### 2.3.4. Explicit Algebraic Reynolds Stress Model

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. The EARSM enables an extension of the current ( and BSL) turbulence models to capture the following flow effects:

- Secondary flows

- Flows with streamline curvature and system rotation.

The implementation is based on the explicit algebraic Reynolds stress model of Wallin and Johansson [188]. Differences from the original formulation by Wallin and Johansson are explained in the following text. The current EARSM formulation can be used in CFX together with either the BSL or model. It is recommended that the current EARSM formulation be used with the BSL model (BSL EARSM).

With EARSM, the Reynolds stresses are computed from the anisotropy tensor according to its definition (see Equation 2–107):

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

| (2–126)

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 [188 ]. 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:

| (2–127)

| (2–128)

where the time-scale is given by:

| (2–129)

In order to arrive at an explicit solution of the Equation 2–126, the anisotropy tensor is expressed as a polynomial based on the strain rate and the vorticity tensors as follows:

| (2–130)

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 2–130 and its coefficients follows the original model by Wallin and Johansson [188] with two differences. First, the fourth order tensor polynomial contribution (the term) is neglected in Equation 2–130. Second, the tensor basis is slightly changed for convenience according to Apsley and Leschziner [219]. 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 [188] also for three-dimensional cases:

| (2–131)

where

A diffusion correction, suggested by Wallin and Johansson [188], is also implemented in CFX as an option. This correction consists in the replacement of in the above written formulas for , and , through :

where is the equilibrium value of , that is a value of achieved under the source term equilibrium =:

For the parameter , which scales the degree of the diffusion correction, Wallin and Johansson suggest a value of 2.2. In the current CFX implementation, it is set by default to zero, but this can be overridden.

In the original model by Wallin and Johansson [188], the diffusion terms in the transport equations for and (or ) were calculated using the effective eddy viscosity, , of EARSM, where . The EARSM model, implemented in CFX, 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 [190].

For the underlying or BSL model, the standard coefficients are used.

#### 2.3.4.1. Streamline Curvature and System Rotation

In order to account for effects of streamline curvature, the non-dimensional vorticity tensor is extended in the following way ([189], [190]):

| (2–132)

where the contribution of the curvature correction based on the work of Wallin and Johansson [189] and Spalart and Shur [191] is given in the following way:

| (2–133)

where

and

In this formulation represents the Levi-Chivita factor, which is equal to 0 if two or more indices are equal, 1 if the indices {i,j,k} form an even permutation of {1,2,3} and –1 if they form an odd permutation. are the components of the coordinate system rotation vector. The coefficient has to be calibrated. The scaling coefficient has been introduced similar to the ‘Curvature Correction of Two-Equation Models’ in order to enable you to influence the effect of the curvature correction term

if needed for a specific flow. The default value of this coefficient is 1.

In a rotational frame of reference, where the coordinate system rotates relative to a steady inertial frame, the vorticity tensor reads:

| (2–134)

Because the anisotropy tensor is computed in terms of nondimensional strain-rate and vorticity tensors, the nondimensional form of this term becomes:

| (2–135)

It should be noted that there are significant robustness issues with the streamline curvature model described in Equation 2–133. This is because one has the second velocity derivative (that is, the gradient of the strain rate) divided by the first velocity derivative (that is, the strain rate). Consequently, numerical noise in the second velocity derivative is amplified when the strain rate is small (for example, in the free stream outside of a shear layer). Because the curvature corrected Reynolds stresses enter directly in the momentum equation, this can create a feedback loop that results in increasing spikes in the Reynolds stresses that are unphysical and can even cause the solver to diverge. The solution to this problem turned out to be to compute two sets of Reynolds stresses with the EARSM, those with and without the curvature term in Equation 2–132. The Reynolds stresses without the curvature term are used in the momentum equations, while the Reynolds stresses with the curvature term are used in the assembly of the production term in the and (or ) equations. Consequently, the change in turbulence production due to curvature is still captured, however the feedback loop that caused the spikes in the Reynolds stresses is avoided because the curvature term does not enter directly into the momentum equations. There is probably some loss in accuracy due to this treatment. However this is preferable to a model, which is not robust and cannot be used in industrial applications. The constant in Equation 2–132. has been calibrated to a value of -0.4.

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