Civil Engineering Reference
In-Depth Information
The element stiffness matrix is observed to be identical to that of the stream
function method. The nodal force vector is significantly different, however. Note
that, in the right-hand integral in Equation 8.40, the term in parentheses is the
scalar product of the velocity vector and the unit normal to an element boundary.
Therefore, the nodal forces are allocations to the nodes of the flow across the
element boundaries. (Recall that we assume unit dimension in the
z
direction, so
the terms on the right-hand side of Equation 8.40 are volumetric flow rates.) As
usual, on internal element boundaries, the contributions from adjacent elements
are equal and opposite and cancel during the assembly step. Only elements on
global boundaries have nonzero nodal force components.
EXAMPLE 8.1
To illustrate both the stream function and velocity potential methods, we now examine
the case of a cylinder placed transversely to an otherwise uniform stream, as shown in
Figure 8.6a. The underlying assumptions are
1.
Far upstream from the cylinder, the flow field is uniform with
u
=
U
=
constant
and
v
=
0
.
2.
Dimensions in the
z
direction are large, so that the flow can be considered two
dimensional.
3
.
Far downstream from the cylinder, the flow is again uniform in accordance with
assumption 1.
y
U
x
(a)
Uy
b
b
c
0
x
Constant
d
a
e
0
(b)
Figure 8.6
(a) Circular cylinder in a uniform, ideal flow. (b) Quarter-symmetry
model of cylinder in a uniform stream.
