Math 801: Spring 2011
Hydrodynamic instabilities, Chaos and Turbulence

	Fabian Waleffe Day  Time  Place  MW  2:30-3:45pm B131 Van Vleck Hall

Course description

Lecture notes on moodle/ecow2 (login required)

References

• D.J. Acheson, Elementary Fluid Dynamics, Oxford U.P. 1990 (Chapter 9)
• P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge U.P. 1981
• P. Manneville, Instabilities, Chaos and Turbulence, Imperial College Press, 2004

Grading

Projects/Exercises

1. Intro to instabilities: Consider a rigid double pendulum of lengths L1 and L2 < L1. They rotate in a horizontal plane (vertical axis of rotation). The arm L1 rotates at fixed rotation rate Ω. Pendulum L2 is attached at the end of L1 and there is a mass M at its end. Let the angle between the arms L1 and L2 be called θ (positive counterclockwise, with θ=0 corresponding to the mass at a distance L1-L2 from the axis of rotation). Derive the equation of motion for this system assuming no friction. Show that θ=0 and θ=π are equilibria. Show that θ =0 is unstable. Investigate the resulting dynamics for initial condition θ ≠ 0. Analyze as much as possible, use numerical simulations to illustrate and complement the analysis.
   • Variant 1: Consider a more realistic compound pendulum (i.e. rigid rods).
   • Variant 2: Pendulum L1 rotates at fixed Ω in a horizontal plane, but L2 rotates in a vertical plane.
2. Everyone: Consider advection-diffusion of a passive scalar (e.g. Temperature): ∂_tΦ + v⋅∇ Φ = κ ∇² Φ for a plane Couette flow v = S y x̂ in a channel of width 2h, between two infinite walls perpendicular to the y direction. The boundary conditions for Φ are that Φ=0 at the walls for all times (e.g. fixed temperature).
   • What are the physical units of S and κ?
   • If the boundary conditions on Φ were Φ=A on one wall and Φ=B on the other wall, with A, B constants, show how to redefine Φ so the boundary conditions are Φ=0 on both walls.
   • Nondimensionalize the equations so that the nondimensional y goes from -1 to 1, and the nondimensional velocity goes from -1 to 1. What are the nondimensional physical parameters of the problem?
   • Look for solutions of the form Φ= Φ(y) exp(i α x) exp(i γ z) exp(λ t). What are α, γ and λ? Are they free parameters or to be determined? What is the relevance of those special solutions?
   • Analyze the resulting equation theoretically, asymptotically (what kind of limits could you consider?) and numerically.