Improvement of transonic buffet knowledge

The buffet phenomenon on a 2D airfoil is different from the one occurring on a 3D wing.

Fig. V.48 shows PSDs for the 2D buffet (left, from ref. [34]) and the 3D buffet (right). One can remark that the 2D buffet phenomenon is characterised by well marked peaks at low frequency (70 Hz or St = f.c/U∞ = 0.06 – 0.07) whereas the 3D buffet phenomenon is characterised by a bump at high frequency (200 Hz or St = 0.26). Moreover, concomitantly with this difference of characteristic frequencies, the shock oscillation is much larger on a 2D airfoil (20% of chord) than on a 3D wing (2% of chord). The 2D buffet phenomenon has been explained by Lee [39] as a self-sustained loop based on the coupling between the shock and the trailing edge through pressure waves. The shock wave generates pressure waves which propagate downstream inside the bound-ary layer. Then, these waves are scattered at the trailing edge, generating new waves that travel back upstream outside the boundary layer, up to the shock. These waves create new pressure waves and close the loop. More recently, Crouch et al. [19] and Sartor et al. [49] have explained the turbulent 2D buffet phenomenon as an unstable global mode. At high Mach number or angle of attack, the analysis of the eigenvalues of the Jacobian of the Navier-Stokes operator exhibit one single eigenvalue with a positive real part at the buffet frequency leading to an amplification of the perturbations. Concerning the 3D buffet phenomenon, there is no explanation on why the Strouhal number is four times higher on a 3D wing than on a 2D airfoil for the same Mach num-ber, Reynolds number and chord length.

During the AVERT European project and an ONERA own-funded project, wind tunnel tests have been performed on two half wing-body configurations in the S3Ch and S2MA wind tunnels

(see pictures Fig. V.78 (a) and Fig. V.71). These models were equipped with unsteady pressure transducers on the suction side of the wing. The analysis of these two experimental databases has been presented in ref. [25]. Signal processing tools like cross-spectra or frequency-wavenumber spectra have been used to search for correlations between sensors and then to determine the con-vection velocities of the 3D buffet phenomenon. It has been shown that the 3D buffet phenome-non consists in the spanwise convection of what has been called later “buffet cells” by Iovnovich

& Raveh [33]. Instead of having synchronized motion of the shock in all spanwise sections like in the 2D buffet case, the shock foot oscillates in the spanwise direction as shown by the purple line in Fig. V.49. Using the streamwise lines of unsteady pressure transducers, convection speeds at the buffet frequency between 45 and 63 m.s^-1 are found. Using a spanwise line of sensors close to the trailing edge, a convection speed of 100 m.s^-1 is computed which means that contrary to the 2D buffet, the 3D buffet has a strong spanwise component. Since these two lines of sensors are not orthogonal, the velocity norm and direction are found using the following system of equa-tions:

⎩⎪

⎨

⎪⎧ 𝑈𝑈𝑐𝑐

cos(𝛽𝛽) = 63 𝑈𝑈𝑐𝑐

cos �𝜋𝜋2 − 𝛽𝛽 − 𝜑𝜑^{𝑇𝑇𝑇𝑇}�= 100

where β is the buffet propagating angle, Uc the convection velocity norm and 𝜑𝜑_{𝑇𝑇𝑇𝑇} is the angle between the two lines of sensors. The solution is (in purple in Fig. V.49):

�𝑈𝑈^𝐷𝐷 = 61 𝑚𝑚. 𝑠𝑠⁻¹ 𝛽𝛽 = 13°

Fig. V.48: PSD on the 2D OAT15A airfoil (left, from [34]) and on the CAT3D model (right).

Fig. V.49: Convection velocities for the buffet phenomenon on the CAT3D model (M = 0.82, α = 4.25°, Rec = 2.83×10⁶).

In summary, the 3D buffet phenomenon is characterised by a convection velocity equal to 61 m.s^-1 or 0.2U∞. The propagation angle is equal to 13° which roughly corresponds to the angle between the spanwise and the streamwise components of the velocity upstream of the shock. The wavelength of the phenomenon is roughly equal to the mean aerodynamic chord or twice the dis-tance the shock and the trailing edge which means a quite long wavelength. The second main con-clusion is that two different instabilities coexist on the suction side of the wing: the 3D buffet phenomenon at St = 0.26 and the Kelvin-Helmholtz instability (1 ≤ St ≤ 4) which develops in the shear layer between the separation downstream of the shock and the outer flow. The spectral anal-ysis has shown that each phenomenon has a different Strouhal number but also different convec-tion velocities: 0.2U∞ for the buffet (with an angle of 13° with respect to x-axis) and 0.6Uloc for the Kelvin-Helmholtz instability (in the x direction).

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Now, in terms of numerical simulation of the 3D buffet phenomenon, it has been comput-ed in ZDES with elsA by Brunet in 2008 [8] and in URANS by Iovnovich & Raveh in 2015 [33].

At ONERA, with elsA, up to now, we were not able to simulate the 3D buffet phenomenon in URANS. In fact, the URANS computations converged to a steady state. Before the publication of ref. [33], it was thought that this phenomenon could not be simulated in URANS for which the turbulence is completely modelled but only in RANS/LES with a part of turbulence resolved. De-spite a lot of simulations with different linear or non-linear eddy viscosity models, different spa-tial and temporal schemes, different meshes, URANS simulations always converged to a steady state.

Since we were able to compute the 2D buffet phenomenon in URANS but not the 3D buf-fet, it was decided to increase step by step the complexity by first performing 2.5D URANS simu-lations like in ref. [33]. The 2D mesh is extruded in the spanwise direction with a sweep angle.

Compared to the 3D wing in Fig. V.49, there is no taper ratio (the chord is constant along the span) and there is no twist or dihedral angle. This time, the URANS simulations with Spalart-Allmaras turbulence model did not converge to a steady state and the 3D buffet phenomenon is

captured. Fig. V.50 shows the instantaneous wall pressure on the wing with a sweep angle of 20°.

The boundary conditions are the following: symmetry on the right hand side and a zeroth order extrapolation on the left hand side. The oscillation of the shock foot in the spanwise direction is clearly visible starting from the lambda shock on the symmetry condition. In the movie of this simulation, the convection of the “buffet cells” in the left direction is clearly visible. These results have been presented in ref. [46] with additional sweep angles.

Fig. V.50: Instantaneous wall pressure on the swept RA16SC1 wing (URANS simulation).

The table below summarizes the results for different sweep angles. In particular, it is noteworthy that the Strouhal number increases with the sweep angle up to a value of 0.27 which is close to the value of 0.26 found previously experimentally on the 3D wing for a sweep angle of 30°. The convection velocity is also the same (0.2U∞). These simulations show that there is a con-tinuous increase of the buffet frequency from the 2D buffet at low frequency to the 3D buffet at higher frequency. The 2D and 3D buffet phenomenon are not two different phenomena as it was thought previously, it is the same with a frequency which increases the sweep angle.

sweep angle (°) St Uc/U∞ λ/c

10 0.046 0.09 1.9

15 0.068 0.1 1.5

20 0.16 0.15 0.9

25 0.27 0.2 0.7

Unfortunately, the same computation with periodic boundary conditions in the spanwise direction gives the 2D buffet phenomenon with no spanwise oscillation of shock but rather a syn-chronised motion for all spanwise sections like a 2D simulation. In the previous computation in Fig. V.50, the presence of a lambda shock on the right hand side allows unsymmetrising the solu-tion but here with periodic boundary condisolu-tions, this is not the case. The use of the Selective Fre-quency Damping (SFD) method [48] in the RANS computations allows the apparition of “buffet cells” instead of a solution with no spanwise oscillation (see Fig. V.51). In fact, during the con-vergence, with SFD, there is a bifurcation to a 3D solution instead of a 2D one. This is not the objective of the SFD which is to help converging unsteady simulations by adding a forcing term in the Navier-Stokes equations towards an averaged field. Once these buffet cells appeared, it is possible to start the URANS simulation without SFD and these buffet cells will remain present and be convected in the swept case in the left direction. In the unswept case (Fig. V.51 (top)), there is no spanwise component of the velocity to force the convection of the buffet cells and so there is a random movement of the buffet cells and a streamwise oscillation of the shock. In fact, the 2D URANS simulations of buffet in the literature are just a slice of the real 3D phenomenon with buffet cells.

Fig. V.51: Instantaneous wall pressure on the RA16SC1 wing (top: no sweep, bottom: 30° swept).

Coming back to the original problem of the buffet on a 3D wing with taper ratio and twist, a URANS simulation without SFD still converges to a steady state. A RANS simulation with SFD allows the appearance of the buffet cells. Then, a URANS simulation is started from this solution and it is now possible to compute the 3D buffet phenomenon in URANS. Fig. V.52 shows the instantaneous skin friction distribution. The presence of two buffet cells is clearly visible. A mov-ie shows the convection of these buffet cells towards the wing tip. The Strouhal number is in very good agreement with the wind tunnel tests (St = 0.26) as well as the Cp and Cprms distributions.

Fig. V.52: Instantaneous skin friction coefficient on the CAT3D model (M = 0.82, α = 4.25°, Rec = 2.83×10⁶).

In fact, what has been called “buffet cells” by Iovnovich & Raveh [33] is a more well-known phenomenon in low speed conditions as “stall cells”. Fig. V.53 shows an oil flow visuali-sation from Schewe [50] at α = 12°, U∞ = 75 m.s^-1. This figure is very similar to Fig. V.51 (top) which is in buffet conditions. The PhD thesis of E. Paladini (in progress) will study the prediction of the spanwise wavelength of these stall cells thanks to a global stability analysis. Once this wavelength λ/c computed and since the convection velocity Uc/U∞ of these cells can be linked to the spanwise component of the velocity upstream of the shock, it is possible to estimate the 3D buffet Strouhal number as:

𝑆𝑆𝑑𝑑 =𝑓𝑓. 𝑐𝑐 𝑈𝑈∞ = 𝑈𝑈_𝑐𝑐

𝑈𝑈∞.𝑐𝑐 𝜆𝜆

Fig. V.53: Oil flow visualisation (from Schewe [50]).

The appearance of stall cells has been described by Gross et al. [30] and Spalart [54]. A necessary condition for their appearance is to have a negative slope in the lift curve (see Fig.

V.54). If in one section of the wing a separation appears, its circulation will decrease, two coun-ter-rotating vortices will be shed at the trailing edge which will induce a lower angle of attack on the neighbouring sections. Now, if the lift curve slope is positive, this decrease of the effective angle of attack will lead to a reduction of lift on this neighbouring sections and thus homogenise the lift repartition in span. On the contrary, if the slope is negative, a decrease of the effective angle of attack will lead to an increase of lift on the neighbouring sections and thus an amplifica-tion of the spanwise lift variaamplifica-tions and the appearance of stall cells.

From a numerical point of view, this phenomenon leads to the appearance of multiple converged solutions in 3D RANS computations [36] depending on the numerical method and ini-tial conditions and so it does not allow predicting precisely the lift in this region around stall.

Fig. V.54: Sketch of a lift curve.

5.1.2. 2D laminar buffet

Since laminarity is seen as one way to decrease aircraft drag in cruise, it is important to look on how evolves the buffet phenomenon when replacing the turbulent shock/boundary layer inter-action by a laminar one. To answer the question “is the laminar buffet less or more intense than the turbulent buffet?”, a European project called BUTERFLI has been launched in 2013. In par-ticular, wind tunnel tests on a laminar airfoil have been performed in ONERA's S3Ch wind tun-nel. The experimental setup allowed varying the Mach number, the angle of attack and the state of the boundary layer upstream of the shock (laminar or turbulent) depending on the presence of artificial tripping. In the turbulent case at M = 0.735, α = 4°, Rec = 3×10⁶, Brion et al. [6] reported a low frequency peak in the wall pressure spectra at about 75 Hz (St = 0.07) in agreement with Jacquin et al. [34]. In the laminar case (same flow conditions without tripping), a peak at a much higher frequency (about 1130 Hz or St = 1) is observed. So, the question is: what can explain this order of magnitude of difference between the turbulent and the laminar buffet frequency? To an-swer this question, a Large-Eddy Simulation of laminar buffet has been performed with the new ONERA FastS solver. The airfoil geometry is the OALT25 supercritical laminar airfoil. For the present simulation, one test case in laminar buffet conditions from the S3Ch experimental data-base has been selected: M = 0.735, α = 4°, Rec = 2.8×10⁶. On the lower side, the flow is tripped at x/c = 7% like in the wind tunnel tests. The mesh size is 400×10⁶ cells.

Fig. V.55 shows a Q-criterion isosurface (in red) with isosurface M = 1 (in grey). On the upper side, since the boundary layer is laminar up to the shock foot, there are no turbulent struc-tures. Then, the laminar/turbulent transition occurs in the separation bubble at the shock foot. On the lower side, since the boundary layer is tripped, turbulent structures are visible.

Fig. V.55: Q-criterion isosurface with isosurface M = 1.

Fig. V.56 (left) shows a comparison of the time-averaged wall pressure distribution with the wind tunnel test result. There is a very good agreement. The small concavity between x/c = 43% and 57% is characteristic of the laminar separation bubble at the shock foot. The sepa-ration of the laminar boundary layer occurs at the middle of this concavity and the reattachment is downstream of the shock. The LES simulation predicts very well the length of this separation bubble as well as the shock location. Fig. V.56 (right) shows a comparison of the PSDs with the wind tunnel test for a sensor located at x/c = 52% on the suction side. The laminar buffet frequen-cy is very well predicted (1200Hz in the LES (St = 1.12) compared to 1140Hz (St = 1.06) in the experiment). At x/c = 52%, which corresponds to the separation point of the bubble, a bump cen-tred around 36 kHz (St = 34) is also visible and is due to the Kelvin-Helmholtz instability in the mixing layer of the separation bubble.

Fig. V.56: Time-averaged wall pressure distribution (left) and PSD at x/c = 52% (right).

To understand from where is coming this high frequency peak in the pressure spectra, phase-averaged flowfields have been analysed. Fig. V.57 (left) shows one of these fields for a phase of 90°. A separation bubble is visible at the shock foot as well as close to the trailing edge.

A movie shows that the separation bubble at the shock foot regularly shed vortices downstream.

By a cross-spectral analysis of wall sensors and by looking at the phase evolution with x at the buffet frequency, it is possible to compute the convection velocity of these vortices which is equal to Uc/U∞ = 0.4. By using these convection velocities and the distance between the shock and the trailing edge (0.4c), it is possible to retrieve the laminar buffet frequency (St = Uc/U∞.c/λ

= 0.4/0.4 = 1). The present LES has allowed explaining the laminar buffet phenomenon as a sepa-ration bubble breathing phenomenon with vortices shed downstream.

When looking at the wall pressure spectra, there is a well marked peak (and not a large bump) like in the turbulent buffet phenomenon for which it is already known that there is a global instability (Crouch et al. [19]). So, the fact that the laminar buffet is characterised by a peak rather than a bump tends to support the hypothesis that there is also a global instability of the separation bubble at the shock foot but this has to be confirmed.

Fig. V.57: Phase-averaged Mach number field (left) and phase of the cross-spectra for a reference sensor at x/c = 48% (right).

5.1.3. 3D laminar buffet

In the previous section, the 2D laminar buffet phenomenon has been described but the real buffet phenomenon has to be studied on a 3D wing. The question which arises is: since for the turbulent buffet phenomenon the frequency increases with the sweep angle, does this frequency increases too for the laminar buffet phenomenon? In the framework of JTI Clean Sky European project, Dassault Aviation has tested a low-sweep business jet model in the ETW wind tunnel.

Since the wing sweep has been decreased as well as the Mach number, the laminar/turbulent tran-sition is not triggered by the crossflow instability and the flow remains laminar up to the shock foot like in the previous section.

Fig. V.58: Dassault Aviation LSBJ model in the ETW wind tunnel.

First, starting with the turbulent regime (tripped case), Fig. V.59 shows PSDs at y/b = 65.5%. A high frequency bump is visible around 300 Hz which corresponds to a Strouhal number of 0.26 like for the CAT3D model in Fig. V.49 but for lower sweep angle (20° instead of 30°).

Fig. V.59: PSDs at y/b = 65.5% for M = 0.78, α = 6°, Rec = 55×10⁶ (turbulent regime).

The same analysis based on cross-spectra as for Fig. V.49 has been done on the present model (Fig. V.60) and the convection velocities and propagation direction are very similar to the CAT3D model. There is just the spanwise wavelength which is smaller:

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Fig. V.60: Convection velocities of the buffet phenomenon for M = 0.78, α = 6°, Rec = 55×10⁶ (turbulent regime).

Now, in the laminar case (without artificial tripping), Fig. V.61 shows the PSDs for the spanwise section y/b = 65.5%. A bump at a higher frequency (2 kHz instead of 300 Hz in Fig.

V.59) is visible which corresponds to a Strouhal number between 1.3 and 2.1 depending on the spanwise section. So, like for the turbulent buffet phenomenon, the laminar buffet frequency seems to increase with the sweep angle from St = 1 in 2D to 1.3–2.1 in 3D.

Fig. V.61: PSDs at y/b = 65.5% for M = 0.78, α = 5°, Rec = 55×10⁶ (laminar regime).

The same analysis as for Fig. V.60 has been done in the laminar case (see Fig. V.62). The 3D laminar buffet convection velocity is slightly higher (0.24U∞) than in the turbulent regime (0.2U∞). The propagation direction is also more directed towards the wing tip. Since the Strouhal number is higher than in the turbulent case, the spanwise wavelength is smaller (0.15c instead of

0.6c). In summary, there is also a spanwise convection of buffet cells towards the wing tip in the 3D laminar buffet like in the turbulent one but the wavelength is much smaller because the fre-quency is higher:

Fig. V.62: Convection velocities of the buffet phenomenon for M = 0.78, α = 5°, Rec = 55×10⁶ (laminar regime).