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RANS
lunedì 26 giugno 2023 12:57
Dissipation anomaly : ϵ = 2νsijsij ∼ U03 / L0
|∇u| ∼ √Re
Multiscale phenomenon: L η ∼ Re3/4
In CFD we use Reduced Order Equations which break the law L η ∼ Re3/4 and because L η ∼ N
RANS
∂Ui / ∂xi = 0
∂Ui / ∂t + ∂UiUj / ∂xj = -1 / ρ ∂P / ∂xi + ν ∂2Ui / ∂xj2 - ∂/ ∂xj(uiuj)
rij = Reynolds stress tensor - uiuj = v'u' + u'v' + u'w'
These equations are exact but we can't solve them: it introduces 6 new unknowns: this is the -turbulence closure problem.
We can follow two strategies:
- First order modeling: Express rij as a function of U and P. However this equation doesn't exist directly so we can use our knowledge of the flow to relate some features of the main flow to some of the turbulent flow.
- Second order modeling: Derive the exact equation of rij we obtain it by doing (NSi)·(CNSj); and the (NSi)·(Uj)(NSi).
∂rij / ∂t + ∂rij Uk / ∂xk = -Γikuk - Γjkuk / ∂xk
Ist ORDER MODELING - BOUSSINESQ ASSUMPTION
Is the most famous first order assumption, the Eddy viscosity:
The goal is to pass from the disturbed solution to the diagonal one. The idea is to add a viscosity that dissipates the unwanted irregularities. rij is called “stress tensor” because the fluctuations are felt by the mean flow as an additional stress. Based on this, we introduce the eddy viscosity assumption:
rij - 1/3 rkkδij = -2νT
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