Instability and low-frequency unsteadiness in a shock-induced laminar separation bubble

Abstract

Three-dimensional direct numerical simulations (DNS) of a shock-induced laminar separation bubble are carried out to investigate the flow instability and origin of any low-frequency unsteadiness. A laminar boundary layer interacting with an oblique shock wave at $M=1.5$ is forced at the inlet with a pair of monochromatic oblique unstable modes, selected according to local linear stability theory (LST) performed within the separation bubble. Linear stability analysis is applied to cases with marginal and large separation, and compared to DNS. While the parabolized stability equations approach accurately reproduces the growth of unstable modes, LST performs less well for strong interactions. When the modes predicted by LST are used to force the separated boundary layer, transition to deterministic turbulence occurs near the reattachment point via an oblique-mode breakdown. Despite the clean upstream condition, broadband low-frequency unsteadiness is found near the separation point with a peak at a Strouhal number of $0.04$, based on the separation bubble length. The appearance of the low-frequency unsteadiness is found to be due to the breakdown of the deterministic turbulence, filling up the spectrum and leading to broadband disturbances that travel upstream in the subsonic region of the boundary layer, with a strong response near the separation point. The existence of the unsteadiness is supported by sensitivity studies on grid resolution and domain size that also identify the region of deterministic breakdown as the source of white noise disturbances. The present contribution confirms the presence of low-frequency response for laminar flows, similarly to that found in fully turbulent interactions.

Analysis of the two-dimensional dynamics of a Mach 1.6 shock wave/transitional boundary layer interaction using a RANS based resolvent approach

“…In this study the medium frequency dynamics is linked to the shedding of vortical structures at the end of the mixing layer. The high frequency dynamics S t ≈ 1-2 is also observed by Sansica et al (2016) who suggest that it is due to the transition process close to the reattachment point.…”

“…Sansica et al (2014) carried out a DNS analysis of a two-dimensional (2-D) interaction at M = 1.5 forced either upstream of the separation, inside the circulation bubble or both, and found this low frequency unsteadiness to be particularly strong near the separation point and present even in the absence of an upstream forcing, indicating that incoming perturbations are not necessary to promote the low frequency unsteadiness in this type of interaction. Sansica, Sandham & Hu (2016) further showed that the flow in the interaction is convectively unstable due to oblique mode growth (on this subject, see also the work of Chang & Malik (1994)) and that the bubble acts like a filter-amplifier of low frequency disturbances at approximately S t ∼ 0.1. A broad band spectrum of low frequency fluctuations is localized in the vicinity of the separation point, around a Strouhal number of S t = 0.04, similarly to the turbulent case of Pirozzoli & Grasso (2006) mentioned above.…”

“…As mentioned in the introduction, Larchevêque (2016) also found three frequencies, the first one at S t = 0.04 in the vicinity of the separation, the second for 0.1 < S t < 0.3 inside the interaction and the last one for S t > 1 near the impingement point. Also Sansica et al (2016) could identify a first frequency at S t = 0.04, a second one for S t 0.1 and a third one for S t = 1 − 2. In the rest of the text, these three modes will consistently be referred to as the low, medium and high frequency modes.…”

“…The question of higher frequency unsteadiness has been investigated by Larchevêque (2016), Agostini et al (2012Agostini et al ( , 2015 and Sansica et al (2016). Based on the previously mentioned LES of a transitional configuration, Larchevêque (2016) finds that the middle of the interaction features large fluctuations in an intermediate range of https://doi.org/10.1017/jfm.2018.932 frequency (0.1 < S t < 0.3) and that the flow past the impingement point features energy at high frequencies (S t > 1).…”

Dynamic linear response of a shock/turbulent-boundary-layer interaction using constrained perturbations

“…This model was later verified experimentally by Poggie & Smits (2001) and more recently derived analytically with supporting assumptions validated through LES by Touber & Sandham (2011). Sansica, Sandham & Hu (2016) also reproduced many of the key features of the oscillation by suitably forcing a laminar SBLI, showing that broadband low-frequency upstream content is not necessary to elicit a low-frequency response at the separation point. Rather the response can result from 'forcing' via transition downstream, which is then low-pass filtered by the interaction.…”