Math 812: Fall 2006 Intro to Stability and Turbulence in Shear Flows

Fabian Waleffe

Day  Time  Place
TR  9:30-10:45pm B317 Van Vleck Hall

Office hours: by appointment.

Prerequisites

Advanced Calculus (through Math 321 at least), Differential equations (e.g. Math 319), Linear Algebra (e.g. Math 340), general physics, elementary fluid dynamics, elementary numerical methods (e.g. CS 412), familiarity with Matlab or a programming language (e.g. Fortran, C, C++,...).

REFERENCES: (no textbook required) I'll pick some inspiration from

• Turbulent Flows, Stephen B. Pope, Cambridge University Press (2000) (Engineering oriented)
• Hydrodynamic Instability, P.G. Drazin and W.H. Reid, Cambridge University Press (1981) (Classic reference for stability)
• Elementary Fluid Dynamics, D.J. Acheson, Oxford University Press. (Very readable, advanced undergrad intro to fluid dynamics)

SUGGESTED READING: (note this is not an endorsement of all of these papers, although most are recommended reading, some are there to test your critical thinking) [I have provided some links, but don't be too lazy and limit yourself to those! do a bit of `research' to get the other papers too]

Navier-Stokes Equations, and a few words about turbulence

• Origin, derivation, assumptions: We did it our way but most of that and some more is in
• Pope, Chapters 1, 2.
• See also Cauchy stress tensor, constitutive laws, Newtonian fluid, Kinetic Theory point of view (and a nice brief review of KT;), Kinetic Energy dissipation rate, Entropy production. Sound waves, speed of sound, Mach number; incompressible flow.

For further information about the derivation of the Navier-Stokes equations see Chapter 6 in Acheson (especially read the 1st two pages of history) and/or the books by Batchelor (An introduction to fluid dynamics, Cambridge U. Press 1967) or Landau and Lifshitz `Fluid Mechanics', 2nd edition, Pergamon Press (1987) (usually pretty complete but a bit hard to read). The book by R.L. Panton `Incompressible Flow', Wiley 1996 is also quite good and well-known in engineering depts. The first 6 (!) chapters cover the material we reviewed in our first 3 lectures. Although it is called `incompressible flows' it contains good discussions/reviews of the continuum hypothesis, thermodynamics, indicial (or index) notation, vector calculus, etc. Any fluid mechanics book should have a derivation of NSE and a discussion of the Cauchy stress tensor. Yep, this is the same Cauchy as in Cauchy-Riemann, Cauchy's theorem, Cauchy sequences, etc.

Some recent papers on boundary conditions for the Poisson equation for pressure in incompressible flow:

• Hans Johnston and Jian-Guo Liu, `Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term', Journal of Computational Physics, Volume 199, Issue 1, 1 September 2004, Pages 221-259
• Dietmar Rempfer, `On Boundary Conditions for Incompressible Navier-Stokes Problems', Applied Mechanics Reviews -- May 2006 -- Volume 59, Issue 3, pp. 107-125
• Jian-Guo Liu, Jie Liu, Robert L. Pego, `On Incompressible Navier-Stokes Dynamics: A New Approach for Analysis and Computation', arxiv.org math.AP/0503212 (Mar 2005)
NOTE: if you are interested in numerical/mathematical issues, reading the papers above and relevant references therein, and trying to sort out the issues would be a good project for this class. This could be a small group project (2 or 3 people).

Part II: Turbulence in wall-bounded flows

• Pope, Chapter 7.
• Stability of shear flows:
• Drazin & Reid, Chapter 1, Chap 4, sects 20, 21, 22, 23, Chap 7, sect 53.1 and 53.2.
• Romanov, V.A., "Stability of plane-parallel Couette flow", Functional Anal. & its Applic. 7 , 137-146 (1973). [Proof of the linear stability of plane Couette]
• Chapman, S.J., Subcritical transition in channel flows, J. Fluid Mech. 451, 35-97 (2002) [Lots of things I don't quite agree with, see `On transition scenarios in channel flows' above, but some nice asymptotics of the linear stability of plane Couette and Poiseuille flow]
• Cathleen Morawetz, The Eigenvalues of Some Stability Problems Involving Viscosity, Indiana Univ. Math. J. 1 No. 4 (1952), 579-603
• Coherent structures
• Robinson, S.K. Coherent Motions in the Turbulent Boundary Layer, Annual Review of Fluid Mechanics, 23 , 601-639 (1991)
• Kerswell R.K. Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity 18 , R17-R44 (2005).
• Waleffe, F. Exact coherent structures in turbulent shear flows, Proceedings of the Conference on Turbulence and Interactions (TI2006), [in production].
• Spectral methods