Interactions between shock waves and boundary layers (SWBLI) are encountered in many industrial applications dealing with supersonic flows (aircraft design, rocket nozzles...). If the shock is strong enough, those interactions may cause the boundary layer separation yielding dynamic loads, increased heat fluxes and pressure fluctuations. Even if the physics of SWBLI is not fully understood, it is well known that the separation zone and the reflected shock are subjected to a low-frequency streamwise motion spreading over several tenth of the boundary layer thickness. The origin of this motion is, however, not completely elucidated but several studies has linked it to the shedding of vortices in the mixing layer downstream of the separation [1-2] and a simple model has even been developed [3] to explain this unsteadiness. In addition, recent experiments conducted in the IUSTI supersonic wind tunnel have shown that the recirculation region is highly three-dimensional [4]. These 3D aspects have been related to two contrarotative vortices developing downstream of the shock. Several numerical simulations using LES have been performed for the same geometry, most of them with periodic conditions in the spanwise direction. While the results are in rather good agreement with the experimental data for a weak shock wave, these simulations have failed to capture the three-dimensional aspect of the separation for a 9.5 degrees shock generator angle, demonstrating that the lateral walls of the wind tunnel are, at least partially, responsible for the 3D modulations of the recirculating region [5]. The number of points being prohibitively high in the case of LES (or DNS) of the full-span wind tunnel with lateral walls, alternative solutions have been tested. For instance, Garnier [6] has computed the entire domain with SDES, an hybrid RANS/LES method and has found that effective section of the wind tunnel is reduced because of the lateral walls, leading to a strengthened interaction. This observation explains the discrepancy between the separation regions obtained experimentally and numerically. This study has also shown that highly unsteady secondary flows develop around the corner of the wind tunnel. No obvious connection between these oscillations and the motion of the separation region has been found.
In this study, the interaction between an oblique shock wave and a laminar boundary layer has been studied by DNS. The simulations are performed with an in-house parallel (MPI) Finite-Volume based DNS/LES solver developed at LIMSI-CNRS [7]. The convective fluxes are discretized by Monotonicity-Preserving shock-capturing scheme, based on a Lax-Wendroff approach through a 7th order accurate coupled space and time approximation [8]. A second order centered scheme is used for the diffusive fluxes.
The shock conditions are identical to those of Pirozzoli and Grasso [9]. The inlet boundary layer conditions are, however, different from the latter study since a laminar boundary layer is considered here. The aim was to investigate if the streamwise motion could somehow be related to the turbulent aspect of the incoming boundary layer. The freestream Mach number is set to M=2.25 and the shock, created using the Rankine-Hugoniot relations, forms an angle of 33.1 degrees with respect to the horizontal plane. A 4th order polynomial approximation of the Blasius profile is imposed on the inlet plane in order to realistically simulate a laminar boundary layer.
2D simulations have first been performed. The extent of the computational domain, normalized with the inlet boundary layer thickness, is D=50x20 and the domain is discretised using M=1000x252 cells. A 3% geometrical stretching is applied in the wall-normal direction. Figure 1 represents an instantaneous numerical schlieren field. It is obvious that the shock is strong enough to make the boundary layer separate. In addition, the shock system, including the incident, reflected and reattachment shocks, as well as the expansion fan, clearly appears.
Fig 1: Instantaneous contours of the numerical schlieren from a 2D DNS
The isosurfaces of the Q-criterion coloured by the streamwise velocity, obtained by a preliminary 3D simulation is shown in Figure 2. The shock system (in translucent grey) is represented using isosurfaces of the divergence of the velocity. Even if the incoming boundary layer is laminar, therefore two-dimensional, the interaction with the shock and the subsequent separation create a fully three-dimensional flow. Just donwstream of the separation, small 3D structures are created. But, the further it goes downstream the more the flow exhibits a strong 3D behaviour. This is confirmed by the presence of large hairpin vortices around the interaction.
Fig 2: Isosurfaces of the Q-criterion coloured by the streamwise velocity and the shock system (grey)
The final paper will discuss the influence of the incoming boundary layer on the streamwise oscillations in the light of these new numerical simulations and will compare the incoming laminar boundary layer results with some initial turbulent boundary results either from numerical simulations [2-9] or experiments [5].
[1] S. Priebe and M. P. Martín (2012) Low-frequency unsteadiness in shock wave–turbulent boundary layer interaction. Journal of Fluid Mechanics Vol. 699:1--49
[2] G. Aubard, X. Gloerfelt, and J.-C. Robinet (2013) Large-Eddy Simulation of Broadband Unsteadiness in a Shock/Boundary-Layer Interaction. AIAA Journal Vol. 51, No. 10
[3] S. Piponniau, J.-P. Dussauge, J.-F. Debiève and P. Dupont (2009) A simple model for low-frequency unsteadiness in shock-induced separation. Journal of Fluid Mechanics Vol. 629
[4] P. Dupont, C. Haddad, J.-P. Ardissone and J.-F. Debiève (2005) Space and time organisation of a shock wave/turbulent boundary layer interaction. Aerospace Science and Technology Vol. 9
[5] E. De Martel, E. Garnier and P. Sagaut (2007) Large Eddy Simulation of Impinging Shock Wave/Turbulent Boundary Layer Interaction at M = 2.3. IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries Vol. 14
[6] E. Garnier (2009) Stimulated Detached Eddy Simulation of three-dimensional shock/boundary layer interaction. Shock Waves Vol. 19
[7] C. Tenaud, Y. Fraigneau and V. Daru (2011) Numerical simulation of the turbulent separation reattachment flow around a thick flat plate. Journal of Physics: Conference Series Vol. 318(4)
[8] V. Daru and C. Tenaud (2004) High order one-step monotonicity preserving schemes for unsteady flow calculations. Journal of Computational Physics Vol. 193
[9] S. Pirozzoli and F. Grasso (2006) Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M=2.25. Physics of Fluids Vol. 18
