Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis

Abstract

A transonic interaction between a shock wave and a turbulent boundary layer is experimentally and theoretically investigated. The configuration is a transonic channel flow over a bump, where a shock wave causes the separation of the boundary layer in the form of a recirculating bubble downstream of the shock foot. Different experimental techniques allow for the identification of the main unsteadiness features. As recognised in similar shock-wave/boundary-layer interactions, the flow field exhibits two distinct characteristic frequencies, whose origins are still controversial: a low-frequency motion which primarily affects the shock wave; and medium-frequency perturbations localised in the shear layer. A Fourier analysis of a series of Schlieren snapshots is performed to precisely characterise the structure of the perturbations at low-and medium-frequencies. Then, the Reynolds-averaged Navier-Stokes (RANS) equations closed with a Spalart-Allmaras turbulence model are solved to obtain a mean flow, which favourably compares with the experimental results. A global stability analysis based on the linearization of the full RANS equations is then performed. The eigenvalues of the Jacobian operator are all damped, indicating that the interaction dynamic cannot be explained by the existence of unstable global modes. The input/output behaviour of the flow is then analysed by performing a singular-value decomposition of the Resolvent operator; pseudo-resonances of the flow may be identified and optimal forcings/responses determined as a function of frequency. It is found that the flow strongly amplifies both medium-frequency perturbations, generating fluctuations in the mixing layer, and low-frequency perturbations, affecting the shock wave. The structure of the optimal perturbations and the preferred frequencies agree with the experimental observations.

Instabilities in oblique shock wave/laminar boundary-layer interactions

“…Finally, one may also remark that the spectrum obtained in the laminar regime differs from those given by Pirozzoli et al (2010) and Sartor et al (2015) associated with supersonic and transonic turbulent shock wave/boundary-layer interactions, respectively. For instance, the least damped mode obtained by Pirozzoli et al (2010) is non-oscillatory and concentrated in the recirculation zone.…”

“…However, such a mode is not observed in the present computation. Furthermore, the least damped global modes associated with transonic channel flow over a bump computed recently by Sartor et al (2015) are acoustic resonance modes and shock-wave low-frequency modes. Sartor et al (2015) conclude that the major part of the stable modes are probably not linked to the unsteady dynamics observed in turbulent SWBLI.…”

“…Furthermore, the least damped global modes associated with transonic channel flow over a bump computed recently by Sartor et al (2015) are acoustic resonance modes and shock-wave low-frequency modes. Sartor et al (2015) conclude that the major part of the stable modes are probably not linked to the unsteady dynamics observed in turbulent SWBLI. In our case, from observations of the impulse response in the linear regime, it seems clear that the space-time dynamics of the laminar OSWBLI is closely associated with global modes.…”

“…However, based on the assumption that coherent structures are uncorrelated to the disorganized ones, recent global stability analyses carried out on a mean turbulent flow (Sartor et al 2013;Sartor 2014) suggest that low-frequency unsteadiness in transonic turbulent shock wave/boundary-layer interaction cannot be explained by the presence of unstable global modes, and is mainly driven by a linear pseudo-resonance mechanism. Hence, it should be an interesting prospect to perform a similar analysis for the turbulent OSWBLI to give a more definitive statement.…”

Effects of Compressibility and Shock-Wave Interactions on Turbulent Shear Flows

“…Another useful way of looking at the problem is provided by the recent work of Sartor et al (2015) , who used a resolvent mode analysis to look at the global response of a transonic bump flow problem (the so-called Delery bump). This linearised analysis considers forcing in the problem formulation and the resulting singular value decomposition provides information about the flow response to forcing and about the forcing that gives the largest gain.…”