Robust reactive control of 3-D instabilities in a supersonic boundary layer

We consider a three-dimensional (3-D) supersonic boundary layer at upper M Subscript normal infinity Baseline equals 4.5 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>4.5</mml:mn> </mml:math> $M_{\infty }=4.5$ and focus on obtaining a robust law by means of localized sensors and actuators to control finite-amplitude 3-D instabilities and delay transition to turbulence. This study extends the methodology developed for spanwise invariant flow in Nibourel et al. (2023 J. Fluid Mech . 954 , A20; Nibourel et al. 2024 AIAA J . 62 (3), 896–914), to the 3-D case: the control law is still built using linear tools, but the multiple-input–multiple-output centralized problem is transformed into several single-input–single-output decentralized problems in the transverse wavenumber space to simplify model identification and controller synthesis. Thanks to the feedback set-up (estimation sensors placed downstream of the actuators) and the structured mixed upper H 2 divided by upper H Subscript normal infinity <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> $H_2/H_{\infty }$ synthesis, nominal performance, stability robustness and performance robustness despite noisy estimation sensors or free stream velocity variations are ensured. Furthermore, by simultaneously targeting the linear growth of both the first and second instability modes, the control law delays transition to turbulence and increases by 20 percent sign <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>20</mml:mn> <mml:mspace width="thinmathspace"/> <mml:mi mathvariant="normal">%</mml:mi> </mml:math> $20\,\%$ the laminar zone from the upstream edge of the plate.