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Critical Flow

Critical flow conditions occur when, for a given discharge, the specific energy of flow is at a minimum. The specific energy, E, in Bentley FlowMaster is computed using the equation:

(5.15)

Where

y

=

Flow depth (m, ft)

 

V

=

Velocity (m/sec., ft/sec.)

 

g

=

Gravitational acceleration (m/sec.2, ft/sec.2 )

The quantity V2/2g is also known as the velocity head. The specific energy equation used in Bentley FlowMaster is valid only for small slopes (< 10%). It also neglects the effects of the velocity variation across the flow section; that is, the velocity coefficient, a, is assumed to equal 1.0.

At critical depth, the velocity of flow is also equal to the wave celerity (i.e. the speed at which waves will ripple outward from a pebble which is tossed into the water). The Froude number, F, is defined as the ratio of actual velocity to wave celerity. This number is only defined for sections that have a free surface; it is undefined for closed conduits or closed top irregular channels when flowing full.

The ratio is:

(5.16)

Where

D

=

Hydraulic depth of channel is equivalent to A/T (m, ft)

 

A

=

Flow area (m2, ft2)

 

T

=

Top width of flow (m, ft)

When F is less than one, the flow is said to be subcritical (velocity slower than wave celerity). When F is greater than one, the flow is said to be supercritical (velocity faster than wave celerity). When F is equal to one, the flow is said to be critical. A diagram showing these flow ranges appears below.

Figure 5-1: Specific Energy Curve

The requirement that wave celerity equals actual velocity at critical flow conditions means that critical depth can be computed by varying depth of flow until F equals 1.0. Specifically, Bentley FlowMaster uses the following function, and solves by iterating over depth until f (y) = 0:

(5.17)


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