Here, the two-parameter turbulence models BSL (baseline) and SST (shear stress
transport) are treated [51]. The
[37] and the
-model [90] are special cases of the BSL-model. The two
parameters are the turbulent kinetic energy
and the turbulence frequency
. The equation for
reads [51]:
![]() |
(713) |
Since
![]() |
(714) |
and
![]() |
(715) |
one can write
![]() |
(716) |
This leads to:
![]() |
(717) |
Notice that
can be written as:
In the above conservation equation is a function of
the flow characteristics through the blending factor
[51] and
the dynamic turbulent viscosity satisfies:
![]() |
(719) |
In fact, the BSL model is a linear combination of the model and the
-model with coefficients
and
, respectively. Near a
wall
tends to 1, thus favoring the
-model which is
particularly good in the near-wall region, far away from a wall
tends to
zero, leading to a pure
-model.
The blending factor used in iteration is
, i.e. its
calculation is mainly based on the results from the previous iteration and satisfies
the following equations:
![]() |
![]() |
(720) |
![]() |
![]() |
(721) |
![]() |
![]() |
(722) |
where is the distance from the element center P to the next solid surface.
Using the blending factor one obtains the correct parameter values, e.g.:
![]() |
(723) |
For completeness the parameters are listed here:
![]() |
![]() |
(724) |
![]() |
![]() |
(725) |
![]() |
![]() |
(726) |
![]() |
![]() |
(727) |
In the conservation equation for k one very easily identifies the time-dependent, convective, diffusive and body terms. They are treated in a completely analogous way to the energy equation. The convective boundary conditions amount to:
For the convective interpolation of the modified smart algorithm has not
shown any advantages, therefore, the upwind difference scheme is always used.
The diffusion boundary conditions are:
![]() |
(728) |
is known (cf. convective inlet boundary conditions), and
![]() |
(729) |
Notice that no facial values are calculated for the turbulent
parameters. Therefore,
in the above equation is approximated
by
.
![]() |
(730) |
![]() |
(731) |
The source terms are treated in the following way:
![]() |
(732) |
in which the term in brackets depends on
and
only,
cf. Equation (718). It can be split in a sum of
![]() |
(733) |
which is positive and corresponds to a source (treated explicitly, i.e. on the right hand side) and
![]() |
(734) |
![]() |
(735) |
or
![]() |
(736) |
respectively.
![]() |
(737) |
This term is treated implicitly ( is evaluated at iteration
and
the term ends up on the left hand side) since it is a negative source.
The equation for the turbulence frequency runs [51]:
![]() |
(738) |
which can be rewritten as:
![]() |
![]() |
|
![]() |
(739) |
One easily recognizes the time-dependent, convective, diffusive and source terms. The convective boundary conditions amount to:
For the convective interpolation of the modified smart algorithm has not
shown any advantages, therefore, the upwind difference scheme is always used.
The diffusion boundary conditions are:
![]() |
(740) |
is known (cf. convective inlet boundary conditions), and
![]() |
(741) |
Notice that no facial values are calculated for the turbulent
parameters. Therefore,
in the above equation is approximated
by
.
![]() |
(742) |
![]() |
(743) |
where
and
is the distance to the next point away
from the wall; same treatment as for inlet (notice, however, hat
at the wall).
The source terms are treated as follows:
![]() |
(744) |
where
is the term in the outer brackets
on the right hand side of Equation (709). Part of this term goes to
the right hand side (sources) and part to the left hand side (sinks) as
discussed extensively for the energy equation.
![]() |
(745) |
if
(sink) and by
![]() |
(746) |
if
(source).
![]() |
(747) |
![]() |
(748) |
if the term is positive (source), and by:
![]() |
(749) |
if the term is negative (sink).