Large-eddy simulation of transonic buffet over a supercritical airfoil by tcm16179

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									Large-eddy simulation of transonic buffet over
a supercritical airfoil

E. Garnier1 and S. Deck2
1
    ONERA, Applied Aerodynamics Department, 8 rue des Vertugadins, 92190
    Meudon, France. eric.garnier@onera.fr
2
    ONERA, Applied Aerodynamics Department, 8 rue des Vertugadins, 92190
    Meudon, France. sebastien.deck@onera.fr

1 Introduction
The transonic buffet is an aerodynamic phenomenon that results in a large-
scale self-sustained periodic motion of the shock over the surface of the airfoil.
The time scale associated to this motion is much slower than the one of the wall
bounded turbulence. It is then an appropriate case for URANS approaches
and first attempts with these methods have been reasonably successful in
reproducing the mean features of such flows. Nevertheless, as shown by Thiery
and Coustols [2] results are very sensitive to the turbulence model. Moreover,
with some models, it is necessary to increase the angle of attack with respect to
the experiment to obtain an unsteady flow. Furthermore, they have evidenced
a significant sensitivity of the results to the confinement due to the wind
tunnel walls. The first hybrid RANS/LES computation on this configuration
was performed by Deck [1] who has demonstrated that zonal DES (ZDES)
generally improves the results with respect to URANS computations carried
out with the Spalart-Allmaras model. In particular, the spectral content of
the pressure fluctuations in the separated zone is much more closer to the
experimental data with ZDES than with URANS. In this latter computation,
the shock/boundary layer interaction was treated in RANS mode and one
of the purpose of the present study is to assess the improvement that can
be obtained from a fully turbulent treatment of the boundary layer on the
suction side of the airfoil by means of LES.
    More generally, the main objective of this study is to assess the capabil-
ities of LES to capture the buffet phenomenon. The large amount of data
provided by these simulations could then support the progress in the physical
understanding of such flows. The validation of the computation is performed
against the very comprehensive experiment performed at ONERA by Jacquin
et al. [3] which was also used by Thiery and Coustols [2] and Deck [1].
2      E. Garnier and S. Deck

2 Description of the computation

The supercritical OAT15A airfoil was computed in the same flow conditions
than in the experiment by Jacquin et al.[3]. This airfoil has a chord of 230 mm
and a relative thickness of 12.3 %. Its angle of attack is equal 3.5 degrees. The
free-stream Mach number was set to 0.73 and the Reynolds number based on
the chord length is equal to 3 106 .
    The flow solver is the structured multiblock code FLU3M developed at
ONERA. It is second-order accurate in space and time. The numerical scheme
dedicated to the computation of the convective fluxes is based on a Roe scheme
which was modified to adapt locally its dissipation using the Ducros et al. sen-
sor [4]. The Selective Mixed Scales Model has been chosen for this study [5].
The time step has been imposed to 3.10−7 s in order to ensure the convergence
of the subiterative process of the temporal implicit scheme using 5 subitera-
tions.
    In order to limit the required computational effort, the flow is computed
in RANS mode on the pressure side of the airfoil and in LES mode on the
suction side and in the wake. Moreover, RANS zones are treated in 2D. The
grid refinement criteria commonly used in LES of attached flows are satisfied
(∆x+ ≈ 50 in the longitudinal direction, ∆z + ≈ 20 in the spanwise direction
         +
and ∆ymin ≈ 1 in the wall-normal direction). Despite the zonal treatment
of the flow, 20.8 millions of cells are necessary to compute a domain width
of only 3.65 % of chord in the grid A (Nz=140). The span and consequently
the number of points were doubled to construct the grid B (Nz=280). This
may be insufficient but the grid size results from a compromise with the long
integration time required to capture few buffeting periods.


3 Mean field analysis

After a transient of 2 periods, the flow has been averaged over only one pe-
riod of the buffet phenomenon for the case A. The span was then doubled to
generate the grid B and, after a transient of one period, the statistics were
collected over one another period. Figure 1 presents an isovalue of the Q cri-
terion colored by the longitudinal velocity. The separation occurs after the
location of the shock identified by one isovalue of the pressure (in purple). On
this snapshot which corresponds to a situation where the shock moves down-
stream, the flow is separated under the lambda shock and near the trailing
edge.
    Figure 2 (left) shows the averaged pressure distribution on the airfoil. The
buffet zone is shifted downstream by 6 % of chord with respect to the experi-
ment. This shift appears more clearly on the pressure fluctuation distributions
presented in figure 2 (right). The use of a doubled span (grid B) significantly
reduces the fluctuations near the trailing edge. The analysis of the instanta-
neous fields obtained on grid A has evidenced that this overestimation was
       Large-eddy simulation of transonic buffet over a supercritical airfoil                                        3




Fig. 1. Q criteria near the wing wall and one isovalue of the pressure to mark the
shock location


due to the presence of intense bidimensional coherent structures developing
when the flow separates from the shock up to the trailing edge. The span of
the grid B allows the tridimensionalisation of these structures which limits
their intensity and subsequently the wall pressure fluctuations.


                -2                                                           0.5


           -1.5
                                                           LES A                                            LES A
                                                           LES B             0.4                            LES B
                                                           Exp.                                             Exp.
                -1

                                                                             0.3
                                                                   Prms/Qo


           -0.5
          Kp




                0
                                                                             0.2

               0.5

                                                                             0.1
                1


               1.5                                                            0
                     0   0.2   0.4         0.6   0.8   1                      0.2   0.4         0.6   0.8
                                     x/c                                                  x/c




Fig. 2. Averaged pressure coefficient distribution (left) and rms pressure distribu-
tion (right)


    The profiles of averaged and fluctuating longitudinal velocity at x/c=0.35
are plotted in figure 3. These data evidence that, upstream from the interac-
tion, the velocity field is well estimated by the LES. This result was far from
being trivial since the flow undergoes a numerically forced transition at the
same station than in the experiment (x/c=0.07).
    Downstream from the interaction (at x/c=0.75), one can observe in figure
4 that the agreement of the LES with both the averaged and the fluctuating
longitudinal velocity profiles is more than satisfactory. It is however worth-
while to notice that between x/c=0.4 and x/c=0.6, both experimental and
numerical velocity profiles differ significantly since the shock is not located at
the correct mean position.
4      E. Garnier and S. Deck

              0.014                                                                       0.014




              0.013                                                                       0.013




          y (m)




                                                                                      y (m)
              0.012                                                                       0.012




              0.011                                                                       0.011




                  0.01                                                                        0.01
                         0       50    100   150   200     250    300   350   400                    0   20            40             60   80
                                                   U(m/s)                                                          Urms (m/s)




Fig. 3. Mean longitudinal velocity (left) and longitudinal velocity fluctuations
(right) profiles at x/c=0.35 (Grid B).


                  0.03                                                                        0.03




                  0.02                                                                        0.02
             y (m)




                                                                                         y (m)
                  0.01                                                                        0.01




                     0                                                                           0
                             0        50     100     150         200    250     300                  0        50                100        150
                                                   U (m/s)                                                         Urms (m/s)




Fig. 4. Mean longitudinal velocity (left) and longitudinal velocity fluctuations
(right) profiles at x/c=0.75 (Grid B).


   Nevertheless, it is believed that the results quality is sufficient to initiate
a physical analysis of the flow.


4 spectral analysis

Due to the short duration of the LES simulations, an auto-regressive (AR)
model method has been used to compute the Power Spectral Density of the
pressure. Indeed, this method is well adapted to study short data that are
known to consist of sinusoids in white noise[8]. The AR parameters are ob-
tained with Burg’s method[9]. The pressure spectrum for x/c = 0.9 is com-
pared to experiment in figure 5. The occurrence of strong harmonic peaks
highlights the periodic nature of the motion. On the experimental side, the
main peak at 69 Hz represents the frequency of the self-sustained motion of
the shock over the airfoil. A higher frequency near 76 Hz is found in the
computation. This 10 % error can be considered here as acceptable.
       Large-eddy simulation of transonic buffet over a supercritical airfoil                                                                               5


                                 180                                               LES
                                                                                   exp. S3Ch



                                 160




                             SPL
                                 140




                                 120




                                 100

                                        1        2                             3                                      4
                                   10       10                         10                                        10
                                                 frequency (Hz)




                       Fig. 5. PSD of pressure fluctuations.


5 Space and time scales
Once, the main statistical and spectral features of the flow have been found,
it is worthwhile to study the kinematics of these pressure waves. To this
end, let us consider the fluctuating pressure at different stations. The two-
                                                                                                                            ′
                                                                                                                                (x1 ,t)P ′ (x2 +ξ,t−τ )
point two-time correlation coefficient: Rx1 ,x2 (∆ξ, τ ) = √P                                                                               √
                                                                                                                          (P ′ 2 (x1 ))       (P ′ 2 (x2 +ξ))
establishes the correlation between two signals located at abscissa x1 et x2 + ξ
and separated by a time delay τ . The convection velocity can be obtained as
the slope of the linear fitting of the ξ versus τmax (τmax represents the delay
where the correlation coefficient reached its maximum), as illustrated in figure
6.


                                   1
                                                                           ∞
                                                             ∞



                                                                    1U
                                                          4 U

                                                                 -0 .2 7
                                                       -0 .3 6




                                 0.8

                                 0.6
                          ∆ξ/c




                                 0.4                                                        6 .0
                                                                                                   81
                                                                                                        0   -2
                                                                                       7.                       U
                                                                                            18                   ∞
                                 0.2                                                             10   -2
                                                                                                        U
                                                                                                            ∞



                                   0
                                            0                                      5
                                                     τmaxU ∞/c


Fig. 6. Propagation velocities obtained by a least square fitting of the linear relation
between the separation distance ∆ξ and time delay τ (filled symbol: exp, solid line:
upper side of the airfoil).


    On the upper-side of the airfoil, a downstream propagation velocity equal
6.08 10−3U∞ is clearly identified for the LES and appears to be slightly lower
than in the experiment. On the lower side of the airfoil, a forward motion at
velocity 0.364 10−3U∞ is evidenced. The latter velocity corresponds exactly
to the upstream travelling acoustic waves on the lower side of the airfoil.
6         E. Garnier and S. Deck

6 Discussion
To assess the frequency of the motion, Lee[6] proposed that the period of
the shock oscillation should agree with the time it takes for a disturbance
to propagate from the shock to the trailing edge added to the time needed
for an upstream moving wave to reach the shock from the trailing edge. A
simplified model has been used in reference [1] to assess the complete duration
                                  c−xs        c−xs
to complete such a loop: T = vdownstream + |vupstream | where c is the chord and
xs is the mean location of the shock wave. xS can be obtained by noting
the first abscissa where the skewness of pressure fluctuations is zero. One
gets (xs /c)LES = 0.52 while (xs /c)exp = 0.45. The velocity of upstream-
travelling acoustic waves is vupstream = a(M − 1) where a is the local speed
of the sound in the field outside the separated area. With M = 0.8 and
a = 330 m/s, the Lee’s equation gives f = 1/T ≈ 110 Hz which is higher
than the frequency FLES ≈ 80 Hz. More recently, Crouch et al.[7] advocated
that transonic buffet results from global instability where the unsteadiness
is characterized by phase-locked oscillations of the shock and the separated
shear layer. Within this scenario, the region downstream from the shock is
not the only region contributing to the feedback loop. Indeed, an upstream
travelling acoustic motion has been highlighted on the lower surface of the
airfoil (see figure 6). A deeper investigation of these phenomena will follow
the present work.


7 Acknowledgments
This work has been partly sponsored by the French National Research Agency
(project ANR-07-CIS7-009-04).


References
    1. Deck, S. (2005) AIAA J. 43:1556-1566
    2. Thiery, M. and Coustols, E. (2004) Flow, Turbulence and Combustion 74:331-
       354
    3. Jacquin, L., Molton, P., Deck, S., Maury, B., Soulevant, D. (2005) AIAA paper
       2005-4902
    4. Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C.,
       and Poinsot, T. (1999) J.Comput. Phys. 152:517-549
    5. Lenormand, E., Sagaut, P., Ta Phuoc, L., and Comte, P. (2000) AIAA J.
       38:1340-1350
    6. Lee, B.H.K. (1990) Aeronautical Journal, 143-152
    7. Crouch, J.D. and Garbaruk, A. and Magidov, D. and Jacquin, L. (2008), Pro-
       ceedings of IUTAM conference, Corfou, GREECE
    8. Trapier, P. and Duveau, P. and Deck, S. (2006) AIAA J. 44:2354-2365
    9. Burg, J.P. (1978) In Modern Spectrum Analysis, Ed. D.G. Childers, 34-41,
       IEEE Press, New-York

								
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