 We're now going to take a look at a method of analyzing the boundary layer that was first proposed by Theodore von Karman in 1921 and this technique is one that results in an approximation for characteristics or description of the boundary layer, mainly the growth rate as well as the skin friction coefficient. And the reason why we're going to spend time looking at it is because it uses the tools that we have developed thus far in this course, an introductory course in fluid mechanics, that being control volume analysis. So what von Karman did is he applied control volume analysis to studying the flat plate boundary layer and he was able to come up with some really quite remarkable results considering the approach that he took and the fact that it really was a very basic analysis and consequently what we'll be doing is looking at that. It's referred to as being von Karman's momentum integral theory. And during this period I didn't check but I would assume that he was a professor at Aachen having obtained his PhD from Ludwig Prantl. But what he was doing here was he was analyzing the flow over flat plate using boundary layer control volume analysis using the momentum equation. And there are more advanced methods that were developed by Prantl and a student of his, Blasius, and we will look at that in a later segment. However, we don't have the tools to analyze that. For that we require an integration of the Navier-Stokes equation and that's left for a more advanced course in fluid mechanics. So what we'll be doing is using the tools that we currently have and studying von Karman's approach to the boundary layer in his momentum integral theory approach. So we're going to begin by drawing out a schematic of the boundary layer and then we'll go through and we'll apply continuity, momentum, and then we'll simplify some of the relations. Okay, so here we have a schematic of the boundary layer as von Karman studied it. And one of the things that he did, and this is a bit of a trick when you're dealing with boundary layer analysis and control volumes, he defined the control volume as going from the plate, the lower plate, he had the inlet at one, but he used the outer streamline as being one of the boundaries of the control volume. And that simplified the analysis that he was performing. So what we're going to do, we're going to begin by looking at the continuity equation and then we'll move into momentum. So we'll take a look beginning with continuity. So it was a steady flow field that von Karman was studying and consequently that term disappears. And then for the mass flux, we're turning back to our schematic, there would be mass flux coming in here and then mass flux exiting at three. And given that he put the streamline being on the upper surface and the wall on the lower, there was no mass flux on either of those surfaces. There is an unknown here, however, and that is this height h, as well as the boundary layer thickness at a given location further downstream, which would be denoted x. And so those were two unknowns, we'll just carry those through the analysis and then try to find a relation for them as we work along. And so using the technique that we have been using throughout this course, whenever we look at mass flux, coming in through surface one, it would be a negative due to the arrangement between the velocity vector and the area vector for that surface. And then for the mass flux leaving through three on the right hand side, that would be a positive, but it is an integral from zero to delta. And delta, by the way, is the boundary layer thickness. And it was the velocity profile Udy. So he performed an integral there. And with this, what we can do is we can simplify it somewhat to begin with a number of the terms cancel. So rho is going to cancel and b, rho and b. And what we're left with then, and we can move this U not h to the left hand side of the equation. So we get that. And now what we're going to do, let's isolate for h. So we get this relationship here. And we're just going to box it and leave it for a moment. We will come back to that when we do analysis with the momentum equation, which is what we're going to do next. So the next step is taking a look at the momentum equation. So what we will be doing is considering the x direction of momentum. And to begin with, on the left hand side, we have some of the forces. And so if we look back at our schematic, the only force that is going to be acting on our control volume is going to be on this lower surface here along the wall. So it's basically the drag on the wall on our control volume. And for that, what I'm going to do, I'm going to draw it a little schematic here of our plate. So this is our plate. And this is the x direction. Now when fluid flows over a plate, there's going to be a drag on the plate in this direction. And that will be our drag force D. But if we look at our control volume, and I'm just going to write out an arbitrary control volume here, there would be an equal and opposite force on the control volume. So this is going to be the drag on the fluid. And consequently, this is going to be a minus D. So we begin by writing out the momentum equation in the x direction with the minus D on the left hand side. And another thing, the time rate of change term that disappears because, again, we're dealing with steady flow. And then what we have is the momentum flux in and out. And so momentum flux in is U naught. That's the velocity there. And then again, our mass flux term being careful of the sign. And momentum flux out, we have a velocity profile there. That's the little U. And the mass flux out term is a positive due to the sign of the area vector and the velocity vector leaving at surface three. And so we obtain that. And what I'll do is I will rearrange that a little bit. So we get this equation here. Hopefully it's not too messy with my zero. Maybe I should move that. I'll put it up there. And that delta is at the top of the integral. So it should be up here. Okay, now what we're going to do, if you recall when we looked at continuity, we came up with a relationship for H. And it was an integral. But what we're going to do, we're going to pull it in and put it into this equation now. Okay, so when we bring in H, what we see is that we have an integral for both of the terms that it looks similar. It's going from zero to delta. At least that's where it's similar. And so let's continue working this equation. Okay, so if we have a length L of plate, this would be the drag force. And it's left in terms of this integral of the velocity profile going from zero to delta. We don't know what delta is. We don't know what the velocity profile is. And consequently, we do have a number of unknowns. But what we're going to do, we're going to rewrite this equation. And I'm going to make a substitution here for this term. And we're going to rewrite this equation with the substitution that I will just show you in a moment. So I have rewritten the equation. And I have this new term here theta. This theta is referred to as being the momentum thickness. And this we will call equation one. And this is the definition of this new term theta that we call the momentum thickness. It's essentially a quantification of the thickness of the shear layer, although not exactly the thickness of the shear layer. But it does give us an indication of that. So what we can do, we have this one relationship for the drag on the plate. And the drag on the plate can also be expressed in terms of the shear stress. If we go way back and look at our schematic, we have the shear stress. And so what we're going to do, let's rewrite drag on the plate by integrating that shear stress function. Right now we don't know what it is, but we will write drag in terms of the shear stress. So we'll do that here. So we have that. I'll take the derivative with respect to x. So we get that. And I will call that equation two. And now what I'm going to do is I'm going to equate this equation, equation one, with this equation here, equation two. And we'll do that in the next slide. So what von Karman was able to build by doing this analysis with an expression for the wall shear stress. And he didn't know what theta was yet. Theta is equal to an unknown. And if you recall, theta is the momentum thickness. And in order to get that, he needed to have an expression for the velocity profile more normal to the wall. He doesn't have that either. But what we'll do will make some approximations in the next segment that enable us to work towards that. But this expression here for the wall shear stress provided he does have theta, and he can take the derivative of it with respect to position along the wall. So that's going along the wall in that direction. Remember y is in this direction. It's normal to the wall. He would then be able to get to the wall shear stress, which is a very important parameter to be able to determine because that gives us the skin friction coefficient, a lot of other things. Another comment that we should make is that we have made no assumptions about the flow field if it is laminar or turbulent. So this relationship is valid for either laminar or turbulent boundary layer flow. So it's a very useful relationship. And what we'll be doing in the next segment is we're going to be zooming in on an approximation that von Karman made for the velocity profile in a laminar boundary layer. Now he probably would have relied on experimentation and a coupling between the theory and the experiments in order to come up with this velocity profile. But you'll see the beauty of the technique. He was able to come up with some very important engineering parameters using very, very basic analysis techniques, things that we've learned in this course thus far. So we'll continue on in the next segment by looking at the velocity profile assumption that he made. And then after that we'll come back to this equation and calculate parameters. The parameters we'll calculate will be delta for the displacement thickness. And we will get tau wall, which that enables us to get the skin friction coefficient as a function of possession along the wall. So that's where we're going. We'll take a look at velocity profile in the next segment.