Open Channel Flow Basics 2 - Supercritical Flow

Supercritical flow, subcritical flow, and critical flow are categories of open channel flow. Any open channel flow must be in one of these three categories. Supercritical flow is generally high flow velocity, shallow flow (less than critical depth) on a steep channel slope that has Froude number greater than one. Subcritical flow, on the other hand, is characterized by slower flow velocity, deeper flow depth (more than critical depth) on a shallow channel slope with Froude number less than one. Critical flow is the dividing condition between supercritical flow and subcritical flow.

Let's take a look at the conditions for critical flow first. It's the dividing line between subcritical and supercritical flow. The parameter, specific energy, is often used to introduce the concepts of critical, subcritical, and supercritical flow. The specific energy of a liquid flowing in an open channel is defined to be the sum of its kinetic and potential energy per unit weight of flowing liquid, relative to the channel bottom.

Thus: E = y + V²/2g, where

click to enlarge

E = specific energy, ft-lb/lb,

y = depth of flow above the channel bottom, ft,

g = acceleration due to gravity = 32.2 ft/sec², and

V = average flow velocity, ft/sec, = Q/A, where

Q = volumetric flow rate, cfs, and

A = cross-sectional area of flow normal to the direction of flow, ft².

The figure at the right above, shows a plot of specific energy versus depth of flow for a particular rectangular channel example. It shows the general pattern for how specific energy, E, varies with depth. At large values of depth, E is large because y is large. For very small values of y, E is large because the flow velocity, V, becomes large, making the kinetic energy large. In between there will be a point at which the specific energy, E, is a minimum. This is the point that is defined to have critical flow conditions (occurring a depth of flow = critical depth). Flows with a shallower depth and higher flow velocity are called supercritical flow and flows with a higher depth and smaller flow velocity are called subcritical flow, as shown on the diagram.

By using a little calculus (setting dE/dy = 0 and solving for y), it can be shown that y_c = (q²/g)^1/3, for a rectangular channel, where

y_c = critical depth (depth of flow at critical flow conditions), ft,

q = Q/b, cfs/ft, and

b = width of the rectangular channel, ft.

The Froude Number is a dimensionless parameter used in connection with open channel flow. For flow in a rectangular channel, the Froude Number is defined as: Fr = V/(gy)1/2, where the variables in the definition are as defined in the previous section. The equation for critical depth in the previous section, y_c = (q²/g)^1/3, can be rearranged after substituting q = Q/b and Q = VA = Vby, to give the equation: V²/gy_c = 1, or in other words, Fr² = 1, or Fr =1 for critical flow conditions. This can be extended to give the following results:

for Fr < 1, flow is subcritical

for Fr = 1, flow is critical

for Fr > 1, flow is supercritical

This gives a set of criteria for determining whether a given flow is subcritical, critical, or supercritical flow.