these are largely manifested in the horizontal plane over most of the water
column. It is expected that the shed vortices will be in approximate similitude
in a geometrically distorted physical model. Experiments conducted to assess
potential turbulence scaling effects in distorted models are discussed in the
following chapter.
Flow at Bend
Finally, in addition to scale effects related to generation of turbulence, any
mechanism that creates significant vertical velocities or accelerations will be
problematic in a geometrically distorted model. For example, at river bends
centrifugal forces pile water up on the outside of the bend, which creates a lateral
slope in the water surface. The force imbalance results in a cross-channel return
flow toward the inside of the curve along the bottom. The resulting secondary
flow resembles a helix as it moves downstream. In a distorted model the curve
geometry will be tighter and the side slopes will be steeper. The same
phenomenon occurs where flow accelerates as it passes a headland or jetty
causing a lowering of the local water surface.
Similitude requirements are formally determined as before, only this time the
following governing equations are presented in cylindrical coordinates as often
adopted for numerical modeling of flow around river bends. Turbulence terms
and viscous shear stress terms have been omitted from the momentum equations
to focus on the distorted model scale effects associated with flow acceleration at
a river bend.
∂vr vr 1 ∂vθ ∂vz
++
+
=0
(18)
r r ∂θ
∂r
∂z
r-direction momentum
2
∂vr
∂vr vθ ∂vr vθ
∂v
1 ∂p
+ vr
+
- + vz r = -
(19)
r ∂θ
ρ ∂r
∂t
∂r
∂z
r
acceleration terms
pressure term
θ-direction momentum
∂vθ
∂v
v ∂v
∂v
1 ∂p
vv
+ vr θ + θ θ + r θ + vz θ = -
(20)
r ∂θ
ρ r ∂θ
∂t
∂r
∂z
r
acceleration terms
pressure term
32
Chapter 4 Turbulence Scale Effect in Distorted Models