Free surface flows 105
where
Ω , is the vector of the earth radial velocity,
Ω =
2
86400
π rad/sec,
and
U ui vj
= +
ˆ
ˆ is the fluid particle velocity vector. The components of the
Coriolis force in the horizontal flow plane (x–y) are given as
C
fv
C
fu
x
y
ρ
ρ
=
= −
and
(5.34)
where f = 2Ωsinφ and φ is the latitude of the geographic location of the flow.
In the case of two-dimensional flows, the unknown variables are the free
surface deviation ζ(x,y,t) from the horizontal still water level h(x,y) and
the depth mean horizontal velocity components u(x,y,t) and v(x,y,t). The
governing equations are formulated on the basis of the principles of mass
conservation and momentum balance along the horizontal x- and y-axes.
The external forces acting on the control volume (a water column extending from the bed to the surface with base dimensions Δx, Δy) are the wind
friction on the water surface (τ sx , τ sy ), and the bottom friction on the solid
bed (τ bx , τ by ). These shear stresses are commonly expressed as quadratic
functions of the velocities: of the wind (
ˆ
ˆ )
w w i w j
x
y
=
+
on the surface and
of the water ( )
U on the bed, along with the corresponding friction coefficients f s and f b :
τ
ρ
τ
ρ
sx
s x
x
y
sy
s y
x
y
f w w
w
and
fw w
w
=
+
=
+
2
2
2
2
(5.35)
τ
ρ
τ
ρ
bx
b
by
b
f u u v
f v u v
=
+
=
+
2
2
2
2
and
(5.36)
The order of magnitude of these dimensionless friction coefficients is O(f b ) =
10 –3 , O(f s ) = 10 –6. Apart from those external forces, internal frictional forces
intervene in the equilibrium, resulting in the diffusion of the momentum in
the water body. Since the Reynolds numbers are high, the flows are turbulent
and the internal friction is regulated, not by the molecular viscosity, μ, but
by the ‘eddy’ or turbulent viscosity, ε. The internal friction effects can be
described, in the simpler possible form, by means of diffusive terms:
D
u
x
u
y
and D
v
x
v
y
x
h
y
h
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
ε
ε
2
2
2
2
2
2
2
2
(5.37)
where ε h is the horizontal eddy viscosity coefficient (Rodi 2000).
where
Ω , is the vector of the earth radial velocity,
Ω =
2
86400
π rad/sec,
and
U ui vj
= +
ˆ
ˆ is the fluid particle velocity vector. The components of the
Coriolis force in the horizontal flow plane (x–y) are given as
C
fv
C
fu
x
y
ρ
ρ
=
= −
and
(5.34)
where f = 2Ωsinφ and φ is the latitude of the geographic location of the flow.
In the case of two-dimensional flows, the unknown variables are the free
surface deviation ζ(x,y,t) from the horizontal still water level h(x,y) and
the depth mean horizontal velocity components u(x,y,t) and v(x,y,t). The
governing equations are formulated on the basis of the principles of mass
conservation and momentum balance along the horizontal x- and y-axes.
The external forces acting on the control volume (a water column extending from the bed to the surface with base dimensions Δx, Δy) are the wind
friction on the water surface (τ sx , τ sy ), and the bottom friction on the solid
bed (τ bx , τ by ). These shear stresses are commonly expressed as quadratic
functions of the velocities: of the wind (
ˆ
ˆ )
w w i w j
x
y
=
+
on the surface and
of the water ( )
U on the bed, along with the corresponding friction coefficients f s and f b :
τ
ρ
τ
ρ
sx
s x
x
y
sy
s y
x
y
f w w
w
and
fw w
w
=
+
=
+
2
2
2
2
(5.35)
τ
ρ
τ
ρ
bx
b
by
b
f u u v
f v u v
=
+
=
+
2
2
2
2
and
(5.36)
The order of magnitude of these dimensionless friction coefficients is O(f b ) =
10 –3 , O(f s ) = 10 –6. Apart from those external forces, internal frictional forces
intervene in the equilibrium, resulting in the diffusion of the momentum in
the water body. Since the Reynolds numbers are high, the flows are turbulent
and the internal friction is regulated, not by the molecular viscosity, μ, but
by the ‘eddy’ or turbulent viscosity, ε. The internal friction effects can be
described, in the simpler possible form, by means of diffusive terms:
D
u
x
u
y
and D
v
x
v
y
x
h
y
h
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
ε
ε
2
2
2
2
2
2
2
2
(5.37)
where ε h is the horizontal eddy viscosity coefficient (Rodi 2000).