Civil Engineering Reference
In-Depth Information
fluid. In general, liquids are most often treated as incompressible but the viscos-
ity effects depend specifically on the fluid. Gases, on the other hand, are gener-
ally treated as compressible but inviscid.
In this chapter, we examine only incompressible fluid flow. The mathemat-
ics and previous study required for examination of compressible flow analysis is
deemed beyond the scope of this text. We, however, introduce viscosity effects in
the context of two-dimensional flow and present the basic finite element formu-
lation for solving such problems. The extension to three-dimensional fluid flow
is not necessarily as straightforward as in heat transfer and (as shown in Chap-
ter 9) in solid mechanics. Our introduction to finite element analysis of fluid flow
problems shows that the concepts developed thus far in the text can indeed be
applied to fluid flow but, in the general case, the resulting equations, although
algebraic as expected from the finite element method, are nonlinear and special
solution procedures must be applied.
8.2 GOVERNING EQUATIONS
FOR INCOMPRESSIBLE FLOW
One of the most important physical laws governing motion of any continuous
medium is the principle of conservation of mass. The equation derived by appli-
cation of this principle is known as the
continuity equation
. Figure 8.2 shows a
differential volume (a control volume) located at an arbitrary, fixed position in
a three-dimensional fluid flow. With respect to a fixed set of Cartesian axes,
the velocity components parallel to the
x
,
y
, and
z
axes are denoted
u
,
v
, and
w
,
respectively. (Note that here we take the standard convention of fluid mechanics
by denoting velocities without the “dot” notation.) The principle of conservation
of mass requires that the time rate of change of mass within the volume must
w
v
v
d
y
y
u
x
d
x
u
d
x
u
d
y
d
z
y
w
z
w
d
z
v
x
z
Figure 8.2
Differential volume element in
three-dimensional flow.
