Physical Modelling in Fluvial Geomorphology

PHYSICAL MODELLING IN FLUVIAL GEOMORPHOLOGY 227
In the case of the Froude number, since acceleration due to gravity, g, remains constant for
both model and prototype, then
and therefore
2
U m Rm
=
2
U R p p
0.
5
m m
u ( 8)
⎟
p p
Equations (6) and (8) can only be resolved if Rm is equal to Rp. Thus, the flow Reynolds
number is commonly relaxed, with the proviso that in the case of open channel flows it
remains within the fully turbulent flow regime (Re > 500). As noted previously, this form
of modelling is referred to as Froude scale modelling (FSM).
⎟
U ⎛ ⎞
⎜
R
λ = =
U ⎜
⎝
R
⎠
Movable-bed modelling (case with mobile sediment)
If a movable rather than fixed bed is considered, the flow can be considered as a
two-phase flow with both fluid and particles. The following set of parameters is used to
describe these flows: µ, ρ, S, R, g and two parameters which describe the sediment, ρs (the
sediment density) and D (the characteristic grain size of the sediment). Some of these
variables can be replaced by other dependent parameters. For example, the shear velocity
U* = (gRS) 0.5 can replace S and the immersed specific weight of grains in the fluid γs =
g(ρs -ρ) can replace g giving µ, ρ, R, D, ρs, U* and γs, These variables also produce n - 3 or
4Π terms
The Π1and Π2 terms represent relative roughness of the sediment and relative density
respectively, while the term Π3 is the grain Reynolds number (Re*), which is a measure of
the roughness of the bed relative to the thickness of the viscous sub-layer. Equation (12)
expresses the Shields relationship which is normally rearranged as
τ =
*
∏
∏
∏
∏
where τ is the bed shear stress responsible for initiating sediment transport for a particular
grain size, D, and τ* is the dimensionless shear stress. Together, τ* and Re* form the axes
of the Shields entrainment diagram (Figure 9.2). The scatter of points on the Shields
1
2
3
4
R
=
D
ρs
=
ρ
ρU*
D
= = Re
µ
s
2
*
ρU
=
γ D
( ρ − ρ )
s
τ
gD
*
( 7)
( 9)
( 10)
( 11)
( 12)
( 13)