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I.K. Marchevsky, V.V. Puzikova
20
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
To obtain accurate quantitative results when simulating unsteady flows characterized
by high-speed airfoils movement, and hence by high values of local Reynolds number,
strong mesh refinement is need. It leads to a sharp increase in computational cost of direct
numerical simulation. The traditional way here is turbulence simulation by using some
well-known approaches and turbulence models. However, the corresponding numerical
schemes haven’t been developed for LS-STAG approach.
In the present research the LS-STAG discretizations for two-dimensional RANS,
LES and DES equations and transport equations from Smagorinsky, Spalart —
Allmaras,
,
k
k
and
k
SST turbulence models are constructed.
Governing equations.
The problem is considered for 2D unsteady case when the
flow around an airfoil assumed to be viscous and incompressible within the framework
of RANS, LES and DES approaches. In contrast to direct numerical simulation (DNS)
based on solution of Navier — Stokes equations and resolution of all turbulent
movement scales, turbulence models usage involves simulation of turbulence scales
contribution to the averaged motion (in case of RANS approach) or simulation of scales
that do not exceed the filter width
(in case of LES approach). In case of RANS
approach one speaks of the Reynolds stress simulation and in case of LES approach one
speaks of the subgrid stress simulation.
The Reynolds-averaged Navier — Stokes equations are being solved in RANS
approach, and the filtered Navier — Stokes equations are being solved in LES approach
instead of the Navier — Stokes equations. DES approach usage means that RANS
equations are being solved in one part of the computational domain, and LES equations
are solved in the other part. It is possible to write down the unified governing equations
in dimensionless variables for all approaches, because the form of LES equations is
similar to the form of RANS equations. So the incompressible flow is described by the
following RANS/LES/DES equations:
ˆ
= 0,
( ) =
.
t
p
t
v
v
v v
v
(1)
Here
= ( , , ) =
x
y
x y t u v
v v
e
e
is the dimensionless Reynolds averaged of filtrated
velocity,
= ( , , )
p p x y t
is the dimensionless Reynolds averaged of filtrated pressure,
=1/ Re
is the dimensionless viscosity,
ˆ
t
is the Reynolds or subgrid stresses tensor.
The relationship between
ˆ
t
and flow Reynolds averaged or filtrated variables is given
by the turbulence model.
Flow around a fixed airfoil in com-
putational domain
with boundary
=
1
2
3
4
is considered (Fig.
1).
In all our simulations, the upstream and
outflow boundaries are set at the distances
8
D
and 15
D
, respectively, from the airfoil
center, and the blockage ratio is equal to
1/12. The previous studies have shown that
such computational domain is sufficiently
wide to obtain results that are don’t depend on the domain size.
Fig. 1.
Computational domain