Turbulent Mass Transport Models
The RANS turbulence models included in the Reacting Flow interfaces are based on averaging of the fluid flow equations. Applying a corresponding decomposition of the fluctuating mass fraction, into mean and fluctuating parts, and averaging the mass transport equations, additional unclosed terms are introduced in the equations. These terms need to be modeled in order to close the set of equations. The most important terms, containing the correlations of the velocity and mass fraction fluctuations, referred to as the turbulent mass transport fluxes, are given by:
Here the double primes denote Favre (density-based) fluctuations. In the case of varying density flow, Favre averaging is favored over Reynolds averaging since it reduces the number of unclosed terms and renders the equation on the same form as the incompressible RANS equations. For more background on averaging, see Turbulence Modeling.
The most common way to model this term is to use a gradient based assumption, where the additional turbulent transport is related to the turbulent viscosity through a turbulent Schmidt number ScT:
(8-57)
Here denotes the Favre averaged mass fraction which is the quantity solved for.
Using a RANS turbulence model, the turbulent mass flux is defined from Equation 8-57, and the equation solved for each species is:
where the molecular diffusion coefficient, Di, is given by the diffusion model (Mixture-Average or Fick’s Law).
When modeling transport of dilute solutes in a solvent, for example using the Transport of Diluted Species interface, the species concentration are solved for and the most important unclosed turbulent transport terms correspond to
Here primes indicate turbulent fluctuations from a Reynolds averaging. Using a gradient based assumption, the additional turbulent transport is modeled as
(8-58)
where represents the average concentration. Using a RANS turbulence model the equations solved for the concentration of each species is:
(8-59)
Kays-Crawford
Assuming that the turbulent transport mechanisms of heat and mass processes are analogous, the turbulent Schmidt number is defined by (Ref. 1):
where the Schmidt number at infinity is ScT∞ = 0.85, and the turbulent Peclet number is defined as the ratio of the turbulent to molecular viscosity times the Schmidt number:
High Schmidt Number Model
In the case of high Schmidt numbers, which is typical for mass transport in liquids, the mass transfer near walls can be significantly different than that for Schmidt numbers of order unity. A diffusion layer near a solid wall, due to, for example, a reaction on the wall, does not have the same properties as the (momentum) boundary layer. Most importantly the diffusion layer thickness is significantly smaller than the boundary layer thickness for high Schmidt numbers. To correctly capture the mass flux at the wall, the wall resolution required is dictated by the diffusion layer rather than by the boundary layer.
The High Schmidt number model is based on the model by Kubacki and Dick (Ref. 2) and is available when using the Low-Reynolds k-ε turbulence model or the SST turbulence model. In this case the fluid flow is resolved all the way to the physical wall and consequently, and species boundary equations are applied directly on the wall (without using wall functions).
In the near wall region, where the species transport is limited by diffusion, the mass diffusivity is modeled using an analytical function of the nondimensional wall distance due to Na and Hanratty (Ref. 3):
The nondimensional wall distance applied, , corresponds to that defined by the turbulence model. The value of the constants b and m are given in Table 8-1.
Table 8-1: Na and Hanratty parameters
Schmidt number
b
m
0.1 < Sc < 5
7.3·10-4
3
5 < Sc < 50
5.3·10-4
3
Sc > 50
4.63·10-4
3.38
Further out from the wall, where the mass transport is governed by the turbulent transport, the transport is modeled using a turbulent Schmidt number of the form (Ref. 2):
In order to combine the two descriptions, the blending function by Kader (Ref. 4) is used:
where