NUMERICAL SIMULATION OF LOW-SPEED STALL AND ... - IAG

Conference on Modelling Fluid Flow (CMFF’12)
The 15 th International Conference on Fluid Flow Technologies
Budapest, Hungary, September 4-7, 2012
NUMERICAL SIMULATION OF LOW-SPEED STALL AND ANALYSIS OF
TURBULENT WAKE SPECTRA
Philipp Peter GANSEL 1 , Sebastian ILLI 2 , Thorsten LUTZ 3 , Ewald KRÄMER 4
1 Corresponding Author. Institute of Aerodynamics and Gas Dynamics, University of Stuttgart. Pfaffenwaldring 21, 70569 Stuttgart, Germany.
Tel.: +49 711 685 63416, Fax: +49 711 685 63438, E-mail: gansel@iag.uni-stuttgart.de
2 E-mail: illi@iag.uni-stuttgart.de
3 Head of branch “Aircraft aerodynamics“. E-mail: lutz@iag.uni-stuttgart.de
4 Head of Institute. E-mail: kraemer@iag.uni-stuttgart.de
ABSTRACT
At the Institute of Aerodynamics and Gas
Dynamics (IAG) numerical studies on the low-speed
stall of aircraft wings, the development of the wake
and its interaction with the empennage are performed.
In the present paper unsteady Reynolds-averaged
Navier-Stokes (URANS) calculations as well as highly
resolved Delayed Detached-Eddy Simulations (DDES)
of the turbulent wake downstream of a boundary
layer separation on an airfoil section are described.
For these simulations a NACA 0012 airfoil test case
with a free stream Mach number of 0.3 and a
Reynolds number of 6 million at a high angle of
attack was chosen. URANS calculations on different
grids and with different turbulence models show that
a fine structured discretisation of the wake region
facilitates the transport of turbulent viscosity with low
dissipation. The DDES approach allows propagation
of resolved turbulent structures far downstream of
the flow separation. The spectral analysis shows a
good agreement of measured and simulated pressure
fluctuations in the junction of energy-containing range
and inertial sub-range of the spectrum. Furthermore
semi-empirical turbulence spectrum models were
used to obtain spectral turbulence information from
URANS solutions. The resulting turbulent velocity
fluctuation spectra are compared to the resolved
velocity fluctuations of the DDES.
Keywords: CFD, DDES, stall, spectra, wake
NOMENCLATURE
E 11 [m 3 /s 2 ] one-dimensional longitudinal
turbulent kinetic energy spectrum
F+ [−] reduced frequency
L 11 [m] longitudinal integral length scale
Ma [−] Mach number
R 11 [m 2 /s 2 ] spatial velocity autocorrelation
Re [−] chord Reynolds number
U ∞ [m/s] free stream velocity
c [m] airfoil chord length
c l [−] lift coefficient
c p [−] static pressure coefficient
k [m 2 /s 2 ] specific turbulent kinetic energy
r [m] distance
t [s] time
u [m/s] velocity inx-direction
x, y,z [m] space coordinates
y 1 + [-] dimensionless wall coordinate
α [ ◦ ] angle of attack
∆ [m] maximal cell edge length
in the structured wake mesh
ε [m 2 /s 3 ] dissipation rate ofk
κ 1 [1/m] longitudinal wave number
κ e [1/m] wave number of the most
energy containing eddies
ω [1/s] turbulent dissipation frequency
Subscripts and Superscripts
¯ temporal mean
′
fluctuation
1. INTRODUCTION
At the boarders of the flight envelope flow effects
occur which still are not completely understood and far
from being reproduced correctly by today’s numerical
methods in industry relevant flow cases. At low Mach
numbers and sufficiently high angles of attack large
boundary layer separations on the wing cause a rapid
lift drop and form a highly unsteady turbulent wake.
This phenomenon called low-speed stall is the subject
of the presented studies, with a focus on the spectral
information in the wake downstream of the separation.
The fluctuations can affect the inflow conditions and
the flow state of the safety critical horizontal tail plane
(htp) [1] or even excite its structural modes.
However, only few published studies concerning
transport aircraft configurations deal with turbulent
fluctuations in the wake. Whitney et al. [2] developed
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an URANS based model to predict htp yaw and roll
moment fluctuations measured near low-speed stall.
In the German HINVA research project in-flight stall
tests with the DLR ATRA aircraft are planned to
collect amongst others unsteady pressure signatures
on wing and htp as a validation basis for unsteady
CFD methods [3]. A couple of investigations of
turbulent wake fluctuations have been published in
the context of active flow control devices. Seifert
and Pack [4] evaluated the spectral content of the
turbulent wake of a stalling airfoil with and without
periodic blowing near the leading edge by means
of unsteady pressure transducers. Wokoeck et
al. [5] used PIV measurements of a stalling tail
plane airfoil to validate RANS results with different
transition prediction methods. Lopez Mejia et al.
[6] explored time-averaged and instantaneous vorticity
fields around an airfoil with tangentially aligned
synthetic jet actuators using an incompressible DDES
code (EDDES) and PIV measurements. Since hybrid
RANS-LES methods as DES were developed to model
massive separation, at first applications concentrated
on flows around blunt bodies or airfoils in deep stall
[7–9]. Beyond that, Onera performed zonal and
delayed DES on three-element airfoils [10–12] and
on a civil aircraft type configuration [13]. Nagy
et al. [14] conducted zonal RANS-LES and LDA
measurements of a fan airfoil with intent to calculate
the flow generated noise. In their numerical studies
Durrani and Qin [15] compared Reynolds stresses in
the boundary layer and the near wake region of an
airfoil from DDES, DES, URANS and experiments.
Probst and Radespiel [16] applied different DES
approaches and showed deficiencies in the prediction
of mild trailing edge separation. Stall of a NACA
0012 airfoil was investigated by Lehmkuhl et al. [17]
by comparing Direct Numerical Simulation and Large
Eddy Simulation (LES) results.
In the presented research three different ways to
assess the spectrum of turbulent motions in separated
airfoil flow are tested and compared: unsteady RANS
calculations with evaluation of the resolved large
scale fluctuations of the flow variables, DDES with
fine resolution of turbulent fluctuations down to the
dissipation range and a tubulence spectrum model
based on URANS turbulence model entities.
2. FUNDAMENTALS
2.1. Turbulent Spectrum Model
Proccesses in turbulent flows at high Reynolds
numbers can be described by the concept of the energy
cascade [18]. It states that turbulence consists of eddies
of different sizes. Energy is entered into turbulence at
large scales driven by the boundary conditions given as
the adjacent flow field, then transferred to successively
smaller eddies and finally dissipated to heat at the
smallest scale, called the Kolmogorov scale. The so
called Kolmogorov hypotheses state that at high Re
i) small scale turbulence is locally isotropic, ii) the
statistics of turbulent motions are of universal form
and uniquely determined by turbulent dissipation rate
ε and viscosity and iii) in a certain range of scale
the statistics of turbulent motions are independent
of viscosity as well and thus solely defined by ε.
Accordingly the turbulent spectrum can be devided
into the energy-containing range, where big eddies are
formed and depend on the flow conditions, the inertial
subrange, where energy is passed on to smaller scales
by inviscid processes and the dissipation range where
viscous effects dominate. The universal functional
form of the turbulent spetrum allows to construct
models independent of the particular class of flow.
Von Kármán developed a turbulent energy spectrum
model, wherefrom Lysack and Brungart [19] derived
the one-dimensional longitudinal spectrum
E 11 (κ 1 ) = 8.73
55
[
k
κ e
(
κ1
1+
κ e
) 2
] 5/6
(1)
with the most energy containing eddies’ wave number
κ e = 1.9 ε
k 3/2 (2)
and the turbulent kinetic energy k. When derived
from a time and space dependent data sample,E 11 (κ 1 )
equals twice the Fourier transform of the velocity
fluctuation two-point auto-correlation
R 11 (r,t) = u ′ (x+r,t)·u ′ (x,t) . (3)
2.2. Numerical Methods
The calculation of the fluid flow describing
incompressible Navier-Stokes equations for industry
relevant cases with complex geometries and high
Reynolds numbers today is still not affordable, since
the number of grid points required for sufficient
resolution and the CPU-time needed for computation
are proportional to Re 9/4 and Re 3 respectively [20].
One approach is to decompose the flow variables in the
Navier-Stokes equations into a temporal mean value
and a time-dependent fluctuation. This leads to the
RANS equations which are solved for the mean values.
The remaining fluctuation-dependent terms (called
Reynolds stresses) are modelled. To determinate the
Reynolds stresses eddy viscosity turbulence models
are widely used. For the URANS simulations
conducted in the presented studies mainly the Menter
SST two-equation turbulence [21] model was used.
It provides transport equations for k and either its
dissipation rate ε (in the free flow) or the dissipation
frequency ω = ε/k (in near wall flow).
If turbulent fluctuations shall be resolved in the
simulations RANS methods are not applicable by
definition. DES became popular for flows with large
scale boundary layer separation in the recent years.
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It combines the advantage of LES to resolve the big
turbulent structures with relatively small grid sizes.
The outer part of the separated flow is calculated with
LES, where the total grid points scale with Re 0.4
[20]. In the boundary layer, where LES would require
much more cells, RANS equations are applied. In
contrast to RANS, in LES the Navier-Stokes equations
are not averaged in time but spatially filtered, where
the filter width is proportional to the biggest cell
extension. In the equations terms depending on the
residual fluctuations filtered out remain according to
the Reynolds stresses in RANS and are named residual
or subgrid scale (SGS) stresses. Again they can be
modelled using eddy viscosity models, of which the
Smagorinsky model is used in the present study.
Spalart et al. [22] presented an implementation of
DDES where the SA turbulence model is used in the
RANS domain and the SGS model is formulated in
such a way that the transport equation can be transfered
into the SA model equation by replacing the local
grid size by the distance to the nearest wall. A flow
dependent blending function controls the application
of the different approaches. Nevertheless the adequate
selection of the required spatial grid resolution is
crucial for correct operation of the hybrid scheme.
3. CONDUCTED SIMULATIONS
3.1. Test Case Description
As a test case wind tunnel tests were chosen
from literature with available unsteady measurements
in the wake [4]. The analysed NACA 0012 airfoil
section with a chord length c of 0.165m was tested
in a transonic cryogenic wind tunnel under free stream
conditions of Ma = 0.3 and Re = 6·10 6 . The data
chosen for the presented simulations were recorded at
an angle of attack of α = 16 ◦ , where a big turbulent
separation area occurs on the airfoil.
Due to the high memory consumption of full flow
field solutions, the spectral analysis, which all time
steps are required for, was conducted only in three
defined points listed in Table 1. Position A is very close
to the trailing edge and position B lies approximately
in the middle of the wake at a downstream position.
In the experiments unsteady pressure transducers were
placed in the wake at position C. The origin of the used
coordinate system is placed at the airfoil leading edge,
the x-axis points towards the trailing edge, the y-axis
in span wise direction and thez-axis upwards.
Table 1. Analysed measurement positions
Position A B C
x/c 1.2121 3.3750 3.3750
z/c 0.12121 0.60606 1.2198
3.2. Numerical Setup
The presented simulations have been performed
using the unstructured finite volume code TAU (version
2011.1.0) from DLR (German Aerospace Center) [23].
It can handle the mixed-type element hybrid meshes
created with the meshing software Gridgen V15.17.
For the URANS calculations the computational
domain is a two-dimensional (2d) hybrid mesh with
a C-type boundary layer mesh, a big structured wake
area and an unstructured farfield region. The boundary
layer grids of the three resolution levels fine, medium
and coarse have 60 cells in wall normal direction with
a first cell size of y 1 + ≤ 1 independently of the grid
resolution. The wake mesh is directly attached to the
boundary layer mesh with a smooth transition of cell
size. Table 2 shows cell sizes in the homogeneously
structured wake region, overall and surface point
counts for the different grid resolutions. To save
computational costs, the wake mesh is restricted to the
part of the flow field, where the measurement positions
are defined and the turbulent wake is assumed to pass
through (Figure 1). The farfield boundary distance
from the airfoil geometry is 100c.
Table 2. Point counts and wake region cell size ∆ of
the used RANS and DES meshes
RANS
DES
fine medium coarse
points 219000 94000 47000 17.8·10 6
surf. pts. 445 243 139 445×81
∆ 0.5%c 1.0%c 2.0%c 0.5%c
U ∞
unstructured
tetrahedral mesh
A
structured
hexahedral mesh
Figure 1. Computational domain with the analysed
measurement positions A, B and C
In the URANS simulations the second order
Jameson central discretisation scheme was used with
matrix dissipation for stabilisation. The dual time
stepping method was applied with a physical time
step size of 25µs which is approximately 1/100 times
the convective time scale c/U ∞ . Amongst others
the one equation turbulence model from Spalart and
Allmaras [24] in its original formulation (SAO), the
shear stress transport model (SST) from Menter [21]
and the Reynolds stress model (RSM) implemented in
TAU [23] were used as turbulence models. For DDES
calculations the structured parts of the fine 2d URANS
mesh were extruded span wise by 40%c with the same
cell size ∆ in y-direction yielding three-dimensional
(3d) isotropic structured cells in the wake region. The
farfield mesh is filled up with tetrahedra. On the
span wise boundary planes periodic conditions were
C
B
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applied to simulate an infinite wing without any tip or
wind tunnel side wall effects. The distance between
the periodic planes was chosen as big as possible
under consideration of limited computational costs to
minimise the error introduced through the forced span
wise periodicity of the flow. The numerical schemes
were the same as applied to URANS. During one time
step of 2.5µs the flow passes the shortest cells in
the wake which, in combination with the isotropically
structured mesh, ensures a low dissipation in the
propagation of the turbulent wake structures.
4. RESULTS
4.1. URANS Simulations
The analysis of the 2d URANS simulations
concentrates on the SST turbulence model because
it is used for the semi-empirical turbulent fluctuation
spectrum model later. The results with the other
eddy-viscosity turbulence models are comparable. In
the simulations the boundary layer flow separates from
the suction side of the airfoil, as expected from the
compared experimental data. The unsteady flow is
exactly periodic with a very long main cycle of about
11.5 convective time scales (see Figure 2). Over a
big part of this period the separated flow reattaches to
the upper surface of the airfoil and forms a separation
bubble (see Figure 3 a)). Gradually the separation point
is moving upstream and the vortex inside the bubble is
getting stronger. This becomes significant in the rising
lift coefficient during this phase. There is virtually no
large-scale turbulent movement in the wake.
1.6
1.4
c l
1.2
1
0.8
0 5 10 15 20 25 30 35 40
t U c ∞
Figure 2. c l time series of a NACA 0012 airfoil at
16 ◦ angle of attack (URANS SST)
At one moment a counter rotating vortex develops
behind the separation bubble and a shedding process
begins which means a drastic decrease of lift (see
Fig. 2). The streamlines in Fig. 3 b) show the flow
state shortly after this event. Finally several alternately
rotating vortices are shed from the airfoil, the flow
reattaches in the upstream part due to the reduced
circulation and the remaining small separation bubble
starts again to grow and increase the lift (see Fig. 3 c)).
The time-averaged pressure distribution on the
airfoil simulated with the three different turbulence
models is compared to experimental data in Figure 4.
The numerical results lie very close to each other.
However, the measurements of Seifert and Pack [4] are
qualitatively different. In contrast to the simulations
a)
b)
c)
Figure 3. Instantaneous streamlines at a) t U∞ c
=
14.5, b) 17.5 and c) 19 as highlighted with◦in Fig. 2
the airfoil flow is separated over the whole chord
length which is expressed in the lower pressure in
the rear part of the airfoil. The drop of circulation
causes a decrease of the leading edge suction peak
and overall lift level. Yet there are older wind tunnel
tests conducted by Ladson et al. [25] in the same
facility that fit the simulations very good. This leads
to the assumption that the test case is exactly at the
boarder between those two flow patterns and little
disturbances can determine it. Since it is known
that eddy-viscosity turbulence models tend to predict
boundary layer separation too late in the sense of
higher incidence, some calculations at increased α
were accomplished. The surface pressure distribution
at α = 18.5 ◦ was found to be in very good agreement
with the experiment of Seifert and Pack (see Fig. 4).
-6
-5
-4
c
-3
p
-2
-1
0
1
16° Exp. Seifert
16° Exp. Ladson
16° C-grid URANS SAO
16° C-grid URANS SST
16° C-grid URANS RSM
18.5° C-grid URANS SST
0 0.2 0.4 0.6 0.8 1
x/c
Figure 4. Fine grid URANS mean pressure
distributions compared to experimental data
In the comparison of the pressure distribution
simulated on the different grids (Figure 5) it can be
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seen that with increasing spatial resolution the pressure
on the suction side in front of about 25%c is increasing
a bit, but decreasing behind that point. On the pressure
side there are no noticeable differences.
-6
-5
-4
-3
c p
-2
-1
0
1
Exp. Ladson
SST ∆ = 0.5% c
SST ∆ = 1.0% c
SST ∆ = 2.0% c
0 0.2 0.4 0.6 0.8 1
x/c
Figure 5. URANS meanc p with diff. resolutions
The resolved pressure fluctuation spectra at
position C in the simulation on the fine and medium
grids are dominated by the main flow oscillation seen
above in the lift characteristics and its higher harmonics
(see Figure 6). The bad results on the coarse mesh
can be explained with high numerical dissipation of the
fluctuations in the insufficiently resolved wake mesh,
since the pressure distributions on the airfoil surface
are almost equal for all URANS grids. The medium
grid simulation seems to match the experimental data
in its peaks, but in terms of grid convergence the
peaks of the fine mesh simulations overestimate the
amplitudes. Compared to the URANS calculations
the experiment shows a smoother spectrum without
harmonic peaks. This fact suggests a more continuous
vortex shedding progress in the experiment than the
URANS simulations show.
c p
’
Position C
Exp. Seifert
10 -1 x = 3.375c
SST ∆ = 0.5%c
z = 1.220c
SST ∆ = 1.0%c
SST ∆ = 2.0%c
10 -3
10 -5
10 -7
10 -9
10 -11
10 -1 10
F+
0 10 1
Figure 6. Pressure fluctuations resolved by URANS
on fine, medium and coarse meshes
4.2. DDES Calculations
The DDES calculation produces a physically more
reasonable flow pattern with continuously shed vortices
of different scales. The 3d vortices can be visualised
using the λ 2 criterion as illustrated in Figure 7. It is
evident that the fine turbulent structures dissipate as
soon as they leave the structured wake mesh and enter
the coarser areas of the tetrahedra.
Figure 7. λ 2 iso-surface of turbulenct wake (DDES)
The time-averagedc p distribution from the DDES
shows an earlier and more clearly visible separation
point (see Figure 8) than the URANS. It is
remarkable that even the RANS simulation with the
SAO turbulence model produces a different pressure
distribution than DDES, where in the boundary layer
enclosing RANS area the same turbulence model is
used. This shows the feedback of the separated flow
on the boundary layer state.
-6
-5
-4
-3
c p
-2
-1
0
1
Exp. Ladson
SAO DDES
URANS SST
URANS SAO
0 0.2 0.4 0.6 0.8 1
x/c
Figure 8. DDES span wise averaged mean c p
compared to URANS and experiment
To give a better insight in the differences of
URANS and DDES simulated flow, the instantaneous
vorticity in a x-z plane is plotted in Figure 9 (for the
URANS plot the same time step was chosen as shown
in Fig. 3 c)). While the URANS results show few
big discrete vortices the DDES generates structures of
different sizes. Furthermore small vortex cores of high
vorticity are preserved longer.
The pressure fluctuation spectrum derived from
DDES time series at position C is in good agreement
with the measurement data (see Figure 10). The
spectrum has no outstanding peaks like the URANS
spectrum in Fig. 6 which confirms the assumption
about the character of the DDES calculated flow.
The spectra derived at positions A and B are plotted
additionally in Fig. 10. The spectrum at position A
shows the highest amplitudes due to its proximity to
the airfoil trailing edge where the shedding turbulent
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medium mesh results in Fig. 6) the same characteristics
are expected as with the fine DDES mesh. According
studies also discussing different DES formulations are
to be published by Illi et al. [26].
Another approach to gain information about the
turbulent spectrum is to perform the relatively cheap
URANS and model a turbulence spectrum from the
turbulent quantities calculated by the turbulence model.
This is done for the presented test case in the following
section.
Figure 9. Vorticity in x-z plane from URANS SST
(upper picture) and DDES (lower picture)
vortices are formed driven by the boundary conditions.
In position B the fluctuation level is already about one
order of magnitude smaller because of the decaying
processes in the energy cascade and the spreading of
the wake while propagated downstream. In point C the
amplitudes are even smaller, although in this position
the flow has passed roughly the same distance from
the airfoil as in point B. However, point C lies close to
the boarder of the wake to the outer free stream region
where the turbulence intensity is significant lower.
10 -3
4.3. Modelled Fluctuation Spectra
For an applicability assessment of the velocity
spectrum model presented by Lysack and Brungart
[19] for separated wake flow first the turbulent kinetic
energy k and its dissipation range ε were extracted
from URANS solutions. For comparison k = 0.5 ·
(u ′2 + v ′2 + w ′2 ) also has been calculated from the
DDES results and listed in Table 3. Here the amount of
turbulence modelled by the SGS model is neglected.
Table 3. Time-averaged turbulent kinetic energy k
from URANS SST and DDES
k [m 2 /s 2 ] pos. A pos. B pos. C
DDES 678.28 43.670 0.041669
URANS fine 264.90 49.921 6.3069
URANS medium 301.34 44.718 0.33582
URANS coarse 398.00 46.848 0.00047421
10 -2 Exp. Seifert Pos. C
c’ p
10 -4
10 -5
10 -6
10 -7
DDES ∆ = 0.5%c Pos. A
DDES ∆ = 0.5%c Pos. B
DDES ∆ = 0.5%c Pos. C
10 -1 10 0 10
F+
1 10 2
Figure 10. Pressure fluctuations resolved by DDES
in wake positions A, B and C
It should be noted in this place that the simulated
physical time covered by the DDES calculation is
much shorter than in the URANS cases. A total of
6000 physical time steps with 200 inner iterations each
were computed within the DDES on 500 computing
cores in parallel mode. The last 2000 time steps
were considered for spectral analysis after the time
averaged flow quantities have reached convergence.
Due to the big grid, the small time step size
and limited computational resources the spectrum is
limited to relatively high frequencies, since the lowest
representable frequency is reciprocally proportional to
the total time length of the evaluated signal.
One possibility to overcome this problem without
adding computational time is to perform the DDES
on a coarser mesh with less points and thus bigger
time steps. Like shown for URANS (compare fine and
The values show that close to the trailing edge
(position A) the URANS simulations give a much
lower turbulence level than the DDES. Additionally
k is increasing with decreasing spatial resolution. At
position B in the middle of the wake downstream
of the airfoil k is very comparable for all presented
simulations. At the boarder of the turbulent wake
in point C k is very low and the relative differences
between the methods and grid resolutions are in the
range of several orders of magnitude.
Utilising Taylor’s hypothesis of frozen turbulence
the time dependent velocity data logged at A, B and C
during the DDES calculation can be transferred into a
space domain [18]. The one-dimensional longitudinal
velocity spectra E 11 (κ 1 ) from DDES are derived by
duplicating and Fourier transformingR 11 (Eq. (3)). For
URANS E 11 (κ 1 ) is calculated via Eq. (1).
In Figure 11 E 11 (κ 1 ) at the different evaluation
positions is plotted. As seen from the turbulent kinetic
energy in Table 3 the modelled spectra in point A
underpredict the DDES calculated amplitudes. At
position B in the middle of the wake it is the other way
around. In both cases the URANS modelled spectra are
almost the same for all grid resolutions. Also the wave
number range of the transition region between large
scale range and inertial subrange which corresponds to
κ e is in good agreement to the DDES spectra.
In position C the large differences of k in the
URANS simulations give a very dissimilar description
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a)
b)
c)
10 0
E 11
10 -1
DDES ∆ = 0.5% c
10 2 SST ∆ = 0.5% c
10 1
SST ∆ = 1.0% c
SST ∆ = 2.0% c
10 -2
10 -3
10 -4
10 -5
10 -1
10
E -2
11
10 -3
10 -4
10 -5
10 -6
E
10 -4
11
10 -5
Position A
x = 1.212c
z = 0.121c
10 0 10 1 10 2 κ
10 3 10 4 10 5
1
DDES ∆ = 0.5% c
10 1 SST ∆ = 0.5% c
10 0
SST ∆ = 1.0% c
SST ∆ = 2.0% c
10 -3
10 -6
10 -7
10 -8
10 -9
Position B
x = 3.375c
z = 0.606c
10 0 10 1 10 2 κ
10 3 10 4 10 5
1
-1 DDES ∆ = 0.5% c
10
SST ∆ = 0.5% c
10 -2
SST ∆ = 1.0% c
SST ∆ = 2.0% c
Position C
x = 3.375c
z = 1.220c
10 0 10 1 10 2 10 3 κ
10 4 10 5 10 6
1
Figure 11. E 11 (κ 1 ) from DDES and turbulent
fluctuation spectrum model at positions A, B and C
of the flow and cannot be used for detailed analysis.
Also the very low velocity fluctuations in the DDES
are not useful for spectral evaluation. Nevertheless the
pressure fluctuation spectrum from DDES matches the
experimental measurement as shown in the previous
section. This fact leads to the assumption that even
if the turbulent wake passes by a certain point in such
a way that the turbulent vortices are not recognisable
in the local velocity, still you can find a significant
footprint of the turbulent structures in the pressure.
5. CONCLUSIONS
The presented studies revealed the ability of
URANS and DDES calculations to propagate turbulent
vortical structures on structured meshes in the wake
of the flow past an airfoil in low-speed stall. The
dissipation of coherent turbulent structures strongly
depends on the local grid cell size in the wake region.
While URANS simulates big discrete periodically
shed vortices, DDES shows a continuously forming
of turbulent 3d structures of different scales in the
wake. Hence evaluated pressure spectra from URANS
simulations base on the fundamental frequency of the
periodic vortex shedding pattern and its harmonics,
whereas DDES calculations give a smooth spectrum
which fits the experimental measurements better.
The turbulent spectrum model based on URANS
turbulence model entities is able to give a rough
estimate of the velocity fluctuation spectrum, at least
the wave number of the most energy containing eddies.
Furthermore it could be observed that on the
boarder of the wake virtually no meaningful velocity
spectrum could be derived from the DDES results,
while the measurable pressure fluctuations fit the
experimental data. This can be of relevance when
simulating the wake of an aircraft with boundary layer
separation occurring at the inner wing sections. Even
when the wake of the separation passes underneath
the htp it can have an effect on the surface pressure
distribution and cause unsteady fluctuations in the
integral forces.
ACKNOWLEDGEMENTS
The presented studies are part of the HINVA
and the ComFliTe project funded by the “German
Federal Ministry of Economics and Technology”.
Computational resources were kindly provided by the
“Leibniz Supercomputing Center” Munich during the
DEISA 2 call and the “High Performance Computing
Center Stuttgart” within the bwGRiD project [27].
REFERENCES
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