Welcome back to Sports and Building Aerodynamics, in the week of basic aspects of fluid flow. In this module, we're going to focus on fluid statics, kinematics and dynamics. We start again with a module question. In fluid kinematics, a pathline is defined as the trajectory of a particle as it travels through space and time. On the other hand, a streamline is a line that at a given moment in time is tangential to the velocity vector in each of its points. What do you think then is shown in the animation below? And this animation is actually a smoke visualization in a wind tunnel test of the flow around an airfoil. Are these A, pathlines, B, streamlines, or C, neither pathlines, nor streamlines. Please hang on to your answer, and we'll come back to this question later in this module. At the end of this module, you will understand some basic concepts of fluid statics, fluid kinematics, fluid dynamics, and you will understand the differences between Navier-Stokes equations, the Euler equations and the Bernoulli equation. So let's start with fluid statics, which is the study of fluids at rest. Then we'll continue with fluid kinematics, which is the study of fluids in motion. However, without considering the forces that bring about this motion. And finally, with fluid dynamics, we'll consider the relationship between the forces and the motion. [BLANK_AUDIO] So, we start with fluid statics here and we only give a very brief summary. It derives actually this word from the Greek statikos, to cause to stand. And we can define the hydrostatic pressure there as the pressure exerted by a liquid, by water in rest. So imagine we have this container here and the point C submerged in the liquid, in the water at the depth h below the water level. And above the water level we have a certain reference pressure. Which can be the atmospheric pressure. Then actually the hydrostatic law says that the pressure, which is static pressure in this case, in point C is given actually by the sum of these external reference pressure and the pressure caused by the fluid column above this point. In an equation form, this is indicated as follows. Where as mentioned before, P0 is the external pressure, rho is the density of the liquid, g is the gravitational acceleration and h is the depth below the water level. [BLANK_AUDIO] Then we can also look at the variation of atmospheric pressure with height. And under isothermal conditions, this is given by the so-called barometric formula, which is indicated here. So the variables you see in this equation are the molar mass of dry air, the gravitational acceleration, the height above the sea level, the universal gas constant for air, and the temperature. And when we show this in a graph, you see the very strong decrease of pressure with altitude. Then the law of Pascal is another important law in fluid statics. Let's focus again on this point C submerged in liquid and then we take an imaginary plane A. And if you integrate the pressures on both sides of this plane over this surface area, then we get the forces, and then this law of Pascal actually says these two forces are the same. The definition I provided here on the slide can be quite confusing. It says that the pressure in a point in a static fluid is the same in all directions, and only dependent on the density of the fluid and the depth of the point below the fluid surface. But, it's a bit tricky to talk about directions in terms of pressure because pressure is a scalar. It is not a vector, so it does not have a direction. On the other hand, if you integrate pressure over a surface you get a force and that is vector. Another important law is Archimedes' principle, or the law of buoyancy. Imagine again this container with the fluid in rest, and then imagine a surface that has a spherical volume inside it. Because the fluid is at rest this also means that this volume is at rest. So if you integrate pressures over this surface area you get forces indicated with the arrows here. And these forces, actually, the net force will be the same as the weight of this water volume. If not, it would be in motion and not in rest, and that was our assumption here. This also means that if you replace this volume by a real volume of another material, then when the density of this material is larger, it will sink. When it is smaller than water, it will float. And the force on this volume can be calculated with this equation, where it is important to realize that the density is the density of water. So this means if you immerse a body in a fluid. It would be buoyed up by a force equal to the weight of the displaced fluid. So not the weight of the volume but the weight of the displaced fluid. Let's briefly focus on fluid kinematics. Also here, just a very brief introduction of a few aspects. So we considering the motion here. This word derives from the Greek word kinema, which means motion. And we only focus on location, speed, acceleration and time and the relationship between those four variables. There is an important distinction to be made that is the Lagrangian versus the Eulerian approach. Lagrangian approach actually comes from Joseph Louis comte de Lagrange which as opposed to what his name would suggest was an Italian mathematician and astronomer and the Eulerian approach comes actually from Leonhard Euler who was a Swiss mathematician and physicist. So the Lagrangian approach actually means that you are going to track the movement of a particular particle, one maybe, in space and time. And that is the so-called particle description. In the Eulerian approach you are going to focus on fixed points in space, and you are going to study the fluid motion as a function of time of the particles that flow through this particular point, or through these particular points. This is a so-called field description. Another important definition is that of the substantial derivative which is a derivative in the Langrangian framework. There are many names for this derivative. Sometimes it's called total derivative, Lagrangian derivative, material derivative, also particle derivative, and there are even more names for it. It's defined as follows, so it is based on this continuum assumption that we made before. And it's the total rate of change of a given variable, a, that varies with time and space when subjected to a velocity field v. And then that is the definition indicated here where D indicates substantial derivative. The round d indicates the partial derivative. x is position, t is time, v is the velocity vector, and del is the gradient operator. So what you have here is actually three terms. This is the total rate of change. This is what you get in terms of rate of change when you follow a fluid particle in its path through space and time. That's according to the Lagrangian approach, this is the local rate of change at a given fixed point, so this is the Eulerian approach, focus on a fixed point and you see what happens, what changes in this point when the particles pass through it, and finally there's a so-called convective rate of change, that's due to the fact that the fluid particle that is convected from one location to another and there the value of a, it can be any parameter, will be different or can be different. [BLANK_AUDIO] Then some definitions. A pathline is a trajectory of a fluid particle over a period of time. This fits within the Lagrangian approach and actually what you see here in this animation, these are pathlines. Because smoke particles are released, and actually what you see here is that we are actually following particles in their path through space and time. Not one, many of them, but still these are pathlines. A streamline, on the other hand, is a line that at any moment in time is tangential to the velocity vector in each of its points. And this fits into the Eulerian framework or the Eulerian approach. Then the streakline is another definition, it's another concept, actually. It's the whole of locations at a certain moment in time of all particles that have passed a certain point before this moment in time. And you can determine that, for example, by injecting smoke or a dye into the flow at a fixed point and over an interval of time and then looking at the particular streaklines as indicated in this drawing. And a nice but unhealthy example of streak lines is what you see here in the cigarette smoke. So, let's turn back briefly to the module question, which actually has been answered just before. Well, what you see here actually indeed are pathlines, not streamlines, also not streaklines. The pathlines are indicated in white, and are actually, the lines, the trajectories of the smoke particles as they move in this flow field in the wind tunnel around this airfoil. A streamtube is a tube that is formed by streamlines starting and ending on a closed surface, as illustrated in the drawing here. There is no flow through the walls of a streamtube, because indeed these walls are composed of streamlines, and there the velocity vectors are tangential to these lines by definition of the streamline. At a certain moment in time, when you have a fixed streamtube, this means that you can apply conservation of mass for the streamtube. So, what does that mean? Well, conservation of mass is also sometimes called the continuity equation, and specifies that the incoming mass in a certain volume during a certain time interval, minus the outgoing mass applied to the same volume and the same time interval, equals the increase of mass in the same volume and in the same time interval. So in differential form, you can describe this as follows. With a local derivative of the density and then the divergence of the product of density and velocity. If you have a steady flow, then the time derivative vanishes. And if you have an incompressible flow then the density can be actually also removed and then you end up with this very simple continuity equation. The divergence of the velocity vector has to equal zero. Let's then turn to fluid dynamics. Here we consider the relationship between forces and motion. This is derived from the Greek word dunamikos, powerful. And important equations here are the Navier-Stokes equations which are named after Claude-Louis Navier and George Gabriel Stokes, respectively from France and from Ireland who were two researchers that actually independently from each other arrived to the same set of equations. And that's why both of their names were given to those equations. And let's have a look at those equations for the simplified case of homogeneous, isotropic, incompressible, Newtonian fluid. And then the momentum equations or the Navier-Stokes equations in the three coordinate directions look as follows where you see that u, v and w are instantaneous fluid velocity components. So those will be indicated with the small letters, p is also the instantaneous pressure also indicated with small letters, and then G with subscript x, y, and z are the body forces. And all these equations might look complex. Actually they are not. Well, they're very complex to solve but they're not that too complex to understand. So let's have a look, a more closer look, at these equations. Actually, what you see here is again, a local derivative. In the Eulerian framework, there is a local change in velocity due to the change in time. Then here you have the convective change, the change in velocity due to the movement in the fluid from one place to another. So actually, this is the acceleration and then on the right-hand side, you have the forces. You have the body force, you have the force by pressure differences, and you have the dissipative viscous forces. So actually what is shown here is nothing more than Newton's second law. Force equals mass multiplied by acceleration. Then we can simplify those equations further if we assume that the fluid is inviscid. And those are then called the Euler equations, there we take out the viscous terms. And then we arrive at a slightly less complex system of equations. Then there is the Bernoulli equation, well there are actually different forms of the Bernoulli equation. Here we are going to look at the Bernoulli equation for steady flow along a streamline. This actually is based on the work of Daniel Bernoulli who was a Swiss mathematician and physicist. So we're going to look at the flow of an inviscid fluid. It is a steady flow along a streamline. And we only consider gravity as external force. And then Bernoulli's law can be expressed as follows. Here you see the capital P because we assume a steady flow. So this is a mean value, also the capital V, this is also the mean speed. You also see the gravitational acceleration indicated there, multiplied by density, multiplied with height. So these first two terms are the static pressure and the third term is the dynamic pressure. So we can apply that to, for example, flow in a tube where we can often clearly define streamtubes. For example, here's a streamline in the middle. Then we can apply the Bernoulli equation between two different points on the streamline, two different positions and calculate some variables based on the knowledge of the others. We can apply the Bernoulli equation also, for example, to the steady non-separated flow around a circular cylinder in a horizontal cross-section. When we have a horizontal cross-section, the difference in height actually is zero, so this term can be left out. So, this means that when you illustrate streamlines, as we do here in this graph, then along the steamline you can apply the Bernoulli equation. So, let's take two points, one point upstream, one point very close to the cylinder, that means that when you're close to the cylinder, you see that actually the streamlines, which also could be considered as streamtubes, as bounding streamtubes, that actually these streamlines get closer to each other. Because of continuity of mass and we also assume incompressibility here. that means that in point 2 we have a higher speed than in point 1. Because of the Bernoulli equation, and because the sum of static and dynamic pressure needs to be constant, a higher speed means that we have to have a lower static pressure. So, in this way, you can also look at pressures, at the surface of this circular cylinder. So, this is actually another illustration of the speed around a circular cylinder. So a high density of streamlines means high local speeds, indicated through the red color here. But it also means, a low speed where the streamlines, actually, are farther away, which is the blue color here. So those are the positions of the highest pressure. And in red you have the positions with a low pressure. If you would integrate the pressures that you find here along the circumference of the cylinder, you would find something that is potentially very wrong with this figure. And if you want to know what is really wrong with this figure please keep watching because we'll address it in the next modules. In this module, we've learned about some aspects of fluid statics, some of fluid kinematics, some aspects of fluid dynamics. We've also seen what the differences are between the Navier-Stokes equations, the Euler equations, and the Bernoulli equation. In the next module, we'll start focusing on boundary layers. We'll explain the concept of the boundary layer, the difference between a laminar and a turbulent boundary layer, and how skin friction is influenced by the type of boundary layer. Thank you for watching and we hope to see you again in the next module. >> [BLANK_AUDIO]
