Water flows over a broad-crested weir

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Flow through nozzles: area change, choking and thrust

Problem

Water flows over a broad-crested weir. Far upstream the specific energy (relative to the crest) is

E = h + q^2/(2gh^2)

, with

q

the discharge per unit width and

g = 9.81\,\mathrm{m\,s^{-2}}

. On the crest the flow becomes critical,

\mathrm{Fr} = u/\sqrt{gh} = 1

, exactly as a choked nozzle becomes sonic at its throat. ((a)) Using

\mathrm{Fr} = 1

on the crest together with the definition of

E

, show that the critical depth is

h_{\rm c} = \tfrac{2}{3}E

and that the discharge is

q = \sqrt{g}\,h_{\rm c}^{3/2}

. Evaluate both for an upstream specific energy

E = 1.20\,\mathrm{m}

. ((b)) Recompute

h_{\rm c}

and

q

for

E = 1.44\,\mathrm{m}

(a 20 per cent increase) and check that

q

scales as

E^{3/2}

. ((c)) Explain, in the language of choking, why the discharge over a critical weir crest grows as

E^{3/2}

while the choked mass flow through a sonic throat grows only in direct proportion to the reservoir pressure

p_0

at fixed

T_0

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