Appendix: Equilibrium Discharge Relationship
Formulation: Assume the vertical velocity profile during maximum discharge through a
tidal inlet can be represented as a steady, fully-developed, rough turbulent boundary layer
extending from the bottom to the free surface. Any contribution by waves is neglected. The
boundary layer velocity profile can be adequately approximated by a 1/8-power curve (Yalin
1971) with the shear stress at the bed given as
2
V
(5)
τo = ρw
1/8
Ck (h/de)
where
ρw
mass density of water
V
depth-averaged velocity
Ck
undetermined constant
h
water depth at maximum discharge
de
median grain-size diameter
The constant Ck is a boundary layer shape factor that includes the unknown relationship
between de and bottom roughness.
The Critical Shear Stress of the noncohesive sand bed is given by the Shields parameter as
τcr = Cs (ρs - ρw ) g de
(6)
with
Cs
constant of proportionality
ρs
mass density of sand
g