where: [rho] is the fluid density (kg/[m.sup.3]), v is the air velocity (m/s), p is the static pressure (Pa), [mathematical expression not reproducible] is viscous molecular stress tensor (Pa), [mathematical expression not reproducible] is turbulent Reynolds stress tensor (Pa), [S.sub.b] is source of forces (N/[m.sup.3]), e is the sum of kinetic and internal energy (J/kg), [mathematical expression not reproducible] is molecular heat flux (J/([m.sup.2]s)), [mathematical expression not reproducible] is turbulent heat flux (J/([m.sup.2]s)), [S.sub.e] is sources of heat (J/([m.sup.3]s)).
This equation includes an additional term in the form of the Reynolds stress tensor. Due to this term, the set of equations is not closed.
The Reynolds stress tensor
-[rho][[bar.u'].sub.i][u'.sub.j]] in (5) is
The key to enhancing the stability concerning the coupling of the velocity ([U.sub.i]) and Reynolds-stress [??] fields is an appropriately blended Reynolds stress tensor entering the divergence operator on the right-hand-side of the momentum equation (Eq.
The turbulent viscosity [v.sub.t] is modeled in terms of the Reynolds stress anisotropy parameter A, the specific turbulence dissipation rate [omega]= [epsilon] /k and the turbulent kinetic energy k (computed directly from the Reynolds stress tensor: [??]; the viscosity effects are introduced via the Kolmogorov length scale [[eta].sub.k] = [([v.sup.3]/(k[omega])).sup.1/4]; see Jakirlic and Maduta (2015).
Such flows are studied by the Reynolds stress turbulence models (RSTM), which are based on the transport equations for all components of the Reynolds stress tensor
and the turbulence dissipation rate.
where [R.sub.ij] is the Reynolds stress tensor
, and it is directly solved with Reynolds stress transport model that is of the following form:
They consider a range of stationary and rotating flows, and explore possible functionalities based on vorticity, rapid distortion, and the anisotropy of the Reynolds stress tensor
, together with a turbulence Reynolds number.
The second term in the bracket indicates stress disorder called Reynolds stress tensor and proves that it is always positive.
The k- [epsilon] model is applied to calculate the Reynolds stress tensor. The values of k and [epsilon] are determined by the following semi-experimental equations:
In order to close the turbulent Reynolds stress tensor
in the momentum equations, the so-called Boussinesque-approximation, which is based on an analogy between molecular diffusion and diffusion turbulent eddies, has been employed.