ERDC/CHL CETN-IV-26
June 2000
LM NbS g OP1/ 2 Nd3/8 N1L/8
N Q
Nv =
(1)
s-1
e
in which the scale ratios are defined as:
Nv = velocity scale [= V p V m Vp /Vm]
N(Ss-1) = sediment immersed specific gravity scale [=(Ss - 1)p/(Ss - 1)m]
Nde = sand grain size scale [=(de)p/(de)m]
NL = geometric length scale [=Lp/Lm]
where the subscripts p and m represent "prototype" and "model," respectively, and the variables
are defined as:
V = depth-averaged velocity
= sediment specific gravity [= ρs /ρw]
Ss
ρs
= mass density of sediment
ρw
= mass density of water
= sand median grain size
de
L
= length
The N(Ss-1) scale ratio allows use of model sediment having different density than the prototype.
However, for practical reasons quartz sand is typically used in models so N(Ss-1) = 1, reducing
Equation 1 to a simpler scaling relationship given by:
N v = N d /8 N 1/8
3
(2)
L
e
The Nde scale ratio in Equation 2 compensates for model sediment that is relatively larger than if
it had been geometrically scaled according to the length scale, NL, as required by strict geometric
scaling. Consequently, the velocity scale in Equations 1 and 2 is a "distorted" velocity scale. In
other words, the sediment grains in the model are larger than they should be, so the model
currents need to be faster than Froude-scaled currents to achieve the same equilibrium depth of
scour. For the extremely rare case when the prototype sediment can be scaled geometrically,
then Nde = NL, and Equation 2 reduces to the Froude velocity scale, given by
N v = N 1/2
(3)
L
Scaling Validation: The scaling relationship given by Equation 2 is validated by
demonstrating that results from small-scale laboratory inlets at (or approaching) equilibrium
follow the same tidal-prism relationship as real inlets when the model results are scaled using
Equation 2.
O'Brien (1931, 1969) established the relationship between inlet equilibrium cross-sectional area
and tidal prism. O'Brien's original work was extended and refined by Jarrett (1976). The solid
line on Figure 1 is Jarrett's Area - Prism relationship for all inlets. The adjacent dashed lines are
the 95 percent confidence limits. The cluster of data points in the lower left corner of Figure 1
2
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