lect_17 - The Vorticity Equation To understand the...

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1 The Vorticity Equation To understand the processes that produce changes in vorticity, we would like to derive an expression that includes the time derivative of vorticity: K = y u x v dt d Recall that the momentum equations are of the form K K = = dt dv dt du Thus we will begin our derivation by taking x-component momentum equation y-component momentum equation x [y-component momentum equation] y [x-component momentum equation] x [y-component momentum equation] y [x-component momentum equation] = = + + + = + + + + x p fv z u w y u v x u u t u y y p fu z v w y v v x v u t v x ρ ρ 1 1 + = + + + + + + + + x y p y x p x f u x u f x w z v z x v w x v y v y x v v x u x v x v u t v x ρ ρ ρ 2 2 2 2 2 2 1 1 + = + + + + + + y x p y x p y f v y v f y w z u z y u w y v y u y u v y u x u y x u u t u y ρ ρ ρ 2 2 2 2 2 2 1
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