This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 KirchhoffRouth 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 Nvortex problem. Analytical techniques . Applied Mathematical Sciences, vol. 145. SpringerVerlag, 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!...
View
Full
Document
This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
 Fall '10
 list

Click to edit the document details