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Elliptic relaxation for near wall turbulence models

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The ϕ − f model<br />

A changes of variable is applied to the original model:<br />

Leading to:<br />

Dϕ<br />

Dt<br />

= f − P ϕ<br />

k<br />

L 2 ∇ 2 f − f = 1<br />

T (C1 − 1)<br />

f = f + 2ν(∇ϕ∇k)<br />

k<br />

2 νt ∂ϕ ∂k<br />

+<br />

k σk ∂xj ∂xj<br />

<br />

ϕ − 2<br />

<br />

3<br />

− C2<br />

+ ν∇ 2 ϕ (2)<br />

+ ∂<br />

∂xj<br />

P<br />

k<br />

νt<br />

σk<br />

− 2ν<br />

k<br />

∂ϕ<br />

∂xj<br />

∂ϕ<br />

∂xj<br />

<br />

∂k<br />

∂xj<br />

(3)<br />

− ν∇ 2 ϕ<br />

Far from the <strong>wall</strong> the last two terms are negligible, ensuring<br />

the correct behaviour.<br />

(4)<br />

<strong>Elliptic</strong> <strong>relaxation</strong> <strong>for</strong> <strong>near</strong> <strong>wall</strong> <strong>turbulence</strong> <strong>models</strong> – p. 14/22

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