Direct numerical simulation and stability analysis of transonic flow around airfoils at moderate Reynolds numbers

The performance of turbomachinery components and the safe flight envelope of next-generation aircraft is often limited by complex transonic flow phenomena. Since the first flights close to sonic speeds, experiments have been carried out to explore the origin of flutter phenomena, supplemented with simulations of the Reynolds-averaged Navier-Stokes equations that are dependent on turbulence models. To date, direct numerical studies of low-frequency phenomena have been limited to low Reynolds numbers. The present work explores the transonic flow regime around Dassault Aviation’s V2C laminar-flow profile at moderate Reynolds numbers, and also analyses boundary-layer instabilities on a high-pressure turbine vane. Direct numerical simulations of an un-swept wing section are carried out at Mach 0.7 and an angle of attack of 4◦ using the in-house code SBLI, which is a well-validated high-order finite-difference flow solver. While the flow at Reynolds numbers of Re = 200,000 is purely subsonic, a significant supersonic region is observed for Re ≥ 500,000. Whereas experimental investigations of the same airfoil at higher Reynolds numbers showed single shock waves, the present reference case at Re = 500,000 exhibits continuously upstream-propagating shock waves. Besides laminar/turbulent boundary-layer transition and acoustic phenomena, a low-frequency phenomenon, known as transonic buffet, is studied. In addition to spectral analyses of the flow, linear stability analysis and a dynamic mode decomposition method are used to study flow phenomena apparent at different frequency ranges between Strouhal numbers of St ≈ 20 (Kelvin-Helmholtz instabilities) and St = 0.12 (transonic buffet). Resolution of small-scale structures is established by a grid convergence study and also employing spectral error indicators to assess the grid quality. The flow characteristics are also confirmed by a simulation with a five times wider spanwise domain size (25% of the chord length) comprising more than five billion grid points. A key observation of this work is the clear distinction between an acoustic mechanism associated with the shock-wave motion (St ≈ 0.5) and a quasi-periodic mode at significantly lower frequencies causing strong fluctuations in the aerodynamic lift (St ≈ 0.12). The Strouhal number of the low-frequency phenomenon agrees well with the buffet frequency in experiments at higher Reynolds numbers