The computation of turbulent natural convection flows - Turbulence ...

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2/3εδij. At moderate Reynolds numbers it is proposed that εij = εuiuj/k. But
this model is not valid immediately adjacent to a wall. Therefore Craft and
Launder[56] suggested more complex modelling for εij.
εij = (1−A)
D
ε ′
ij = ε ulun ∂
uiuj+2ν
k k
√ k∂
∂xl
√ k
δij+2ν
∂xn
ului
k
ε ′
ij +ε′′
ij +Aδijε (3.68)
∂ √ k∂
√ k
∂xj
∂xl
+2ν uluj
k
∂ √ k∂
√ k
∂xi
∂xl
(3.69)
ε ′′
ij = ε 2 uluk
k dal dakδij − ului
k dal da uluj
j −
k dal da
i (1−A) (3.70)
D =
d a i =
Ni = ∂
′
ε kk +ε′′
kk
2ε
Ni
0.5+(NkNk) 0.5
∂xi
3/2 1/2 k A
ε
(3.71)
(3.72)
(3.73)
3.3.8 Differential Stress Closure Used withθ 2 andεθ Transport
Equations
In this research, both versions of the differential stress closure (Basic and
TCL) have been employed, which calculates turbulent heat fluxes based on
the equation 3.20. Approximation of turbulent heat flux uit was explained in
detail in Section 3.3.4. In order to calculate the equation 3.20, it is required
to solve differential transport equations of θ 2 and εθ. Another approach is to
solve a differential transport equation forθ 2 and calculateεθ using an assumed
constant ratio of τθ/τ ≈ 0.5. The equations are the same as those shown in the
previous sections apart from cθ in the equation for uiθ which is now taken
as 0.3, and the turbulent diffusion terms in the θ 2 and εθ differential transport
equations which are formulated in terms of the Generalized Gradient Diffusiv-
ity Hypothesis (GGDH) instead of the effective eddy diffusivity hypothesis.