16100lecture7_cg

16100lecture7_cg - Compressible Equations Conservation of...

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16.100 1 Compressible Equations Conservation of mass ( ) ( ) 0 0 0 0, volume surface d dv u ndS dt u t D u Dt u incompressible ρ ρ ρ ρ ρ ρ + = +∇• = + ∇• = ∇• = ∫∫∫ ∫∫ K K K K K Conservation of Momentum ( ) ( ) ( ) ( ) ( ) Net viscous force in : volume surface surface surface xx yx zx xy yy zy xz yz zz xx xx zx d udv uu ndS pn idS idS dt Note ds dx dy dz i dx dy dz j dx dy dz k u p uu t x x y z ρ ρ τ τ τ τ τ τ τ τ τ τ τ ρ τ τ τ ρ + = − + = + + + + + + + + + ∇• = − + + + ∫∫∫ ∫∫ ∫∫ ∫∫ K K K K K K K K K K K x yx xx zx Du p Dt x x y z τ τ τ ρ = − + + + ±²²²³²²²´ Recall: 2 ( ), 3 j i ij ij j i u u u x x T τ µ δ λ µ µ λ µ = + + ∇• = = − K Incompressible Equations in Cartsian Coordinates 2 2 2 0, i.e. 0 1 1 1 u v w u x y z Du p u Dt x Dv p v Dt y Dw p w Dt z ν ρ ν ρ ν ρ + + = ∇• = = − + ∇ = − + ∇ = − + ∇ K
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Compressible Equations 16.100 2002 2 Where 2 2 2 2 2 2 2 , kinematic viscosity D u v w Dt t x y z x y z µ ν ρ = + + + ∇ = + + Incompressible Equations for Cylindrical Coordinates
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