fluid_mechanics

6.4 Inviscid Flow 281
inviscid flows, but for compressible fluids the variation in r with p must be specified before the
first term in Eq. 6.56 can be evaluated.
For inviscid, incompressible fluids 1commonly called ideal fluids2 Eq. 6.56 can be written as
p
r V 2
gz constant along a streamline
2
(6.57)
and this equation is the Bernoulli equation used extensively in Chapter 3. It is often convenient to
write Eq. 6.57 between two points 112 and 122 along a streamline and to express the equation in the
“head” form by dividing each term by g so that
p 1
g V 2 1
2g z 1 p 2
g V 2 2
2g z 2
(6.58)
It should be again emphasized that the Bernoulli equation, as expressed by Eqs. 6.57 and 6.58, is
restricted to the following:
inviscid flow
steady flow
incompressible flow
flow along a streamline
You may want to go back and review some of the examples in Chapter 3 that illustrate the use of
the Bernoulli equation.
The vorticity is zero
in an irrotational
flow field.
6.4.3 Irrotational Flow
If we make one additional assumption—that the flow is irrotational—the analysis of inviscid
flow problems is further simplified. Recall from Section 6.1.3 that the rotation of a fluid
1
element is equal to 21 V2, and an irrotational flow field is one for which V 0 1i.e.,
the curl of velocity is zero2. Since the vorticity, Z, is defined as V, it also follows that in
an irrotational flow field the vorticity is zero. The concept of irrotationality may seem to be
a rather strange condition for a flow field. Why would a flow field be irrotational? To answer
1
this question we note that if 21 V2 0, then each of the components of this vector, as
are given by Eqs. 6.12, 6.13, and 6.14, must be equal to zero. Since these components include
the various velocity gradients in the flow field, the condition of irrotationality imposes specific
relationships among these velocity gradients. For example, for rotation about the z axis to be
zero, it follows from Eq. 6.12 that
and, therefore,
Similarly from Eqs. 6.13 and 6.14
v z 1 2 a 0v
0x 0u
0y b 0
0v
0x 0u
0y
0w
0y 0v
0z
0u
0z 0w
0x
(6.59)
(6.60)
(6.61)
A general flow field would not satisfy these three equations. However, a uniform flow as is illustrated
in Fig. 6.13 does. Since u U 1a constant2, v 0, and w 0, it follows that Eqs. 6.59, 6.60, and
6.61 are all satisfied. Therefore, a uniform flow field 1in which there are no velocity gradients2 is
certainly an example of an irrotational flow.
Uniform flows by themselves are not very interesting. However, many interesting and
important flow problems include uniform flow in some part of the flow field. Two examples are