 So we're looking at experimental external flows and the first force that we're going to look at on objects is going to be drag and we often characterize drag in terms of the drag coefficient and so the way that we refer to that is c with the subscript d and the drag coefficient we saw this in dimensional analysis earlier on in the course quite often it will be a function of the Reynolds number because things happen within the boundary layer around objects that have implications on the pressure distribution and that leads to impact on the drag coefficient which we will see in this segment and the Reynolds number here and the drag coefficient would be the drag force whatever it might be and we divide by the dynamic pressure and you'll notice in both the Reynolds number as well as in the drag coefficient we have a number of things v is the free stream velocity so that's pretty easy but we have this characteristic length and we have some area and so you have to be a little careful with that and so we're going to spend a little bit of time looking at how those are defined so to begin with the length in the Reynolds number so it is some characteristic length of the body that is being investigated and it could be examples could be the diameter it could be the cord length if you're looking at an airfoil the main thing is whenever you're using experimental data and drag coefficients make sure you know what the characteristic length that they have been doing their experiments at or using the other thing is the area that is in the drag coefficient so let's take a look at that so the area can be referred to as being the frontal area and that refers to the area as seen from somebody in the fluid moving towards whatever body you're characterizing so that could be as examples a sphere or a car it's basically the projected area as seen from somebody coming towards the body so another form of area is the plan form area and that refers to the body as seen from above and so examples where you may use the plan form area could be a wing you might use the plan form area so the area as looking down on the wing a nautical application would be hydrofoils and it's looking at the area from the top of a hydrofoil so that's plan form area it's another area that could sometimes be used and a final area that could be used is wedded area and again that would be from a nautical application so for ships or barges it would essentially refer to the component or the area of the ship that is in contact with liquid so the main point here is just to be careful in terms of understanding what length or area has been used to produce the drag coefficient data that you might be using okay so that is the area and the length scale and the Reynolds number and the drag coefficient now when we look at drag drag on an object for the most part well it consists of two main parts and we already talked about this earlier on in the last segment where we talked about the forces we had shear force and pressure distribution acting on a body and that resulted in all of the forces lift drag side force and then the the rolling moments that might be impacting the body but when you look at drag we have drag associated due to pressure and this is sometimes called pressure drag or form drag and what the pressure or form drag refers to is the low pressure zone that develops behind the body what will happen we'll see in a moment you get separation and there's low pressure in the separated fluid zone and that leads to a force acting in the direction of the velocity which is represented as a drag and the other form of drag is attributed to skin friction or viscous shear along the wall and this is referred to as being viscous drag and we studied this rather extensively when we looked at the flat plate boundary layer so we've looked at viscous drag the wall shear stress so those are the two forms of forces that that can result in drag we have viscous forces and we have form drag or pressure drag so what we're going to do now we're going to take a look at a video clip that illustrates some of these aspects and and so what we are going to be looking at here is the flow over a cylinder and so here we can see flow over a cylinder and we begin we have a dividing streamline on the upstream side and and then I'm going to note a couple of points that on the front of the cylinder those are separation points so you'll notice the flow is separating at those locations that's where the boundary layer is separating from the cylinder downstream we get a strong recirculation zone that would have negative pressure low pressure and that's what leads to the form drag and then further downstream we have the von Karman vortex street where there's a lot of oscillations and that leads to different types of forcing and instability that can occur downstream that's why if you're traveling down the highway and you're behind a semi-trailer truck you will find your car has side force going side to side and it's due to that vortex street behind the semi-trailer truck so looking at the flow of flow over a cylinder let me just sketch it out and we'll denote some of the regions that we just saw okay so here we have a schematic showing what was happening within that cylinder we come along we have a leading edge or a front stagnation point boundary layer forms and starts to grow as we go around the cylinder either up or down and then once we get to a location it's probably around 80 degrees I think that's typically what is found you get to the point where there's an adverse pressure gradient and so the boundary layer lifts off it separates there's zero shear along the wall without location when that happens all of the vorticity that is in the boundary layer goes into this free shear layer that's forming and and then you get these large-scale structures that are downstream and that's what leads to the what we call the von Karman vortex street now where are the forces here well we have the boundary layer viscous dried so there is shear where the flow is attached and so consequent we have shear force there and we have viscous drag and then the other thing that's happening is we have a pressure distribution on the cylinder and in this region here downstream of the separated flow region we have very low pressure and that's due to the fact we have separated flow there's a strong recirculation zone there and that is one of the main contributors to the drag on a cylinder and consequently those are the two forms of drag that would be present in a cylinder or pretty much any bluff body where you get separated flow like you do with the cylinder so if we were to try to analyze this and let's say we were to try to use the steady flow energy equation and we've seen the steady flow energy equation from our earlier analysis and I'm going to neglect the elevation term gz and if we were to try to do this between a point far upstream and let me go back so if we were try to apply this from this point here so let's say some point downstream in here which would be where we have p low v low first of all it would be very difficult to do but the fact that the flow is not irrotational doesn't mean that this exists we have to have something else on the right hand side and what is on the right hand side would be the loss term so there would be losses associated with all of the turbulence downstream of in the separated flow area and the losses we saw when we looked at pipe flow we had the major losses for a pipe or another way of thinking of this was the when we were looking at the minor losses in pipe flow things like elbows or expansion so minor losses so if you really wanted to apply this what you would need to have is a way to be able to quantify all of the turbulence that is occurring in the wake of the cylinders so you would need to have a way of being able to account for all of the energy being lost in this area in order to be able to apply that and that's not a very good case you probably would never apply the Bernoulli equation between this point infinity and this point low but if you were to try to what you would need to do is find a way to be able to represent the loss of energy and so that's kind of what that is referring to there so that is flow over a cylinder and also some of the drag characteristics we'll spend a little bit more time in an X segment looking in more detail at the drag characteristics and what they might look like and then we'll move on and we'll look at left on external flow