The realizable - model  [591] differs from the standard - model in two important ways:

The term "realizable" means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard - model nor the RNG - model is realizable.

To understand the mathematics behind the realizable - model, consider combining the Boussinesq relationship (Equation 4–14) and the eddy viscosity definition (Equation 4–41) to obtain the following expression for the normal Reynolds stress in an incompressible strained mean flow:

(4–51)

Using Equation 4–41 for , one obtains the result that the normal stress, , which by definition is a positive quantity, becomes negative, that is, "non-realizable", when the strain is large enough to satisfy

(4–52)

Similarly, it can also be shown that the Schwarz inequality for shear stresses (; no summation over and ) can be violated when the mean strain rate is large. The most straightforward way to ensure the realizability (positivity of normal stresses and Schwarz inequality for shear stresses) is to make variable by sensitizing it to the mean flow (mean deformation) and the turbulence (, ). The notion of variable is suggested by many modelers including Reynolds  [552], and is well substantiated by experimental evidence. For example, is found to be around 0.09 in the logarithmic layer of equilibrium boundary layers, and 0.05 in a strong homogeneous shear flow.

Both the realizable and RNG - models have shown substantial improvements over the standard - model where the flow features include strong streamline curvature, vortices, and rotation. Since the model is still relatively new, it is not clear in exactly which instances the realizable - model consistently outperforms the RNG model. However, initial studies have shown that the realizable model provides the best performance of all the - model versions for several validations of separated flows and flows with complex secondary flow features.

One of the weaknesses of the standard - model or other traditional - models lies with the modeled equation for the dissipation rate (). The well-known round-jet anomaly (named based on the finding that the spreading rate in planar jets is predicted reasonably well, but prediction of the spreading rate for axisymmetric jets is unexpectedly poor) is considered to be mainly due to the modeled dissipation equation.

The realizable - model proposed by Shih et al.  [591] was intended to address these deficiencies of traditional - models by adopting the following:

One limitation of the realizable - model is that it produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones (for example, multiple reference frames, rotating sliding meshes). This is due to the fact that the realizable - model includes the effects of mean rotation in the definition of the turbulent viscosity (see Equation 4–55 – Equation 4–57). This extra rotation effect has been tested on single moving reference frame systems and showed superior behavior over the standard - model. However, due to the nature of this modification, its application to multiple reference frame systems should be taken with some caution. See Modeling the Turbulent Viscosity for information about how to include or exclude this term from the model.

The modeled transport equations for and in the realizable - model are

(4–53)

and

(4–54)

where

In these equations, represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated as described in Modeling Turbulent Production in the k-ε Models. is the generation of turbulence kinetic energy due to buoyancy, calculated as described in Effects of Buoyancy on Turbulence in the k-ε Models. represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, calculated as described in Effects of Compressibility on Turbulence in the k-ε Models. and are constants. and are the turbulent Prandtl numbers for and , respectively. and are user-defined source terms.

Note that the equation (Equation 4–53) is the same as that in the standard - model (Equation 4–39) and the RNG - model (Equation 4–42), except for the model constants. However, the form of the equation is quite different from those in the standard and RNG-based - models (Equation 4–40 and Equation 4–43). One of the noteworthy features is that the production term in the equation (the second term on the right-hand side of Equation 4–54) does not involve the production of ; that is, it does not contain the same term as the other - models. It is believed that the present form better represents the spectral energy transfer. Another desirable feature is that the destruction term (the third term on the right-hand side of Equation 4–54) does not have any singularity; that is, its denominator never vanishes, even if vanishes or becomes smaller than zero. This feature is contrasted with traditional - models, which have a singularity due to in the denominator.

This model has been extensively validated for a wide range of flows  [309],  [591], including rotating homogeneous shear flows, free flows including jets and mixing layers, channel and boundary layer flows, and separated flows. For all these cases, the performance of the model has been found to be substantially better than that of the standard - model. Especially noteworthy is the fact that the realizable - model resolves the round-jet anomaly; that is, it predicts the spreading rate for axisymmetric jets as well as that for planar jets.

As in other - models, the eddy viscosity is computed from

(4–55)

The difference between the realizable - model and the standard and RNG - models is that is no longer constant. It is computed from

(4–56)

where

(4–57)

and

where is the mean rate-of-rotation tensor viewed in a moving reference frame with the angular velocity . The model constants and are given by

(4–58)

where

(4–59)

It can be seen that is a function of the mean strain and rotation rates, the angular velocity of the system rotation, and the turbulence fields ( and ). in Equation 4–55 can be shown to recover the standard value of 0.09 for an inertial sublayer in an equilibrium boundary layer.

The model constants , , and have been established to ensure that the model performs well for certain canonical flows. The model constants are