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

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One option is to derive and solve modelled transport equations for the turbu-
lent heat fluxes, in a similar way as outlined above for the Reynolds stresses.
The other way is to employ algebraic relations for the turbulent heat fluxes,
similar to those adopted within an eddy-viscosity based scheme. These two
alternatives are presented below.
The correlation uiθ represents the heat flux in the direction of xi by tur-
bulent fluctuations. The exact transport equation of uiθ can be derived by
multiplying the transport equation for the temperature fluctuations by the ui
velocity component adding it to the xi-component of the Navier-Stokes equa-
tions multiplied by θ, and then averaging. The exact form of the uiθ transport
equation is then obtained as:
D ρuiθ
Dt
= ρPiθ +ρGiθ +ρφiθ +diθ −ρεiθ
(3.38)
Piθ expresses the rate of generation ofuiθ by mean velocity and mean scalar
gradients. The mean velocity gradient tends to increase velocity fluctuations
and the mean scalar gradient increases the magnitude of the temperature fluc-
tuations:
Piθ = −
uiuk
∂Θ
∂xk
+ukθ ∂Ui
∂xk
(3.39)
TheGiθ term represents the generation ofuiθ as a result of buoyancy forces:
Giθ = −βgiθ 2 (3.40)
The pressure temperature gradient correlation,φiθ, is similar to the pressure
strain term in the stress equations:
The diffusion term is:
The dissipation rate ofuiθ is:
φiθ = p ∂θ
ρ∂xi
diθ = − ∂
∂θ
ρukuiθ +ptδik −Γui
∂xk ∂xk
−µθ ∂ui
∂xk
(3.41)
(3.42)