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 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 can be derived as,
The expressions for the various ε based and
ω based turbulence models differ only in terms of constants and damping functions. Note that the v2-f turbulence model uses the wall distance to determine the relation between the wall-normal turbulent fluctuations and the acceleration of gravity. When the buoyancy contribution is determined from a multiphysics node, density variations with respect to the variable in the coupled physics interface are considered. For example, in nonisothermal flow the coupled interface is a Heat Transfer in Fluids interface, and,
Equation 3-211 is replaced by,