fluid_mechanics

268 Chapter 6 ■ Differential Analysis of Fluid Flow
The three components, v x , v y , and v z can be combined to give the rotation vector, , in the form
v x î v y ĵ v z kˆ
(6.15)
An examination of this result reveals that is equal to one-half the curl of the velocity vector. That is,
1 2 curl V 1 2 V
(6.16)
Vorticity in a flow
field is related to
fluid particle rotation.
Wing
since by definition of the vector operator V
î
1
2 V 1 2 ∞ 0
0x
u
ĵ
0
0y
v
kˆ
0
∞
0z
w
1 2 a 0w
0y 0v
0z b î 1 2 a 0u
0z 0w
0x b ĵ 1 2 a 0v
0x 0u
0y b kˆ
The vorticity, Z, is defined as a vector that is twice the rotation vector; that is,
Z 2 V
(6.17)
The use of the vorticity to describe the rotational characteristics of the fluid simply eliminates the
1 1 22 factor associated with the rotation vector. The figure in the margin shows vorticity contours of
the wing tip vortex flow shortly after an aircraft has passed. The lighter colors indicate stronger
vorticity. (See also Fig. 4.3.)
We observe from Eq. 6.12 that the fluid element will rotate about the z axis as an undeformed
block 1i.e., v OA v OB 2 only when 0u0y 0v0x. Otherwise the rotation will be associated
with an angular deformation. We also note from Eq. 6.12 that when 0u0y 0v0x the rotation
around the z axis is zero. More generally if V 0, then the rotation 1and the vorticity2 are zero,
and flow fields for which this condition applies are termed irrotational. We will find in Section 6.4
that the condition of irrotationality often greatly simplifies the analysis of complex flow fields.
However, it is probably not immediately obvious why some flow fields would be irrotational, and we
will need to examine this concept more fully in Section 6.4.
E XAMPLE 6.1
Vorticity
GIVEN For a certain two-dimensional flow field the velocity
is given by the equation
V 1x 2 y 2 2î 2xyĵ
FIND
Is this flow irrotational?
SOLUTION
For an irrotational flow the rotation vector, , having the components
given by Eqs. 6.12, 6.13, and 6.14 must be zero. For the prescribed
velocity field
and therefore
u x 2 y 2 v 2xy w 0
v x 1 2 a 0w
0y 0v
0z b 0
v y 1 2 a 0u
0z 0w
0x b 0
v z 1 2 a 0v
0x 0u
0y b 1 312y2 12y24 0
2
Thus, the flow is irrotational.
(Ans)
zero, since by definition of two-dimensional flow u and v are not
functions of z, and w is zero. In this instance the condition for irrotationality
simply becomes v z 0 or 0v0x 0u0y.
The streamlines for the steady, two-dimensional flow of this example
are shown in Fig. E6.1. (Information about how to calculate
y
COMMENTS It is to be noted that for a two-dimensional flow
field 1where the flow is in the x–y plane2 and will always be
v x
v y
F I G U R E E6.1
x