Hydrology EESC BC 3025

Flow in open channels

we need to know some background regarding flow in channels, because of the following questions:

How are water depth and discharge in a stream related?
How does the velocity in a stream change as the amount of water carried by the stream increases?

streams are different from pipes because they do not have a lid!
streamline: a path defined by the motion of fluid elements in a flow; at any point along a streamline, the flow direction is tangent to the streamline. Conceptually, water may not cross streamlines. The density of streamlines is proportional to the flow velocity.

Specific Energy

Bernoulli equation is a good starting point, although it is dealing with a frictionless fluid; over short distances no friction can be assumed
recall the Bernoulli equation, for a frictionless fluid,
for horizontal flow the velocity is the same everywhere (=average velocity); also we can assume that the pressure will follow the hydrostatic equation, p= ρ*g*d, then we can simplify the equation by: U²/(2*g) + h +z_b = H, U being the average flow velocity (Fig4.2)
the first two terms are the specific energy: U²/(2*g) + h = E, the dimension of E is [L]!
by inserting Q=U*h*w (w is width of stream), we are getting Q²/(2*g*w²*h²)+h=E
for a certain discharge rate of the stream, we can plot depth of the stream versus its specific energy: there are two depths possible for each value of specific energy! (Fig4.3)
supercritical, subcritical flow and hydraulic jump (Fig. 5.6)

as shown in the previous examples, we can control the criticality of flow by using obstructions
when installing a weir, we will move from subcritical through critcal to supercritical flow (spillway) (Fig4.7)
we would take advantage of this and derive the equation for discharge as a function of the depth of the water above the weir
the following equation describe this relationship for two geometries of weirs:

in many cases, friction does play a big role in stream flow, so we are actually loosing head in the flow (Roughness of a stream)
flow in a channel is described by Manning's equation (Fig4.8):

R_H=wh/(2h+w): Hydraulic radius
k=1 m^1/3s^-1
S: slope
n: roughness coefficient

for open channels and using 4*the hydraulic radius for the diameter D, the transition between laminar and turbulent flow occurs at the same range of Reynolds numbers (between 2300 and 4000)
velocity distribution as a function of depth (Fig4.10), most flow is turbulent, average flow velocity is at a depth of 0.6*total depth

weirs can be used to measure discharge rates as we did in the lab
we can look impirically at the discharrge rate vs. height relationship for a stream without a well-defined weir and can can use this to estimate the discharge rate (Fig4.13, for this particular creek the exponent in the empirical relationship between Q and h is 2.85).
schematic diagram of a stream gaging station (Fig5.2)
velocity distribution as a function of depth (Fig4.10), most flow is turbulent, average flow velocity is at a depth of 0.6*total depth
we can measure the discharge rate by taking measurements at 0.6* depth (from surface) wading through the river (Fig4.11)
Q = U*A = S w_i*h_i*U_i
Small stream gaging
Gaging larger streams and rivers
stream gaging on the Colorado River at Lee's Ferry

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