where gi is the gravitational acceleration and the last approximation is applicable for small Froude numbers. Applying a gradient-diffusion modeling approach,
Equation 3-210 can be recast into,
where σT is the turbulent Schmidt number. The production term in the
ε equation is derived from the production term in the
k equation. Hence, the corresponding term in the
ε equation becomes,
where Cε1 is the corresponding constant for every model (
1.44 for the Realizable k-
ε model). For the v2-f model
ε/k is changed to
τ-1. θ is the angle between
u and
g, accounting for the difference between buoyant vertical shear layers and buoyant horizontal shear layers:
Using the relation between k,
ε, and
ω, the buoyancy production term in the
ω equation of the k-
ω model can be derived as,
Hence, the expressions for the various ε based and
ω based turbulence models differ only in terms of constants and damping functions.
The v2-f turbulence model uses the gradient of the wall distance variable lw to compute the wall-normal direction, and relating the wall-normal turbulent fluctuations to the direction of gravity results in the following expression for buoyant production of
ζ,
