Delft University of Technology Transition mechanism behind a backward-facing step in a supersonic flow

The path of laminar-to-turbulent transition behind a backward-facing step (BFS) in the supersonic regime at Ma = 1.7 and Reδ0 = 13718 is investigated using a very well-resolved large eddy simulation (LES). Five distinct stages are identified in the transition process by the visualisation of instantaneous flow. The transition is initiated by a Kelvin-Helmholtz (K-H) instability of the separated shear layer, followed by secondary modal instabilities of the distorted K-H vortices, leading to Λ-shaped vortices, hair-pin vortices and finally to a fully turbulent state around the reattachment location. Spectral analysis and proper orthogonal decomposition (POD) reveal that the low-frequency breathing dynamics also plays a major role in the transition process. INTRODUCTION The flow dynamics over a BFS has attracted extensive attention in the past decades, as it is not only an appealing prototype for studying the separation, recirculation and reattachment behavior in the view of its geometrical simplicity, but also for investigating the transition from laminar to turbulent flow without artificial disturbances (Kostas et al., 2002; Lanzerstorfer & Kuhlmann, 2012). There is a considerable amount of experimental and numerical work about the development of flow instability in the subsonic flow (Marquet et al., 2008; Duncan Jr, 2014). The conclusion drawn is that several distinct mechanisms, like centrifugal, lift-up effects or K-H instability, play a major role in the transition behind the step (Theofilis, 2011). In the supersonic regime, however, there are additional mechanisms related to compressibility and the occurrence of compression waves at flow reattachment (Tihon et al., 2001; Sriram & Chakraborty, 2011). Thereby, it is reasonable to conjecture that a different mechanism may contribute to the transition process in the supersonic case. The main objective of the current investigation is to characterize the transition process behind a BFS by scrutinizing the evolution of instantaneous vortical structures. NUMERICAL SETUP The setup for the current study is an open BFS (no upper wall) with supersonic laminar inlet boundary conditions at Ma = 1.7 and Reδ0 = 13718 based on the inlet boundary layer thickness δ0 and free-stream viscosity. The size of the computational domain is (Lx, Ly, Lz) = ( [−40,70], [−3,30], [−2.5,2.5] ) δ0 with a step height of h= 3δ0. The main flow parameters are summarized in Table 1. At the domain inlet, a clean laminar boundary layer profile is imposed. The step and wall are modeled as non-slip adiabatic surfaces. All flow variables are extrapolated at the outlet of the domain. On the top of the domain, nonreflecting boundary conditions based on Riemann invariants are used. Periodic boundary conditions are enforced in the spanwise direction. We employ the implicit large eddy simulation (ILES) method of Hickel et al. (2014) for solving the compressible Navier-Stokes equations, which has been successfully applied to various supersonic cases, such as shock wave/boundary layer interactions (SWBLI) on a compression ramp (Grilli et al., 2012) and a flat plate (Pasquariello et al., 2017). More details about the numerical method can be found in Hickel et al. (2006, 2014). RESULTS AND DISCUSSION Mean flow organization The main flow features are visualized by the timeand spanwise-averaged flow field in Figure 1. The incoming flow experiences a centered Prandtl-Meyer expansion and separates at the step corner. Then the free shear layer develops towards the downstream wall and finally impinges on the wall surface. Compression waves are generated at the reattachment location and then coalesce into a reattachment shock (white solid line). The low-speed recirculating flow forms a separation bubble underneath the dividing line (here defined for convenience as the isoline of u = 0 indicated by the black dashed line), while the high-speed flow proceeds downstream by overcoming the reattachment pressure rise. The mean reattachment length is about Lr = 10. 9δ0 (3.6h), which is consistent with the previous results at similar con-