Welcome back to Sports and Building Aerodynamics, in the week on basic aspects of fluid flow. In this module we'll start focusing on boundary layers. And well this module might be experienced as quite tough. It's a lot of information in a rather short period of time so I think this quote by Aristotle is again very applicable. The roots of education are bitter, but the fruit is sweet. So please hang on as we walk together through these modules because they will really benefit you in understanding the rest of this MOOC. We start again with the module question. Imagine that you have a ball with a smooth surface and it is launched into the air. First, you do that with a flow Reynolds number that yields a laminar flow in the boundary layer. Next, you do it again, but then with a flow Reynolds number that yields a turbulent flow in the boundary layer. The question is, in which case is the friction at the surface of the ball the highest? Is it A, in the laminar case, B, in the turbulent case, or C, equal in both cases? [BLANK_AUDIO] Please hang on to your answer, and we'll come back to this later in this module. At the end of this module, you will understand the concept of the boundary layer. You will understand the difference between a laminar and a turbulent boundary layer, and you will understand how skin friction is influenced by the type of boundary layer. [BLANK_AUDIO] Part of the information here in this module was obtained from these two books, which are definitely recommended reading material. Let's start with the so-called paradox of d'Alembert. Let's focus again on the Bernoulli equation that we explained in the previous module. And when we apply that to a horizontal section of a column, for example, in a cross-flow which can be a flow of water, then the height is the same along this cross-section, this horizontal section. And you can visualize the streamlines and the contours of the speed around this column as follows. Where actually the place where the streamlines come together indicates that the speed increases. And where they go farther apart the streamlines, so wider streamtubes, the speed decreases. If you look at the pressure distribution around the circumference of the cylinder where you have the angle theta as indicated in this drawing. And we make the pressure non-dimensional so we subtract the reference pressure from the pressure, so the reference pressure is the pressure upstream in the flow in the undisturbed field and we divide that by the dynamic pressure in the upstream flow. Well, then we get this illustration here, and if you visualize that along the circumference of the cylinder, then you get these curves for the overpressure. So the larger pressure than the reference case. So where the size of the pressure is indicated here by the distance between the line and the point on the surface. And this is then the underpressure, the negative pressure, compared to the reference pressure. And if you integrate those pressures around the circumference of the cylinder, you find the force, and actually the net force is zero. But this is not what is found in experiments. It is also not what you would experience if you're standing in a river that is flowing and actually you will feel that your body is being dragged down by the flow. And this is called the paradox of d'Alembert. So, something is wrong here with this pressure distribution. Actually, what is wrong here is that the concept of the boundary layer has not been taken into account. And this concept actually is due to Ludwig Prandtl, who was a German engineer who did quite some work in his research water tunnel in Hannover. And in 1904, he presented an important paper that was called: Uber Flüssigkeitsbewegung bei sehr kleiner Reibung. Which means, on the motion of fluids with very little friction. And he did that at the third the International Congress of Mathematicians in Heidelberg, Germany. And actually, in this presentation, in this paper, he launched the hypothesis that for low viscosity, the viscous forces are negligible everywhere in the flow, except close to the solid surfaces where the no-slip condition applies. And the no-slip condition means that the speed is zero at the surface and that the stress is not zero at the surface. The opposite to that is the slip condition, where the shear stress would be zero and the speed would not be zero at the surface. So, in the boundary layer concept, we assume the no-slip behavior. The boundary layer concept actually also included statements that the thickness of boundary layers approach zero when the viscosity goes to zero. That outside the boundary layer, the flow is essentially inviscid, and that this concept explains drag, flow separation, and so on. Quite important practical consequences for flows in engineering. There are different ways in which you can define the thickness of the boundary layer and this is one of them. Based on a certain value that is reached that is a fraction of the outside flow, outside the boundary layer. So here, the height of the boundary layer is indicated by the place where the speed is 99% of the speed outside the boundary layer. This is an arbitrary value. We could have also taken 95%, or 90% or 85%, but generally 99% is used. Then, in the boundary layer, also the thickness of the boundary layer varies with the flow Reynolds number. This is indicated here, where the average boundary layer thickness is indicated with d and the overbar. L is a characteristic dimension of the body, which can be an airfoil or a ball in the streamwise direction, and you also see the Reynolds number indicated here. So this means indeed that when you increase the speed of the flow, that then, the thickness of the boundary layer will become less. This is important. This is also something that you can experience when you're walking outside in very cold weather and you would blow air over your skin. Then you would feel the cold or you would feel the increased heat flux from your body to the outside air. The reason is that by blowing over your skin, you decrease the thickness of the boundary layer because you increase the flow Reynolds number. And actually the boundary layer offers the largest resistance to heat and mass transfer. So well, when you decrease this layer, you decrease your insulation against the cold. And the same will happen when you're in a sauna, when you blow air over your arm, you will feel, you will experience that the heat transfer will be augmented substantially. [BLANK_AUDIO] Let's focus now on the flow over a semi-infinite flat plate with a zero thickness. Then this is a typical drawing that shows the height of the boundary layer. I have to mention that this drawing is not to scale, so the vertical scale is enlarged here very substantially and the line that is drawn above this plate, that indicates the thickness of the boundary layer. What you see also here is a leading edge, so the place where the flat plate starts. Then you can define the critical Reynolds number, based on the distance along the plate. And what you see here in this Reynolds number is indeed the distance along the plate. You also see the approach flow speed indicated here with U subscript infinity. Then what you will get in the beginning is a laminar boundary layer, where the boundary layer will grow as you move further along the plate. Then at some point, you will get instable behavior, sporadic vortex-like instabilities, and at some point later on the flow will transition to a turbulent boundary layer, a fully turbulent boundary layer, where you will have a much more rapid growth of the boundary layer thickness. So again, I have to stress that this figure is not to scale. Let's then focus on the differences between the laminar and the turbulent boundary layer. Well, first of all, there is a different velocity profile. There is also a different boundary layer growth rate, as mentioned before. And there is a different shear stress. And this is indicated by these tangential lines here. And if you take the derivative of the speed as a function of the vertical coordinate, you will indeed find, based on the definition of shear stress, that the shear stress in the turbulent case is substantially larger than in the laminar case. [BLANK_AUDIO] Let's look at the structure of the boundary layer. You see here the wall indicated at the bottom and then the flow over this wall from left to right and also vortical structures indicated, and actually these structures get smaller as you move closer to the wall. Because, simply, close to the wall, there is no room for large structures to exist and very close to the wall, actually you will have no vortical structures at all anymore. You will have a laminar flow there. So, we distinguish first of all, between an outer layer that is fully turbulent and an inner layer where we have viscous effects. Then we can subdivide the inner layer even further in a log-law layer, a buffer layer, and a linear sub-layer. In the linear sub-layer, the viscous effects are important and you have almost no turbulent effects. In the buffer layer, they keep each other more or less in equilibrium. And in the log-law layer, the inertial effects are much more important than the viscous effects. These are a few illustrations of flow in the vertical cross-section over a semi-infinite flat plate. And this is a turbulent boundary layer indicated here. This is another illustration of a turbulent boundary layer, again, flow from left to right. And this is visualized by laser-induced fluorescence. And this is a perspective view of a boundary layer that exhibits these streaks, these sporadic disturbances, instabilities, as the boundary layer transitions to turbulence. [BLANK_AUDIO] This is a graph that illustrates again, the semi-infinite flat plate and also some typical values of the Reynolds numbers where transition to turbulence might take place. But this value where the transition occurs, the critical Reynolds number, depends largely on the surface roughness. If you have a larger surface roughness, the transition to turbulence will occur faster. Also, if you have large intensity of fluctuations in the free-stream, so turbulence intensity, also then the transition will occur faster. And the shape of the leading edge is also important. And here we have assumed a zero thickness flat plate. So actually an infinitely sharp leading edge. So then these are some typical values. If the Reynolds number is below 60,000, then you will have a laminar boundary layer irrespective of the type of disturbances that you will apply, roughness or approach flow fluctuations. If it's larger than 4 million, then the transition to turbulence will be complete, but often, transition to turbulence will be completed already much earlier. We can then also look at the drag coefficient, or the skin friction of a flat plate. And here it's important to realize that this is a flat plate with a zero thickness with an approach flow speed parallel to the plate. So the aerodynamic drag here is only composed of skin friction. The form drag is zero. Then we can have these typical values of the drag coefficient. Illustrated here in green, is the case when you have the completely laminar boundary layer. In purple, the completely turbulent boundary layer, and indeed at some case, in this case 500,000 you get transition to turbulence, then the black curve in between should be followed to find the corresponding drag coefficient. So what you see here indeed is that skin friction is substantially larger for the turbulent boundary layer and this is indeed directly related to the shape of the fluid speed close to the wall. Again, I have to stress this is for a flat plate with zero thickness where the form drag is zero, and the only drag component is the skin friction drag. So let's turn back, then, to the module question. With a smooth ball that we launch into the air first with a flow Reynolds number that yields laminar flow, and then with a flow Reynolds number that yields turbulent flow. In which case is the friction at the surface the highest? Well, this is a turbulent case. This is due, as mentioned before, to the shape of the fluid speed profile close to the wall. Actually, in the point at the wall, where also the surface shear stress will be different. In this module, we've learned about the concept of the boundary layer. We've seen what the differences are between a laminar and a turbulent boundary layer, and how the skin friction or the skin friction drag, not the form drag, is influenced by the type of boundary layer. In the next module, we're going to focus on the concept of flow separation. We're going to see how flow separation is influenced by the type of body, bluff versus streamlined bodies, how it's also influenced by the type of boundary layer, laminar versus turbulent boundary layer. How the flow around a circular cylinder changes as a function of the Reynolds number. And we'll also look at two very interesting counter-intuitive aspects on boundary layers and flow separation. Thank you again for watching and we hope to see you again in the next module. [BLANK_AUDIO]
