 In the last segment we came up with an expression for the friction factor for laminar flow and that enabled us to figure out the head loss and laminar pipe flow. What we're now going to do we're going to go for a couple of the segments. This segment and the next one we're going to be looking at turbulent shear flow and what we're after is a relationship for the shear stress along the wall along a pipe wall with turbulent flow but we'll look at turbulent shear flow generically. So this could be shear flow in either a boundary layer or in a pipe and we're going to come up with a relationship that will then enable us to do a very similar thing and come up with the friction factor for turbulent pipe flow. So one disclaimer that I should say is that this works for both internal flows and external the theories that we're going to be looking at here. So what we'll be looking at when we talk about turbulent shear theory we're looking for both internal or external flow so in the case of an external flow that can be on a flat plate boundary layer for example. Now we're going to make a couple of assumptions here. So in our analysis we're going to assume that the flow is incompressible there's constant viscosity and no heat transfer and we're going to begin with the governing equations continuity and momentum as well as boundary conditions and the boundary conditions that we apply. No slip at the walls and no inlet and exit conditions. So we've been talking quite a bit about laminar flows and with laminar flows we can come up with closed form solutions for velocity profiles using the Navier-Stokes equation. When we get to turbulent flows it becomes a little more difficult and so I just want to make a little bit of a comment about that. Okay so the thing about fluid mechanics is we have our governing equations and we can have either a laminar flow or a turbulent flow and they both satisfy the governing equations and so in a way you could say that the equations are non-unique you can have multiple solutions and so we can have either the laminar or the turbulent state. Now the Reynolds number would change but it's also related to instabilities that can develop in the flow as you go through the transition process. The other thing is that for laminar flow there are a number of solutions and when I say solutions I'm referring to solving for the velocity profile because that's usually what we're after in fluid mechanics and so there are limited number of geometries we have closed form analytic solutions for and if you take a graduate class in fluid mechanics you come up with a lot of those solutions. With computers now we're able to solve this using CFD so we have the ability to solve for quite complex geometries using CFD however when we get to turbulent flow there is no way to solve it analytically at current understanding. CFD is making progress using things such as large eddy simulation and direct numerical simulation but as I mentioned earlier those are for fairly basic flow fields so things of industrial importance or interest it's very very difficult to do using direct numerical simulation for example and so we make different assumptions using turbulence models but that's a different story I'm not going to get into that today but what I want to do now is I kind of want to do a bit of a quick dive into turbulent shear flow theory and what we'll do is we'll look at a process by which the people who studied fluid mechanics years and years ago so back in 1920s 1930s they they came up with some dimensional arguments and so we're kind of going to go through the logic that they followed and we're going to talk about people like Theodore von Karman and Ledwig Prandtl and they came up with using dimensional analysis and by doing a lot of experiments a lot of insight deep insight into what's happening within a turbulent flow field and that then became the basis of what we'll be using for determining the friction factor in turbulent pipe flow so that's kind of where we're going with this and what we're going to do we're going to begin with looking at mean quantities in a turbulent flow so if all you're interested in are the mean quantities which is what we'd be interested in for the friction factor then we're going to go through a bit of an analysis here and I'll show you the statistics that come in with turbulent flow fields so when we're dealing with turbulence the velocity is going to be a function of the three spatial dimensions as well as time and quite often what we will do is we express the time as a time average sorry we express velocity as a time average quantity and a lot of the different parameters that we look at the velocity pressure things like that we go through this process and the way that we take the time average it's over some period of time that we would collect velocity fluctuation data or velocity data and then we do a time average of it and what I'm going to do I'm going to show a couple of profiles here or time signals of velocity and pressure and then I will relate that to the time average and what we call turbulent fluctuations so we have our two time signals here and for the velocity we're going to say that we have some average u over bar and similarly for pressure we have p over bar and then we have our turbulent signals on top of those so this would be if you put a hot wire animometer into the flow or high frequency response pressure transducer that's how you get these two measurements so the thing with the turbulent signal it is quite random and chaotic looking however there is order behind it but the one thing is that it will have a mean value and so statistically we can compute statistics on it and that would be the average value if this does look periodic it shouldn't be periodic it's just the way that I've drawn it and what I'm going to do for the velocity we would normally go in and look with respect to the average value so I'm going to try to pick a value here this one looks good so I'm going to draw a line in here and this here represents u prime so that is going to be the turbulent fluctuation and then the velocity at any given point in time we will denote that by u of t is going to be u bar plus u prime where u prime is the turbulent velocity fluctuation at that point in the flow field and I'm going to do the same thing for pressure I'll find a location in here where we have something that we can draw on that that's a big one there hopefully I don't mess it up so I'm going to draw a line in there that should be a vertical line and that is p prime and then the pressure at that point is going to be p of t is the average plus p prime and that would be the pressure fluctuation within the flow field so that's typically what you'll get if you do an experiment involving a turbulent flow you get these signals that are quite dynamic but what we normally do is we say that the turbulent fluctuations are defined by and they usually have a prime to denote the fluctuation it's going to be the value as a function of time minus the time average value for that particular parameter and so that would be turbulent fluctuation and so what people did years ago is they did these thought experiments where they took these fluctuations and they put them into the governing equations and they did a lot of time averaging and then as experimentation advanced they became or were able to measure a lot of these parameters and quantities so we're going to look at some of the statistical analysis tools that they used the time average of a fluctuation so if we do the time average of a fluctuation within turbulence theory that would be the velocity and i'm using velocity here minus velocity bar so the average and what we would get is the average of the velocity minus the average average of the velocity and that turns out to be zero so what that tells us is a fluctuation has zero time average to it and by definition it should be that way another thing let's say you take a fluctuation and you square it and you take the time average let's see what happens there so if we do that by the definition of our time average u prime dt what we find if we were to do this experiment and actually measure this and compute it we would find that that doesn't equal zero and that that's one of the things the fluctuations either squared or multiplied with one another are not zero and and that leads to a result in the governing equations that we'll see in a moment so we can write that products of fluctuations are not zero so if you take your fluctuating velocity in the x multiplied by the y and you take the time average that may not be zero it wouldn't be zero and u prime p prime similarly is not necessarily zero and so the what people like Prandtl and von Karman they they did years and years ago they they went through a process or procedure referred to as Reynolds time averaging and for this what they did is they substituted into the governing equations a mean and a fluctuating value so they would substitute for all of the different variables that are in the governing equation and when you do that and then you take the time mean of each equation so when you do that for continuity you end up with basically just the continuity equation but now expressed in time average form and so that's not really all that interesting the momentum equation is a little more interesting and here we're going to look at the x direction of the momentum equation and when you do this so when you do this you get kind of an interesting new form of the governing equation everything looks like it was to begin with with the exception of these terms that start coming in and these are basically products of the fluctuating quantities or turbulence quantities and they we have the squares we have the fluctuating components and they the way that i've drawn the equation here written it out they appear to be stress terms because they're on the right hand side of the equation so if you recall this is inertia inertia term and then this is pressure this is body force and then we have shear and and so these terms appear to be shear terms and and indeed when people found them years and years ago they they call them turbulent stresses because they have the units of of the stress but really when you look at the way that the derivation goes about here and what we did is we substituted these values into our equation it turns out that these turbulent stresses actually arise out of the inertia term so they arise or arise out of the left hand side of the equation where we have non-linearities we have things like u du by dx and when you put in the the average and the fluctuating component you get these stress terms coming out and and and so these are terms that make it difficult to solve for the mean velocity profile in a turbulent flow okay so we have these terms and these are turbulence fluctuations the solution of the governing equations people have worked with these for years and years and there are different theories trying to come up with a relationship between these and mean gradients that would be within the flow field because in order to solve for the velocity profile you need to know these but you need to know the fluctuating components before you can solve for them which it kind of makes it what they call the turbulence closure problem and is how you come about determining what these different fluctuating components would be and and many many people worked on these Prandtl worked on this he came up with mixing length theory in order to solve for the velocity profile for a turbulent boundary layer and things like that but that is what has basically stopped researchers from being able to come up with closed form solutions to a turbulent flow field it's the fact that we get these fluctuating components entered in so what we're going to do in the next segment is we're going to take a look at these turbulent stresses and it turns out that for the flows that we would be interested in so either external flat plate boundary layer flows or turbulent pipe flow some of the terms are larger than others and so you can make a little bit of a argument and cancel out some terms and simplify the governing equations which is exactly what people like Prandtl and Longkarm and Deb years and years ago so that's what we're going to do in the next segment we're going to dive in there and and we'll come up with some velocity formulations and then we'll work eventually towards friction factors