Accounting for velocity distributions in water surface profiles

Actual velocities in a cross section are distributed from highest, generally in the center at a depth that is some small proportion beneath the surface, to much lower values in overbanks and at flow boundaries (fig. 11). A velocity meter measures velocities related to the vertical flow area close to the instrument.

This elementary phenomenon is responsible for the fact that an average cross-sectional velocity cannot provide a precise measure of the kinetic energy of the flow; the alpha and beta coefficients therefore are needed as modifiers.

When the flow velocity in a cross section is not uniformly distributed, the kinetic energy of the flow, or velocity head, is generally greater than V2/2g, where V is the average velocity. The true velocity head may be approximated by multiplying the velocity head by alpha (α), the energy coefficient. Chow (1959) stated that experiments generally place alpha between 1.03 and 1.36 for fairly straight prismatic channels. The nonuniformity of velocity distribution also influences momentum calculations (as momentum is a function of velocity).

Beta (β) is the momentum coefficient that Chow indicates varies from 1.01 to 1.12 for fairly straight prismatic channels. Beta, also called the Boussinesq coefficient, is also described in Chow (1959). Both coefficients may be calculated by dividing the flow area into subareas of generally uniform velocity distribution. 

However, for natural channels, the calculation is better made using conveyance. HEC–RAS uses the following formulas:

Every cross section is only a two-dimensional slice of a three-dimensional reality. Cross sections change along the stream profile, inevitably setting up transverse velocity vectors, and the flow is induced into a roughly spiral motion. This flow behavior leads to point bars, pools and riffles, meandering patterns, and flood plains. Further information on the velocity and shear in the design of streambank protection in bends is given in NEH654.14, Stabilization Techniques.