 In this lecture what we're going to be looking at is flows within confined flows within fluid mechanics and so we're going to be looking at the topic that is referred to as being internal viscous flows. So what we'll be looking at internal viscous flows, these are quite often flows that would be confined within a duct or a pipe or a diffuser. The most popular one within fluid mechanics obviously would be pipe flow and so we'll spend quite a bit of time analyzing pipe flow and coming up with the equations enabling us to figure out the drop in pressure as fluid flows through a pipe. But before we get to that what we want to discuss are the two main regimes that a flow can exist in with any kind of flow that is but we'll look specifically at pipe flow here. So one of the two main flow regimes that we have is not of laminar flow. Now laminar flow has the characteristics of the flow being very smooth and orderly and if you were to look at it you would not see a lot of change with time. The flow would be moving in a smooth manner if you had flow visualization you'd see all of the flow moving parallel to one another and consequently that would be a very smooth flow and that's characterizing laminar flow. The other state that the flow can exist in is turbulent flow. So turbulent flow counter to laminar the fluid flow is fluctuating at all times and so consequently if you were to put something like a velocity probe within a turbulent flow you would notice that the velocity would have an average value so there'd be a mean value but it would be fluctuating with time about that mean value and we live in a turbulent boundary layer the planetary boundary layer of the earth during the day is is obviously very very turbulent. If you've ever landed at an airport on an aircraft you'll feel as you come through the cloud layer as you approach the earth at about a kilometer, kilometer up that's when you start entering a planetary boundary layer and you'll encounter a lot more turbulence. Now at night time sometimes what can happen is that turbulence will all kind of die out and and you get a very very calm boundary layer that would be at night time. However, daily we encounter turbulence so turbulence is actually the more common flow state that we encounter in engineering applications at least but also within nature and and so which state you exist in has a big implication in terms of when we look at internal viscous flow the pressure drop within a conduit or a pipe and and the state that you exist at is a function of one of the non-dimensional numbers that we looked at when we did dimensional analysis and that is the Reynolds number and so what we will do we'll take a look at the Reynolds number okay and so if we were to take a velocity sensor and there are many many different ways that you can measure velocity in a fluid but the one that is often used for being able to study turbulent flows it's an intrusive sensor but it's called a hot wire animometer and it is capable of measuring high frequency fluctuations in a flow. There are other techniques that have been developed in more recent years optical based techniques laser Doppler velocity symmetry particle image velocity symmetry time resolved particle image velocity symmetry those are starting to give us the ability to be able to get a time resolved velocity measurement but the hot wire is one that's really quite good it's easy to do and if you were to put it into a flow and let's say we had laminar flow regime over here and way over here on this side would be turbulent and in the middle is what we call a transitional flow and in this region the equations get a little more difficult to come across because sometimes you're laminar sometimes you're turbulent you get these turbulent bursts coming out but if you were to put a sensor hot wire animometer into a fluid flow and measure the velocity what you would find for laminar flows you'd have something like this now it wouldn't be perfectly flat there'd be some fluctuations there and that would be associated with the background turbulence that would be in the flow field there there may be some instabilities in in the fluid there there may be something driving the flow you might get a little bit of vibrations off of a pump or a fan and consequently it's very very difficult to get very very clean flow without any kind of disturbances in it but anyways that would be laminar it would have some average value here and then if we were to look at transitional what you'd find is it'd be nice and steady and then once in a while you get these bursts coming up but they would decay back down and then you might get another burst and it would decay back down so the nature of transitional flow is instabilities can arise but they will damp out with time and and then you return back to the laminar state and and then finally with the turbulent flow regime again you do have an average but but the nature of the signal is is quite scattered and it's it's very very dynamic and so a turbulent signal could look something like that now just looking at a signal like that it's very difficult to determine whether or not it is turbulent we use a technique called spectral analysis where we will look at a power spectra of our signal and by looking at that there are certain telltale characteristics that we won't go into in this course but would give you an indication whether or not you do truly have a turbulent flow or if you don't and and then the same with laminar you you can do power spectral analysis on that and by looking at at the power spectra plotted in certain ways versus wave number wave number of of the flow I won't get into details with the wave number but you would have a certain slope for laminar versus turbulent flow so that would be what the signal would look like and but again with turbulent flow you are going to have some average values you do have statistics with turbulent flow that enable you to determine things and and they will be stationary they they don't necessarily change the time unless your flow field is changing with time so that is laminar and turbulent flows now when we look at duct flows or pipe flows like we're talking about here so if you were to plot pressure drop in a standard pipe as a function of the velocity of the fluid going through it what you would find and and if we were to say this here in this part is laminar and this part here is turbulent as velocity increases so this would be transitional now what you'll find if you were to plot a delta p as a function of the velocity in this section here where it's laminar we find that delta p is proportional to the velocity and as we go through transitional the slope here changes and then we get a change in slope characteristic and when we get up into turbulent we have delta p is proportional to velocity to the 1.75 so what that tells us is that when the flow becomes turbulent we have a we have a higher pressure drop along the length of the pipe and consequently it takes more energy to transport a fluid if we have turbulent flow now there is an experiment that was done years and years ago by Osborne Reynolds and he was doing these experiments and what he did is he had a pipe and it was gravity-fed and so it was a very very smooth pipe very calm there was no pump pumping the fluid that would lead to any kind of disturbances but he had a very carefully designed bell mouth entry and he introduced a die a laminar die into the flow and he could track that die and when the flow was laminar he would get something that looked like that and when he would do the experiment by increasing the flow rate so he would increase the flow rate which would drive the velocity the average velocity in the pipe up and when he would do that i'll draw this again so as he increased the flow rate and he went into the turbulent regime what he would find again when he would introduce the die it would come in this way but then it would start going into instabilities and and then it got very very difficult to see and so it basically looked like the the die had diffused in the pipe but actually what it was is there was a turbulent flow and so if you take a spark photography you would see the stream lines and they're very convoluted and very complicated that's characteristic of turbulence if you're looking at the smoke coming up off of a cigarette for example you'll notice that it's very nice and smooth and laminar and then it goes into instabilities and eventually becomes kind of turbulent as it ascends up same sort of thing with Reynolds's experiment interesting thing is they went back and tried to replicate the experiment and they found that they could not repeat it with the same Reynolds number in terms of when you would transition from laminar to turbulent and the what i have heard it goes that the disturbances from modern infrastructure led to disturbances on the side of the experimental facility which made it harder to get the same Reynolds number that Reynolds originally had showing that external disturbances can play into the way that experiment goes but typically what we do for pipe flow we say that there is a critical Reynolds number and we always have a critical Reynolds number for any kind of flow that we're looking at and what that indicates is when the flow will start to transition from laminar to turbulent but for pipe flow the number that we use is 2300 and here red that's Reynolds number based on diameter so Reynolds number is the density of the fluid the velocity some characteristic velocity a characteristic length scale that being diameter and then the dynamic viscosity so in this case v is equal to an average velocity and we have to say that because we're going to have a velocity profile if you look at the velocity and pipe flow depending if it's laminar or turbulent it may look like that or it may look something like that depending upon what state you're in and d little d is equal to the pipe diameter now quite often you may find in books people saying re greater than 4000 is turbulent now that's more of a conservative estimate because between 2300 to 4000 or 2100 to 4000 you could be in the transitional regime so this would guarantee that you certainly are within the turbulent flow regime however I have also heard that experiments can be conducted where you can get Reynolds numbers as high as 10,000 and you still have laminar flow but I think that would be a very very specially designed experimental apparatus where you minimize all instabilities all disturbances and it would be very very difficult to replicate anything like that and best really and so that's more just a scientific type investigation so for our purposes typically if you see anything around here 2000 2100 2300 you're starting to go into a turbulent flow scenario and certainly by the time you get to about 4000 you can say you have turbulent flow and that enables you then to determine which equations you need to use to solve for whatever problem you're looking at so that's Reynolds number laminar turbulent flow what we're going to do now is we're going to continue on looking at pipe flows