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Turbulence, Heat and Mass Transfer 4
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©2003 Begell House, Inc.
Ten Years of Industrial Experience with the SST
Turbulence Model
F. R. Menter1, M. Kuntz1 and R. Langtry1
1Software Development Department, ANSYS – CFX, 83714 Otterfing, Germany
florian.menter@ansys.com; martin.kuntz@ansys.com, robin.langtry@ansys.com
Abstract – This document describes the current formulation of the SST turbulence models, as well as a
number of model enhancements. The model enhancements cover a modified near wall treatment of the
equations, which allows for a more flexible grid generation process and a zonal DES formulation, which
reduces the problem of grid induced separation for industrial flow simulations. Results for a complete aircraft
configuration with and without engine nacelles will be shown.
1. Introduction
The starting point for the development of the SST [1,2] model was the need for the accurate
prediction of aeronautics flows with strong adverse pressure gradients and separation. Over
decades, the available turbulence models had consistently failed to compute these flows. In
particular, the otherwise popular k-ε [3] model was not able to capture the proper behaviour
of turbulent boundary layers up to separation [4]. The Johnson-King model [5] was the first
formulation, which allowed the accurate prediction of separated airfoil flows. Unfortunately,
the model was not easily extensible to modern three-dimensional Navier-Stokes codes due to
its algebraic formulation.
The k-ω model is substantially more accurate than k-ε in the near wall layers, and has
therefore been successful for flows with moderate adverse pressure gradients, but failes for
flows with pressure induced separation [1]. In addition the ω-equation shows a strong
sensitivity to the values of ω in the freestream outside the boundary layer [6]. The freestream
sensitivity has largely prevented the ω-equation from replacing the ε-equation as the standard
scale-equation in turbulence modelling, despite its superior performance in the near wall
region. This was one of the main motivations for the development of the zonal BSL and SST
models.
The zonal formulation is based on blending functions, which ensure a proper selection of
the k-ω and k-ε zones without user interaction. The main additional complexity in the model
formulation compared to standard models lies in the necessity to compute the distance from
the wall, which is required in the blending functions. This is achieved by the solution of a
Poisson equation and is therefore compatible with modern CFD codes.
The SST model was originally used for aeronautics applications, but has since made its way
into most industrial, commercial and many research codes. This is in agreement with the
present authors experience that the need for accurate computations of flows with pressure-
induced separation goes far beyond aerodynamics. The SST model has greatly benefited from
the strength of the underlying turbulence models. In particular, the accurate and robust near
wall formulation of the Wilcox model has substantially contributed to its industrial
Turbulence, Heat and Mass Transfer 4
usefulness. As well, all the model additions developed by Wilcox for rough walls and surface
mass injection etc. can be used with minor modifications [7].
While the original model formulation has largely stayed unchanged from the formulation
given in [1] (small modifications see bellow), there have been several areas of improvement
carried out within the CFX codes. Robustness opimisation have brought the model to the
same level of convergence as the standard k-ε model with wall functions. An improved near-
wall formulation has reduced the near wall grid resolution requirements, which has resulted
in a substantial improvement for industrial heat transfer predictions [8]. Finally, the zonal
formulation of the model has been beneficial in the formulation of an industrial Detached
Eddy Simulation (DES) model. A large number of model validation studies and applications
can be found on the internet.
2. SST Model Formulation
In this section, the complete formulation of the SST model is given, with the limited number
of modifications highlighted.
(
)
(
)
( )
∂
∂
+
∂
∂
+−=
∂
∂
+
∂
∂
i
tk
i
k
i
i
x
k
x
kP
x
k
4
t
k
µσ
*
~ (1)
(
)
(
)
( )
( )
ii
w
i
t
ii
i
x
x
k
F
x
x
S
x
U
t∂
∂
∂
∂
−+
∂
∂
+
∂
∂
+−=
∂
∂
+
∂
∂1
12 21
22
µσβαρ
ωρ
ω
Where the blending function F1 is defined by:
=
4
2
2
2*
1
4
500
maxmintanh yCD
k
,
y
,
5
k
F
k
(2)
with
∂
∂
∂
∂
=−10
210,
1
2max
ii
kw xx
k
CD
ω
ω
ρσ
ω
and y is the distance to the nearest wall.
F1 is equal to zero away from the surface (k-
ε
model), and switches over to one inside the
boundary layer (k-
ω
model).
The turbulent eddy viscosity is defined as follows:
( )
21
1,max FSa
ka
t
ω
ν
= (3)
Where S is the invariant measure of the strain rate and F2 is a second blending function
defined by:
=
2
2*
25002
maxtanh y
,
6
k
F (4)
A production limiter is used in the SST model to prevent the build-up of turbulence in
stagnation regions:
(
)
ωρβµ
kPP
x
U
x
U
x
U
Pkk
i
j
j
i
j
i
tk *
10,min
~⋅=→
∂
∂
+
∂
∂
∂
∂
= (5)
F. R. Menter et al.
All constants are computed by a blend from the corresponding constants of the k-
ε
and the
k-
ω
model via
(
)
FF
−
+
=
1
21
α
α
α
etc. The constants for this model are: β*=0.09, α1=5/9,
β1=3/40, σk1=0.85, σω1=0.5, α2=0.44, β2=0.0828, σk2=1, σω2=0.856.
The only modifications from the original formulation are the use of the strain rate, S,
instead of the vorticity in Equation 3 and the use of the factor 10 in the production limiter,
instead of 20 as proposed in [1,2].
3. Near Wall Treatment
One of the essential features of a useful industrial turbulence model is an accurate and
robust near wall treatment. In addition, the solutions should be largely insensitive to the near
wall grid resolution. For complex industrial flows the requirement 2<
+
y is excessive and
can in most cases not be satisfied for all walls. On the other hand, the strict use of wall
functions, which allow the use of coarser grids, limits the model accuracy on fine grids. A
new near wall treatment was therefore developed [8], which automatically shifts from the
standard low-Re formulation to wall functions, based on the grid spacing of the near-wall
cell.
Figure 1 shows velocity profiles for Couette flow simulations on three vastly different grids
( 100~;9~;2.0~ +++ yyy ). Despite the large differences in the near wall spacing, the
computed wall shear-stress varies by less than 2% and all solutions follow the logarithmic
profile. As a result, the new wall formulation has significantly improved the predictive
accuracy of general industrial applications, as the user influence via the grid generation is
drastically reduced.
Figure 1 Velocity profiles for three different grids using the automatic wall treatment of CFX-5
4. Application of the SST Model to Aerodynamic Flows
The SST model was selected by CFX for its contribution to the testcases of the 2nd AIAA
drag prediction workshop (http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw/). Two geometries
have been selected by AIAA and the grids have been provided by the organizers. Figure 2
shows the geometries simulated by the participants.
Turbulence, Heat and Mass Transfer 4
Figure 2 Geometries selected for AIAA drag prediction workshop
The low-Re grids had 5.83m (WB) and 8.43m (WBNP) hexahedral cells and have been
provided by ICEM. Convergence for the drag (most sensitive variable) has been achieved
typically with around 120-150 time steps.
Figure 3 shows the drag polar for the mandatory runs against the experimental data, as well
as the convergence history. The simulated results are in very good agreement with the
experimental data.
Figure 3 Drag polar for AIAA drag prediction testcases (left). Lift and Drag convergence (right)
This is a strong indication that optimized RANS models/codes can accurately simulate
complete aircraft configurations. More information can be found on the web-page of the
workshop and follow-up AIAA publications.
5. Zonal SST-DES Formulation
Recently, Spalart [9] has proposed a hybrid model formulation that utilises the RANS
equations inside the boundary layer and an LES-like formulation for free shear flows. The
model is termed Detached Eddy Simulation (DES) and is currently used in combination with
the Spalart-Allmaras and the SST turbulence model [10]. The main reason why these models
have been selected as the underlying RANS models lies in their improved separation
prediction capability. The DES modification in the SST model is applied to the dissipation
term in the k-equation as follows:
WB
WBNP
F. R. Menter et al.
∆
=⋅→= 1,max
**
DES
t
DESDES C
L
FwithFkk
ωρβωρβρε
(9)
where ε is the dissipation rate, ∆ is the maximum local grid spacing (
(
)
zyx
∆
∆
∆
=
∆
,,max in
case of a Cartesian grid), β* is a constant of the SST model,
ωβ
*
k
Lt= is the turbulent length
scale and CDES= 0.61 is a calibration constant of the DES formulation.
For fine grids, the switch from RANS to DES can take place somewhere inside the
boundary layer and produce a premature (grid-induced) separation [11]. Figure 4 shows the
effect for a 2-dimensional airfoil simulation. In this case the grid spacing in the spanwise
direction is assumed to be of the same order as the chordwise spacing (this is usually the case
for unstructured meshes or for flows where the flow direction is unknown during the grid
generation phase). It can be seen that the original DES limiter affects the RANS model and
moves the separation point upstream relative to the original SST model, which was in good
agreement with the data (upper right picture).
Figure 4 Region of flow separation on airfoil for different models. Lower right refined grid. Separation
indicated by arrow.
In order to reduce the grid influence of the DES-limiter on the RANS part of the boundary
layer, the SST model offers the option to “protect” the boundary layer from the limiter. This
is achieved again with the help of the zonal formulation underlying the SST model [11]. The
following modification significantly reduces the influence of the DES limiter on the boundary
layer portion of the flow:
( )
21,,0;1,1max FFFwithF
C
L
FSSTSST
DES
t
CFXDES =
−
∆
=
− (10)
SST
-
RANS
SST
-
DES Strelets
SST
-
DES
-
CFX
SST
-
DES
-
CFX
Turbulence, Heat and Mass Transfer 4
In this equation, FSST can be selected from the blending functions of the SST model. For
FSST =0, the model of Strelets [10] is recovered. Figure 4 shows also the effect for the same
2D airfoil, using FSST =F2 . It can be seen that with this modification, the boundary layer is not
affected and the separation point predicted with the SST model is unchanged, even under
more severe grid refinement (lower right picture).
Note that the zonal DES formulation does not completely eliminate the problem of grid
sensitivity in the RANS region, as the F2 function does not cover 100% of the boundary
layer. It does however reduce the critical limit by one order of magnitude.
Another interesting effect of the zonal DES formulation can be seen in Figure 5 for the
flow around a cube mounted inside a 2D channel. At the inlet, a fully developed channel flow
enters the computational domain. For this type of flow, the maximum grid spacing is smaller
than the turbulent length-scale over most of the domain. For the original SST-DES model,
this would mean that the DES limiter is activated over most of the domain, which would
essentially require a simulation carried out in LES modus. For the zonal SST-DES model, the
inlet part is covered by the F2 limiter and can be treated by the RANS model. The DES
limiter is only activated downstream of the cube, where the large turbulent structures are
resolved.
Figure 5 Flow around cube in channel flow. Solution SST-DES-CFX model.
Figure 5 shows the flowfield using iso-surfaces of the invariant
(
)
(
)
ijji xUxU
∂
∂
⋅
∂
∂
//
coloured by the ratio
µ
µ
/
t. The flowfield upstream is covered by the SST model and is
close to steady-state (except for pressure disturbances from the separated zone) and the flow
downstream is covered by the DES formulation.
Figure 6 Velocity profiles in symmetry plane of cube. Comparison of SST and SST-DES-CFX model with
experimental data.
F. R. Menter et al.
Figure 6 shows a comparison of the velocity profiles computed with the SST and the SST-
DES-CFX zonal model. The main difference is that the DES formulation captures the flow
recovery downstream of the separation zone in good agreement with the experimental data.
6. Future Directions
It has been observed for a long time that RANS turbulence models underpredict the level of
the turbulent stresses in the detached shear layer emanating from the separation line [13].
This in turn seems to be one of the main reasons for the incorrect flow recovery predicted by
the models downstream of reattachment. It was found it the 9th ERCOFTAC/IAHR/COST
Workshop on Refined Turbulence Modelling for the flow over a periodic hill, that models
with improved separation prediction capabilities, like the SST and the SA model did
overpredict the extent of the separated region. This is a matter of concern and is an area of
current research. The problem is shown for the asymmetric diffuser testcase of Obi [14]. The
SST model gives a significantly improved separation compared to the k-ε model, but predicts
a flow recovery that is slower than observed in the experiments. Note that the better
comparison of the k-ε model in this region is an artefact of the underpredicted separation.
Figure 7 Velocity profiles for asymmetric diffuser flow
While an improved flow recovery could be computed with the DES formulation, as shown
in Figure 6 this is not always possible. For pressure induced separation bubbles from smooth
surfaces, the original DES model cannot be applied due to the danger of grid-induced
separation. Alternatively, the zonal DES formulation would stay in RANS mode and would
have no influence on the results. A possible alternative to current DES formulations is the
extension of the Scale-Adaptive Simulation (SAS) approach [15] to the SST model.
Another interesting future development is the combination of the BSL model (underlying
the SST model) with explicit algebraic stress models (EASM) as proposed by Helsten and
Laine [16]. This allows the inclusion of secondary motions and the effects of streamline
curvature and system rotation.
7. Summary
This paper gave an overview of the current state and direction of development of the SST
turbulence model. The standard model formulation has been repeated and extensions for
improved wall treatment and a zonal DES formulation have been presented. Simulations for a
complete aircraft without and with engine nacelle have been briefly discussed. Directions for
future developments have been outlined.
Turbulence, Heat and Mass Transfer 4
Acknowledgement
Part of this work was supported by research grants from the European Union under the FLOMANIA
project (Flow Physics Modelling - An Integrated Approach) is a collaboration between Alenia, Ansys-
CFX, Bombardier, Dassault, EADS-CASA, EADS-Military Aircraft, EDF, NUMECA, DLR, FOI,
IMFT, ONERA, Chalmers University, Imperial College, TU Berlin, UMIST and St. Petersburg State
University. The project is funded by the European Union and administrated by the CEC, Research
Directorate-General, Growth Programme, under Contract No. G4RD-CT2001-00613.
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