Objetivos específicos

The results obtained with the DDES method differ substantially from those of the URANS simulations using the same baseline model. Remarkable differences exist in the shock wave/boundary layer interaction region, in the zone of intermittent separation over the rear part of the airfoil and particularly near the trailing edge, where the DDES overes- timates the flow properties fluctuation levels. More specifically, the statistical pressure distributions in figure 6.9 show that the DDES tends to depart from the experimental data as one approaches the trailing edge, whereas the URANS results maintain a good

(a) x/c= 0.28 (b) x/c= 0.45 (c) x/c= 0.60 (d) x/c= 0.75

Figure 6.16: Wall profiles with the shock at its most upstream location (t∗

= 3.56).

(a) x/c= 0.28 (b) x/c= 0.45 (c) x/c= 0.60 (d) x/c= 0.75

Figure 6.17: Wall profiles at minimum lift (t∗

behavior. Nevertheless, the analysis of the RANS and LES modes distribution during buffet indicates that the DDES is effective in preventing MSD even when the boundary layer gets very thick before the onset of separation.

In the sequence of flow snapshots presented in figure 6.3, it can be noted that resolved structures in the shear layer and in the separated region arise only over the rear part of the airfoil. This phenomenon is seen more clearly in figure 6.18, which illustrates isosurfaces of the Q-criterion, Q = 1

2 |Ω| 2

− |S_|2

, for two instants during separation. Despite the shock-induced separation occurring on the first half of the upper surface, a long distance is required for the formation of small eddies in the flowfield. Such effect may have been intensified in the DDES as the RANS-mode layer around the airfoil remains relatively thick during separation, which can be noted by comparing the fd-function distributions of figures 6.15 and 6.16. Hence, one possible reason for the results obtained is that, when the flow separates, the turbulent content in LES regions is not sufficiently rich due to a severe gray area. Such delay in the generation of resolved turbulence would result in a more unstable flowfield and appears to be at least partially responsible for the generation of the strong and regular alternate vortex structures when the flow separates. These are intimately related to the high flow unsteadiness levels observed in figures 6.9(b) and 6.13 near the trailing edge.

(a) t∗

= 3.56 (b)t∗

= 4.45

Figure 6.18: Instantaneous Q-criterion isosurfaces during separation (Q(c/U)2 _{= 100).} In hybrid RANS-LES methods, the ‘gray area’ issue can become especially critical in flows where separation occurs over smooth geometries instead of being triggered by some geometry feature such as the edge of a backward-facing step or in the case of a spoiler, for example. Because of the intermittent separation in the present transonic buffet case, the gray area issue may have been more serious as the RANS-LES interface evolves over time. Therefore, a given point in the flowfield can be treated by both RANS and LES modes during buffet depending on the flow phase. To some extent, another possible contribution to the lack of small resolved structures may have been the dissipation resulting from the use of a third-order upwind scheme. The global grid refinement may also have influenced the results as it is not clear whether the cutoff length scale remains always within the inertial subrange of the turbulence spectrum. The three-dimensional flow patterns re- ported in the experiments of Jacquin et al. [39] may also be mentioned. However, they

do not seem to explain the DDES results since the URANS predictions near the trailing edge are in good agreement with the wind tunnel measurements.

Some results of the Zonal Detached-Eddy Simulation (ZDES) of the same test case per- formed by Deck [4] are compared with the present DDES in figure 6.19 and support the previous considerations. Both approaches guarantee a proper switching between RANS and LES during buffet. In the case of that particular ZDES, MSD was avoided by ex- plicitly imposing RANS-mode (using the original SA model) in regions where the grid spacing in the direction normal to the wall was smaller than that in the spanwise direc- tion. This ensured that the whole shock wave/boundary layer interaction was treated in RANS. Besides, the LES mode adopted the standard subgrid length-scale formula- tion of LES (i.e., ∆ = (∆i∆j∆k)1/3) and suppressed all near-wall functions of the SA model. Despite the many differences in the modeling approaches, numerical methods, time steps and grids, the mean pressure distributions in figure 6.19(a) present the same basic characteristics, showing large separation regions compared to the experiments and low trailing-edge pressures. The present DDES provides a better prediction of both the shock-wave motion range and the pressure recovery region. In fact, the method is closer to the experimental data over the whole mean pressure distribution. Nevertheless, be- cause of the aforementioned differences in the simulation conditions, this result should not be regarded as the affirmation of the superiority of one approach over the other. Fig- ure 6.19(b) reveals that the maximum pressure fluctuation levels predicted by the DDES and ZDES are very similar despite the somewhat different shock-wave motion ranges and mean locations. In the trailing edge region, the fluctuations obtained with the DDES are more intense than those with the ZDES. This result might be related to the fact that, in that ZDES, the hybrid model formulation was not used around the lower surface, which was completely treated in RANS mode. Therefore, the overall trailing-edge unsteadiness is attenuated, potentially preventing the development of alternate vortex shedding.

(a) Mean pressure coefficient (b) RMS pressure on the upper surface

Chapter 7

Numerical Study of a Laminar

Transonic Airfoil

Contents

7.1 The V2C profile . . . 122 7.2 Grid generation . . . 123 7.3 Preliminary investigation . . . 124

7.3.1 Freestream Mach number 0.70 . . . 125 7.3.2 Freestream Mach number 0.75 . . . 126 7.3.3 Code-dependence study . . . 128

7.4 Effect of the transition location . . . 129

7.4.1 Pre-buffet condition . . . 130 7.4.2 Fully-developed unsteady regime . . . 132

7.5 Scale-resolving simulation . . . 134

7.5.1 Introduction . . . 134 7.5.2 Flowfield dynamics . . . 135 7.5.3 Statistical flow properties . . . 138

7.1

The V2C profile

To achieve faster cruising speeds and reduce emissions, new generation aircraft will need to have high performance, which will require the association of advanced aerodynamic design with more efficient propulsion systems and materials. Regarding the contribution of aerodynamics, performance can be improved by means of efficient lift generation and by maintaining drag as low as possible. For the latter, an obvious way to minimize par- asite drag is to reduce its skin-friction component by obtaining a maximum extent of laminar flow. However, as the freestream Mach number increases in the transonic range, the generation and progressive strengthen of shock waves may lead to detrimental effects as discussed in chapter 2. In such scenario, laminar boundary layers are less resistant to shock-induced separation than turbulent ones and, for this reason, it is desirable that transition occurs upstream the shock to avoid laminar separation. This can be achieved through proper airfoil design for natural transition or by means of some boundary layer tripping method. Therefore, it is important to know how shock wave/boundary layer interaction properties such as the shock strength, the separation position and flow un- steadiness (e.g., buffet) are affected by the transition location.

A particular question addressed in the TFAST Project is how far from the shock must transition occur for the interaction to exhibit a purely turbulent behavior. Concerning the application to transonic wings, the project has selected a two-dimensional ‘laminar’ airfoil developed by Dassault Aviation as one of its test cases. The profile has been specifically designed in such a way that laminar flow is supposed to be maintained from the leading edge to the shock wave on the upper surface up to buffet onset. The technique employed was based on the eN _{method for transition prediction (see, for instance, Ref.} [103]) and the airfoil surface was generated in such a way that theN-factor remains small for low-to-moderate turbulence intensity levels, thus providing laminar flow. The design was validated numerically by Dassault by means of RANS computations for various angles of attack at freestream Mach numbers of 0.70 and 0.75, yielding chord-based Reynolds numbers of approximately 3.245_×106and 3.378_×106respectively. The study was performed using a compressible Navier-Stokes code (see Ref. [104] for details) adopting a two-layer

k-εmodel, with the transition location being determined from the fully-turbulent flowfield using a three-dimensional compressible boundary-layer code (see Ref. [105]) by means of the N-factor amplification with a parabola method (see Ref. [106]). The analysis of the flowfield around the airfoil indicated that the boundary layer is supposed to remain laminar up to the shock wave for angles of attack between 1◦

and 7◦

. At Mach 0.70, the flow separated between α = 6◦

and 7◦

. The amplification factor N was shown to be smaller than 3 up to the shock wave, thus guaranteeing laminar flow. At Mach 0.75, the value ofN remained smaller than 2 up toα= 7◦

. The final profile was named ‘V2C’ and is sketched in figure 7.1 (solid line), which compares it with the OAT15A supercritical airfoil (dashed line).

Figure 7.1: Geometries of the V2C (solid line) and OAT15A (dashed line) airfoils. maximum thickness is located around the mid-chord, yielding a long region of favorable pressure gradient. As in supercritical airfoils, a slight camber exists in the rear part of the V2C to produce additional lift.