Explicit Algebraic Reynolds Stress Models (EARSM) represent an extension of the standard two-equation models. They are derived from the Reynolds stress transport equations and give a nonlinear relation between the Reynolds stresses and the mean strain-rate and vorticity tensors. Due to the higher order terms, many flow phenomena are included in the model without the need to solve transport equations for individual Reynolds stresses. The WJ-BSL-EARSM allows an extension of the baseline (BSL) turbulence model to capture the following flow effects:

For information on enabling the WJ-BSL-EARSM model, see Setting up the WJ-BSL-EARSM Model in the Fluent User's Guide.

The BSL model blends the robust and accurate formulation of the - model in the near-wall region with the freestream independence of the - model in the far field. The BSL model was developed by Menter [428] and is described in Baseline (BSL) k-ω Model). There is also an option GEKO which allows the combination of the EARSM with the Generalized (GEKO) Model (see Generalized k-ω  (GEKO) Model for more information on the GEKO model).

The implementation of the EARSM in Ansys Fluent is based on the explicit algebraic Reynolds stress model of Wallin and Johansson [683]. Differences from the original formulation by Wallin and Johansson are explained in the following text.

With EARSM, the Reynolds stresses are computed from the anisotropy tensor according to its definition:

where the anisotropy tensor is searched as a solution of the following implicit algebraic matrix equation:

(4–220)

The coefficients in this matrix equation depend on the -coefficients of the pressure-strain term in the underlying Reynolds stress transport model. Their values are selected here as =1.245, =0, =1.8, =2.25.

The values of , , and are the same as those used in the original model by Wallin and Johansson [683]. As for the value of , it is increased from 1.2 to 1.245 in the course of calibrating EARSM for its use together with the BSL model.

and denote the non-dimensional strain-rate and vorticity tensors, respectively. They are defined as:

(4–221)

(4–222)

where the time-scale is given by:

(4–223)

In order to arrive at an explicit solution of the Equation 4–220, the anisotropy tensor is expressed as a polynomial based on the strain rate and the vorticity tensors as follows:

(4–224)

The -coefficients are evaluated to:

where the denominator Q is:

The invariants, which appear in the formulation of the anisotropy tensor and the coefficients, are defined by:

The model representation of the anisotropy tensor Equation 4–224 and its coefficients follows the original model by Wallin and Johansson [683] with two differences. First, the fourth order tensor polynomial contribution (the term) is neglected in Equation 4–224. Second, the tensor basis is slightly changed for convenience according to Apsley and Leschziner [23]. Although the tensor basis is changed, the model remains algebraically equivalent to the original model of Wallin and Johansson. The latter change results in correspondingly changed expressions for the coefficients .

In three-dimensional flows, the equation to be solved for the function is of sixth order and no explicit solution can be derived, whereas in two-dimensional mean flows the function can be derived from a cubic equation, an analytic solution of which is recommended by Wallin and Johansson [683] also for three-dimensional cases:

(4–225)

where

In the original model by Wallin and Johansson [683], the diffusion terms in the transport equations for and were calculated using the effective eddy viscosity, , of EARSM, where . The EARSM model, implemented in Ansys Fluent, uses the standard eddy viscosity for the diffusion terms. This model change helps to avoid the problems with the asymptotic behavior at the boundary layer edge, which were reported by Hellsten [237].

When in combination with the BSL model, the standard coefficients of the underlying BSL - model are used. When in combination with the GEKO model, then the EARSM model uses the GEKO model coefficients.