Vortices are commonly observed aspect in many fluid flows, particularly
at high Reynolds numbers. Their structure and evolution are crucial to
problems in aerodynamics, geophysical fluid dynamics and other areas.
This class will cover vortex dynamics, starting from basics and moving
on to topics such as point vortices, vortex rings, vortex filaments,
vortex patches and other things vortical.
Prerequisites
Fluid mechanics (MAE101A/B or MAE210A/B), ODEs and some PDEs (e.g.
MAE105). Experience with Matlab (e.g. MAE107) will be very useful.
Graduate level classes are not required, but I shall be progressing at
a pace appropriate for a graduate class.
Computer resources
Access to matlab is crucial for the final project. Access to LaTeX is
needed for the notes.
Grading
10% Writing up lecture notes in LaTeX. Style files will be provided.
20% Review of 1 landmark paper (3-4 pages, including some historical
context).
80% Matlab project (including 20% for documentation).
Watch the two movies on vorticity at the NCFMF website (you will need
RealPlayer).
Lectures notes
You will be writing these in LaTeX. I will post them as they are
written. You will need the class file classnotes.cls,
the style file pdffig.sty (which is actually
probably redundant but never mind), the main file notes.tex
and an auxiliary file content.tex. Each
lecture then is a single
LaTeX source file. See the first two chapters c1example.tex
and c2example.tex as examples. This
exampls also uses a number of .m and .pdf files: f21.pdf,
f22.pdf,f23.pdf, p21.m, p22.m, p23.m.
For a primer on LaTeX, see here
and here.
Paper review
I have chosen 21 papers. You should agree amongst yourselves which
paper you will each review. I just want one review per student on
Tuesday May 19. I am giving only authors and dates. You will need to
get hold of the paper yourselves (if you have questions on how to do
this, ask me). You should also check you have the correct title (some
authors have several papers in one year). I have divided the papers up
into groups.
Basics: Tait (1867) - this is a translation of Helmholtz (1858) Po-Ting;
Moffatt (1969) Silvana.
Point vortices: Routh (1881); Thomson (1883) Ingeborg; Lin (1941) Morten; Onsager
(1949); Rott (1956) Emile.
Vortex patches: Zabusky et al. (1979); Kida (1981).
Vortex sheets: Moore (1979).
Vortex column: Kelvin (1880); Crow (1970) Warren; Moore &
Saffman
(1975) Kristoffer; Tsai
& Widnall (1976).
Vortex filaments/3D structures: Hill (1894); Hasimoto (1972); Norbury
(1973).
Instability: Rayleigh (1880) Jaemo;
Bayly (1986); Pierrehumbert (1986).
Turbulence models: Lundgren (1982)
Eric.
Imagine you are writing a report for an editor with an undergraduate
background in fluid mechanics but who does not know about vorticity.
What does the paper do that is new? Explain some of the key results
(this usually means going through the algebra) and applications. Your
report should be 3-4 pages long in pdf. Use the template rep_template.tex and the class style proposalnsf.cls.
For some more ideas about how to write a review, see here,
and here.
Note that a fair amount of the issues discussed are about experiments,
which are not the focus of the papers above. Some idea of the
historical evolution of vortex dynamics may be useful. Wikipedia has an
entry here.
The Annual
Review of Fluid Mechanics has one or more historical articles per
volume, some on vortex dynamics. There are few good histories of fluid
mechanics. I have placed the interesting book by Darrigol on reserve.
Matlab project
The goal is to produce a stand-alone Matlab suite of programs that
illustrate models and concepts in vortex dynamics that are used in thic
class. The program should be called up via a single command. This
should give a GUI like that say of demo or pdetool
with a number of different topics to investigate.
You will be working together to produce this software. You should
divide up into groups, each concentrating on certain aspects of the
project. The following students are taking the class for credit:
Abrahamsen, Arobone, Chan, Chen, Christofferson, Kuon, Lien, Park,
Saitori, Sandøy, Suehiro. The following are auditing but come to
class regularly: de Stadler, Elfring, Pak, Peng. I encourage them to
act as testers and consultants.
How you do this is up to you, but here is a possible list of tasks (you
could have fewer or more):
GUI design.
Documentation.
Pedagogical examples (e.g. illustration of what vorticity is,
some very simple pictures/flows)
Point vortices
Vortex patches
Vortex filaments
Vortex rings
2D turbulence
Testing
Grading: you should as a team come up with categories that I grade.
These should minimally include the contents of the list above. Note
that documentation will count for 25% of the grade at least.
For each topic, you should provide an implementation of the problem,
i.e. solve the equations of motion for point vortices and plot the
trajectories. The user should be able to provide input data, and there
should also be demonstration cases which are particularly interesting.
You could consider other topics for vortex dynamics, so feel free to
look at books and papers. However Matlab is not the language of choice
for 3D CFD and seeing gigabytes of output is not very pedagogical, so
you will probably want to limit yourselves to situations descrbied by
ODEs.
Point vortices. Integrate equations of motion for n vortices. Specify initial
positions in window by clicking. Canned demos: 3 vortex collapse, 4
vortes scattering. Monitor: invariants. Different domains: plane,
half-plane, circle, strip (need log(sinh) Green's function), arbitrary
domain (difficult). Blinking vortex model of Aref.
Vortex patches: Kida and Kirchhoff vortices (see here
as prototype). Vortex merger using contour dynamics. Piecewise linear
shear layer contour dynamics (Pullin 1981, Stern &
Pratt 1985). Complex domains (Crowdy &
Surana 2007 hard).
Vortex filaments: LIA, Majda model (see Knio et al. 2007).
2D turbulence: periodic domain, formation of coherent structures.
You should all come and see me later this week to talk about the
specific problems, and probably frequently thereafter. There are
references and examples on the internet, use them. Whatever you code up
should be your own.
Reserves
Saffman and a number of other books have been placed on reserve
at the
S&E library (need to be on UCSD campus or to be using a Proxy
server/VPN for the link to work).
Grading policy
I remind you of UCSD's policy
on academic integrity. I may rescale the components to arrive at the
final
grade.