Flow Characteristics: 1-D, 2-D, 3-D Flows, Steady and Unsteady Flows

Okay, so when you first see the title over here, 1D, 2D, or 3D, you may be asking myself, why are you wasting my time, and you're also wasting bandwidth with this YouTube video as well. I know what this is. Well, my answer is, do you really? Maybe you do, maybe you don't. I want to make sure that everybody's on the same page, okay? So I'll give you an example. Let's say that I have something like expanding like this nozzle, okay? So this is the starting point of it, right? At some point my velocity like this, and you'll see what will happen to the velocity ratios in module five. But so far, you don't know why this gets lower value. We will cover this, and it will make a lot of sense to you why this velocity is smaller than that’s velocity. Then I'm gonna ask you a very simple question and say that, hey, is this a 1D or 2D flow? And I want you to realize it's in my channel, okay? Look at how it is, right? So it's expanding out. And if I ask this question to students in my class, typically what happens is I ask, is this 1D, 2D, or 3D? 3D is out of the question for this particular case, but the majority of the students answer this is a 2D, okay? Which is wrong by the way. This is a 1D flow, so that's why I had this segment for you. It's a pretty short segment, and you may be telling yourself, okay, yeah, I know this is 1D for sure. Well then, this segment is not for you; move on to the next segment. But if it is not, well, hang on with me. So basically the way that I define, once I write this, this will make sense. The way that I define my 1D, 2D, 3D is, I write my velocity and I look at what it's a function of. It can be a function of x, y, z, right? Obviously, it can be a function of time; that's a different segment, not this segment, okay? So now I'm looking at the coordinate axis. Well, let's go ahead and write x is this way, y is that way. So now I'm gonna ask you a question. So I'm over here, okay? If I travel in the x direction, do my velocity changes in this direction, in the positive x direction? That's a yes, it does, okay? Note that the velocity here is more than the velocity there. So for that reason, my velocity is reducing as I go in that direction. So far I have established that V is a function of x. Let's continue. Also, one thing that you may not realize is when I am asking you, what is is this a function of x, I'm fixing the y, okay? So I'm not going up and down because then it's hard to judge, right? So I want to fix something, so I'm fixing y, so I'm the only like this. So now let's do the reverse of it. Let me fix x, let's say to this point, and go up and down. So if I go up and down, do you think my velocity changes? The answer is no; the velocity is not a function of y. I can pick another x value, this one that I plot over here, and you can see what? You can see that the velocity still constant. So now the velocity is not a function of y then. So you can see over here what happened was now V is just a function of x. It's not a function of y; it's not a function of z. So this is a definition of 1D. I can have the exact same; obviously, I will not be able to replicate very well, okay? But this will do. Doesn't have to be exactly the same. Again, this is the beginning of this. Let's say a pump is connected here or something; we'll talk about these things. And let's take a look. Let's say that over here, the velocity profile is like this, and over here my velocity is like this, and you can see this is getting lower, right? The maximum velocity, that's not a coincidence. We'll talk about it. So my question is, what is this velocity function of? Okay, let's take the same x this way, y is pointing up. If I travel in the x, what will happen is, oh, my velocity is getting lower. So it's obviously a function of x. What about y? Let me fix my x, I go up and down. Huh, this this time around I have a parabola, so a maximum velocity at the center. We'll derive these things, and then it goes over here. So then I get myself y variance as well. So I can repeat the analysis here, and you will see the same conclusion. Still a function of y, okay? So this is called 2D flow. So let me give you additional information over here. You may very well ask me, hey, you showed me pretty much the same, except I can't draw that well, geometry, but the physical velocities, how they look, is completely different. So what are you talking about? Are these both possible in nature, or are you giving something to me in a theoretical sense but it has absolutely nothing to do with real life? And as an engineer, I don't really care about it; tell me something that exists in real life. Good news, both are possible in real life, okay? There's a small caveat, and we talked about this actually, so I can mention over here. But look at the boundary conditions over here. I said that, hey, this is a stationary, right? So the velocity has to be zero over here and not a finite value; it's called no slip condition. In real life, the no slip condition, there are cases where it doesn't hold, but for this type of things that I'm showing here, it should hold off, okay? So the velocity has to be zero here. So disregard that; we'll talk about these things. But let's say I'm a few millimeters away from this surface, then this is realistic like that, okay? We'll talk about these things down the road. So this is called inviscid flow, okay? And this is much more realistic, called viscous flow. Depending on the size, depending on the velocity, depending on the boundary conditions, you will find out that both can exist. This is inviscid, this is viscous, and I'll even confuse you further and I will tell you that at a cross section I can have both of them. I know your head is about to explode when I just said that, but again we’ll discuss those. We're at the beginning of these modules, okay? So this segment is going to analyze whether the flow is steady or unsteady, okay? In theory, this is pretty simple, but in reality I see confusions, okay? Let me tell you right away, steady doesn't mean the velocity is constant. Let me repeat, steady does not mean that the velocity is constant. That has nothing to do with it. If you tell me that, hey, the velocity is not a function of time, then that's time of steady, okay? So what I mean by that is this. Let me draw from the previous segment. I have something opening up. People tell me that, already students tell me that steady means that the velocity is constant everywhere here, here, here, here, here. It's the same. It's five mile per hour over here, this is five mile per hour over here. No, that's not going to fly. That's not steady or unsteady, okay? That's wrong, right? So let's talk about what steady and unsteady is. The way that I explain this actually, about this, in my opinion, makes it fairly straightforward. So I have a velocity in here. I don't want that to be a function of time, or I do; it depends on my application. This determines whether this is steady or unsteady. In the previous segment, I looked at x, y, z, right? Now this is another parameter that I have over here, which is called the time. In real life things, are they steady or unsteady? That is an important point, okay? But well, my answer is, it really depends, okay? So I can tell you this. My hair, I can tell you unfortunately, it is unsteady. It is not steady. It's getting less and less as the years go by, okay? So that's not good, but it is what it is. Let's say that I have this analysis that I'm doing over here, and this is going with, I think I said five mile per hour. Let's try to be consistent if I can. And let's say that over here, now, you'll see this, it will reduce to some other value. Let's say three mile per hour over here, okay? So here is what it says. Steady means, you know right now I'm actually feeling fairly hungry myself, okay? Been doing this for a while now, the recording since the morning, so I need to eat something, okay? And I have my experimental setup right over here. What I do is, I want to collect the data too, though, okay? Before I go to lunch, I want to run my system to collect data. I can connect this to the computer, and the computer is collecting this data, okay? I go to lunch, I come back, and I look at this velocity, or whether I look at this entire system, and I see that nothing has changed. That is the definition of steady. It is not a function of time. You see, I gave the example of lunch. Basically, this is an excuse for time variance. I will also give you another example, and this will be actually beyond the scope of an undergraduate fluid mechanics course, but there's something called steady in the mean. What it means is this. I have a mountain, okay, and there's a river over here. The flow is coming down, right? So if I look at it over the, let's say a month going from January till July, right? So in January, everything is frozen, so I don't have much flow over here, right? And come March or April, whatever, what will happen is things are starting to deice, and I will have some flow coming down over here. The velocity will be or volumetric flow rate will be kind of high, right? But then after it comes to, let's say June or July and the summer comes over, well there's no more ice, and there's no more flow going on. It's completely dried. The riverbed is completely dry. Is the flow rate steady? The answer is no, right? But just hang on a second, it really depends the time scale you're looking at. There's something called steady in the mean. What it means is this. If I look at it per monthly basis, obviously it's not going to be steady. It's changing over time, no question about it. Let me analyze this on a yearly basis. If I go from year 2020 to 2021 and I still have Jan to July and Jan to July, is this flow rate going to be the same over this average? Let's call this the average of these over the time or the total amount, right? This can be steady depending on whether this total flow rate is constant or not, okay? Something to note.