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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