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:

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