Free surface flows 127
The friction coefficients f s , f i and f b depend collectively on the wind, flow
intensity, fluid properties and the bed roughness. Average values are of the
order O(f s ) = 3 × 10 –6 , O(f i ) = 10 –3 and O(f b ) = 10 –2 .
For a two-dimensional x–y horizontal domain, the unknown variables
are the layer total depths h o and h u , and the layer mean velocities u o , u u ,
v o and v u , all functions of x, y and t. The equations formulated for the
estimation of the aforementioned functions are based on the principles of
mass continuity and the forces equilibrium along the horizontal directions
(Savvidis et al. 2004).
5.4.1 Governing equations for horizontal
two-dimensional stratified flows
By assuming small water surface and interface slopes, the equations for the
upper and lower layers can be derived similarly to the ones for quasihorizontal two-dimensional geophysical flows. Thus, by following the
same procedure as for the depth-averaged free surface flows, the governing
equations for the two layers in the two horizontal directions are
Upper layer
∂
∂
+
∂
∂
+
∂
∂
=
h
t
h u
x
h v
y
o
o o
o o
(
)
(
) 0
(5.70)
∂
∂
+
∂
∂
+
∂
∂
= −
∂
+ +
(
)
∂
+
+
∂
u
t
u
u
x
v
u
y
g
h h z
x
fv
u
o
o
o
o
o
o
u
b
o
h
ε
2
o o
o
sx
o o
ix
o o
x
u
y
h
h
∂
+
∂
∂





 +
−
2
2
2
τ
ρ
τ
ρ
(5.71)
∂
∂
+
∂
∂
+
∂
∂
= −
∂ + +
∂
−
+
∂
v
t
u
v
x
v
v
y
g
h h z
y
fu
v
o
o
o
o
o
o
u
b
o
h
(
)
ε
2
o o
o
sy
o o
iy
o o
x
v
y
h
h
∂
+
∂
∂





 +
−
2
2
2
τ
ρ
τ
ρ
(5.72)
Lower layer
∂
∂
+
∂
∂
+
∂
∂
=
h
0
u
u u
u u
t
h u
x
h v
y
(
)
(
)
(5.73)