4.6-Rotation and Vorticity.pdf

Rotation and Vorticity

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4.6 Rotation and Vorticity

Concept of Rotation

In many applications of fluid mechanics it is important to establish the rotationality of a flow field. There are

many closedform solutions for fluid flow fields based on irrotational flow. To make use of these solutions, the

engineer must be able to determine the degree of flow rotation in his or her application. The purpose of this

section is to introduce the concept of rotation.

The idea of fluid rotation is clear when a fluid rotates as a solid body. However, in other flow configurations it

may not be so obvious. Consider fluid flow between two horizontal plates, Fig. 4.16, where the bottom plate is

stationary and the top is moving to the right with a velocity V. The velocity distribution is linear; therefore, an

element of fluid will deform as shown. Here it is seen that the element face that was initially vertical rotates

clockwise, whereas the horizontal face does not. It is not clear whether this is a case of rotational motion or not.

Figure 4.16 Rotation of a fluid element in flow between a moving and stationary parallel plate.

Rotation is defined as the average rotation of two initially mutually perpendicular faces of a fluid element. The

test is to look at the rotation of the line that bisects both faces ( a a and b b in Fig. 4.16). The angle between this

line and the horizontal axis is the rotation, θ.

The general relationship between θ and the angles defining the sides is shown in Fig. 4.17, where θ A is the angle

of one side with the x axis and the angle θ B is the angle of the other side with the y axis. The angle between the

sides is

, so the orientation of the element with respect to the x axis is

Figure 4.17 Orientation of rotated fluid element.

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Rotation and Vorticity

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The rotational rate of the element is

(4.24)

If , the flow is irrotational .

An expression will now be derived that will give the rate of rotation of the bisector in terms of the velocity

gradients in the flow. Consider the element shown in Fig. 4.18. The sides of the element are initially

perpendicular with lengths x and y . Then the element moves with time and deforms as shown with point 0

going to 0′, point 1 to 1′, and point 2 to 2′. The lengths of the sides are unchanged. After time t the horizontal

side has rotated counterclockwise by θ A and the vertical side clockwise (negative direction) by θ B .

Figure 4.18 Translation and deformation of a fluid element.

The y velocity component of point 1 is ν + (∂ν/∂ x ) x , and the x component of point 2 is u + (∂ u /∂ y ) y . The net

displacements of points 1 and 2 are *

(4.25)

Referring to Fig. 4.18, the angles θ A and θ B are given by

(4.26)

Dividing the angles by t and taking the limit as t → 0,

(4.27)

Substituting these results into Eq. (4.24) gives the rotational rate of the element about the z axis (normal to the

page),

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Rotation and Vorticity

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This component of rotational velocity is defined as z , so

(4.28a)

Likewise, the rotation rates about the other axes are

(4.28b)

(4.28c)

The rateofrotation vector is

(4.29)

An irrotational flow ( = 0) requires that

(4.30a)

(4.30b)

(4.30c)

The most extensive application of these equations is in ideal flow theory. An ideal flow is the flow of an

irrotational, incompressible fluid. Flow fields in which viscous effects are small can often be regarded as

irrotational. In fact, if a flow of an incompressible, inviscid fluid is initially irrotational, it will remain

irrotational.

Vorticity

Another property used frequently in fluid mechanics is vorticity, which is a vector equal to twice the rateof

rotation vector. The magnitude of the vorticity indicates the rotationality of a flow and is very important in flows

where viscous effects dominate, such as boundary layer, separated, and wake flows. The vorticity equation is

(4.31)

where × V , from vector calculus means the curl of the vector V .

An irrotational flow signifies that the vorticity vector is everywhere zero.

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Example 4.10 illustrates how to evaluate the rotationality of a flowfield.

EXAMPLE 4.10 EVALUATIO OF ROTATIO OF

VELOCITY FIELD

The vector V = 10 x i 10 y j represents a twodimensional velocity field. Is the flow irrotational?

PROBLEM DEFINITION

Situation: Velocity field given.

Find: If flow is irrotational.

PLAN

Flow is twodimensional, so w = 0 and . Use Eq. (4.30a) to evaluate rotationality.

SOLUTION

Velocity components and derivatives

Thus flow is irrotational.

The calculation to determine the amount of rotation of a fluid element in a given time is shown in Example 4.11.

EXAMPLE 4.11 ROTATIO OF A FLUID ELEMET

A fluid exists between stationary and moving parallel flat plates, and the velocity is linear as shown.

The distance between the plates is 1 cm, and the upper plate moves at 2 cm/s. Find the amount of

rotation that the fluid element located at 0.5 cm will undergo after it has traveled a distance of 1 cm.

Sketch:

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PROBLEM DEFINITION

Situation: Flow between moving, parallel, flat plates.

Find: Rotation of fluid element at midpoint after traveling 1 cm.

Assumptions: Planar flow

PLAN

1. Use Eq. (4.28a) to evaluate rotational rate with ν = 0.

2. Find time for element to travel 1 cm.

3. Calculate amount of rotation.

SOLUTION

1. Velocity distribution

Rotational rate

2. Time to travel 1 cm:

3. Amount of rotation

REVIEW

Note that the rotation is negative (in clockwise direction).

Rotation in Flows with Concentric Streamlines

It is interesting to realize that a flow field rotating with circular streamlines can be irrotational; that is, the fluid

elements do not rotate. Consider the twodimensional flow field shown in Fig. 4.19. The circumferential velocity

on the circular streamline is V, and its radius is r . The z axis is perpendicular to the page. As before, the rotation

of the element is quantified by the rotation of the bisector, Eq. (4.24), which is

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