Gamma-Re Transition Model

Gamma-Theta (y-Re) transition model is a two equation model used in Computational Fluid Dynamics (CFD) to modify turbulent equations to simulate laminar, laminar-to-turbulet and turbulence states in a fluid flow. The Gamma-Theta does not to model the physics of the problem but attempts to fit a wide range of experiments and transition methods into its formulation. The transition model uses an intermettincy factor that create turbulence by slowly introducing turbulent productio at the laminar-to-turbulent transition.

Principle

The goal of developping the gamma-theta transition model was to develop a transition model based on local variables which could be easily implemented into modern CFD code with unstructured grids and massive parallel execution. The majority of earlier transition models such as the en model needs to know the strucutre of the boundary layer and the integration along it; both concepts are hard to implement in three-dimensions along many subdivisions of a grid. Another key insight to the formulation of this model is that the Reynolds Voriticy Number can be related to the Reynolds Transition Onset Number so there is a local way to calculate the transition number. The Gamma-Theta Transition model has two equations and is based on the two equation turbulence models. This way both local and global trends can be modelled. The Intermittency or Gamma determines the percentage of time the flow is turbulent( 0 = completely laminar / 1 = completely turbulent). The intermittency acts on the production term of the kinetic turbulent energy in the SST model to simulate laminar/turbulence.

Standard Gamma-Theta model

For turbulent kinetic energy k

${\frac {\partial (\rho k)}{\partial t}}+{\frac {\partial (\rho ku_{i})}{\partial x_{i}}}={\frac {\partial }{\partial x_{j}}}\left[{\frac {\mu _{t}}{\sigma _{k}}}{\frac {\partial k}{\partial x_{j}}}\right]+2{\mu _{t}}{E_{ij}}{E_{ij}}-\rho \epsilon$

For dissipation $\epsilon$

${\frac {\partial (\rho \epsilon )}{\partial t}}+{\frac {\partial (\rho \epsilon u_{i})}{\partial x_{i}}}={\frac {\partial }{\partial x_{j}}}\left[{\frac {\mu _{t}}{\sigma _{\epsilon }}}{\frac {\partial \epsilon }{\partial x_{j}}}\right]+C_{1\epsilon }{\frac {\epsilon }{k}}2{\mu _{t}}{E_{ij}}{E_{ij}}-C_{2\epsilon }\rho {\frac {\epsilon ^{2}}{k}}$

Other models

Following are some more models which are usually employed.

en low-Reynolds Number