 As we start talking about external flow, we're going to be taking our analysis from internal flow and applying it to everything else. The modeling approach we use isn't going to be very different. We're going to try to characterize our flow as generally as possible, and then apply correlations to that characterization. But the first thing we need to do is limit our scope a little bit. External flow encompasses analysis of flow around everything. Even internal flow can be roughly approximated with an external flow analysis. There are three general ways to approach an external flow problem. The first is computational fluid dynamics, where we let a computer perform calculations on a whole bunch of little tiny cells and then group them together to come up with an analysis of the whole. The second method is the application of experimental correlations to a general case, like we did in our internal flow analysis. The third is an old-fashioned method where you simplify an analysis to flat shapes and use their boundary layer to simplify the Navier-Stokes equations. The best sort of analysis that we can do is actually a combination of options one and two. You build a CFD model, you support it with experimental data, and then you apply it to a slightly different situation. But before anyone can get into that, they have to start to understand where those equations come from. And for the simple case, we're going to be using boundary layer theory. For the purposes of this class, we will talk primarily about option three and a little bit about option two. CFD will be outside the scope of this class. If you are interested in that, and I would very much encourage you to try to get interested in it, as that is the future of fluid mechanics and heat transfer. I would encourage you to take the computational fluid dynamics tech collective or the aerodynamics tech collective, which involves a lot of CFD. In our boundary layer theory analysis, we're going to be talking about the development of boundary layers across primarily flat bodies. Like with internal flow, we are going to be developing flow characterizations for different situations and the way that the boundary layer develops in those situations. For our purposes, we're going to be splitting our flow characterization into laminar and turbulent analysis again. But this time, we're going to be using a different critical Reynolds number. You'll remember that for internal flow, we can pretty broadly characterize the flow as transitioning at a Reynolds number of about 2,300. That critical Reynolds number didn't really change because we were talking about flow through primarily round shapes. Even square or rectangular ducts can be pretty safely approximated as a round pipe. But for external flow, there are many more shapes that we need to consider. And as a result, the Reynolds number will vary a lot more. The critical Reynolds number is going to be specific to a type of case or to a shape. For flat plates, the Reynolds number we use for the critical transition from laminar to turbulent is 5 times 10 to the fifth or 5e5. You can see here the approximations we're going to be using for the development of the boundary layer for laminar and turbulent flow. That boundary layer thickness delta is expressed as a non-dimensional parameter here as delta over x, where x is your position back from the leading edge of the plate. And for laminar flow, that's equal to 5 over the Reynolds number to the one-half power. And for turbulent flow, it's about 0.16 divided by the Reynolds number to the one-seventh power. You'll also notice that the Reynolds number here is expressed with x as the characteristic length. So when you're calculating the Reynolds number for a given position, instead of using diameter like we did in internal flow, we are using the position back from the leading edge of the plate. Furthermore, we need to understand that when we talk about flat plate theory, we are neglecting the effects of the shape of the body itself. If you consider around cylinder in cross flow, the boundary layer development around the cylinder will have an effect on the drag, but so will the negative pressure region on the trailing edge of the shape. So when we talk about flat plates, we are talking about their drag primarily in the form of friction, also known as skin friction. When we start adapting that analysis to broader bodies, we are going to be developing both friction drag and pressure drag. When considering our flat plate drag, we will develop equations for the laminar and turbulent cases. For the laminar case, we are using this simplification developed by Blasius. We can see here that this analysis develops a coefficient of drag, which is a non-dimensional parameter, which represents the proportion of the drag force to one-half times the density times the fluid velocity at the free stream squared times the area of effect. And that's approximately equal to 1.328 divided by the Reynolds number where the length of the plate is the characteristic length raised to the one-half power. Note while we're here that area is the area of effect of the flat plate, if you have drag on both sides of a plate, that is going to be double the area of the plate itself, because you're accounting for the area on both sides of the plate. For smooth turbulent flow, we can develop another representation for the coefficient of drag that's going to be 0.031 divided by the Reynolds number where the length of the plate is the characteristic length raised to the one-seventh power. And I will point out while we're here that this equation is only for the turbulent part of the analysis. You cannot always assume that the turbulent flow starts from the leading edge of the plate. In situations where you have developing boundary layers with laminar conditions on the very front of the plate and turbulent conditions everywhere else, we can use this relationship. This will give us the coefficient of drag that includes the laminar and turbulent part of the plate. For rough turbulent flow, we have these equations, which are correlations from experimental data. Note that the epsilon value here is the same roughness we used in internal flow. And unlike the Colbrook equation, we don't need a fancy calculator to be able to handle this math. It's a lot easier to perform this calculation on a regular calculator. But, just like the Colbrook equation, we also have access to the graphed data, which is convenient in certain circumstances. The blue lines here represent the proportion of the length of the plate divided by the roughness of the surface of the plate. The x-axis is the Reynolds number, again where the length of the plate is the characteristic length for determining the Reynolds number itself. And the y-axis is the coefficient of drag, which again is the drag force divided by one half times the density times the free stream velocity squared times the area of effect. The only difference in our analysis of the pressure drag instead of the friction drag is where the coefficient of drag comes from. The textbook gives us some common drag coefficients for some common geometrical shapes. We have a table full of coefficients of drag for two-dimensional flow and a table full of coefficients of drag for three-dimensional flow. For any other shape, we would have to determine the coefficient of drag from experimental data, or we would have to look up the coefficient of drag published by the manufacturer. So in conclusion, we're going to use a coefficient of drag or a coefficient of lift to determine a drag force or a lift force. And for the coefficients of drag, we are going to consider either the drag coefficient on the surface for flat plates, or around the body for blunt objects. And those will either come from a table, from a calculation we make from experimental data, or from the calculations made from general correlations for flat plates. Let's try an example.