Principles of Operation:
Channel Flow

This section introduces some of the physical concepts that play roles in flow monitoring with an EasyQ.

Turbulence and Stratification  

It is good that rivers are turbulent, because this is why we can measure flow with a simple velocity and stage sensor at the side. Turbulence mixes and redistributes everything in the river, including the flow itself. So even though turbulent velocities fluctuate widely, the constant mixing by turbulence makes the river's mean velocity structure predictable.

Stratification, on the other hand, makes it much harder to estimate flow from a velocity/stage sensor. Stratification is the process in which water forms layers of different density, the heaviest layers lying below the lightest layers. Free-flowing rivers are mostly unstratified and fully turbulent. Slow-moving flows in backwater conditions can stratify in hot weather, with warmer water on top and colder near the bottom. Estuaries can also stratify where salty ocean water meets fresh river water.

You could consider stratification and turbulence as being at odds with one another. Turbulent mixing tends to break down stratification, and stratification tends to inhibit turbulence. In fact the two do not coexist. Stratified water does not mix much, and that is why velocity measured in one part of the flow may not represent the velocity elsewhere.

Vertical Velocity Structure in Turbulent Flow  

Flow in a river or channel is related to flow in a turbulent boundary layer, and in fact you could consider the river or channel to be nothing more than one large boundary layer. Physicists model the mean or steady flow in boundary layers as a log layer--the flow increases with the log of the distance from the boundary. Without going into too much detail, there are some ramifications:

• velocity changes rapidly near the boundary and more slowly in the interior of the flow
• the equations are not time dependent, which is a way of saying that velocity in one part of the flow will maintain a constant relationship with the velocity in a different part.

The flow in the interior of a wide, flat channel is typically modeled as a log layer (Figure 1). The underlying physical concept that leads to a log layer is the idea that eddies can get larger as you get further from the wall. Over time, physicists have become convinced that this rule works well. Many people have found it more convenient to substitute a power law for the log law. The power law is not based on physics; it is just a curve fit. But, as you can see in Figure 1, the power law closely matches the log law.

Figure 1. Log law vs. power law. The 1/6 power law is a standard profile, commonly applied to river flow.

The log layer's parameter z₀ represents bottom roughness. Larger bottom roughness corresponds to larger z₀, with the result that currents vary more from top to bottom. Man-made channels are smoother on the bottom, and the flow in such a channel is more uniform, top to bottom, than the flow in a river. Figure 2 shows a family of log-law curves, each with the same mean velocity, but with different bottom roughnesses. The blue line is the velocity profile one would find in a smooth channel, while the red line is the profile in a channel with a very rough bottom.

Figure 2. Family of curves representing log-law profiles for varying roughnesses. All have the same mean velocity. The red curve uses the roughness parameter z₀ = 1/10 of the depth, while the blue line corresponds to z₀ = 10^-6 times the depth (i.e. a very smooth bottom). The magenta line (z₀ = 0.0005 times the depth) corresponds approximately to a 1/6 power law, while the blue line corresponds approximately to a 1/12 power law.

A standard USGS practice allows hydrographers to estimate the average flow velocity by either measuring the velocity at 0.6 times the depth or by averaging the velocities measured at 0.2 and 0.8 times the depth. Figure 3 shows that the USGS practice is pretty good. both the 0.2/0.8 and 0.6 methods produce velocities that are within around 0.1% of the depth-averaged velocity, and the method works over a wide range of bottom roughness. Another message of Figure 3 is that it is important to get the right depth when you use these methods!

Figure 3. Standard USGS practice. The red and blue lines give the velocities computed using the 0.6 depth and 0.2/0.8 depth methods, as a fraction of the true mean velocity. The resulting velocity is close to 1.00 over a wide range of bottom roughnesses. The arrows for the 1/6 and 1/12 power laws show the bottom roughness that give log profiles most closely matching these power laws. The 1/6 power law corresponds to typical river flow and the 1/12 power law might apply in a man-made channel.

Secondary Flows  

Friction at channel walls induces secondary flows, as sketched in Figure 4. The water loses momentum as the secondary flow moves it up past the side of the channel. Then when it flows out into the interior of the flow at teh top of the channel, it has a lower down-channel velocity than the water it is flowing into. This reduces the velocity at the top of the flow, with the result that the maximum velocity occurs beneath the surface. Figure 5 illustrates how secondary flows will change the vertical profile of velocity.

Figure 4. Secondary flow. Top: pattern of cross-channel velocity. Bottom: Down-channel velocity contours. The depth of maximum velocity (dashed line) is below the surface, and deepens closer to the side.

Figure 5. How secondary flows change the vertical profile of velocity.

Because secondary flows require the presence of a side wall, their effects are localized near the wall. At a distance into the channel or river equal to several times the depth, secondary flows should disappear.

Flow Separation  

Recirculation zones often form at the sides of rivers, typically (but not always) behind obstacles such as boulders, headlands or bridge supports. These recirculation zones are the result of turbulent flow separation. Figure 6 illustrates this circulation. The boundary between the interior flow and the flow separation (shown with the red dashed line) has a nominal location, but it can move back and forth.

Figure 6. Flow separation causes recirculation at the side of a river. The dashed line indicates the boundary between the interior of the turbulent flow and the region of flow separation.

Acoustic Propagation  

Figure 7 shows how an acoustic beam propagates into a river from the side. This example shows a 3 degree beam oriented horizontally. At about 10 m from the side wall, the sides of the beam begin to touch the bottom and surface. Echoes from the surface are unlikely to change the measurements much because the surface acts like a mirror and reflects most of teh sound forward, and because the surface is moving at nearly the same speed as the center of the water column. Echoes from the bottom, however, will tend to bias the velocity toward zero. The bottom tends to be acoustically rough, so it scatters sound back, and the signal it sends back biases the measured velocity toward zero becuase the bottom is still. A narrower beam will propagate further into the river without being contaminated by the bottom. A standard 2 MHz EasyQ has a beamwidth of 1.7 degrees and a 1 MHz EasyQ has a beam width of 3.5 degrees.

Figure 7. Beam propagation into a river.

Stratification complicates beam propagation, as shown in Figure 8. The beam in Figure 8 begins to touch the bottom much closer to the transducer. Using a narrower beam helps a little, but not a lot. The assumed stratification in Figure 8 is a bit extreme: 12 C colder at the bottom than at the top. but even weaker stratification can have a similar effect. Stratification tends to be the biggest concern in situations where you use narrower beams in an effort to see further into the flow.

Figure 8. Beam propagation in temperature-stratified water. The assumed temperature difference is 12 C, top to bottom.