### 2.3.1. The Reynolds Stress Model

The standard Reynolds stress model in ANSYS CFX is based on the -equation. The CFX-Solver solves the following equations for the transport of the Reynolds stresses:

| (2–89)

where is the pressure-strain correlation, and , the exact production term, is given by:

| (2–90)

The production due to buoyancy is

| (2–91)

where the second term represents the buoyancy contribution from the pressure-strain term (Launder [196]), and is given by

| (2–92)

If the Boussinesq buoyancy model is used, then is modeled as

| (2–93)

where is the thermal expansion coefficient. Otherwise the term is modeled as (full buoyancy model based on density differences)

| (2–94)

As the turbulence dissipation appears in the individual stress equations, an equation for is still required. This now has the form:

| (2–95)

In these equations, the anisotropic diffusion coefficients of the original models have been replaced by an isotropic formulation, which increases the robustness of the Reynolds stress model.

The Reynolds stress model is also available with anisotropic diffusion coefficients. In this case, the CFX-Solver solves the following equations for the transport of the Reynolds stresses:

| (2–96)

where is the pressure-strain correlation, and , the exact production term, is given by Equation 2–90.

In this case the production due to buoyancy is for the Boussinesq approximation modeled as:

| (2–97)

Otherwise the term is modeled as (full buoyancy model based on density differences):

| (2–98)

The equation for is:

| (2–99)

The model constants are listed below for each model.

#### 2.3.1.1. Pressure-Strain Terms

One of the most important terms in Reynolds stress models is the pressure-strain correlation, . It acts to drive turbulence towards an isotropic state by redistributing the Reynolds stresses.

The pressure strain term can be split into two parts:

| (2–100)

where is the ‘slow’ term, also known as the return-to-isotropy term, and is called the ‘rapid’ term.

There are three varieties of the standard Reynolds stress models based on the -equation available. These are known as LRR-IP, LRR-QI and SSG. The LRR-IP and LRR-QI models were developed by Launder, Reece and Rodi [4]. In both models, the pressure-strain correlation is linear. “IP” stands for Isotropization of Production and is the simplest of the 3 models. The two terms read:

| (2–101)

| (2–102)

The values of the two coefficients are and . is given by .

“QI” stands for Quasi-Isotropic and differs from the IP model in the formulation of the rapid term:

| (2–103)

where is given by:

| (2–104)

The SSG model was developed by Speziale, Sarkar and Gatski [5] and uses a quadratic relation for the pressure-strain correlation.

In order to compare the pressure-strain correlations for the three models, a general form can be derived based on the anisotropy tensor and the mean strain rate tensor and vorticity tensor, and respectively. The general form reads:

| (2–105)

| (2–106)

where

| (2–107)

| (2–108)

| (2–109)

This general form can be used to model linear and quadratic correlations by using appropriate values for the constants. The constants are listed in the table below for each model.

Model |  |  |  |  |  |
--- | --- | --- | --- | --- | --- | ---
LRR-IP | 0.1152 | 1.10 | 0.22 | 0.18 | 1.45 | 1.9
LRR-QI | 0.1152 | 1.10 | 0.22 | 0.18 | 1.45 | 1.9
SSG | 0.1 | 1.36 | 0.22 | 0.18 | 1.45 | 1.83

Model |  |  |  |  |  |  |
--- | --- | --- | --- | --- | --- | --- | ---
LRR-IP | 1.8 | 0.0 | 0.0 | 0.8 | 0.0 | 0.6 | 0.6
LRR-QI | 1.8 | 0.0 | 0.0 | 0.8 | 0.0 | 0.873 | 0.655
SSG | 1.7 | -1.05 | 0.9 | 0.8 | 0.65 | 0.625 | 0.2

Selection of the appropriate model is carried out on the Fluid Models panel of the Domains form in CFX-Pre. The following options correspond to the types of models listed above:

- Reynolds Stress Model - LRR-IP

- SSG Reynolds Stress Model - SSG

- QI Reynolds Stress Model - LRR-IQ

The coefficients of the pressure strain term can be specified in CFX-Pre in the general form for the LRR-QI and SSG models. The LRR-IP model has been added for completeness in the table above. In CFX-Pre the pressure strain coefficients of the LRR-IP are asked in the form as given in Equation 2–101 and Equation 2–102. For the Boussinesq buoyancy model the default value of is 0.9 for LRR-IP and LRR-QI and 2/3 for SSG. In the full buoyancy model based on density differences the default value of is 1.0 for all models.

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