finally dynamic similarity where the forces on the prototype in the model scale with respect to one another. So those are the conditions for similarity that you would want to have if you're going to use sub-scale model testing and then be able to translate that into full-scale prototype development. And that is it for our analysis looking at similarity and dimensionless variables. We've looked at Buckingham Pi, a way to come up with the Pi parameters. We've looked at non-dimensionalization, some very important non-dimensional numbers that we've looked at use in fluid mechanics. And finally we've looked at similarity, which is what you need to do if you're doing any kind of model testing, be it in a towing tank, a wave tank, in a wind tunnel and then taking those results and extrapolating them to real world systems.