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Turbulence lecture 15 - Turbulence Lecture 15 The K Model...

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Turbulence Lecture 15 The K ε Model Relate v τ to , K ε v τ 2 K ε units 4 2 3 u m Lu u s L 2 3 j i i j ij j i U U u u K v x x τ δ = − + + (Discussion of ε will follow later) First have to consider turbulent energetics. We previously derived an equation for the energetics of the mean flow. Want an equation for the turbulent energy 1 2 i j K u u = i i i u U u = + i Derive an equation for u by subtracting equation for U from equation for i i i i u i ( ) i i u U t t = + i u i .... 2 i i i i j i j i j i j j j i u U u t t t u p U u u U u u t x x x x x ρ ρ ρ ρ µ = + + + + = − + ±²²²²³²²²²´ ±²³²´ ij j s Result: equation for fluctuations Coupling with mean flow non-linear Then take the inner product with u and average: contract on i . i (see equation 4.35 Matthieu & Scott) 2 2 2 2 2 2 j i i j j i ij ij ij i j ij j j u u u U pu u s s s t x x ρ ρ µ µ + = +
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