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
gravitational acceleration
de
median grain-size diameter
and 6 results in the expression
V2
ρ
4
h
1
w
=
(7)
(Ce)8
ρs - ρw
de
g de
where the two unknown constants, Ck and Cs, have been combined into Ce. The term in
square brackets on the right-hand side of Eqn. 7 is the ratio of grain-size Froude number to
the immersed specific gravity of the sand, and it is defined as the Grain Mobility Number
(Yalin 1971).
A more useful form of Eqn. 7 is obtained by multiplying both sides by h8 and rearranging
to get an expression for the equilibrium discharge per unit width, i.e.,
1/2
de3/8 h9/8
qe = Ce [g (Ss - 1)]
(8)
where the qe is defined as the Equilibrium Maximum Discharge per unit width, given
by
qe = V h
(9)
and Ss = ρs/ρw is the sediment specific gravity (about 2.65 for quartz sand). As expected
Eqn. 8 indicates that the equilibrium maximum discharge is primarily a function of water
depth with sediment size having a relatively minor effect.
16
Hughes/Schwichtenberg
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