Problem 58324. Compute the critical depth of a channel

Problem statement
Write a function to compute the critical depth of a channel with discharge Q. The unit system will be specified in units as either ‘SI’ or ‘USCS’ (U.S. customary system); take the acceleration of gravity g to be either 9.81 m/s2 or 32.2 ft/s2. The geometry of the channel’s cross section will be specified by a structure channelStruct as in Cody Problem 58314.
Specific energy for a flow is E = y + V^2/2g, where y is the water depth and V is the velocity averaged over the cross section. The critical depth is the depth of minimum specific energy. Using the definition of the average velocity V = Q/A and differentiating with respect to depth gives
dE/dy = 0 = 1+(-2) (Q^2/2gA^3)dA/dy = 1-(Q^2/gA^3)dA/dy
Then using the definition of the top width T = dA/dy gives the condition for the critical depth in terms of a dimensionless parameter called the Froude number Fr:
Fr^2 = Q^2T/gA^3 = 1
Flows with depths smaller than critical are supercritical—fast and shallow (Fr > 1), and flows with depths greater than critical are subcritical—deep and slow (Fr < 1). A flow changes from supercritical to subcritical with a hydraulic jump, which can be observed in the field, in the laboratory, and even in a kitchen sink.

Solution Stats

71.43% Correct | 28.57% Incorrect
Last Solution submitted on Jun 20, 2023

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