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Review of Hydrodynamic Theory Chapter | 2 43

∂η ∂(¯uh) ∂(¯vh)
+ + = 0 (2.74)
∂t ∂x ∂y
2
∂(¯uh) ∂(¯u h) ∂(¯u¯vh) ∂η ρ a
2
2
+ + =−gh + CW x W + W + ···
y
x
∂t ∂x ∂y ∂x ρ
√
2
¯ u ¯u +¯v 2 1 ∂(¯ xx h) ∂(¯ yx h)
τ
τ
2
− gn + + (2.75)
h 1/3 ρ ∂x ∂y
2
∂(¯vh) ∂(¯u¯vh) ∂(¯v h) ∂η ρ a
2
2
+ + =−gh + CW y W + W + ···
x
y
∂t ∂x ∂y ∂y ρ
√
2
τ
τ
¯ v ¯u +¯v 2 1 ∂(¯ xy h) ∂(¯ yy h)
2
− gn + + (2.76)
h 1/3 ρ ∂x ∂y
These equations are known as the Saint-Venant, or SWEs, and are the basis
of tidal and surge modelling. Considering η, ¯u, and ¯v as state variables, three
equations can be used to find the solution of a hydrodynamic field; note that
total water depth h can be computed based on η, and bathymetry (b).
We still have the problem of the water column shear-stress terms (i.e.
τ
τ
τ
τ
∂(¯ xx h) ∂(¯ yx h) and ∂(¯ xy h) ∂(¯ yy h) ) in the momentum equations, as they
∂x + ∂y ∂x + ∂y
introduce additional unknowns. In many problems (e.g. open channel flow),
shear stresses in the water column are either neglected or specified by very
simple laws (e.g. constant eddy viscosity). These terms represent some forces
such as turbulent shear stresses, wave radiation stresses, etc., which act in the
water column. They are usually replaced by additional equations. More details
about turbulence and wave radiation forces will be presented in later chapters.
2.4 HYDRODYNAMIC EQUATIONS IN 1D STEADY CASE
The equations of the fluid that was described in the previous sections are
generally solved by numerical methods. On the other hand, for a very simple
case in which the flow is incompressible, approximately 1D, and steady (i.e.
no change in time), the conservation laws can be applied directly (without
resort to numerical methods) to a problem. For instance, we will use these
equations to study the basic hydrodynamics of wind/tidal turbines.
Starting from the continuity equation (2.14); N = m, because the flow is
steady, the Reynolds transport equation becomes

ρu · dS = 0 ⇒ u · A = 0 (2.77)
S
where A = An is the surface vector, and n is the normal vector which
is perpendicular to the surface. Eq. (2.77) indicates that the summation of
volumetric fluxes through the surfaces of a control volume should be zero in
the steady-state case. For a special case of 1D flow, where the velocity is normal
to the control surface, we have