Geoscience Reference
In-Depth Information
3.4
HYDRODYNAMICS
The hydrodynamic behavior is governed by mathematical equations that have been
known for more than 100 years. It is believed that these equations have universal
character, and the only reason that we are not able to exactly predict the dynamical
evolution of a water body is that we do not possess analytical solutions to these
equations and our knowledge of boundary and initial conditions is incomplete. These
points will be discussed later in the sections that follow.
The basic hydrodynamic equation can be derived in its full form through the
application of conservation laws to the basic variables of the system. More specif-
ically, the general conservation equation derived in the last chapter can be directly
applied to the variables under consideration.
In the oceanic environment, including coastal lagoons, there are seven variables
that completely define the state of the fluid: the water density
ρ
; the three velocity
1,3; the pressure p ; the temperature
T ; and the salinity S . If only freshwater systems are considered, salinity is not a
variable, reducing the number of state variables to six.
Remarkably, the same set of variables is used also for the description of the
atmosphere. However, in this case, humidity (water vapor content) replaces salinity
as a state variable. Moreover, the equations applicable to atmospheric motion differ
only slightly from the ones used in the oceans.
components u i , i
=
1,3, in the direction of x i , i
=
3.4.1
C ONSERVATION L AWS IN H YDRODYNAMICS
As explained previously, the basic hydrodynamic equations can be relatively easily
deduced from conservation equations of the single state variables. A rigorous deduc-
tion of these equations will not always be shown, but the equations in their original
form will be presented and their meanings and implications noted. It will also be
shown how in all possible cases the equations can be derived using the general
conservation equation (Equation (3.8)), noted earlier in this chapter.
3.4.1.1
Conservation of Mass
The first equation can be deduced through the application of the well-known law of
mass conservation. In this case for the quantity to be conserved in Equation (3.8),
we use the density
which is just mass per unit volume. Diffusion can clearly be
neglected in this case (there is no diffusion of water dissolved in water) and with
the assumption that there is no sink or source in the control volume:
ρ
+
ρ
u
x
ρ
i
=
0
(3.11)
t
i
The conservation equation of mass is also called the continuity equation. It expresses
the fact that a mass flux into a fixed volume will lead to an increase of mass in the
volume due to an increase in density.
 
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