From a fundamental standpoint, all the currently used two-equation models suffer from lack of an underlying exact transport equation, which can serve as a guide for the model development on a term by term basis. The reason for this deficiency lies in the fact that the exact equation for does not describe the large scales, but the dissipative scales. The goal of a two-equation model is however the modeling of the influence of the large scale motions on the mean flow. Due to the lack of an exact equation, the - and the -equations are modeled in analogy with the equation for the turbulent kinetic energy, , using purely heuristic arguments. A more consistent approach for formulating a scale-equation has been developed by Rotta (1968, 1972). Instead of using purely heuristic and dimensional arguments, Rotta formulated an exact transport equation for turbulent kinetic energy times length scale, . Rotta’s equation represents the large scales of turbulence and can therefore serve as a basis for term-by-term modeling. The transport equations for the SST-SAS model as implemented in Ansys Fluent are based on transforming Rotta’s approach to - (SST) and defined as:

(4–263)

(4–264)

For the detailed derivation see Egorov and Menter  [431]. The transport equations of the SST-SAS model Equation 4–263 and Equation 4–264 differ from those of the SST RANS model  [428] by the additional SAS source term in the transport equation for the turbulence eddy frequency , Equation 4–264. In Equation 4–264 is the value for the - regime of the SST model.

The additional source term reads (see Egorov and Menter  [431] for details):

(4–265)

This SAS source term originates from a second order derivative term in Rotta’s transport equations  [430]. The model parameters in the SAS source term Equation 4–265 are

Here is the length scale of the modeled turbulence

(4–266)

and the von Kármán length scale is a three-dimensional generalization of the classic boundary layer definition

(4–267)

The first velocity derivative is represented in by , which is equal to , a scalar invariant of the strain rate tensor :

(4–268)

Note, that the same also directly participates in (Equation 4–265) and in the turbulence production term . The second velocity derivative is generalized to 3D using the magnitude of the velocity Laplacian:

(4–269)

So defined and are both equal to in the logarithmic part of the boundary layer, where is the von Kármán constant.

The model also provides a direct control of the high wave number damping. This is realized by a lower constraint on the value in the following way:

(4–270)

This limiter is proportional to the mesh cell size , which is calculated as the cubic root of the control volume size . The purpose of this limiter is to control damping of the finest resolved turbulent fluctuations. The structure of the limiter is derived from analyzing the equilibrium eddy viscosity of the SST-SAS model  [431].