EXERCISES
3.5. KINEMATICS OF SIMPLE PLANE FLOWS
element. A spherical element deforms to an ellipsoid whose axes coincide with the principal axes of the local strain-rate tensor.
so that along the first principle axis there is a linear extension rate ofg/2, along the second principle axis there is a linear compression rate ofeg/2, and there is no shear. This can be seen geometrically inFigure 3.14by examining the deformation of an element PQRS oriented at 45, which deforms to P0Q0R0S0. It is clear that the side PS elongates and the side PQ contracts, but the angles between the sides of the element remain at 90. In a small time interval, a small spherical element in this flow would become an ellipsoid oriented at 45 to thex1,x2-coordinate system.
In summary, the element ABCD in a parallel shear flow deforms via shear without normal strain, whereas the element PQRS deforms via normal strain without shear strain. However, both elements rotate at the same angular velocity.
Now consider two steady vortex flows having circular streamlines. In (r,q)-polar coordi- nates, both flows are defined by ur¼0 and uq¼uq(r), with the first one being solid-body rotation,
ur¼0 and uq¼u0r, (3.22)
where u0 is a constant equal to the angular velocity of each particle about the origin (Figure 3.15). Such a flow can be generated by steadily rotating a cylindrical tank containing a viscous fluid about its axis and waiting until the transients die out. From Appendix B, the vorticity component in thez-direction perpendicular to the (r,q)-plane is
uz¼1 r
v
vrðruqÞ 1 r
vur
vq ¼2u0, (3.23)
which is independent of location. Thus, each fluid element is rotating about its own center at the same rate that it rotates about the origin of coordinates. This is evident inFigure 3.15, which shows the location of element ABCD at two successive times. The two mutually perpendicular fluid lines AD and AB both rotate counterclockwise (about the center of the
FIGURE 3.15 Solid-body rotation. The streamlines are circular and fluid elements spin about their own centers at the same rate that they revolve around the origin.
There is no deformation of the elements, only rotation.
3.5. KINEMATICS OF SIMPLE PLANE FLOWS 83
element) with speedu0. The time period for onerotationof the particle about its own center equals the time period for onerevolutionaround the origin of coordinates. In addition,S¼0 for this flow so fluid elements do not deform and each retains its location relative to other elements, as is expected for solid-body rotation.
The circulation around a circuit of radiusrin this flow is G¼
I
C
u,ds¼ Z 2p
0 uqrdq¼2pruq ¼2pr2u0, (3.24) which shows that circulation equals the vorticity, 2u0, times the area contained byC. This result is true for any circuit C, regardless of whether or not it contains the origin (see Exercise 3.23).
Another flow with circular streamlines is that from an ideal vortex line oriented perpen- dicular to the (r,q)-plane. Here, theq-component of fluid velocity is inversely proportional to the radius of the streamline and the radial velocity is again zero:
ur ¼0 and uq ¼B=r, (3.25)
whereBis constant. From (3.23), the vorticity in this flow at any point away from the origin is uz¼0, but the circulation around a circuit of radius rcentered on the origin is a nonzero constant,
G¼ Z 2p
0 uqrdq¼2pruq¼2pB, (3.26)
independent ofr. Thus, considering vorticity to be the circulation per unit area, as in (3.18) whenn¼ez, then (3.26) implies that the flow specified by (3.25) isirrotational everywhere except at r¼0 where the vorticity is infinite with a finite area integral:
½uzr/0¼lim
r/0
1 A Z
A
uzdA¼lim
r/0
1 pr2
I
C
u,ds¼lim
r/0
2B
r2: (3.27)
Although the circulation around a circuit containing the origin in an irrotational vortex flow is nonzero, that around a circuitnotcontaining the origin is zero. The circulation around the contour ABCD (Figure 3.16) is
GABCD¼ Z
AB þ Z
BC þ Z
CD þ Z
DA
u,ds:
The line integrals ofu,dson BC and DA are zero becauseuanddsare perpendicular, and the remaining parts of the circuit ABCD produce
GABCD ¼ ½uqrrDqþ ½uqrrþDrDq¼0,
where the line integral along AB is negative becauseuanddsare oppositely directed, and the final equality is obtained by noting that the productuqr¼Bis a constant. In addition, zero circulation around ABCD is expected because of Stokes’ theorem and the fact that the vorticity vanishes everywhere within ABCD.
Real vortices, such as a bathtub vortex, a wing-tip vortex, or a tornado, do not mimic solid-body rotation over large regions of space, nor do they produce unbounded fluid velocity magnitudes near their axes of rotation. Instead, real vortices combine elements
3. KINEMATICS
84
of the ideal vortex flows described by (3.22) and (3.25). Near the center of rotation, a real vortex’s core flow is nearly solid-body rotation, but far from this core, real-vortex-induced flow is nearly irrotational. Two common idealizations of this behavior are the Rankine vortex defined by
uzðrÞ ¼
G=ps2¼const: forrs
0 forr>s
and uqðrÞ ¼
ðG=2ps2Þr forrs G=2pr forr>s
, (3.28) and the Gaussian vortex defined by
uzðrÞ ¼ G ps2exp
r2=s2
and uqðrÞ ¼ G 2pr
1exp
r2=s2
: (3.29)
In both cases,sis a core-size parameter that determines the radial distance where real vortex behavior transitions from solid-body rotation to irrotational vortex flow. For the Rankine vortex, this transition is abrupt and occurs at r¼s where uq reaches its maximum. For the Gaussian vortex, this transition is gradual and the maximum value ofuq is reached at rz1.12091s(see Exercise 3.26).