Section 6.4 National Committee for Fluid Mechanics Films: Vorticity Parts 1 and 2
The excellent fluid dynamics films produced by the National Committee for Fluid Mechanics Films include two films on vorticity. These are available to view online at the above link. Please watch both films, and use the notes below to guide your viewing.
If you are short of time, you may watch the Part 1 up to 18’35’’ and then watch Part 2 from 3’00’’ to 13’25’’ and 17’25’’ to the end, as these parts relate most strongly to the material in this course. However, it is recommended that you watch both films in their entirety
Subsection 6.4.1 Vorticity Part 1
Here is a list of points to look out for while watching Vorticity Part 1:
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0’0’’: The film starts by defining vorticity, c.f. Subsection 6.1.1.
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1’10’’: Vortex lines are defined, c.f. Subsection 6.1.3.’
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2’00’’: There is a description of a `vorticity meter’, which can be used to illustrate the vorticity of various flows.
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3’00’’: Vorticity in solid body rotation of a fluid (a bit like the cup of tea in Example 6.2.5).
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4’30’’: Plug hole vortex (Example 6.2.8). This will nicely illustrate an important concept in this chapter: a line vortex Example 4.3.13.
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6’50’’: Crocco’s theorem applied to the channel flow. The theorem as presented is related to the Bernoulli equation Section 3.5. This section is beyond the scope of this course.
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9’20’’: Back to the plughole vortex. We will derive an equation for the surface profile, the ’hyperboloid’. There is also a really nice illustration that the vorticity is zero everywhere except on the axis of the vortex, where it is infinite (in an ideal fluid).
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11’10’’: Definition of the circulation, c.f. Subsection 6.1.2, and Kelvin’s circulation theorem, c.f. Theorem 6.1.2. Several examples of circulation are given, which should help you to conceptualise this important quantity, although note that doing calculations on some of these (e.g. the developing boundary layer, the creation of a wingtip vortex) is beyond the scope of this course.
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18’15’’: Vorticity dissipates due to friction (viscosity). Viscosity is a way of describing the internal friction in the fluid and will be studied in Chapter 7.
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18’35’’: This section of the film describes Kelvin’s circulation theorem in the presence of a non-conservative body force. They consider the Coriolis force due to the Earth’s rotation; this has only a small effect on scales of a few metres, but a big effect on long lengthscales such as occur in atmosphere and ocean dynamics, so it is important in weather forecasting). in more detail than we do it in this course, so this is non-examinable. However, it is worth watching to get a better understanding of the theorem and fluid dynamics in general.
Subsection 6.4.2 Vorticity Part 2
Here is a list of points to look out for while watching Vorticity Part 2:
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0’0’’: The film starts by continuing to discuss Kelvin’s circulation theorem, this time in the case of non-constant density. Again, this is beyond the scope of this course, although watching this will help you to understand the subject better.
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3’00’’: Aerofoils and how they generate lift. This is at a higher level than most of the material on this course, so if you can follow it, it will really help your understanding of circulation and vorticity.
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5’15’’: Helmholtz’s vortex laws are introduced, As mentioned in the lecture, these are equivalent to the theorems we had Theorem 6.2.2.
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6’45’’: Smoke rings, see Example 6.3.10. This should be understandable. It also includes a verbal description of the image vortex, c.f. Remark 6.2.12.
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7’55’’: Wingtip vortices are described.
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8’55’’: Another vorticity meter is introduced and initially used to measure wingtip vortices.
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9’40’’: Application to propellors.
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10’25’’: Application to flying birds.
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11’20’’: Vortex stretching, see Example 6.2.5, which includes a description of a figure skater, as mentioned in the lecture. At 4:00, the second Helmholtz vortex theorem is described. This states that the strength of a vortex tube is constant along its length and in time. Note that this is a direct consequence of the vorticity equation (3.5.8).
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13’25’’: Channel flow around a bend. This is beyond the scope of this course, although, as with other parts of the film, it would help your understanding to watch it. It also includes a description of secondary flow, both in a channel bend, such as a river, and in a closed pipe bend, such as an artery.
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17’25’’: Another example of vortex line stretching, see Example 6.2.5.
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18’20’’: Generation of vertical vorticity and tornadoes. This includes visual explanations of some of the properties we discussed in the lecture.
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19’55’’: Importance of the Coriolis force in large-scale atmospheric and oceanic flows.
