Convection with Vorticity
Prof Bruce R. Morton
31 July - 11 August 2000

Monash University
Melbourne, Australia.

This series of 10 classes will be held from 10am-noon daily from Monday 31 July through Friday 11 August 2000. We will meet in Room 529 Walker. Interaction is encouraged!

Course outline
Announcements
Vorticity Questions - for Thursday, 10 August
List of homeworks

Announcements

The paper "Numerical experiments on the formation of a tornado funnel under an intensifying vortex" by Wipperman, Berkofsky and Szillinsky (Quart. J. Royal Meteor. Soc., 95, 689-702) should be available after class Tuesday.

A new problem set (the splitter plate ahead of a cylinder) can be found in the mailroom in Walker Building. This should be completed and handed in before class on Friday.

An original of the paper The generation and decay of vorticity by Professor Morton (1984: Geophys. Astrophys. Fluid Dyn., 28, pp277-308) is available in the mailroom on the 5th floor of Walker. This folder will be updated daily.Please make copies and return this archive copy to the folder immediately.

Also to be found in the folder are today's thought experiment and a copy of Isaacs et al.'s (1975) paper on anthropogenic influences on tornado generation.

List of Homework Assignments

Comprehension exercises (Tuesday)
Split Couette flow* (Wednesday)
Mechanics exercises (Thursday)
Modeling tornadoes (Friday)
Plate upstream of a cylinder (Monday)
Turbulent plumes flow analysis (Tuesday)
Questions on Vorticity (Thursday)

In addition, each student should have written up 2 classes.

*Work through the thought experiment given on the effect of removing a piece of wall between two laminar flows. If you're unsure of the question, see the folder in the mailroom (5th floor Walker Building) for a copy of it.

Questions on Vorticity

Think through the following questions. Answers to these and all other homework exercises are due by Friday, 11 August.

1. What is the mechanism for vorticity generation at boundaries?
2. Does pressure play any role in vorticity dynamics?
3. Does the Helmholtz equation hold at boundaries?
4. What are the boundary conditions on vorticity?
5. Does wall stress generate vorticity? If vorticity is fluid rotation, shouldn't it be generated by tangential stress?
6. What, if any, role does torque play in fluid dynamics?
7. What are the mechanisms for decay of vorticity?
8. Can vorticity be lost by diffusion to walls?
9. Can vorticity diffuse out of walls?

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Background for Summer Lecture Series Convection with Vorticity

Convection, which I take to be the transport of any property with moving fluid (while advection is the transport of a property to a particular neighborhood), necessarily involves vorticity without which there can be no lasting fluid penetration. Although we can in principle solve all problems from the Navier-Stokes (momentum) equation without introducing the Helmholtz (vorticity) equation, the vorticity variable often provides a more direct description of problems and reveals the basic physics more effectively than the velocity. For example, when a vortex ring is incident normally on a plane wall it is observed that the diameter of the ring starts to increase as it nears the wall and its cross-section to decrease (somewhat in the manner of the inviscid solution), shortly after which propagation of the ring appears to cease suddenly and completely by which stage the dyed core fluid of the ring has come to rest (not at all like the inviscid solution). Moreover, when the plane wall is replaced by a fine gauze (such as a fly screen) the motion of the ring is little affected at low Reynolds numbers but at higher Reynolds numbers the ring passes through the screen and continues as a modified vortex ring in its lee. This behaviour can readily (?) be explained in terms of vorticity, but less readily in terms of velocity. [In either case you might care to think of an explanation which will include what happens to the impulse (momentum) and energy of the ring.]
As you pursue the problem of the incident vortex ring, you might come upon a curious feature. Although dozens (hundreds?) of books on fluid dynamics introduce and discuss the Helmholtz equation, scarcely any so much as mention the boundary or initial conditions on vorticity. Batchelor is one of the very rare exceptions and he concludes that the boundary conditions on vorticity are given in effect by those on velocity. This appears to raise significant questions concerning the roles of initial and boundary conditions which we shall discuss. In many cases the problems we shall consider can be regarded alternatively as partially turbulent flows, often driven internally as by buoyancy or from a brief or small boundary source.

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Draft Syllabus

1) The equations of motion and energy equation for a particle; initial conditions; relations between force, motion, energy and dissipation. The Navier-Stokes, energy and Helmholtz equations for a fluid; initial and boundary conditions.
2) Parallel flows, plane Couette and plane Poiseuille laminar flows of homogeneous fluids.
3) The generation and decay of vorticity at fluid boundaries and in the fluid interior.
4) Aerofoils, lift on an infinite inclined plate and the Kutta-Joukowski condition; lift on a finite aerofoil and trailing vortex wakes
5) Line vortices in inviscid and viscous fluids; a cylinder set impulsively into steady rotation; bound vortices; circular cylinders with and without circulation in a uniform inviscid stream
6) Similarity and the line vortex in a viscous fluid; termination of a line vortex at a normal boundary; modelling tornadoes; anthropogenic vorticity.
7) Steady and instantaneous mass sources and sinks in inviscid and viscous fluids.
8) Laminar and turbulent jets from steady sources and from sources started impulsively in otherwise still environments; jet entrainment.
9) Laminar and turbulent plumes from steady and impulsively started sources in homogeneous and stably stratified environments; plume entrainment.
10) The vortical structure of and entrainment into jets discharged through a side wall into a cross stream.
11) Steady flow past surface-mounted obstacles, hills and buildings; horseshoe vortices.
12) The propagation of and entrainment into neutral and buoyant vortex rings in inviscid and viscous fluids.
13) Turbulent puffs (from instantaneous sources of momentum) and thermals (from instantaneous sources of buoyancy) in uniform and stably stratified environments; entrainment; similarity and self similarity.
14) The relationship between thermals and buoyant vortex rings.
15) Models for cumulus entrainment.

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Last Updated: 9 August 2000