 In the last segment of what we did is we took the governing equations and we performed Reynolds averaging to them and we came up with expressions and came across the turbulent stresses and these are things that make it difficult for us to solve for the velocity profile, mean velocity profile in turbulent flow fields. But it turns out that we can make orders of magnitude estimates here and we can eliminate some terms and so that's what we're going to look at now and we're going to come up with velocity profiles that enable us then to move in the direction of getting the friction factor for turbulent pipe flow. And so it turns out that when if we're looking at duct flow such as pipe flow or boundary layer flow the minus rho u prime v prime term is the most dominant turbulent stress and consequently the governing equation specifically the Navier-Stokes equation can be written the following manner as an approximation. And what I've done here is I've left the last term in terms of a shear stress and the reason for that is when we're dealing with turbulent flows such as duct flow or boundary layer flow we'll approximate the shear stress as being a combination of a laminar contribution and a turbulent stress contribution. And so we have tau laminar plus tau turbulent. So a lot of this work took place back in the 1920s, 1930s and what the researchers found was that the laminar shear is mainly right near the wall and for that reason they often refer to this as being the wall layer or wall region and it's how turbulent that dominates further out and that is referred to as being the outer region or outer layer. And so if you look at the velocity profile with a turbulent velocity profile and you look at the shear stress you can break it into a laminar and a turbulent component and so that's what the researchers did years and years ago and so what we're going to do let's draw that out. So if we're to look at the turbulent stresses in the velocity profile for either pipe flow or a flat plate boundary layer what we find is that there's the viscous wall layer and that's where the viscous shear dominates then we move into an overlap region which is in here and that's where you both have turbulent stress as well as viscous stress being important and then you get into the outer turbulent layer where the layer where the turbulent shear stress is the thing that dominates. So if we were to write this out we have wall layer we have the outer layer and that would be where the turbulent shear dominates and then we have the overlap region or overlap layer and that's where both of them are important. So both laminar and turbulent shear is important. So what we're going to do is we're going to proceed forward and I'll show you three different velocity profiles that were developed back in the 1930s by Millican, von Karman and Prandtl and in order to do this we're going to drop the bar from u-bar so from the velocity tau wall equals wall shear stress delta which I've shown in this picture here is the boundary layer thickness and capital U is velocity at edge of the boundary layer. Okay so we're going to use those things and what Prandtl and von Karman and Millican did is they went through and they came up with velocity profiles a lot of this was using dimensional analysis as well as experimentation but a velocity profile for the wall layer was developed by Prandtl, Ludwig Prandtl, he was at Göttingen when he did this and this was in 1930 and he came up with an argument saying that the velocity in the layer was a function of the viscosity which it would be in the wall layer the wall shear stress the density of the fluid and the distance away from the wall and using dimensional analysis he was able to come up with a thing called the law of the wall and he used transform coordinates so he took the velocity and he divided by this u star which is called the friction velocity which I'll define in a moment and he found that that was equal to a function of y the friction velocity again and then the kinematic viscosity and this term here he denoted as y plus so he gave that a transform coordinate and then u star was equal to tau wall so the wall shear stress divided by density to the one half and this is the friction velocity and so this here is equation one and it's also known as law of the wall and so that was Ludwig Prandtl 1930 his student was Theodore von Karman and and they were competing with one another quite often in terms of deriving these equations and and they von Karman he was probably at Aachen when he developed this so it was after he left being a student of Ludwig Prandtl and von Karman incidentally eventually came to the U.S. and started the Guggenheim Aeronautical Laboratory at Caltech where he was very instrumental in the starting of it but he came up with the velocity profile for the outer layer so this is where turbulence shear is important and again dimensional analysis that's really all that they had at their disposal other than that and experimental data which at the time was not the greatest but they accomplished great things with what they had so dimensional analysis and von Karman was able to come up with this relationship and he did scaling based on the boundary layer thickness that's delta there this is equation two and this is referred to as being the velocity defect law so we have law the wall we have velocity defect law and that was in the outer region or the outer layer and then in the middle the overlap region this was investigated by a fellow named Milliken and he looked at pipe flow as well as channel flow and in about 1937 he came up with a velocity profile again using Ludwig Prandtl's friction velocity and y prime coordinate so he came up with this relationship and here k is equal to 0.41 and b he said was 5.0 and that was based on the experimental data that he had these numbers are constantly being fine tuned as people get more and more data for boundary layers turbulent boundary layers but in here the he also defined a u plus which was velocity divided by the friction velocity and he had a y plus which was y times the friction velocity divided by nu remember the friction velocity was our shear stress at the wall divided by density square root of that and consequently what Milliken was able to come up with was this expression and this we will call equation three now it turns out that this is the equation that applies for quite a bit of the boundary layer and it's a quite representative of the velocity profile for a turbulent boundary layer and consequently that is what we are going to use as we move into the next segment where we will look at developing a relationship for the friction factor for turbulent flow but it is quite remarkable when you look at all the things that von Karman Milliken and Prandtl were able to do this was all based on dimensional reasoning dimensional analysis and experimentation but give them credit they did not have the fancy tools that we have nowadays with time-resolved particle image of law symmetry laser Doppler of law symmetry even hot wire anemometry and if they did it was very very primitive electronics that were involved in those systems so those are the regions we are going to use this one as we move into coming up with a relationship for the friction factor which we'll do in the next segment