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: