Theory for Buoyancy-Induced Turbulence
When there are density variations, an additional contribution to the production of turbulent kinetic energy may be added. In the case of incompressible flow in combination with the Boussinesq approximation for the buoyancy terms, and using conventional time-filtered fluctuations, this contribution can be expressed as
(3-279)
where gi is the gravitational acceleration. Applying a gradient-diffusion modeling approach gives
(3-280)
where σT is the turbulent Schmidt number.
When the flow is compressible or weakly compressible, the additional contribution can be expressed in terms of Favre-averaged fluctuations or conventional time-filtered fluctuations according to
(3-281)
where the last approximation stems again from a gradient-diffusion modeling approach. A further approximation of the production term can be made by replacing the gradient of the averaged pressure by the gradient of the hydrostatic pressure:
(3-282)
These cases can be summarized as
(3-283)
with
(3-284)
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
(3-285)
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. The angle θ is the angle between u and a, which is the vector with components ai as defined in Equation 3-284, and which accounts for the difference between buoyant vertical shear layers and buoyant horizontal shear layers:
(3-286)
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
(3-287)
For the SST-model, the buoyancy production term is
(3-288)
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:
(3-289)
This modeling assumes that the buoyancy-induced turbulence kinetic energy is pumped into the component of k aligned with a.
When the buoyancy contribution is determined from a multiphysics node, density variations with respect to the variable in the coupled physics interface are considered. Currently, this option is available for nonisothermal flow when the coupled interface is a Heat Transfer in Fluids interface. In this case Equation 3-283 is then replaced by
(3-290)
When coupling to other physics, a user-defined Schmidt number should be applied.
Buoyancy-induced turbulence in a rotating frame
In a rotating frame, the effective acceleration vector a, with components ai as defined in Equation 3-284, which is used in the expressions for the different production terms, should also take into account the fictitious forces arising from the non-inertial frame of reference. In this case, the effective acceleration a is defined as follows:
(3-291)
Note that in the incompressible case the terms in the definition of a arise from the Reynolds averaging of the momentum equation, such that the acceleration terms from all volume forces appear, including the ones from the Coriolis and Euler forces. In the (weakly) compressible case, the momentum equation is Favre averaged, and the gradient of the pressure appears in the definition of a. Approximating the gradient of the pressure by the hydrostatic pressure caused by gravity and the centrifugal force leads to the form of a as in the second case in Equation 3-291. See also the theory section for Rotating Frame.