 So we saw in the last set of lectures, segments, we looked at dimensional analysis and the Buckingham Pi method for ways of coming up with the Pi parameter, which are non-dimensional groups that are important for collapsing data within experiments and experimental fluid mechanics. What we're going to do now is we're going to spend a little bit more time looking at some of the most common dimensionless groups. Now the Buckingham Pi method enabled us to come up with dimensionless groups without knowing a lot about the system that we're looking at. We can go through other techniques of coming up with these non-dimensional groups. For example, if you take the Navier-Stokes equations and go through a non-dimensionalization procedure, and what I'll do I'll outline how you would non-dimensionalize the different terms in the Navier-Stokes equations. So if you were to take the Navier-Stokes equations and make substitutions for velocities, length scales, time, and pressure, and replace everything with the star terms, we would end up with an equation that looks something like this. So this would be our total or substantial derivative. We would have a gradient of pressure, plus we would have our viscous dissipation term, and it would be multiplied by some pre-multiplier here. So we've transformed the Navier-Stokes equations into this new equation, but what we notice is that we have this pre-multiplier term here. And what this is, it's 1 over a very common non-dimensional number in fluid mechanics, the Reynolds number. And so by going through this process this gives another example that there are other mechanisms or ways that we can get these non-dimensional parameters that are very, very important within fluid mechanics. So that is another technique that we can obtain these parameters. Now the physical significance of the dimensionalized groups, let's take a little bit of time looking at that. So if we look at a lot of the non-dimensional numbers that are important to fluid mechanics, it turns out that there are ratios of forces in the fluids, and so we're going to look here at six different types of forces, and then we'll look at a number of ratios and the resulting very well-known non-dimensional numbers that result. Okay, so these are some of the forces that are involved in fluid mechanics. We have the inertia force, that is rho v squared times L, viscous force, we find it's the viscosity times velocity times some characteristic length scale L, pressure force is a pressure differential multiplied by an area, so we get delta P L squared, gravity force is mass times gravity which works out to be density times some length scale cube times gravity, surface tension would be the surface tension coefficient multiplied by some length L, and finally the compressibility force is the bulk modulus multiplied by some area, and so what we're now going to do, we're going to form ratios of some of these different forces, and we'll find that when you do that some very very well-known non-dimensional numbers drop out. So we're going to begin with the Reynolds number, and it turns out that the Reynolds number is a ratio of inertia forces to viscous forces, and so if we perform the ratio using those forces we get rho vd over mu, and this is a number that is used to classify the state that the flow is in, and it typically has a rule of thumb, a low Reynolds number, although low is relative it depends upon the type of flow that you're looking at, be it a flow over a cylinder, a sphere, a flat plate, because there the characteristic dimension will change, but typically if it's a low Reynolds number for the application the flow would be laminar, and if it's a higher Reynolds number it could be turbulent, and between the two you go through a transition zone, but again these numbers exactly what does low and what does high mean depends upon the nature of the flow field that you're looking at. So that is the Reynolds number, another one is the Euler number, although quite often it's referred to as being the pressure coefficient, and this is the ratio of pressure forces to inertia, so Euler number eu, although I never use that and very rare that I see it it's often represented as being the pressure coefficient C sub p, and that would be our delta p divided by, and we introduce a one half rho v squared, and the thing in the denominator, well first of all cp is the pressure coefficient, and that can be expressed often as p minus p infinity divided by one half rho v infinity squared. Now the term in the denominator here, this is often referred to as being dynamic pressure, and it is sometimes given the symbol q infinity, and the other term in here p infinity that is pressure free stream, or free stream static pressure, and then p in this equation obviously would be the pressure that is of interest at some given point in the flow field. So that is the pressure coefficient, and a lot of this derives from Bernoulli's equation. If you recall at a stagnation point, we had v equals zero, and with that Bernoulli's equation, if you're following a streamline, so let's assume one is the stagnation point. With that we can say p1 is p0 for the stagnation pressure, v1 is equal to zero, and with that we get on the left hand side is p0, and on the right hand side we have p infinity plus one half rho v infinity squared, and this here is our total pressure, and then we have the static, and then we see also the term that is in the denominator of our pressure coefficient, that is the dynamic pressure, and so if you're wondering where we get that one half, it's coming out of Bernoulli's equation, and it just makes it easier when we do the scaling, so that's where this here is the q infinity that we saw in the previous slide here, when we look at our pressure coefficient, we have dynamic pressure in the denominator, and it's coming out of Bernoulli's equation, so that is the pressure coefficient. Other non-dimensional numbers that we have, we have non-dimensional numbers that are related to wave activity, and the Freud number is the one that we use, and this is a ratio of inertia to gravity, and if we write the Freud number squared, and we look at the ratio of these terms, we get something like this, which then collapses down to something like this, and if we don't write it as the square, we get velocity over gl square root, where g is a gravitational constant, and l is some length scale, but this is for surface flow, so on the surface of the ocean or in a river, wave activity, and open channel flow, the hydraulic jump, and here l denotes the depth of liquid, and I'll say water, but it depends upon the fluid that you're dealing with, so that's the Freud number. Another one, the Weber number, this is something that we use if we're dealing with bubbles or meniscus, and it is inertia forces to surface tension, and it has a symbol, capital W, little e, rho v squared l over sigma for surface tension, so you'd use this for droplets, you could use it for bubbles, and droplet would be if you have a liquid in a continuous phase of gas, bubbles would be if you have a continuous phase of liquid with a dispersed phase of a gas, meniscus, etc., so there are other applications, and another one that we'll look at, the one that is involved with compressible flow is the Mach number, and this is inertia forces to compressibility forces, and here the Mach number, quite often we'll see it expressed as v over c, v would be the local velocity and c is the speed of sound at those conditions, and the place for that is coming from, goes back to our definition for the speed of sound in gases or in a fluid, and we have the bulk modulus, which is e v, and that is our bulk modulus, so that's one way of coming up with the Mach number, another one we can write just the ratio of the forces that we're looking at here, so that would be a ratio of inertia to our compressibility forces, and from that the L's would cancel out, and we basically get the same sort of relationship that we've just derived in the previous equation, so that is the Mach number, as you approach Mach 1, that's when you get to the sonic condition, and a lot of interesting things happen, when you get to Mach 1, you can get shocks, expansion waves, all kinds of things going on, and that is compressible flow, which we will not study in this course, it would be in a more advanced fluid mechanics course. Struel number, this is for flows wherever you have some sort of natural frequency occurring, if you recall back to a very, very early video of why study fluid mechanics, I had the cavity modes of the Ford Fusion vehicle car, and the cavity mode was caused by vortex shedding off the front of the window, and you would have been able to calculate some sort of struel number, probably based on momentum thickness of the shear layer on that window, but a struel number is used in a lot of different flows where you have instabilities, oscillations, and it is given the symbol ST, and it would be the frequency of that oscillation, some characteristic scale divided by some characteristic velocity, and so this is often an oscillating flow. An example of this that is well studied is the wake behind a cylinder, if you ever follow behind a semi-trailer truck on the highway, what's happening is we have large scale vertical structures coming off, and that is oscillating, so the vehicle is moving in that direction, that's why your car, as you try to pass one of these big vehicles, you feel it pulling left to right, and you get instabilities as you drive through this wake, and then as you get by, you get cleaner air when you get up along parallel. So that is a struel number, drag coefficient, and here we have drag force divided by inertia, and we're going to introduce one of those one half just like we did with the pressure coefficient. We do this for a lot of the coefficients, lift coefficient, drag coefficient, pressure coefficient, but what we do is we take the force itself, and this is one that we studied earlier on when we looked at Buckingham Pi, but you get one half, you get the dynamic pressure in the denominator, so one half rho, some characteristic velocity, squared times some projected area, and this would characterize things like drag on bodies. Okay, so those are a lot of the different non-dimensional numbers that are important and used. We can have lift coefficients as well as drag coefficient, but those are things that we use when we do experiments, and so it enables us to be able to collapse the data, to understand the processes that are occurring, and to be able to scale the data such that other people can use it in their investigations or for engineering design purposes, and that's usually what we're doing as engineers. So experimental fluid mechanists go in and they conduct these experiments, they create curves, and those are used by design engineers then when they're going through and doing the design for whatever system they are designing, and so those are non-dimensional numbers.