 Non-linear k-e-v2 model

[Durbin (1995), Lien & Durbin (1996)] Original form

The turbulence model uses the standard equations:  On no-slip boundaries, , The transport equation is where k f represents redistribution of turbulence energy from the streamwise component. Non-locality is represented by solving an elliptic relaxation equation for f: where The Boussinesq approximation is used for the stress-strain relation: where the eddy viscosity is given by The constants of the model are:  As --- y being the minimum distance to walls --- and , Eq. (4) becomes: The viscous and kinematic conditions at the wall show that should be as . In the original model, n=1, yielding the boundary condition for f on no-slip walls. Code-friendly modification

Eq. (11) works fairly well for coupled, implicit solvers [e.g. INS2D code of Rogers Kwak (1990)]. However, for explicit and uncoupled schemes, numerical instability, arising from in the denominator of Eq. (11), sometimes occurs. Therefore, a code-friendly modification is made here by setting n=6, which allows to be imposed as the boundary condition. In addition, and are replaced by where , and the other model constants are:   Non-linear constitutive relation

A general constitutive relation of the type proposed by Pope (1975) can be written as: where . Truncating at the third term for simplicity, gives rise to where Two constrains for parallel flow will be imposed: where . These yield where or ( , in general) and T is defined in Eq. (6). The remaining unknown, , can be evaluated from DNS data of channel flow (Kim et al, 1987), boundary-layer flow (Spalart, 1988) and flow over a backward-facing step (Le et al, 1993). As seen in the following figure, fits DNS data reasonably well. The algebraic model was initially by Durbin (1995) used as an a postiori formula for evaluating . In order to apply Eq. (11) to mean flow prediction, while preventing computational intractability, the coefficients and are modified as:  [UW Home][Engineering Home Page][Mechanical Home Page] 