 Okay so we've been talking about Bernoulli's equation. What we're going to do in this segment is we're going to take a closer look at Bernoulli's equation and we're going to think about the interpretation or physically what does it tell us about the flow. So what Bernoulli's equation does is it says that if we evaluate p over rho plus v squared over 2 plus gz that is going to be equal to a constant provided that we are on the same streamline. So along a streamline. So if we have a flow field that looks something like this and these are our streamlines, what that tells us is that if we want to apply Bernoulli's equation between two points, say point one and point two, and point one is here, then point two needs to be on the same streamline. It cannot be on a different streamline within the flow field. Now there's a more general form of Bernoulli's equation that enables us a little bit more flexibility and that applies if we have a condition where the flow is what we call irrotational. We haven't talked about irrotational flow yet but we will. So this is a more general form of the Bernoulli's equation. If you have irrotational flow it gives you a little bit more flexibility because what we can write is evaluating the Bernoulli's equation. We know that that is equal to a constant, only now that can be anywhere in the flow. So again if we have a flow field that looks something like this, these are the streamlines in that flow field. What this tells us is that if this is our point one where we're evaluating Bernoulli's equation, we can take another point up here point two that is no longer on the same streamline and this form of Bernoulli's will hold provided that this flow satisfies del cross v equals zero and we'll look at the definition of del cross v in a later lecture. But when we look at Bernoulli's equation what we have is p over rho plus v squared over two plus gz is equal to a constant. Now the thing about Bernoulli's equation what it provides us with, it provides us information and a relationship between what happens if you change velocity and pressure. Now usually when we're dealing with flows usually the potential change is small unless you're going over very large distances and you're dealing with a very dense fluid and so consequently we can write a couple of points in terms of interpreting Bernoulli's. So what we see with Bernoulli's equation is that when the velocity increases so let's say the velocity was to go up and let's assume that the potential change term wasn't really changing that much we're dealing with not a very dense fluid. What we can see is that if the velocity goes up the pressure is going to go down and that's a very important component or interpretation of Bernoulli's and application of it and what we're going to do we're going to look at a case that is often looked at with Bernoulli's equation and that's from aerodynamics and we're going to consider the case of an airfoil. So that is the cross section of a wing we're assuming that it's two-dimensional so I'm going to draw out an airfoil here let me close up the trailing edge so it looks a little better than that and what we're going to do typically with an airfoil you'd have your stream line coming in there's a point here that's called the stagnation point we'll talk more about that later on and then the flow off the trailing edge might look like that you're going to have stream lines coming over the top over the bottom so that would be the flow around an airfoil and we have a couple of other points here we have the leading edge that is given the acronym LE the trailing edge is given the acronym TE and then we also have the angle of attack of the airfoil that relates to the difference between the free stream velocity which would be v infinity and a line going from the trailing edge to the leading edge with respect to that free stream that is alpha and sometimes it's AOA for angle of attack this is obviously an airfoil with a lot of camber but what we're going to do now is we're going to take a look at a little experiment and and so what I have is an airfoil section that has pressure taps in it and so there are pressure taps there's one on the top near the leading edge one on the top near the trailing edge and then two more that are here and then on the bottom there are three pressure taps one right in the middle one right about there and one right about there so what these pressure taps are where they're what we would call a static pressure tap and they are measuring the static pressure in the flow so we're going to take a look at a little video clip of an airfoil going placed in a flow and we'll look at what happens with the pressure so we're going to have four pressures on the top so that's top one top two top three and then top four and then in a similar manner we'll have bottom one bottom two and bottom three and we're going to take a look at the pressure and so let's take a look at this video and see what happens to the pressure when you have an airfoil in this situation so here is a picture from end of the airfoil and there you can see the location of the pressure taps there are four on the top three on the bottom and those pressure taps are connected to tubes which go to in a manometer it's a water manometer so we have a number of different sections in the manometer where we can measure the pressure and so I'll label it here and there you can see the four top the three bottom and one at atmospheric and what we're going to do let's put this in a wind tunnel it's actually not a wind tunnel it's a fan but what you can see is the pressure is changing at different positions along the airfoil at the front they're the top one two and three you can see that the pressure is lower and consequently the manometer lifts up and on the bottom the pressure is higher and consequently those move down and closer to the end there's not really much of a change and so you might be wondering what's going on near the trailing edge of the airfoil in order to understand that a little bit more thoroughly within fluid mechanics we have a method called flow visualization and so now what we're going to do we're going to take a look at what is happening on an airfoil as the angle the tack is changing and so there we have a picture of an airfoil so let me begin the video and you'll notice I'm going to change the angle the tack of the airfoil watch what happens to the stagnation point the point where the streamline comes to zero on the front and there's also a separation point on the back of the airfoil so we're going to run this so there you can see at a low angle the tack the flow comes along and it hits the stagnation point that's the point where we have zero velocity on the leading edge of the airfoil and then on the top of the airfoil as we go to higher angle of the tack you'll notice that there's separation from the upper surface and as we go to higher and higher angles of attack that separation point moves forward so what you're noticing here and here we're going to run through that clip again but what you're seeing is that near the back of the airfoil we start losing the effect of Bernoulli's because we have separation and consequently that's why from the earlier clip when we were looking at the performance of of the airfoil in the wind tunnel we were noticing that the pressures near the trailing edge were really not changing that much and the only place where we saw a significant change was on the leading edge and and that's that's because there the flow remains attached and Bernoulli's equation applies further downstream where you get separated flow you no longer can effectively apply Bernoulli's equation and and also the pressure there is is not low because the velocity is not high so looking back at our airfoil that's a bit of an application of Bernoulli's equation and it gives us an idea as to how Bernoulli's equation tells us how velocity changes and how pressure changes within a flow field so looking back at this airfoil we would have a case where on the top this was a high velocity in here high high and then here we have the separated flow so you're not really seeing a lot and that's why that tap 4 didn't change much and on the lower surface we would have a lower velocity and consequently you didn't see the effect of Bernoulli's where you would have a lower pressure you actually had a higher pressure on the lower surface and ultimately that is what results that pressure differential results in lift on an airfoil although it doesn't act at the middle it usually acts up here at the quarter chord location let me erase that because that is not correct in terms of where you would have left it would not be there lift would be at the quarter chord enough at some place like that so anyways that's an interpretation of Bernoulli's equation a little bit of an experiment to show you what the pressure distribution looks like on an airfoil