The k-e turbulence model is based on the isotropic eddy-viscosity concept for closure of the Reynolds stresses.
In some flow situations, such as when body forces or complex strain fields are present, this assumption is too simple.
RSTMs allow not only for both the transport and different development of the individual Reynolds stresses, they also have the advantage that terms accounting for anisotropic effects are introduced automatically into the stress transport equations.
These non-isotropic characteristics of the turbulence play a very important role in flows with significant buoyancy, streamline curvature, swirl or strong recirculation.
Models employing transport equations for the turbulent stresses and fluxes are often called second-moment closures, and such models have been proposed by several research groups, most notably by Launder and his associates ( see for example Launder et al  ).
A review of second-moment computations for engineering flows has been provided fairly recently by Leschziner  and Launder [1989a, 1989b]. The results of these computations demonstrate the superiority of RSTMs over eddy-viscosity models for curved flows, swirling flows, buoyant flows and recirculating flows.
However, a full RSTM closure consists, in general, of 6 transport equations for the Reynolds stresses, 3 transport equations for the turbulent fluxes of each scalar property ( for example energy ) and one transport equation for the dissipation rate of turbulence energy EP.
The solution of all these complex equations together with those of the mean flow is not a trivial task, and it is also computation- ally expensive.
In addition there is a considerable numerical disadvantage arising from the use of a RSTM in that the stabilising effects of an eddy- viscosity field are absent in the mean-flow equations.
Thus, although RSTMs can provide a more realistic and rigorous approach for complex engineering flows, they may be too expensive in terms of storage and execution time for 3 dimensional flows.
The second-moment closure embodied in PHOENICS is essentially that proposed by Launder and his associates ( Launder et al , Launder  and Gibson and Launder  ). The default model in PHOENICS employs the simplest of the pressure-strain models suggested by Launder et al, namely the Isotropisation-of-Production model (IPM).
The more complex Quasi-Isotropic model (QIM) for the pressure strain is also provided as an option because it provides superior predictions in some flow situations. Both models employ the wall-correction terms to the pressure-strain model described by Gibson and Launder .
The performance of the QIM and IPM models for swirling flows has not been found entirely satisfactory, and significant improvements can be obtained by adopting the alternative set of model coefficients proposed by Younis  for the IPM ( see also Gibson and Younis  ). Therefore, this model variant ( hereafter denoted IPY ) is also provided as an option in PHOENICS.
In more recent years, improved models of the pressure-strain process have been developed by Launder's group ( see for example Launder [1989a, 1989b], Leschziner  ) and Speziale et al . In PHOENICS, an option is provided for the SSG pressure-strain model of Speziale, Sarkar and Gatski  which has the advantage of not requiring wall-correction terms.
The turbulent fluxes of energy and any scalar properties such as mass concentration, may be represented by simple or generalised gradient-diffusion models, or alternatively by a full transport closure for each of the individual flux components.
The transport model employed in PHOENICS for the turbulent fluxes is essentially that proposed by Gibson and Launder , and subsequently adapted and used by many other workers ( see for example Malin and Younis [1989, 1990] ).
The implementation of the RSTM in PHOENICS is such that it may be employed for high-Reynolds-number turbulent flow simulations which can be categorised as:
The model may be applied to flows involving solids represented through porosities, but the user is advised to deactivate the wall- reflection model, and also that the stress/flux production rates will be overestimated in near-wall cells.
Since the present implementation is restricted to high-Reynolds- number flows and flow regions, wall functions are required to bridge the viscous sublayer next to walls.
The energy equation may be solved in terms of enthalpy H1 or the temperature variable TEM1. In addition, equations may be solved for up to 9 scalar concentration variables. The turbulent Reynolds fluxes may be represented by one of 3 different models of increasing complexity:
The latter may be used with all 4 pressure-strain models, as PHOENICS automatically ensures that the model coefficients in the turbulent flux equations are set to their appropriate values.
For variable-density flows, two types of statistical averaging are encountered in the literature; either conventional unweighted ensemble averages, or the density-weighted decomposition suggested by Favre ( see for example Jones  ). The PHOENICS RSTM may be interpreted as: (a) using unweighted ensemble averages and neglecting correlations involving fluctuating density, including those due to buoyancy; or (b) using density-weighted averages and neglecting the mean-pressure-driven stress/flux production terms associated with this form of decomposition. In addition it is assumed that the turbulence modelling devised for the unweighted framework carries over without change to the density-weighted formulation.
In future releases the Reynolds-stress/flux transport equations will be extended to include the additional stress/flux production terms associated with variable density. The required extensions for incompressible buoyant flow have been described by, for example, Malin and Younis [1989, 1990]; while those for compressible flow have been presented by, among others, Hogg and Leschziner .
Finally, the present implementation does not include any additional terms to account for coordinate rotation in the turbulence transport equations ( see Launder [1989b] ). This extension is planned for future releases.