Reading the River Current: Basic Hydraulics

I am haunted by waters.

― Norman Maclean, A River Runs Through It

I have been fascinated by flowing water for as long as I can remember. The smooth and gentle, ever shifting landscape of the water surface, in places turning into a noisy, frantic, ragged dance while elsewhere becoming smooth and transparent as glass, brings out my most artistic leanings along with childhood yearnings for play. We have all watched moving water. But, the more you watch it, the more you really pay attention to it, the more its patterns stand out and beckon you to learn more about it. I’d like to start with two of the places where I played in water as a child: parking lots, and sandy beaches by the ocean. When it rains on a parking lot, the water flows is a sheet over the asphalt, gathering in some places into deeper streams. Compared to flow in a river, parking lot flow is extremely shallow, and also quite slow moving, although it might not seem so at first. You can check out the velocity at which the water flows by yourself. Just find a tiny piece of something that will float in this shallow stream, perhaps part of a leaf, or a piece of the ubiquitous garbage you find in parking lots. Then, using your phone as a stopwatch, measure the time it takes for the object to move, say, 3 feet, or a meter. Divide this distance by the time in seconds, and you have velocity. Try this again when you are near some deeper moving water, perhaps a ditch or a small creek. Which one is larger? I am oversimplifying things a bit here, because there are other factors along with depth that influence water velocity. The slope of the stream or parking lot, and the roughness of the ground over which the water flows, are important. But, generally, very shallow flows, other things being equal, have slower velocity than deeper flows.

We can make sense of this with a thought experiment. Thought experiments, which are much easier to perform the real experiments out in nature, have been a source of inspiration to scientists since the beginning of history. So imagine a cube sitting on an inclined plane. This cube represents a cubic foot, or cubic meter, of water. Then imagine lifting the upper end of the plane to increase the slope. Eventually, the cube will begin to slide downhill, reaching some steady, constant, velocity. It slides because of the force of gravity, acting straight downward on the cube, is pointed at an oblique angle to the surface of the plane, partially directing its force along or parallel to it. Now do this again, only this time lift the end of the inclined plane twice as high. Does it make sense that the cube will now slide faster down this steeper surface? Now, I’ve been assuming that you were visualizing a smooth plane, like a board from the hardware store. But what if we coat the surface of this plane with something rough, such as gluing sand onto it. This will slow down the velocity of the cube as it slides. Finally, what if we take a second, identical cube and glue it on top of the first one. Will this double cube slide faster, given that the slope of the plane, and it’s roughness, or the same? In our thought experiment, we have brought together all of the factors that a scientist or engineer would use to calculate the velocity of the water. Velocity increases with increasing depth, with increasing slope, and decreases with increasing roughness.

Now let’s add some details. Just like the block on the inclined plane, when water flows across the surface, it is actually scraping along on that surface. There is friction between the surface and the moving water. In the case of moving water, that frictional force increases with the water velocity. When the flow is very thin, as it is over a parking lot, the frictional force is proportional to the velocity. That is, if you doubled velocity, the frictional force would double. And just like the block sliding down the inclined plane, the water scraping its way down the parking lot increases its speed until that frictional force exactly balances the force of gravity pushing it downhill. Then, velocity stays constant, as long as the surface on which the water is flowing doesn’t change slope or roughness, and the depth remains the same. When the water is flowing at a constant velocity like this, scientists call that “uniform flow.” This is in contrast to “nonuniform flow,” which means that the velocity is changing as the water flows downstream.

Uniform flow also implies that the amount of water flowing, that is, the volume of water per second passing over a given line drawn on the parking lot, stays constant. Volume of water per second is what we call discharge, and it can be measured in units such as cubic feet per second (abbreviated CFS) or cubic meters per second (CMS). Discharge is what scientists talk about when they describe river conditions, like floods or droughts. But let’s get back to our parking lot. If the volume of water discharge is constant over time, and thus the velocity is constant over time, scientists call this condition “steady flow.” “Unsteady flow” means that the discharge changes over time, which means that the velocity, depth, or both, are also likely changing over time. Imagine the flow over the parking lot once the rainstorm has stopped. From that instant on, the flow is unsteady, as the source of water has now been cut off, causing the flow to gradually taper off as the rain water drains away. If the flowing water encounters a steeper slope, using the argument developed above it will speed up. If the water speeds up, what happens to its depth? If the flow is steady, then the same volume water has to pass over a given place, say, a line drawn perpendicular to its flow, every second. So if the water is moving faster, the only way that this can happen is if it gets shallower. This is an intuitive concept called the “conservation of mass,” which simply means that you can’t get more water out of nothing. The amount of flowing water is “conserved.” There is a very simple proportionality, called the continuity of flow equation, which states that water discharge is equal to velocity times the cross-section area of the flow. So if, for example, we double the velocity, the cross-sectional area must be reduced by one half, meaning that the flow on the parking lot will be half as deep.

In comparing the parking lot to the flow in a stream or ditch which is much deeper, there is another characteristic of water which we must discuss, and that is “viscosity.” Viscosity is the tendency of a liquid to resist deformation when its shape is changing. Intuitively, viscosity is the “thickness” or “gooiness” of the fluid. Water has a fairly low viscosity. Syrup or honey or motor oil have high viscosity. Viscosity is what creates the friction between the moving water and the rough surface over which it is flowing. The higher the viscosity, the more friction, and consequently the lower the velocity. Viscosity is actually a property that gives a fluid internal friction. The fluid continually deforms as it flows. That is, if the fluid is moving, it is as if a stack of layers of fluid are sliding on top of one other. Each layer, as it slides, drags the adjacent layer along, through friction. The larger the viscosity, the more each layer drags on the next one. Imagine stirring a bowl full of water with a big spoon, and comparing this with stirring the same bowl, but full of honey. That internal resistance to sliding, the viscosity that is, is much stronger in the honey than in the water. As a fluid is flowing over a surface, the fluid molecules in contact with that surface are essentially stuck to it. These molecules have zero velocity, matching the velocity of the unmoving surface. This layer of zero velocity molecules drags on the next layer of fluid moving above it which in turn drags on fluid layers further out , etc. But the influence of the surface dragging on the layers of fluid decreases as we move further out from that surface, into higher velocity fluid. This region, close to the surface, where the velocity increases sharply as we move out into the flow, is called the “boundary layer.” The boundary layer thickness will be, of course, larger for more viscous fluids. In fact, if that boundary layer extends all the way to the surface of the flowing water, such that the entire depth of water acts like layers sliding past each other, dragging on each other as they move, we have the situation described by scientists as “laminar flow.” The word laminar means layered. This is exactly the situation we have in shallow flow of water over a parking lot. Scientists like to describe this as the case where the “viscous force” (i.e. internal frictional force) overcomes the “inertial force” of the water. Here, inertia refers to the combined mass and velocity of the water (technically, mass times velocity), which has a resistance to change in its state of motion simply by virtue of its mass or “bulk.” Think of the difference between pushing on a teaspoon of water versus the cubic foot or cubic meter of water; a teaspoon has much less inertia, so it is easy to change its velocity. The cubic meter will take a lot more force. So in laminar flow over a parking lot, viscosity is strong enough to control that flow from bottom to top.

In a stream channel a foot deep or a meter deep, this is not the case. Flow in a stream channel is deep enough that the layers of water dragging on each other due to viscosity as they slide past are confined to a region close to the streambed. Above that, this layered movement of water breaks up. Instead of the layers of water orderly sliding along each other, the difference in velocity from layer to layer causes the water to begin to swirl and roll, with small eddies, that carry low velocity water from near the streambed up into the middle of the channel, and carry high velocity water from near the surface back down to near the streambed. This vertical movement of water, created by eddies, means that even though there may be a steady, average velocity which is constant over time, the velocity at individual points in the water column fluctuates, and has both a vertical and a downstream component to its movement. These flow conditions, which are the most common found in any stream or river, are called “turbulent flow.”

In turbulent flow, the water is still scraping against the streambed as it moves down stream. But, it turns out that the friction force between the streambed and the moving water is proportional to the square of the average velocity. That is, if the velocity doubles, this frictional force will increase fourfold. Recall that in the case of laminar flow on the parking lot, frictional force was directly proportional to the water velocity so a doubling of velocity would double the force. Scientists characterize turbulent flow as the condition where the “inertia” or “inertial force” is much greater than the viscous force.

Prediction of water velocity from the characteristics of a stream channel, namely, it’s slope, depth, and roughness, is fairly easy, as long as we assume that the flow is uniform and steady, as described above. The equations used by hydraulic engineers to predict water velocity will be discussed in a later section. This is probably enough material for now, to get you thinking about flowing water. But getting back to the comparison of the parking lot the ditch or stream channel, notice how steady the surface of the water flowing on the parking lot is, in comparison to that of the stream channel. That is the magic of laminar flow. In the stream channel, even when the surface seems relatively steady, it is actually chaotic, with upwelling and downwelling and eddies and small waves gliding from side to side. In addition to friction between the water and streambed, there is friction between the sides of the channel and the moving water. This creates a zone of low velocity or unmoving water near the bank, sometimes with an eddy that acts to locally carry the water back upstream. Moving outward from the bank, there is often an abrupt transition, an “eddy line,” marked by small swirls at the surface, which separate the water flowing rapidly down the middle of the channel with the stagnated water influenced by the friction of streambank. And sometimes within this relatively still water zone near the streambank, you can see pillow like pulses of upwelling water on the surface. This is the magic of turbulent flow. Next section: critical flow