Theory for Buoyancy-Induced Turbulence
When the flow is compressible or weakly compressible, an additional contribution to the production of turbulent kinetic energy may be added. This contribution can be expressed in terms of Favre-averaged fluctuations or conventional time-filtered fluctuations according to,
(3-210)
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,
(3-211)
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,
(3-212)
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,
(3-213)
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,
(3-214)