Adaptation of Eddy-Viscosity Turbulence Models to
Unsteady Separated Flow Behind Vehicles
F.R. Menter and M. Kuntz
Ansys-CFX
Introduction
Turbulence model development for aerodynamic applications has for many
years concentrated on improving the capabilities of CFD methods for separation prediction. Validation studies of turbulence models in the ‘80th have
clearly shown that most turbulence models were not capable of predicting the
development of turbulent boundary layers under adverse pressure gradient
conditions. Based on that observation, new models were developed to specifically meet this challenge, resulting in a series of models capable of capturing
boundary layer separation in good agreement with experimental data (Johnson
and King 1984, Menter 1993, Spalart and Allmaras 1994).
From the experience with the SST turbulence model (Menter 1993), the
present authors would argue that the capability of the model with respect to
the prediction of the onset of separation is within the accuracy of the available
experimental data and that no systematic deviation between the simulations
and the data is observed. Based on the experimental evidence (which is admittedly limited for three-dimensional flows), there is currently little need for
model improvements for that type of flows. Off course, this does not imply
that aerodynamic flows can be predicted within experimental uncertainty levels, as these flows involve other effects, which pose additional challenges to the
turbulence model. The main areas of concern are the behavior of the flow
downstream of the separation line, including the flow recovery after reattachment (Johnson et al. 1994), the proper simulation of vortex flows and
questions related to laminar-turbulent transition. Particularly the flow development downstream of separation is of major importance from an aerodynamic standpoint and can have a significant effect on the characteristics of
aerodynamic bodies. This is particularly true for ground vehicles, as they generally exhibit significant regions of separated flows, even at design conditions.
From a modeling standpoint, 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 (Johnson et al. 1994).
This in turn seems to be one of the main reasons for the incorrect flow recovery predicted by the models downstream of reattachment. The issue is some-
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times masked by the tendency of models of under-predicting the onset and
therefore the strength of the separation zone, which in turn results in an acceptable agreement in the recovery region. However, the improvement is only
the result of a cancellation of errors, as one cannot trade separation prediction
capabilities against improved velocity profiles in the recovery region. The delayed recovery of the boundary layer downstream of the reattachment line can
lead to a premature separation under a second adverse pressure gradient. A
second, and even more disturbing uncertainty resulting from the incorrect
prediction of the detached shear-layer, concerns the flow topology downstream
of the separation line. Current turbulence models cannot reliably answer the
question, whether the flow is forming a closed separation bubble, or a fully
stalled flow regime. This question is of major importance for the prediction of
the aerodynamic characteristics of automobiles, which almost always exhibit
regions of separated flow. The topology of these regions has a strong influence
on the drag and more pronouncedly on the lift of the car.
In an attempt of improving the predictive capabilities of turbulence models
in highly separated regions, Spalart (1997) proposed a hybrid approach, which
combines features of classical RANS formulations with elements of Large Eddy
Simulations (LES) methods. The concept has been termed Detached Eddy
Simulation (DES) and is based on the idea of covering the boundary layer by a
RANS model and of switching the model to a LES mode in detached regions.
Ideally, DES would predict the separation line from the underlying RANS
model, but capture the unsteady dynamics of the separated shear layer by
resolution of the developing turbulent structures. Compared to classical LES
methods, DES saves orders of magnitude of computing power for high Reynolds number flows, due to the moderate costs of the RANS model in the
boundary layer region, but still offers some of the advantages of an LES
method in separated regions.
There are two main concerns with the current DES formulation. The first is
how quickly the unsteady turbulent structures develop after the model has
switched from the RANS to the LES mode. This is of significance for the prediction of separated shear layers, as a delayed onset of resolved turbulent
structures would aggravate the underprediction of the turbulent stresses due to
a reduction of the unresolved turbulence level by the DES formulation.
The second concern is with the switching mechanism employed by the current DES methods. In order to prevent the activation of the DES limiter in
attached boundary layers, it is typically required to ensure a lower limit on the
local surface grid resolution. If this condition is violated, the integrity of the
RANS model is severely compromised resulting in most cases in grid induced
separation. This issue will be addressed in the section on turbulence model
formulation. For an alternative of a hybrid turbulence model without an explicit grid dependency see Menter et al. (2003).
The capabilities and limitations of advanced aerodynamic RANS and DES
turbulence models for automotive applications will be evaluated and discussed.
Alternatives to the current RANS/DES switch will be discussed. The models
will be applied to the flow around a simplified generic car shape, known as the
Ahmed car body (Ahmed et al. 1984, Lienhart and Becker 2003). The RANS
Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow
341
simulations have been presented at the ERCOFTAC workshop on Refined
Turbulence Modelling (Durand et al. 2002) in a comparison study of different
CFD methods for specific testcases. A detailed report is available from the
authors upon request.
All simulations have been computed with the commercial CFD method
CFX-5 of ANSYS.
Flow Physics
The geometry of the Ahmed car body is shown in Figure 1. It consists of a box
with rounded edges and a slanted back. The angle of the slant is adjustable and
is the main variable model-parameter in the experimental investigations. Two
sets of experiments have been carried out for this geometry (Ahmed et al.
1984, Lienhart and Becker 2003).The present comparison is mainly based on
the data of Lienhart and Becker (2003). In this experiment, the Reynolds
number with respect to the length of the car was ReL=2.6x106. Slant angle of
25° and 35° were investigated. The emphasis of the experiments was on the
flow structure in the slant region and downstream of the body.
Figure 1: Ahmed car geometry and drag forces on body
It was observed by Ahmed et al. (1984) that a change in the slant angle gave
a significant change in the drag coefficient, as shown in Figure 1 taken from
Ahmed et al. (1984). An increase in the slant angle from zero to around 30°
results in a gradual increase in the drag coefficient with a significant shift in
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F.R. Menter and M. Kuntz
the contributions from the different surfaces. While the viscous drag forces
show little sensitivity to the slant angle, there is a distinct increase in the drag
portion coming from the slant surface combined with a decrease in the contribution from the back surface. At a critical slant angle of around 30°, the drag
coefficient drops significantly, mainly due to a reduction in the contribution
coming from the slant. At higher slant angles, the drag coefficient stays virtually constant. The change in drag coefficient is due to a change in the flow topology in the aft region of the car body (Lienhart and Becker 2003).
Figure 2: Flow topology in the aft region of Ahmed car body (left: 25° and right: 35°). Courtesy
Lienhart, LSTM Erlangen.
Figure 2 shows a sketch of the flow topologies identified by Lienhart and
Becker (2003) for 25° and 35° of slant angle. One of the main differences is
that the flow at 25° shows a separation bubble at the onset of the slant,
whereas the 35° case is fully stalled over the entire slant.
A possible interpretation of the flow dependence on the slant angle is as
follows: the initial increase in the slant angle results in a change of the direction of the free flow (similar to the effect of a flap on an airfoil). The change in
flow direction requires an opposing force on the body, resulting in an increase
of lift. An increase of lift in turn results in an increase of the induced drag and
a strengthening of the trailing vortices (tip vortices). The effect of the induced
drag results in an increased drag on the slant, as shown in Figure 1. At a critical slant angle, the flow over the slant can no longer follow the contour, resulting in a fully stalled flow with no reattachment on the slant. At this point,
the induced lift breaks down and the drag comes mainly from the base drag of
both the slant and the back surface. Note that the flow topology has to be
identical for 0° and 90°. The present interpretation is consistent with the significantly reduced vortex strength between the 25° and the 35° case measured
in the experiments.
This flow poses a severe challenge to RANS turbulence models, as it requires models which are capable to correctly predict separation and reattachment at the lower slant angles in order to capture the correct flow topology.
Otherwise, the overall flow characteristics will be missed resulting in an incorrect prediction of drag and more severely of lift.
Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow
343
Turbulence Model
RANS simulations have been carried out with the standard k-ε model (Launder and Spalding, 1974) with wall functions and the SST model (Menter
1993) with automatic wall treatment (Esch et al., 2003). In addition, the SSTDES model proposed by Strelets (2001) has been modified and applied to
overcome some of its deficiencies in the RANS regions.
SST-DES Formulation Strelets et al. (2001)
The idea behind the DES model of Strelets (2001) is to switch from the standard SST-RANS model to an LES model in those regions where the turbulent
length, L t, predicted by the RANS model is larger than the local grid spacing.
In this case, the length scale used in the computation of the dissipation rate in
the equation for the turbulent kinetic energy is replaced by the local grid
spacing, ∆.
ε = β *kω =
k 3/ 2
k 3/ 2
→
Lt
CDES ∆
for CDES ∆ < L t; ∆ = max(∆ i ); Lt =
k
βω
*
The practical reason for choosing the maximum edge length in the DES
formulation is that the model should return the RANS formulation in attached boundary layers. The maximum edge length is therefore the safest estimate to ensure that requirement. The DES modification of Strelets can be
formulated as a multiplier to the destruction term in the k-equation:
 Lt

,1
ε = β *kω → β *kω ⋅ FDES with FDES = max
 CDES ∆ 
with CDES equal to 0.61, as the limiter should only be active in the k-ε model
region. The numerical formulation is also switched between an upwind biased
and a central difference scheme in the RANS and DES regions respectively.
SST-DES Formulation CFX
The main practical problem with the DES formulation (both for the Spalart
Allmaras and the SST-DES model) is that there is no mechanism of preventing
the limiter of becoming active in the attached portion of the boundary layer.
This will happen when the local surface grid spacing ∆s is less to the boundary
layer thickness ∆ S < cδ with c of the order of one. This is not a situation unlikely to occur, especially when unstructured grids are used in the simulation.
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In case that the limiter is activated in the boundary layer, the result will in
most cases be grid induced separation. In other words, the separation onset
and therefore the flow topology can be altered by the grid provided by the
user. Figure 3 shows an example of a grid-induced separation based on the grid
spacing shown by the grid lines. It can be argued that the grid induced separation could be avoided by a larger grid spacing in the lateral direction, but that
implies that the flow direction is known at the grid generation stage, which is
not the case in most complex three-dimensional simulations. Furthermore, unstructured prism/tetraeder based grids are typically isotropic on the surface,
eliminating this option. It should also be noted that a grid-spacing in the lateral direction (spanwise for a wing) which is solidly on the “safe” side of the
DES limiter, would prevent the DES mode in the region downstream of separation and thereby limit the effectiveness of the DES model to produce unsteady structures in the separating shear layer.
Figure 3: Regions of negative U-velocity for NACA4412 simulation for SST model (left) and
SST-DES model by Strelets (right) – separation point indicated by arrow
One way of alleviating the grid induced separation problem is to “shield”
the boundary layer from the DES limiter, thereby avoiding/reducing the
problem. As the SST model is based on a zonal formulation, differentiating
between the boundary layer and the rest of the flowfield, the blending functions of the model can also be used to formulate a zonal DES limiter. The following modification is therefore proposed for the SST-DES model:
 Lt

(1 − FSST ), 1; with FSST = 0, F1 , F2
FDES −CFX = max
 C DES ∆

FSST=0 recovers the Strelets et al. model. F 1 and F 2 are the two blending
functions of the SST model. F2 shields more of the boundary layer and is
therefore the prefered default. It should however be noted that even F2 does
not completely eliminate the problem, but reduces it by an order of magnitude, ∆ S < cδ where c is now of the order of 0.1.
Figure 4 shows the same simulation as for the standard SST-DES formulation computed with the SST-CFX-F2 model. The influence of the DES limiter
Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow
345
is avoided and the DES model does not affect the separation point. It can be
seen that even a more severe grid refinement does not lead to separation (right
picture). However, refinement of the surface grid below ∆ S = 0.1δ should be
avoided.
Figure 4: Regions of negative U-velocity for NACA4412 simulation for SST-DES-CFX-F2
model– separation point indicated by arrow. Right – locally refined
Numerical Results
Simulations for the two experimental cases with 25° and 35° slant angle have
been computed using three different grids with 0.65, 1.3 and 2.6 million
nodes for the half-model. The results were grid converged to plotting accuracy
on the medium grid. For the 35° case the k-ε and the SST model predict a
fully separated flow at the slant region, which is in good agreement with the
experimental data. In the following, the flow for the 25° slant angle is analyzed, as it is the more challenging configuration. A detailed report for the 25°
and 35° slant angle, including a grid refinement procedure, can be obtained
upon request from the authors. Note that both cases have been computed by
several authors for a recent ERCOFTAC workshop (http://labo.univpoitiers.fr/informations-lea/Workshop-Ercoftac-2002/Index.html).
RANS Simulations 25°° Slant Angle
For the 25° case, the agreement between the numerical simulations and the
experimental data was much less satisfying than for the higher slant angle and
severe differences between the solutions have been observed. It was also found
that this case was very sensitive to numerical details – a first order upwind
simulation with the SST model resulted in an attached flow, whereas a second
order solution produced a separated flow.
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Figure 5 shows a comparison of the velocity profiles in the symmetry plane.
For this case, the models produce very different results. The k-ε gives a fully
attached flow, whereas the SST model results in a stalled flow. Both solutions
are not in agreement with the data, which show a separation and subsequent
reattachment along the slant. The SST-DES results are discussed in the following chapter.
For this case, the differences in the turbulent stresses are even more pronounced, as can be seen in the turbulent kinetic energy shown in Figure 6,
which is underpredicted by an order of magnitude by the SST model. The k-ε
model does not predict an increased level of turbulence at all, as it fails in
capturing the flow separation. Shear layer experimental data suggest however,
that there is an excessive level of turbulence in the experimental data, which
seems to be a result of a large-scale unsteady behavior of the flow. Spalart
(2003) argues that the observed level of turbulence could be obtained by a repeated switch in the flow topology from attached to separated and back. The
large level of the experimental turbulence intensity in the spanwise direction
suggests that this motion should be accompanied by a strong lateral movement. None of these effects is apparently predicted by any of the RANS models.
Figure 5: Velocity profiles computed with different turbulence models for the 25° case
Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow
347
Figure 6: Profiles for the turbulent kinetic energy for the 25° case
From the standpoint of the steady state flow topology, it appears that the
k-ε model is in closer agreement with the experimental data than the SST
model. For the 25°, the experiments show a topology associated with high lift
and strong trailing vortices. The confined separation bubble does not alter the
overall flow topology. RANS models, which predict no separation, are therefore in closer agreement with the experiments. This does however not imply
that the k-ε model is as superior turbulence model for this type of flow. The
agreement in flow topology is a result of the failure of the model to predict the
separation and not of its ability to accurately predict reattachment. The ability
of a turbulence model of predicting the onset of separation is still the first priority for aerodynamic flow simulations and cannot be traded against other
model characteristics.
DES Simulations 25°° Slant Angle
The 25° testcase is a good candidate for the evaluation of the improvement
achieved by the DES formulation in the separated flow region. The goal is the
resolution of the unsteady features of the shear layer separating from the edge
of the slant. The vortex-shedding and break-up is expected to be the main
mechanism to increase the turbulent energy in the separated shear layer,
thereby forcing a reattachment of the flow. However, as the high levels of turbulence in the measurements cannot be explained by classical shear layer
physics, the next question is, if the large-scale unsteadiness observed in the experiments can be triggered by the turbulent structures emanating from the
slant.
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The DES simulations require a significantly refined grid in the slant region,
particularly in the spanwise direction, to activate the DES limiter. In order to
keep the total number of nodes at an acceptable level, two steps of local grid
refinements have been introduced in the slant region. The total number of
nodes is 1.783.000, and the number of spanwise cells is 70 on the slant. Off
course, the entire body is now computed without a symmetry assumption. The
time step for the simulation is ∆t=10-4[s], which corresponds to 250 time steps
for one passage of the car at freestream velocity. A large number of steps were
required to ensure proper statistical behavior of the solution. Due to the high
computing costs of DES, only one grid could be analyzed. Care was taken,
that the resolution in the RANS region was sufficient to capture the attached
boundary layers. In the separated region, local grid refinement was applied to
ensure an optimal distribution of nodes.
Figure 7 shows the time history of the forces acting on the body in the three
space dimensions. It can be seen that the forces are non-periodic and even after
~7000 time steps do not follow a regular pattern. Nevertheless, the unsteady
structures in the separating shear layer have changed the flow topology as can
be seen in Figure 8, showing the wall-shear stress vectors on the slant for the
SST and the SST-DES-CFX-F1 formulation. Instead of the fully separated
flow topology, with low lift and weak trailing vortices, the DES topology is
closer to the experiments with a confined separation zone, increased lift and
strong trailing vortices. Note that the level of fluctuations in the lift forces is
more than 50% of the mean lift.
Figure 7: Time history of forces on Ahmed car body
Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow
349
Figure 8: Flow structure on the slant for SST (left) and SST-DES-CFX-F1 (right)
More details can be seen in Figure 9 showing the velocity profiles at a plane
at 180 [mm] (the half width of the body is 194 [mm]). In the symmetry plane
(see Figure 5 and Figure 6), the advantage of the DES simulation is not very
pronounced, but in the off-symmetry plane, the change in flow topology and
the improvement in the predicted results is apparent. The unsteady simulations give a first indication of the underlying mechanism of the large-scale unsteadiness of the flowfield. It is observed that the separation zone has a strong
lateral movement, which at some instances interacts with the side vortices,
leading to a vortex-breakdown. This is shown for a certain instance in time in
Figure 10. The right side vortex is unaffected by the separation, whereas on the
left side of the body the separation interacts with the vortex leading to a vortex-breakdown. The time value correlates with a maximum of the side force
(compare Figure 7). While it is likely that this is the main mechanism for the
experimentally observed global unsteadiness, it is not (yet) of sufficient
strength to reproduce the experimental fluctuation level. It is not clear if this is
a question of an insufficient length of the time integration, or a shortcoming
of the DES model.
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Figure 9: Comparison of velocity profiles at Y=180[mm] for SST and SST-DES-CFX-F1 model
for the 25° case
Figure 10: Vortex structures on slant of Ahmed car body (25°)
Grid Induced Separation - Revisited
After the simulation of the Ahmed car body using DES, it is worthwhile revisiting the issue of grid induced separation. The present application required a
significant increase in the lateral grid spacing (compared to RANS) in order to
activate the DES limiter in the separating shear layer emanating from the car
roof. As the 2D vortices originating from the separation are smaller than the
boundary layer thickness and have to be resolved with several grid nodes, it is
also required to have a streamwise grid spacing lower than the boundary layer
thickness. If these grid resolution requirements would not be satisfied in the
region close to the separation line, the onset of the DES limiter would be delayed to a significantly further downstream location, thereby reducing the
Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow
351
chances of capturing the essential flow features. For the present application,
the flow separation takes place at a corner - it is therefore not of major consequences if the DES limiter is activated already in the region upstream of the
separation line, as the boundary layer there is not exposed to an adverse pressure gradient. Grid induced separation is therefore not a problem for the present geometry.
However, for a general car geometry, where the separation can be induced
by an adverse pressure gradient from a smooth surface, standard DES would
face severe difficulties. One choice is generating a fine grid, which would allow
the activation of the DES formulation in the separation zone, at the danger
that the DES limiter will change the RANS part and produce grid-induced
separation. The second choice is the use of a coarser grid and thereby delay the
DES impact far downstream of the separation line and miss the physics of the
flow. As the separation line is not known during grid generation, it is difficult
to imagine how a suitable DES-grid could be generated for such a flow.
The use of the proposed zonal DES formulation, based on the SST model
blending functions, will at least reduce the risk of a grid induced separation
occurrence, however, it still has to be tested if the blending function will
switch quickly enough from RANS to DES to activate the DES mode for a
pressure induced separation. Nevertheless, this approach is preferred, as it reduces the influence of the user (grid) on the solution.
Conclusions
CFD simulations have been carried out for the generic Ahmed car body at
25°. Strong turbulence model differences where observed for this case. The k-ε
model produced an attached flow over the entire slant, whereas the SST model
predicted a fully stalled flowfield. Both solutions are in disagreement with the
experiment, where the flow separates and reattaches at about 50% of the slant.
DES simulations based on a modified version of the SST-DES formulation,
resulted in a significant improvement of the solution compared to the SSTRANS model. Instead of a fully stalled flow, the time-averaged DES solution
shows a confined separation zone and strong trailing vortices, associated with
the experimentally observed flow topology. The DES solution also gives first
insight into the mechanism driving the strong unsteadiness of the flow, as observed in the experiments. The most likely explanation is a strong lateral
movement of the separation zone, which interacts with the side vortices, leading to periodic vortex-breakdown. It could not be determined if the full unsteady effect could be obtained by further continuation of the simulation, due
to constraints in computing power.
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F.R. Menter and M. Kuntz
Acknowledgment
This work was supported by research grants from the European Union under
contracts GRD1-2001-40199 (Flomania) and EVG1-2001-00026 (EXPRO).
The authors want to thank H. Lienhart from the University of Erlangen for
the provision of some of the figures concerning the physical interpretation of
the flow.
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