Turbulence lecture 10

# Turbulence lecture 10 - Turbulence Lecture 10 Mean Flow...

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Turbulence Lecture 10 Mean Flow Dynamics Start with N-S equations Make Reynolds decomposition Totals Tennekes & Lumley Notation \ i i i u U u T T T p P p ρ ρ ρ = + = + = + = + ± ± ± ± Imcompressible flow, no body forces not variable density. 4 equations, 4 unknowns becomes 1. ( ) ( ) [ ] ( ) 2 i i j j i i ij ij j i j U u U u U u P p S s t x x ρ ρ µ x + + + + = − + + + 2 i u ij S is deformation rate tensor 1 2 1 2 j i ij j i j i ij j i U U S x x u u s x x = + = + 2. ( ) 0 j j j U u x + = Continuity Average (ensemble or time) the momentum equation 1

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3. 2 i j i ij i j i j P U U U S u u t x x x ρ ρ µ ρ j + = − + Average Continuity 4. 0 0 i i i U x x = = i u By subtracting 4 from 2 Note j i i j j u u u x x ρ ρ = j u i=1 is x -momentum equation ˆ i j j u u v u w u u u uu uv
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