A closely related topic to dimensional analysis is Similitude, which I will talk about here. Now, the question is Similitude is that how do we design a small scale model of a flow. For example, a flow over a dam spillway, or through a pipe. In the laboratory, how do we correctly design our model, and after running the model, how do we scale the results up to the full size situation? For example, let's suppose we have the flow over of a dam spillway, and we want to make a model of this in the hydraulics laboratory, which of course, looks just the same, but is smaller. And the usual terminology in hydraulic engineering is that the full-scale situation we call the prototype and the small-scale model we call the model. Although the model could actually be larger than the prototype in some situations. Now, for the prototype to represent the model, we must have similarity or similitude between the two. And, the two words, similarity or similitude will be used interchangeably, they mean the same thing. But, generally, we require three types of similitude to be obeyed for the model to be correct. The first one is geometric, which formally means that all linear dimension ratios are equal everywhere, or more simply, the shapes are the same regardless of size. The second requirement is dynamic similitude, which means that the ratios of different forces at similar points in the model and prototype are equal. And the third type of similitude is kinematic, which means that the velocity and acceleration ratios at similar points are the same in model and prototype. Now, for force similitude we have to satisfy all three of these constraints. However, because velocities and accelerations are related to forces of geometry, it turns out that we only have to satisfy the first two of these constraints. And if we satisfy geometric and dynamic similitude, then kinematic similitude will automatically follow, and we don't have to worry about it. To illustrate what we mean by dynamic similitude, let's continue with this discussion of the flow over a dam spillway. And I'll denote the prototype by p, here. And let's consider some point of the flow, as it goes over the crest of the spillway, at this point. The forces which are acting on this are, say, if the streamlines are curved, we might have a pressure force here which is acting in the outwards radial direction. We have the gravity force which is of course acting straight downwards. And we would have viscous or friction forces which are acting to slow the fluid particle down. So I add these forces vectorially. In other words, head to tail, in the polygon over here. And then I end up with an opening here, and the opening is the resulting force, which I denote by Fr. And the result in force here is what produces the acceleration of the fluid particle according to mass times acceleration. So, this result in force, we usually denote as the inertia force, Fi. So, the inertia force. If I do the same thing for the model, then I get a similar thing, but all of the forces are somewhat reduced, and for dynamic similitude, means that the ratios of all force types between the model and the prototype must be equal. So, for example, the ratio of inertia in the prototype to gravity in the prototype must be equal to the ratio of inertia in the model to gravity in the model. But what does that mean? Well, the inertia force is proportional to mass times acceleration. Which in turn is proportional to, mass is proportional to rho l cubed, and acceleration is proportional to velocity squared over length. The gravity force is just the force of gravity or the weight of the fluid which is proportional to m times g. Which is in turn proportional to rho l cubed times g. Therefore the ratio of these, the ratio of inertia to gravity force is rho l V squared, rho l squared V squared, divided by rho l cubed g. Which reduces to V squared over lg. So therefore, for similitude, we must have the ratio V squared over lg be the same in model and prototype. Commonly though, in hydraulic engineering, we don't use that definition. We use the square root of that. V over square root of gl which is called the Froude number. So, for similitude we require that the Froude number in the prototype, be equal to the Froude number in the model. And again, this is the same table that I showed earlier. You can do this for all of the other force types that might be acting, for example friction forces compressibility, surface tension, etc. And the dimensions ratios which are most commonly used in fluid mechanics as showed here. The Reynolds number which is important in situations where friction or viscous forces are important. The Froude number which is important where flows gravity forces are important. Which are often flows with a free surface. The Mach number is important when compressibility is important. The Weber number is important, where you have surface tension forces. Pressure coefficient, where pressure differences are important, and Drag coefficient, when you're interested in drag forces. So, these are the most common numbers that are used in fluid mechanics. And strictly speaking for similitude, we must have all of these dimensionless groups equal in model and prototype. In other words, Reynolds number model equals Reynolds number prototype, etc. However, normally many of these will not be important. For example, for an incompressible fluid, the mach number is not going to be important. If you don't have a free surface or if you only have admissible fluids surface tension is not important. So almost certainly most of the problems that we will encounter will be involved only with Reynolds number and Froude number. So in other words, we will generally require that the Reynolds number in the prototype be equal to Reynolds number in the model and also the Froude number in the prototype be equal to the Froude number in the model. What you find, however, in practice is that this is almost impossible to accomplish, to simultaneously satisfy both those equations. So, generally the problems we encounter will either be Reynolds number problems or Froude number problems, but not both at the same time. So, here again is the table from the handbook with the full list. And this is somewhat unconventional. For example, the Froude number here they give us V squared over lg, whereas the more common definition is V over square root of lg. The ratio which expresses compressibility effects is the Cauchy number here, which again is not commonly used. More commonly we use the mach number, which is related that. And the dimensions group that contains pressure here, rho V squared over p, is also not commonly used, and gives the unfortunate impression that it's pressure what matters in the flow field. But in reality it's not pressure, it's the difference in pressure, Delta P, which is important. And this ratio is also not a conventional one. Usually we take the inverse of that, Delta P over one-half of V squared is known as the pressure coefficient. So, the dimensional scripts are actually given in the reference hand book are not always the common ones that are used in engineering.
