Environmental Engineering Reference
In-Depth Information
Table 3.2 Classification of
boundary conditions,
overview
Type
Name
Condition for variable
u
(s)
First
Dirichlet
u ¼ u
1
specified
Second
Neumann
@u=@s
specified
Third
Cauchy or Robin
a
0
u þ a
1
@u=@s ¼ j
a
0
c þ a
1
@c
@n
¼ j
for mass transport or
a
0
T þ a
1
@T
(3.32)
@n
¼ j
e
for heat transport
with given coefficients
a
1
and given mass flux
j
or heat flux
j
e
. In flow
problems third type boundary conditions are formulated analogously in terms of
hydraulic head, pressure, pressure head or streamfunction. The third type condition
includes first and second type conditions as special cases. Third type boundary
conditions are connecting advective and diffusive fluxes (Table
3.2
).
Values of boundary conditions may change with time. There are applications
where even the type of the boundary condition changes with time.
In transient problems, another form of conditions appears in addition to bound-
ary conditions: the
initial conditions
. As the name tells, an initial condition
concerns the knowledge of a variable at the beginning of the simulation, usually
at time
t ¼
a
0
and
0. It is necessary to know the starting position if the temporal develop-
ment for
t >
0 is to be simulated.
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