ERDC/CHL CETN-IV-26
June 2000
Appendix I: Derivation of Scaling Relationship
An empirical expression for equilibrium scour depth as a function of sediment size and
maximum discharge per unit width was presented in CETN-IV-18 (Hughes 1999). This
relationship was formulated as a balance between the shear stress at the bottom and the critical
shear stress of the noncohesive sediment. An unknown coefficient in the theoretical formulation
was found using maximum discharge measurements from two inlets. The resulting formula for
equilibrium scour depth is restricted to those portions of the inlet channel where currents alone
are responsible for movement of sediment by bed load. The equilibrium scour depth relationship
is given in terms of discharge per unit length as:
0.234 qe /9
8
he =
(10)
b
g
4/9
de /3
1
g Ss - 1
or in terms of velocity as:
b
g 1/2 de3/8 he1/8
V = 5.124 g Ss - 1
(11)
where
he
= water depth at equilibrium
qe
= maximum equilibrium discharge per unit width
g
= gravitational acceleration
= sediment specific gravity [=ρs /ρw ]
Ss
ρs
= mass density of sediment
ρw
= mass density of water
de
= sand median grain size
V = depth-averaged velocity
Similitude of equilibrium scour depths is achieved by assuring that the relationships given by
Equations 10 and 11 have the same effect in the movable-bed model as in the real world. The
necessary scaling relationship is found by first rearranging Equation 11 into the form of a
nondimensional number, i.e.:
V
= 5.124
(12)
b
g
1/2
de /8 he/8
3
1
g Ss - 1
Similitude requires maintaining the same value of the nondimensional number in both the model
and prototype, or:
F
I F
I
GG
J = GG
J
V
V
1/2 3/8 1/8 J
1/2 3/8 1/8 J
(13)
H g bSs - 1g de he K p H g bSs - 1g de he K m
9
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