Adjoint-based linear sensitivity of a hypersonic boundary layer to steady wall blowing or heating

Arthur POULAIN¹, Denis SIPP¹*, Cédric CONTENT¹, Georgios RIGAS², Eric GARNIER¹

¹ DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France – denis.sipp@onera.fr, arthur.poulain@onera.fr, cedric.content@onera.fr, eric.garnier@onera.fr

² Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom – g.rigas@imperial.ac.uk

* Corresponding author

 

Keywords: Boundary layer, Global stability, Linear sensitivity, Adjoint-based optimization.

 

Abstract

Flow control efficiency depends on the location of the actuators. Instead of performing a computational costly parametric analysis, we use an adjoint-based optimisation technique to find the linearly optimal actuator for steady open-loop control achieved through base-flow modification.

Exploiting the benefit of Algorithmic Differentiation to ease the computation of high-order state derivative operators, it relies on the sensitivity of the most predominant modes predicted by the resolvent analysis, on two-dimensional or axi-symmetrical configurations for the present work.

The method is applied on a Mach 4.5 boundary layer over an adiabatic flat plate for steady wall-normal blowing/suction control and wall heat flux control. Through the sensitivity of the resolvent gain to base-flow variations, the linear gradient predicted for the first and second Mack modes are studied in detail.

The resolvent optimal gain decreases when suction is applied upstream of Fedorov’s mode S/mode F synchronisation point leading to stabilisation and conversely when applied downstream. The largest suction gradient is in the region of the branch I of mode S neutral curve.

For heat flux control, strong heating at the leading edge stabilises both the first and second Mack modes, the former being more sensitive to wall-temperature control.

Streaks are less sensitive to any boundary control in comparison with the Mack modes.

Eventually, we show that an optimal actuator consisting of a single steady heating strip located close to the leading edge manages to damp all the instabilities together.