Joris_LectWk4

Joris_LectWk4 - Hamiltonian aspects of fluid dynamics CDS...

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Unformatted text preview: Hamiltonian aspects of fluid dynamics CDS 140b Joris Vankerschaver [email protected] CDS 01/29/08, 01/31/08 Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 1 / 34 Outline for this week 1. Dynamics of point vortices; 1.1 Vorticity; 1.2 Fluid dynamics in 2D; 1.3 Dynamics of N vortices; 1.4 The Kirchhoff-Routh function; 1.5 Dynamics of N = 1 , 2 , 3 vortices; 2. Chaotic advection; 2.1 Arefs stirring mechanism; 2.2 The ABC flow. Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 2 / 34 Vortex dynamics Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 3 / 34 References 1. P. Newton: The N-vortex problem. Analytical techniques . Applied Mathematical Sciences, vol. 145. Springer-Verlag, 2001. 2. H. Aref: Point vortex dynamics: A classical mathematics playground . J. Math. Phys. 48 , 065401 (2007). 3. P. G. Saffmann: Vortex Dynamics . Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, 1992. Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 4 / 34 Dynamics of an inviscid flow 1. Euler equations: d u dt := u t + u u =- p , together with the incompressibility condition u = 0. Pressure p acts as a Lagrange multiplier for this constraint, and satisfies 2 p = 0. 2. Take the curl of Euler, and put = u : d dt = u . ( vorticity form of Euler eqns). Due to the presence of p , system (1) is much more complicated than (2). Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 5 / 34 What is vorticity? Intuitively: vorticity is a measure for the amount of rotation of the fluid. I Suppose given a flow with velocity field u ( x , y , z , t ). I Mathematically, vorticity is a vector field given by = u . Why study vorticity? I Localised patches of vorticity appear quite often in nature; I numerically, vortex methods are very attractive; I vorticity equation contains just as much information as the Euler equation; I vortices are a classical mathematics playground (Aref). Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 6 / 34 Hurricane Rita Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 7 / 34 Example (point vortex) u = 1 2 p x 2 + y 2 (- y , x , 0) = (0 , , ( x , y )) . This will be the building block of our subsequent treatment. Think of a point vortex as being similar to a point mass . Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 8 / 34 Fluid dynamics in 2D We will only be concerned with 2D flows in these lectures!...
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.

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Joris_LectWk4 - Hamiltonian aspects of fluid dynamics CDS...

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