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Unformatted text preview: Chapter 21 2D Vortex Methods Vortex methods are powerful numerical methods to study the evolution of inviscid flow fields. The idea behind the method will be developed in the following notes. 21.1 Vorticity and Streamfunction The 3D vorticity is defined as the curl of the velocity vector: ω = ∇ × u (21.1) The velocity vector, for an incompressible fluid, must satisfy the continuity equa- tion ∇ · u = 0 . (21.2) This condition is immediately satisfied by introducing a vector streamfunction Ψ such that the velocity is its curl: u = ∇ × Ψ (21.3) since vector identities guarantees that the divergence of a curl is always zero ∇· u = ∇ · ∇ × Ψ = 0. The vorticity can now be written in terms of the streamfunction guaranteeing that the resultant flow field is divergence-free: ω = ∇ × ∇ × Ψ = ∇ ( ∇ · Ψ ) − ∇ 2 Ψ (21.4) The above equation can be further simplified by requiring that the streamfunc- tion vector be divergence-free. We then obtain a vector Poisson equation for the streamfunction for a given vorticity field: ∇ 2 Ψ = − ω (21.5) 199 200 CHAPTER 21. 2D VORTEX METHODS The situation is considerably simpler in 2D where the streamfunction is a scalar, ψ , and where only the vorticity component perpendicular to the plane is non-zero ω = ω k . The vector Poisson equation reduces to a scalar Poisson equation: ∇ 2 ψ = − ω (21.6) The important point about vorticity is that for two-dimensional inviscid flows, and in the absence of vorticity sources, the equation governing the evolution of the vorticity is simple and states that the vorticity is conserved following the flow: d ω d t = ∂ω ∂t + u · ∇ ω = 0 (21.7) The idea behind vortex method is then simply to represent the vorticity fields as a collection of vortex patches carrying an intrinsic amount of vorticity. This vorticity has associated with it a flow field which in turn will advect these patches around without changing their intrinsic vorticity. The flow field will of course change once the patches change position, and one merely has to track them in order to simulate the flow evolution. 21.2 Flow due to point vortices Here we derive the relationship between a vorticity source and the induced stream- function and velocity in an infinite plane with no boundaries. This derivation uses the Green function approach where the forcing, the vorticity source, is concen- trated at points and has infinite strength in a pointwise sense, but finite strength in an area-averaged sense. We first look at the impact of a single point vortex then use the linearity of the equations to superpose solution when multiple vortex points are located. 21.2.1 Flow due to single point vortex An analytical expression for the streamfunction equation can be derived for the special case where there is a single “impulse” vorticity in an infinite domain ∇ 2 ψ = − ω δ ( r ) 2 πr , r = x − x , r = | x − x | (21.8) where ω is now simply the strength of a concentrated vortex point. The Greenis now simply the strength of a concentrated vortex point....
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- Fall '08