Basic Hydraulics 2: Critical Flow
Shallow laminar, supercritical flow in a parking lot. Notice the gridwork of intersecting shock waves!
Water against stone-
Dimples, then leaps into white;
Rapids call to me
-River Haiku 1021,
Paul Bakke
I grew up in the city, but just a few blocks from my home was a shallow stream with a rocky bottom. Above it, on one side, was the road. On the other side, was a compacted dirt sidewalk, which led down the street to my school. But there was another path, marked by bare spots in the grass and rocks to hop over, below the sidewalk, right next the water. This is where I liked to go. I never dared walk this path when the rain was strong, because the little creek became a raging torrent. In fact, the path was mostly under water then. There was one place where, on sunny days, the water trickled around the sides of a large rock. But when it rained, that rock was submerged. I can still remember how the water flowed over that rock, even though I never dreamed, back then, that knowing about this would become part of my life’s work.
As the water flowed over that rock, it seemed to smooth out and drop, and then formed a stationary wave, capped with white, just downstream. What was going on here? In the last article, I introduced the idea of laminar versus turbulent flow. To recap, in laminar flow, the water acts as if it consists of many horizontal layers sliding past one another, but not mixing. Each layer partly sticks to, and drags upon, the faster-moving layer above it, causing the friction from the streambed to influence layer upon layer all the way to the surface. The shallow flow in a parking lot is laminar (see photo above). In turbulent flow, these horizontal layers break up and intermix, as eddies of various sizes move water upwards and downwards. This vertical motion brings slower, bottom water upwards, and sends faster water down towards the bottom. The channel becomes well-mixed vertically, with only a thin layer near the bottom where the velocity rapidly drops to zero at the contact with the streambed.
Turbulent flow is what we almost always see in rivers and streams. Some rivers, like the ones with whitewater and dancing, foaming, rugged surfaces are quite obviously turbulent. However, even streams that seem to be slow moving, with glassy, smooth surfaces, are actually turbulent. This is paradoxical to many people, but its just the degree of turbulence that is different: whitewater means large, energetic eddies that extend from top to bottom. Smaller eddies, or lazy, slow-moving ones, can look smooth – but on closer scrutiny, you will see subtle dimples or swirls on the water surface that show upwelling and downwelling.
But there is another even more fascinating characteristic of flowing water: critical flow. To understand this, we need to digress and talk about waves. We all know that bodies of water of any size can develop waves. Drop a stone into a smooth place in a river, and you will see waves radiating out from the splash, becoming ever wider circles, the crests moving at a particular speed. Better yet, touch the river surface with a sick that vibrates to generate a series of circular waves. It turns out that the speed of these waves is proportional to the square root of the water depth. So that means, that if you were to double the water depth, the wave would move about 1.4 times as fast, since the square root of 2 is about 1.4. But there is another important consideration, which is that the water in the river is moving. So the waves moving upstream are slowed because they are bucking the current, and as they slow down, they bunch up. The waves moving downstream are being swept along, and end up moving faster than they would in still water, by an amount equal to the velocity of the river current. So, instead of concentric expanding circles, the circles are bunched up on the upstream side and spread out on the downstream side. So what happens when the water is flowing at exactly the same speed as the velocity of a wave at that depth of water? Scientists call this condition “critical flow.” If the water is flowing even faster than the critical flow velocity, the waves from the splashing stick are swept downstream faster than they can move outward and upstream. And if the water is flowing faster than critical flow, instead of a series of expanding bunched up circles, we now see and upstream pointed “V.” the sides of the “V” represent the crests of all those waves, bunched up and stationary because they are moving at exactly the same speed as the water current. The faster the water, the narrower the “V.” Scientists call this condition “supercritical flow.” And as you might have guessed by now, if the current is moving slower than the critical velocity, this is called “sub critical flow.”
The significance of this phenomenon is profound. Waves of one sort or another are essentially the way that changes to the flow, that is, anything that disturbs the uniform flow we talked about in the last article, are communicated upstream and downstream within the flowing water. If conditions are sub critical, then the water surface upstream can smoothly adjust to any changes to the cross-section or velocity downstream, and this adjustment is a response to gradual, step-by-step, incremental changes to the water surface between the location of the change and the location upstream. If the water is flowing at supercritical velocity, this change cannot be communicated upstream. The “message” that a change has occurred cannot travel upstream, only downstream. Supercritical flow, if it occurs at all, is not continuous over long distances in natural rivers, most of the time,. The moving water accelerates, for instance, as the slope of the river increases, reaching critical flow velocity. Usually this occurs at the point where the streambed slope abruptly increases or the streambed suddenly drops away, such as a ledge, or over a submerged log or rock. Downstream from here, the flow accelerates and becomes supercritical. But, soon, it creates a turbulent fixed-position or standing wave called a hydraulic jump. This is literally a place where the water depth suddenly jumps from shallow to deep, accompanied by a sharp decrease in velocity downstream, and furious, noisy, boiling wave that is stationary. Transitions back to sub critical flow usually happen where there is some trigger, such as a subtle change in depth, width, or slope that breaks up the smooth, supercritical flow.
This is what was happening with my rock. The water was accelerating as it went around the rock, due to the reduced cross-section. Acceleration produced a transition to supercritical flow. And then just downstream, was a standing wave, and sub critical flow. Supercritical flow is odd in a couple of ways. First of all, any transition in flow velocity or direction cannot take place gradually, but must happen abruptly, as with the hydraulic jump described earlier. Changes in direction are marked by abrupt lines on the surface, at oblique angles to the flow. These lines are essentially shockwaves, places where the water bunches up as it changes direction or speed.
In popular culture, we think of shockwaves as an air phenomenon, and associate them with supersonic speeds. Bullets, or rockets or airplanes moving at supersonic speeds produce shockwaves at their tips which represent abrupt changes in air pressure, temperature, and air velocity. When these shockwaves sweep past our ears, we hear a loud bang. Explosions also generate shockwaves in the air. And the reason, intuitively, is the same in air as on the water: an object moving faster than the speed of sound causes the sound waves to bunch up onto one line, adding their individual energies together into a shockwave. Sound waves are invisible, moving increases in air pressure that can be sennsed by our ears. A sound shockwave, or “sonic boom,” is a sudden, large increase in pressure and temperature.
In supercritical flowing water, there are small shock waves criss-crossing the surface, which appear stationary if the flow is steady. Each shock wave forms at a place where the water changes direction or speed, such as a rough point on the bottom or edge, a stick, a pebble.
The second way that supercritical flow is odd is that changes to the water surface as it flows over obstacles in the streambed are the opposite of what occurs in the more-familiar subcritical flow. When water subcritical flows over a rock, the flow is squeezed into a smaller cross section, since the rock takes up some of the space. So, the flow must speed up. And as it speeds up, the water surface drops, forming a dimple. In supercritical flow, the opposite occurs: the water surface actually bulges up as the water velocity slows down. Look at the photo in the introduction. The ripples visible there are from water bulging up as it flows over small bumps in the pavement. That photo also exhibits the intersecting grid of shock waves. These are each connected to some small obstacle caused a change in velocity (speed or direction).