 Another topic with pipe flow is that of the entrance length. Now it turns out that whenever you initiate flow within a pipe, it takes time before you get to what we call fully developed flow within that particular section of pipe. And so in terms of entrance lengths, quite often what we do is we draw the scenario or a schematic where you have a section of pipe and let's see how the Belmouth housing coming in. So you would have your section of pipe and here the axis along the length of the pipe is X and in the radial direction is little R. Now what happens is as the flow comes in, you have a boundary layer developing and growing along the wall of the pipe and a boundary layer is a region where viscosity is starting to play an important role and consequently you would have a velocity deficit within the boundary layers. The velocity is lower than it is at other points in the flow. And so coming in you might have quite often we show the realization of a top hat velocity profile, although it won't look exactly like that because we have this Belmouth here, but that's kind of an idealization that we quite often approximate in fluid mechanics. And then along the wall we have the zero slip condition and so the presence of viscosity diffuses out away from the wall giving us a velocity profile that goes from zero to some free stream amount or the amount that we would be seeing at the inlet. And so that's what the velocity profile would look like and this boundary layer obviously is going to grow all the way around so we have a round pipe and so it's kind of coming in. And eventually you're going to get to a point with that boundary layer meets and so the boundary layer around the pipe meets and that is when you have what we call the development of fully developed flow. So on the other side here obviously we would also have a similar sort of velocity profile coming along going from zero at the wall out to some free stream value and it would be symmetrical. And then you get into this region here and that's where the presence of viscosity has not caused the velocity to slow down. And we call this middle part here the inviscid core where sometimes you'll hear it referred to as being the potential core. And on the sides we have growing boundary layers. And then once you get beyond where the boundary layers meet that is when we have what we call fully developed flow. And here the velocity, let me try to sketch it in assuming it's laminar for plus cell pipe flow. It'd be a parabolic shape. That doesn't quite look parabolic but it should be. Let me clean that up. That line is not good so it should be a parabola. That looks a little better. And the velocity there u is a function of r alone so it's only a function of the radial. Here as we come through this region the velocity is a function of r and x. So it's a function of radial location and position. And we refer to this here. This region from the inlet to there that is the entrance region. And then beyond that is what we would refer to as being fully developed flow. And so the significance of this is that first of all it has an influence on the approximations it will be making as we derive the equations. But it also has implications in terms of flow metering and other things like that. We show here entrance lengths for just coming from let's say a large reservoir. But when you go around an elbow for example you have secondary flow that develops and it takes time before the flow returns back to being what we would call fully developed flow where the velocity is then only a function of r and not a function of position x. So we have a thing that we refer to as being an entrance length. And that is given the symbol l e. And I'll give you approximations for that based on whether or not you have a laminar flow or a turbulent flow. And so let's take a look at that on the next slide. So if you have laminar flow in an entrance region we normally quantify based on the number of pipe diameters. So looking back at our schematic I didn't draw the diameter on here. But that would be the diameter of the pipe. And so we determine an entrance length based on the number of pipe diameters. And for laminar flow you get a relationship that is approximately this where this is Reynolds number based on diameter. And for turbulent pipe flow you get a different relationship raised to the power of one-sixth. So those are two different equations that you use. You'd have to calculate your Reynolds number to determine if you're a laminar or a turbulent. If you're laminar you would have one way to calculate the entrance length and turbulent you would have another. So what we're going to do now we're going to take a look at a quick example involving the entrance lengths. And we're going to consider the case of a wind tunnel. Okay so here we have a problem in the case of a wind tunnel. Wind tunnel is basically just a big pipe. Although it might not have a round cross section. It could have square, rectangular, very very different cross sections. But when we look at a wind tunnel now this is going to be a very poor drawn wind tunnel. But you have your contraction on the front. Sometimes you'll have screens there that will clean up the flow. But then you get into your test section region. And then eventually you'll have a diffuser out here. And you'll have a fan. That's if it's an in-draft. A wind tunnel is what I've drawn here. So it's not a closed-circuit wind tunnel. But what happens is you get a boundary layer that grows along the wall. And you usually don't want to have your models sitting in that boundary layer. You want to have your model in a nice clean flow. And that's why you put the screens here to clean the flow. But what we're going to do we want to figure out is this going to be a problem if we make a wind tunnel. And we'll have some numbers for that. So let's take a look at those now. So there you go. It turns out we are dealing with a wind tunnel that has a round cross section. So it is a big pipe that we're looking at here. Alright so what do we do? We have these numbers. The length of the test section we're told is 5 meters. V is 30 meters per second. That would be the average velocity coming through. Let's calculate the Reynolds number based on diameter. And so that would be, let's row Ud, but we're given the kinematic viscosity. So it would be V times diameter divided by our kinematic viscosity for air. You plug in the numbers 1.99 times 10 to the 6. So we know from our calculation with pipe flow that we're dealing with turbulent pipe flow. So we'll use the relationship for a turbulent entrance region. And when we do that for turbulent flow entrance region, Le over D, we get a number on the order of about 49. So that means about 49 diameters are required. And when we look at this particular problem, we have L over D for the wind tunnel is 5 meters divided by 1 meter. So L over D is 5. And if it takes 49 entrance lengths for that potential cord to collapse, we can safely assume that our model is probably in pretty good error. So we would have something like this. The boundary layer is slowly growing. It's going to take all the way, we'd have to get all the way to 49 diameters to get to the point where that potential core will collapse. I haven't really drawn it well on here, but it's way down there. You get a collapse of the potential core. And we're dealing up here in the front 5 diameters. So what we can assume is that our model will be in nice clean flow. And it will not have the influence of the viscous boundary layer impeding upon whatever kind of measurements we might be making. So that would be a case where we've used entrance length to show that in a wind tunnel for this particular circumstance, the entrance region is long enough that we would be in the potential core. Now, in other cases, you might be doing things such as flow metering. And if you have a pipe and you have your entrance region and then let's say you're going to put an orifice plate meter, that's a way that we measure sometimes volumetric flow rate. We get P here and P here. So we're measuring a delta P across the orifice plate. And then we have a V in a contracted coming through. But they're calibrated. We can figure out exactly what the flow rate would be based on the delta P measurement. That would be a case where you certainly want your potential core to close because you want to have uniform flow. And because that's how that flow meter has been calibrated, be it laminar or turbulent to it have different conditions. But certainly when you're doing flow metering, you want uniform flow. You don't want to be sitting up in the potential core. Whereas the case of the wind tunnel, you would actually want to be in there because you'd have nice clean flow in that section. So just don't get confused with that why we would be wanting to go in the potential core. Usually we don't for engineering applications, just in this particular case we did. So anyways, that concludes entrance links and the example problem.