### 2.4.1. Two Equation Gamma Theta Transition Model

#### 2.4.1.1. Overview

The full transition model is based on two transport equations, one for the intermittency and one for the transition onset criteria in terms of momentum thickness Reynolds number. It is called ‘Gamma Theta Model’ and is the recommended transition model for general-purpose applications. It uses a new empirical correlation (Langtry and Menter) that has been developed to cover standard bypass transition as well as flows in low free-stream turbulence environments. This built-in correlation has been extensively validated together with the SST turbulence model ([101], [102], [103]) for a wide range of transitional flows. The transition model can also be used with the BSL or SAS-SST turbulence models as well.

The Gamma Theta Model has the following limitations:

- The Gamma Theta Model, like all other engineering transition models, is only valid for wall boundary layer flows. It cannot be applied to free shear flow transition. The model will predict free shear flows as fully turbulent.

- Due to the specific formulation of the Gamma Theta equation, only classical boundary layers with a defined freestream region can be handled. This means that the model is not applicable to situations such as pipe or channel flow.

- The model is currently not linked to the buoyancy production terms because buoyancy driven flows are outside the calibration space of the model. Therefore the model should not be used in conjunction with non-zero buoyancy forces.

- The model is not Galilean invariant. This means it is only valid with respect to walls that are not moving relative to the coordinate system.

It should be noted that a few changes have been made to this model compared to original version (that is, the CFX-5.7 formulation, [101],[102]) in order to improve the transition predictions. These include:

1. An improved transition onset correlation that results in improved predictions for both natural and bypass transition.

2. A modification to the separation induced transition modification that prevents it from causing early transition near the separation point.

3. Some adjustments of the model coefficients in order to better account for flow history effects on the transition onset location.

You should use the new formulation [103], although the original version of the transition model (that is, the CFX-5.7 formulation) can be recovered by specifying the optional parameter “`Transition Model Revision = 0`” in the CCL in the following way:

          FLUID MODELS:
            …
            TURBULENCE MODEL:
              Option = SST
              TRANSITIONAL TURBULENCE:
                Option = Gamma Theta Model
                Transition Model Revision = 0
                TRANSITION ONSET CORRELATION:
                  Option = Langtry Menter
                END
              END
            END
            …
          END

In order to capture the laminar and transitional boundary layers correctly, the mesh must have a of approximately one. If the is too large (that is, > 5) then the transition onset location moves upstream with increasing . You should use the High Resolution discretization scheme (which is a bounded second-order upwind biased discretization) for the mean flow, turbulence and transition equations.

Note: The default production limiter for the turbulence equations is the ‘Kato-Launder’ formulation when the transition model is used.

As outlined in CFX Transition Model in the CFX-Solver Modeling Guide, two reduced variants of the transition model are available besides the two-equation Gamma Theta transition model:

- Specified Intermittency transition model — a zero-equation model.

- Gamma transition model — a one-equation model.

#### 2.4.1.2. Transport Equations for the Gamma Theta Transition Model

The transport equation for the intermittency, , reads:

| (2–136)

The transition sources are defined as follows:

| (2–137)

where is the strain rate magnitude. is an empirical correlation that controls the length of the transition region. The destruction/relaminarization sources are defined as follows:

| (2–138)

where is the magnitude of vorticity rate. The transition onset is controlled by the following functions:

| (2–139)

| (2–140)

| (2–141)

is the critical Reynolds number where the intermittency first starts to increase in the boundary layer. This occurs upstream of the transition Reynolds number, , and the difference between the two must be obtained from an empirical correlation. Both the and correlations are functions of .

The constants for the intermittency equation are:

| (2–142)

The boundary condition for at a wall is zero normal flux while for an inlet is equal to 1.0.

The transport equation for the transition momentum thickness Reynolds number, , reads:

| (2–143)

The source term is defined as follows:

| (2–144)

| (2–145)

| (2–146)

| (2–147)

The model constants for the equation are:

| (2–148)

The boundary condition for at a wall is zero flux. The boundary condition for at an inlet should be calculated from the empirical correlation based on the inlet turbulence intensity.

The model contains three empirical correlations. is the transition onset as observed in experiments. This has been modified from Menter et al. [101] in order to improve the predictions for natural transition and is defined as follows:

Parameters entering the empirical correlations for are the local turbulence intensity, , and the Thwaites pressure gradient coefficient, , defined as:

| (2–149)

where is the acceleration in the streamwise direction.

is used in Equation 2–144. is the length of the transition zone and goes into Equation 2–137. The correlation for is a function of and is defined as:

| (2–150)

A modification to is used in order to avoid a sharp increase in the skin friction in the boundary layer shortly after transition:

is the point where the model is activated in order to match both and ; it goes into Equation 2–140. The correlation between and is defined as follows:

| (2–151)

can be thought of as the location where turbulence starts to grow while is the location where the velocity profile first starts to deviate from the purely laminar profile.

#### 2.4.1.3. Separation-induced Transition Correction

The modification for separation-induced transition is:

| (2–152)

The model constants in Equation 2–152 have been adjusted from those of Menter et al. [101] in order to improve the predictions of separated flow transition. The main difference is that the constant that controls the relation between and was changed from 2.193, its value for a Blasius boundary layer, to 3.235, the value at a separation point where the shape factor is 3.5 (see, for example Figure 2 in Menter et al. [101]).

#### 2.4.1.4. Coupling to the SST Model

The transition model interacts with the SST turbulence model, as follows:

| (2–153)

| (2–154)

| (2–155)

where and are the original production and destruction terms for the SST model and is the original SST blending function. Note that the production term in the -equation is not modified. The rationale behind the above model formulation is given in detail in Menter et al. [101].

#### 2.4.1.5. Gamma Theta Transition Model and Rough Walls

If the two-equation transition model is used together with rough walls, then an equivalent sand-grain roughness height must be specified at the wall and in addition the option Roughness Correlation must be selected on the Fluid Models tab of the Domain details view. Further information on the rough wall modifications can be found in Treatment of Rough Walls.

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