Joris_LectWk4 - Hamiltonian aspects of uid dynamics CDS...

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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
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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 Aref’s stirring mechanism; 2.2 The ABC flow. Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 2 / 34
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Vortex dynamics Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 3 / 34
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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
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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 ω = ∇ × u : d ω dt = ω · ∇ 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
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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 ω given by ω = ∇ × 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
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Hurricane Rita Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 7 / 34
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Example (point vortex) u = 1 2 p x 2 + y 2 ( - y , x , 0) ω = (0 , 0 , δ ( 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
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Fluid dynamics in 2D We will only be concerned with 2D flows in these lectures! I Consider a fluid with velocity u ( x , t ) = ( u x ( x , t ) , u y ( x , t )) in 2 D . I Fluid is incompressible if ∇ · u = u x x + u y y = 0 . I Incompressibility: there exists a stream function ψ such that u x = ∂ψ y and u y = - ∂ψ x . I If u is independent of t : steady flow .
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