Lesson Video: Steady and Turbulent Flow Physics

Video Transcript

In this video, we’re talking about steady and turbulent flow. These two types of flow are opposites of one another. And we may already have an intuitive sense for what they mean. We might guess, for example, that the flow in this river, where the boat currently is, is more steady, while the flow farther on after the water has gone down this incline is more turbulent. That’s accurate. And in this lesson, we’re going to see what makes for steady and turbulent flows.

Now, as we mentioned, these two types of flow are opposites. The steadier a flow of fluid is, the less turbulent it is, and vice versa. But then what did these terms steady and turbulent really mean? The way we define these terms comes down to how the speed and direction of the fluid in these flows changes. We can say that steadier flow occurs when fluid’s speed and direction change less, while we would say more turbulent flows occur when speed and direction change more.

It’s important to see that these terms refer to the change in the speed and direction of flowing fluid. To see why this is so, imagine that we have a stream of water. We can say that we’re looking down on this stream, flowing along from left to right, as indicated by this velocity vector. And let’s say further that all the water in this stream is moving along with this same speed in the same direction, left to right. Because neither the speed nor the direction of the water in the stream is changing, we can say that this is a very steady flow.

But then let’s compare this original stream of water with this one shown here. In the second stream, the water is again moving in the same direction, but this time with a much greater speed. Now, here’s a question. Which of these two flows is steadier?

Well, that’s kind of a trick question because both flows are actually equally steady. Even though water moves at different speeds in the two flows, that speed, as well as the direction the water moves, doesn’t change within each one. So then, what would be an example of flow where speed does change?

We can think of it this way. Before, we were looking at this stream here and this stream here from an aerial perspective. It was like looking down on a flow of water as it moved along the ground, say. But now let’s imagine that these are layers of fluid flow stacked on top of one another and that there are three layers. We could think of this as looking from the side on as water flows by, where this up here is the surface of the water. And then as we go down, we increase our depth until we reach the bottom.

So we have these three vertically stacked layers of water. And each layer is moving at its own particular speed. The layer on top is moving slowest. And then speed increases as we increase our depth. So what does this increase in speed as we increase our depth cause?

Well, imagine being at the interface between two of these three layers. At that location, the water in the lower layer is moving faster than the water in the upper layer. And these two layers are in contact. We can imagine what happens here. There’s friction between these two layers moving at different speeds. The faster-moving water tries to speed up the slower-moving water, while the slower water tries to slow down the faster water. This friction between layers causes the flow of water to be disrupted. Both the speed and the direction of the water near these layers changes. And we start to see signs of turbulent fluid flow.

Whenever we see fluid direction changing rapidly and especially when we see fluid moving in closed loops, those are signs of strong change in fluid direction, which indicate turbulent flow. In a qualitative sense, it’s possible to identify steady and turbulent flow by looking at what we can could the streamlines of the fluid as it moves along.

To see this, let’s change our perspective once more and imagine that now we’re looking downward on the surface of a smoothly flowing river. These lines we’ve drawn in represent streamlines of the water in the river as it flows along. And we can see that there’s no change in the spacing between these lines at any point. So if we also assume that the flow is at a steady rate from left to right, then we can say that what we’re looking at now is an example of a very steady fluid flow.

But then, let’s say that we drop a rock into the river, right here. If we do that, we know that this will disrupt the streamlines. Rather than looking like this, the lines would change to look something like this. Notice that now these streamlines are no longer straight. That is, they do change direction. And right behind the rock in the stream, there’s even a closed loop. This indicates water that’s going through an extreme change in direction as it moves through this loop. And we saw earlier that this is a hallmark of turbulent flow.

It’s important to see though that the level of turbulence is not the same everywhere in this section of the river we’re looking down on. If we were to focus in on the region right behind the stone in the river, this would indeed be the highest-turbulence part of the flow we see. But what about another section, say, this one over here?

In this case, we see streamlines entering and leaving this section we’ve highlighted, basically moving in the same direction. And the spacing between the streamlines doesn’t change much at all, indicating that the speed is fairly constant as well.

Just to explain that last point a bit, about the distance between streamlines staying fairly constant, indicating a fairly constant speed, let’s imagine that the speeds of the water moving along these two streamlines was in fact very different. Let’s say, for example, that the speed of the water in this streamline — we’ll call it 𝑆 sub 𝑡 for the top of the two streamlines — is much greater than the speed of the water in this bottom streamline — we’ll refer to it as 𝑆 sub 𝑏.

If this is true, if the water on the top streamline is moving much faster than the water on the bottom one, then just like we saw in our last example, there will be friction between these sections of water moving at different speeds. The faster-moving section would exert a stronger frictional force and tend to push its way downward towards the lower streamline. This push would have an effect on the direction of the lower streamline. But, nonetheless, if there was this large speed difference between the fluid and each one, then over this patch of area that we’ve identified, that would show up as a change in the distance between these streamlines.

We see this going on by the way in the more turbulent section of our flow. All that to say, when we see streamlines, which keep the same distance apart from one another as they flow along, this means that there’s very little or no speed difference between the fluid flowing along those lines. Now that we know that steadier fluid flow happens when fluid speed and direction change less, while more turbulent flow occurs when these properties change more, let’s look at an example exercise about steady and turbulent flow.

The diagram shows the flow of the fluid past a circular obstacle. The gray lines represent the direction of the fluid flow. Black regions represent solid obstacles to the flow. In which of the two regions within the dashed lines is the fluid flow faster?

So we’re imagining here that the fluid in this diagram is flowing left to right. And as it does so, a solid obstacle to that flow, right here, comes up. We could imagine this as being something like a rock in a riverbed. And we know that such a rock causes the fluid around it to flow differently. And indeed, we see that the gray lines indicating the direction of fluid flow do change as the fluid avoids this solid obstacle. Downstream of that obstacle, we see these two regions marked out one and two. And we want to know in which of them is the fluid flow faster on average.

Before we answer this question though, of in which region the fluid flow is faster, let’s consider in which of the two regions is the fluid flow more turbulent. We can recall that fluid flow is more turbulent when a fluid’s speed and direction change more. So if we wanna figure out in which region, one or two, the flow is more turbulent, we can look to see in which one do the fluid speed and direction change more.

We can figure this out by considering the gray lines, what we could call the streamlines of the fluid in this flow. Notice how, in region one, these two flow lines approach one another across this region, while in region two, the flow lines are nearly parallel with one another. This might seem like a small difference. But the changes in fluid direction we see in region one greater than the direction changes we see in region two indicate a more turbulent flow in region one. And indeed, this agrees with our intuition.

We would expect a region in our fluid right behind a large solid obstacle to be more turbulent, whereas farther downstream, farther away from this obstacle, we expect the flow to smooth out. Okay, so if the flow in region one is more turbulent than that in region two, how does that help us answer this question of in which region does the fluid flow faster?

Consider again the streamlines in region one. If we were to draw exaggerated velocity vectors for parts of these streamlines in this region, they might look like this. These vectors that we’ve drawn in, even though they may overstate the motion of the fluid in this dimension, show us that, nonetheless, because of these streamlines getting closer and closer together, fluid in one part of this region will be pushing against fluid in the other part. That is, fluid on this side of our dashed line will be pushing on fluid on this side of the line, and vice versa.

Because the fluid in region one isn’t only moving left to right but is instead also moving up and down we could say from this perspective, some of that fluid velocity will be negated as the fluid flows against itself. This will lead on average across the whole region to a slowing down of this fluid. This is in contrast to the fluid in region two. Instead of some fluid in the region pushing against fluid in other parts of it, it’s all essentially moving in lockstep and therefore fairly rapidly from left to right. Because the fluid in region one resists its own motion more than the fluid in region two, we can say that it’s the fluid in region two which flows faster.

Let’s look now at a second example exercise.

The diagram shows the flow of a fluid past a line perpendicular to the flow. The gray lines represent the direction of fluid flow. Black regions represent solid obstacles to the flow. In which of the regions within the dashed lines is the fluid flow faster? In which of the regions within the dashed lines is the fluid flow steadier?

Okay, so we see in our diagram fluid flowing along. And we can say that it’s moving from left to right as we’re looking at it. As the fluid moves along, there’s an obstacle in its path right here, something solid in the way that won’t move. In response, and we can see this in the flow lines, the fluid changes direction to avoid this obstacle. Downstream of the obstacle, we have these two regions marked out one and two. We want to know, first, in which of the two regions is the fluid flow faster and, secondly, in which of the two is it steadier.

Now, actually, it’s a bit easier to answer this second part of this question before we answer the first. Let’s consider whether fluid flow is steadier in region one or in region two. And to help us figure that out, we can recall that fluid flow is steadier when the fluid’s speed and direction change less. So in a given region, if the fluid there isn’t changing much in speed or direction, then that indicates a fairly steady fluid flow.

If we look at region one in our diagram, we can see right away that the fluid direction is changing significantly in this region. In this region, there is part of a closed loop of fluid flow. These closed loops show the direction of fluid changing drastically and are hallmarks of turbulent flow. By contrast, the streamlines in region two are nearly parallel with one another. They don’t change direction much at all and also show us that the speed of fluid in this region doesn’t change much either.

This clarifies our answer to the second part of this question. We can say that it’s in region two that the fluid flow is steadier because it’s in this region that fluid speed and direction change less.

And now for the first part of our question, which asks in which of the two regions the fluid flow is faster, now it’s a bit counterintuitive. But actually, there’s a correlation between steady fluid flow and faster fluid speed. The reason for this is that when fluid is flowing steadily along, that means its direction isn’t changing very much. It’s not interfering with itself. The effect of that interference, the fluid pushing against itself, would be to slow down the average speed of the fluid.

For example, consider the fluid in region one in our diagram. We can assume that in this closed loop we see, the fluid is flowing in one direction in one part of the loop and in the other direction in the other part. The fact that within this region some fluid is flowing one way and some is flowing in the opposite direction shows us that fluid in this region is strongly interfering with itself. It’s not able to establish a steady flow. And therefore, it doesn’t have much opportunity to pick up speed. And that means that the average speed of the fluid in this region is lower than that of a region within the same flow but where the fluid is not pushing against itself quite so much.

So not only is region two the place where fluid flow is steadier, it’s also the region where fluid flow is faster. In general, the steadier a fluid flow is, the faster it can move along.

Let’s summarize now what we’ve learned about steady and turbulent flow. In this lesson, we saw that steady and turbulent fluid flows are opposites. We saw further that steady flow happens when fluid speed and direction change less, that is, are fairly constant, while turbulent flow is indicated where fluid speed and direction change more. Along with this, we saw that in diagrams of fluid flow, regions that display closed loops, closed streamlines indicating fluid direction, are likely to be regions where turbulent flow is taking place, whereas regions with nearly parallel flow lines that don’t change direction much indicate regions of relatively steady fluid flow.

And lastly, we emphasized that these two types of flows, steady and turbulent, are defined in terms of the changes that do or do not occur in fluid speed and direction. This is a summary of steady and turbulent flow.