4.4.2. Baseline (BSL) k-ω Model

 

4.4.2.1. Overview

The main problem with the Wilcox model is its well known strong sensitivity to freestream conditions. The baseline (BSL) - model was developed by Menter  [428] to effectively blend the robust and accurate formulation of the - model in the near-wall region with the freestream independence of the - model in the far field. To achieve this, the - model is converted into a - formulation. The BSL - model is similar to the standard - model, but includes the following refinements:

  • The standard - model and the transformed - model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard - model, and zero away from the surface, which activates the transformed - model.

  • The BSL model incorporates a damped cross-diffusion derivative term in the equation.

  • The modeling constants are different.

 

4.4.2.2. Transport Equations for the BSL k-ω Model

The BSL - model has a similar form to the standard - model:

(4–101)

and

(4–102)

In these equations, the term represents the production of turbulence kinetic energy, and is defined in the same manner as in the standard k- model. represents the generation of , calculated as described in a section that follows. and represent the effective diffusivity of and , respectively, which are calculated as described in the section that follows. and represent the dissipation of and due to turbulence, calculated as described in Modeling the Turbulence Dissipation. represents the cross-diffusion term, calculated as described in the section that follows. and are user-defined source terms. and account for buoyancy terms as described in Effects of Buoyancy on Turbulence in the k-ω Models.

 

4.4.2.3. Modeling the Effective Diffusivity

The effective diffusivities for the BSL - model are given by

(4–103)

(4–104)

where and are the turbulent Prandtl numbers for and , respectively. The turbulent viscosity, , is computed as defined in Equation 4–74, and

(4–105)

(4–106)

The blending function is given by

(4–107)

(4–108)

(4–109)

where is the distance to the next surface and is the positive portion of the cross-diffusion term (see Equation 4–117).

 

4.4.2.4. Modeling the Turbulence Production

 
4.4.2.4.1. Production of k

The term represents the production of turbulence kinetic energy, and is defined in the same manner as in the standard - model. See Modeling the Turbulence Production for details.

 
4.4.2.4.2. Production of ω

The term represents the production of and is given by

(4–110)

Note that this formulation differs from the standard - model (this difference is important for the SST model described in a later section). It also differs from the standard - model in the way the term is evaluated. In the standard - model, is defined as a constant (0.52). For the BSL - model, is given by

(4–111)

where

(4–112)

(4–113)

where is 0.41.

 

4.4.2.5. Modeling the Turbulence Dissipation

 
4.4.2.5.1. Dissipation of k

The term represents the dissipation of turbulence kinetic energy, and is defined in a similar manner as in the standard - model (see Modeling the Turbulence Dissipation). The difference is in the way the term is evaluated. In the standard - model, is defined as a piecewise function. For the BSL - model, is a constant equal to 1. Thus,

(4–114)

 
4.4.2.5.2. Dissipation of ω

The term represents the dissipation of , and is defined in a similar manner as in the standard - model (see Modeling the Turbulence Dissipation). The difference is in the way the terms and are evaluated. In the standard - model, is defined as a constant (0.072) and is defined in Equation 4–92. For the BSL - model, is a constant equal to 1. Thus,

(4–115)

Instead of having a constant value, is given by

(4–116)

and is obtained from Equation 4–107.

Note that the constant value of 0.072 is still used for in the low-Reynolds number correction for BSL to define in Equation 4–78.

 

4.4.2.6. Cross-Diffusion Modification

The BSL - model is based on both the standard - model and the standard - model. To blend these two models together, the standard - model has been transformed into equations based on and , which leads to the introduction of a cross-diffusion term ( in Equation 4–102). is defined as

(4–117)

 

4.4.2.7. Model Constants

All additional model constants (, , , , , , , , and ) have the same values as for the standard - model (see Model Constants).