 What we're going to do now is we're going to look at the flat plate boundary layer for a turbulent flow. Now, Blasius's solution came out in 1908 and we could follow the technique we were looking at in the previous segment where we had the Reynolds stress terms for the shear stress and we'd have to make approximations there and go through a long analysis with the boundary layer equations. However, a much simpler technique is to use von Karman's momentum integral technique which was developed later in the 1920s but it provided kind of a quick and clean way of doing some of these calculations. Although it's an approximation, it's going to be an approximation no matter what, be it using boundary layer equations with turbulence or using his integral techniques. That's what we're going to do and in doing this we're going to use some relationships that we've seen earlier in the course, mainly the velocity profile and the overlap region that Milliken proposed and this is something that we looked at when we looked at internal viscous flows for pipe flow. So what we were going to do, we're going to use von Karman's momentum integral technique to be able to come up with expressions for the thickness and for the skin friction and that will be the approach we take for the boundary layer equations for a turbulent flow. So we're going to begin with by going back to the overlap region velocity profile that Milliken proposed and this became one of the things that we used when we were doing pressure drop inside of pipes and if you recall the discussion when we were looking at pipe flow we said that the turbulent shear flow theory could apply to either internal or external flow and so here we're using it for an external flow and using the relationship that Milliken proposed, he had a number of empirical constants, he had a friction velocity and then a transform coordinate that was called y plus, I won't get into those here, however the friction velocity u star was defined as being the wall shear stress divided by the density raised to the power of one half and this is friction velocity and the constants people debate them but we'll use those for now and this will get you part way, I'm not going to go through the derivation but this will take you part way and then what happened was Prandtl actually came up with a suggestion for proceeding through this analysis and he suggested a mean velocity profile for the turbulent boundary layer and so we'll also use that and it was a power law relationship that Prandtl proposed so if you have u not being your outer flow field he said that it could be approximated as y being the wall normal coordinate divided by the boundary layer thickness at that point raised to the power one seventh and so this is sometimes called the one seventh power law and it is attributed to Ludwig Prandtl so coupling that with Milliken's overlap region again I'm going to skip over all the details but the main results that came out of this were an expression for the boundary layer thickness and the friction coefficient and finally it's just like when we looked at the flat plate there was an approximation between the friction coefficient evaluated at the end of the plate and the drag coefficient and an approximation for a turbulent flat plate is its seven sixth of the value at x equals l and so those are relations that you can use the last thing that I want to say about this is that there are plots that exist that are very much like the moody diagram if you recall we looked at the moody diagram for pipe flow but what these plots have is the friction drag coefficient plotted as a function of Reynolds number and then we'll have a curve for laminar now unlike the moody diagram it's not always plotted log log sometimes it's log linear but we have the laminar curve we'll have a smooth pipe or sorry smooth plate curve and then you can have wall roughness in your plate and that is quantified by epsilon over l where l is the length of the plate and just like with the moody diagram we'll have these increased curves that at a certain Reynolds number they become asymptotically flat and so that's something that you can look up in tables but this would be data for a rough plate and that enables you to calculate and drag on flat plates that are not smooth turbulent and and with different amounts of roughness so that is the turbulent boundary layer estimates of the boundary layer thickness as well as the friction coefficient or the drag coefficient on those flat plates