As introduced at the beginning of this chapter, the RANS approach aims aims to simulate the ensemble-averaged flow field. The most widely used approach is to model the turbulent transport processes with gradient-diffusion assumptions. For the momentum equation, the deviatoric components of the Reynolds stress are assumed to be proportional to the mean deviatoric rate of strain. The Reynolds stress tensor , is defined as

(2–13)

in which is the turbulent kinematic viscosity, and is the turbulent kinetic energy, defined by:

(2–14)

The turbulent viscosity is related to the turbulent kinetic energy and its dissipation rate by:

(2–15)

where is a model constant that varies in different turbulence model formulations, shown in Table 2.1: Constants in the standard and RNG k - ε models[93] .

The turbulent flux term 𝚽 in the species transport Equation 2–1 is modeled as:

(2–16)

in which is the turbulent diffusivity. Similarly, the turbulent flux term H in the energy Equation 2–5 is modeled as:

(2–17)

in which is the turbulent thermal conductivity and is related to the turbulent thermal diffusivity and heat capacity by . The turbulent mass and thermal diffusivity are related to the turbulent viscosity by:

(2–18)

(2–19)

where and are the turbulent Schmidt and Prandtl numbers, respectively. As seen in Equation 2–15 , the calculation of turbulent viscosity requires that the turbulent kinetic energy and its dissipation rate to be modeled. In Ansys Forte, both the standard and the advanced (based on Re-Normalized Group Theory) k-ε model formulations are available. These consider velocity dilatation in the ε- equation and spray-induced source terms for both k and ε equations.

The standard Favre-averaged equations for k and E are given in Equation 2–20 and Equation 2–21 :

(2–20)

(2–21)

In these equations, , , , , are model constants, which are listed and described in Table 2.1: Constants in the standard and RNG k - ε models[93] .

The source terms involving are calculated based on the droplet probability distribution function (cf. Ref. Amsden 1997 [5] ). Physically, is the negative of the rate at which the turbulent eddies are doing work in dispersing the spray droplets. was suggested by Amsden [5] based on the postulate of length scale conservation in spray/turbulence interactions.

The advanced (and recommended) version of the k - ε model is derived from Re-Normalized Group (RNG) theory, as first proposed by Yakhot and Orszag [105] . The k equation in the RNG version of the model is the same as the standard version, but the ε equation is based on rigorous mathematical derivation rather than on empirically derived constants. The RNG ε equation is written as

(2–22)

where the in the last term of the right-hand side of the equation is defined as

(2–23)

with

(2–24)

(2–25)

and is the mean strain rate tensor,

(2–26)

Compared to the standard ε equation, the RNG model has one extra term, which accounts for non-isotropic turbulence, as described by Yakhot and Orszag [105] .

Values of the model constants , , , and used in the RNG version are also listed in Table 2.1: Constants in the standard and RNG k - ε models[93] . In the Ansys Forte implementation, the RNG value for the variable is based on the work of Han and Reitz [30] , who modified the constant to take the compressibility effect into account. According to Han and Reitz [30] ,

(2–27)

where m =0.5, n =1.4 for an ideal gas, and

(2–28)

with

(2–29)

Using this approach, the value of varies in the range of -0.9 to 1.726 [30] , and in Ansys Forte is determined automatically, based on the flow conditions and specification of other model constants, η ₀ and β . Han and Reitz [30] applied their version of the RNG k- ε model to engine simulations and observed improvements in the results compared to the standard k- ε model. For this reason, the RNG k- ε model is the default and recommended turbulence model in Ansys Forte.