 We're now going to move into the third method of doing analysis and fluid mechanics and in order to begin this what we're going to do is we're going to take a look at the topic of dimensional analysis and similitude. So if you recall from earlier lectures what we've been looking at thus far we've looked at control volume analysis or integral analysis. We then went through the process of doing control volume analysis on a differential element and that led to the method of doing differential analysis and when we did I should go back control volume we looked at mass momentum and energy and then when we did differential analysis we came up with a number of different equations we had the Euler equation which is for an inviscid flow field and we had the Navier-Stokes equations and that would be for a viscous flow field and as a subset of Euler we also looked at things such as potential flow modeling and we just briefly talked about that. That would be another method and usually what you're after you're either after the velocity profiles or pressure distributions things like that in fluid mechanics and then from the pressure distributions you can calculate forces and moments on objects but that was differential analysis and we found that there really it's hard to come up with solutions for these. Quite often this leads to computational fluid dynamics CFD or very very basic solutions are what we're able to come up with using the differential analysis techniques. So we've looked at integral we've looked at differential now what we're going to move into is experimental analysis and experiments are just what they sound like we we do experiments with real-world objects obviously we can't test very very large objects all the time when we're doing experiments so we do scale modeling and that's where similitude comes in it gives us a method of being able to scale what we're doing in an experiment versus what might actually be full-scale systems so and the other thing that we're going to look at is dimensional analysis that enables us to make experiments more efficient and I'll give you a little bit of an example in this segment where we talk about that so we're moving into the third form and that is experiments and experimentation. Okay so this is kind of a long paragraph but this kind of sums up the purpose of looking at dimensional analysis and similitude so what we have seen is we've gone through our analysis and we came up looking at some analytic solutions and some do exist to things such as the Navier-Stokes equations and the Weiler equations. There aren't that many and quite often the flows are not really of industrial importance or of industrial interest and consequently that brings in CFD and so a lot of people have moved to CFD taking the Navier-Stokes equations and solving them however even with CFD you always want to be able to verify the results of computational fluid dynamics using experimental data and consequently that brings on the motivation for conducting experiments and prior to CFD that was really the only way that we'd be able to get really good quantitative information and it still happens and subjects like heat transfer the only real way that you get correlations for heat transfers by conducting experiments and fluid mechanics is a big part of heat transfer experiments and so what that brings us to is the fact that experiments are not cheap they're very expensive it costs money to fabricate models it costs money to conduct the experiments and have all the equipment it takes a lot of time quite often when you set up an experimental apparatus it doesn't always work the way you think it will and you have to tweak it so they're very expensive and what we're going to do we're going to take a look at an example of looking at the flow over a sphere and so we'll go through a bit of a thought experiment in terms of what it might cost to set this experiment up so let's look at this as an example so this is something pretty simple what we have is we have a sphere maybe diameter d and it is in a uniform flow so we have some uniform flow coming over it and we'll say the free stream velocity here is v and we're interested in the drag force on that sphere so force is equal to what we're interested in that but when we look at that and we we think about what parameters are important in order to conduct an experiment to understand how the drag force relates to velocity diameter and anything else so we look at the drag force and let's say that is what we're interested in so we have drag force f and other parameters while I've drawn a couple of them here obviously velocity in the diameter of the sphere so we have d we have the velocity the free stream velocity other parameters well if we did this experiment high up on top of Mount Everest versus at sea level we would have a difference and the reason is is because the density of the air so that would be another one density and finally if we did this experiment in honey versus in air the drag force would obviously be very different and so we know the viscosity of the fluid is going to be important so typically when you do an experiment you should be doing this type of thing quite often we don't because experiments have been done before and so we just look at the the variables but those are some of the the parameters that this experiment will be dependent upon so if we want to understand the characteristics of the drag force on this sphere what we can say is that the force is going to be a function of a number of different variables we can say it's going to change with the diameter it'll change with the velocity flowing over the sphere with the density of the fluid flowing over it as well as the viscosity and and and those are some of the parameters and all of these can be measured so what does that mean that means that if you really want to understand how the force the drag force varies due to these other variables you have to vary those variables and so let's do a thought experiment here and let's assume that we're going to do 10 points data points for each test or experiment and an example of that let's say you do an experiment where you can fix the velocity the density and the viscosity and you vary d 10 times so you have 10 different diameters spheres and you evaluate the drag force due to 10 different diameters spheres after you fix velocity density and viscosity so if you look at this and the fact that we have these four parameters and then this fifth one that we're interested in measuring what that leads to and if we were to do 10 tests for every one of these combinations you would end up with 10 to the four tests that you need to conduct so that is quite a large number and and if you think about it these tests aren't going to take place very very quickly it will take time to conduct them and so let's budget and say maybe one half hour per test and let's say you're working an eight hour work day eight hour work day half hour per test 10 to the four tests that leads to 2.5 years of experiments investigating the drag characteristics on this sphere this is not efficient so what does that tell us and this is something that experimentalists and fluid mechanists years ago came up with they realized that this was not efficient and they needed better ways to be able to plan their experiments so we need a more efficient way we need a more efficient method and the more efficient method that we will look at is dimensional analysis and what dimensional analysis is going to rely on is a thing called dimensional homogeneity and what that says and we'll be using it as we go through dimensional analysis so what dimensional homogeneity says is that all equations and physics will have the same units on both sides and this is a common technique for any engineering student whenever you're solving a problem if if you want to check your answer you check the units to ensure that it is consistent on the left and right it doesn't guarantee that you've done it right but it means you're moving in the right direction so an example let's say you have distance is equal to velocity times time looking at the units here we know distance units of length and then on the right hand side we have length time to the minus one and that is being multiplied by time and so then what we get is length equals length so I mean that's a very simple one oops sorry not time length is equal to length and and that's a very simple one but what it does is it provides a mechanism by which we can check things and so when we go into dimensional analysis we're going to be coming up with non-dimensional numbers and the way that we're going to get these non-dimensional numbers is through dimensional homogeneity and and the non-dimensional numbers are numbers that are very important for the experiments that we're looking at and essentially what we're doing is we're taking a bunch of data so instead of taking like we showed in this example 10 to the four tests and all these data points you do fewer experiments but you collapse your data collapse your data onto curves that have your non-dimensional variables so you'll then be able to come up with we'll use pi they are going to be the variables that we use but by collapsing all of your data points would then fall into some relation and and that's what the experimentalist is always after and they can get a linear relationship like that even better and then they have a trend that says from this they then know pi one is equal to some function of pi two and pi will be our non-dimensional variables and an experimentalist can get something like that they're very happy and and by doing this they don't have to do thousands of experiments they only have to do let's say 50 experiments or or 10 experiments even and then all the data collapses and they get a beautiful relationship like that life is not always that simple and clean but but sometimes it is and that's what we look for as experimental fluid mechanists so we're going to move on to dimensional analysis and i'll show you the techniques that are used to determine these pi variables which enable us to collapse the data and make it a lot more efficient when we do experiments