Wall-temperature effects on second-mode kinematics and mechanics

The effects of wall temperature on hypersonic boundary layer transition are investigated by analysing the kinematics (acoustic ray trajectories) and mechanics (fluctuation energy production and transport) of second-mode instabilities. The disturbance energy formulation is taken from Roy & Scalo ( J. Fluid Mech. 2025, vol. 1007, A49). Flow conditions are taken from a Mach 6 boundary layer over a $3^\circ$ cone, with varying degrees of wall-to-adiabatic temperature ratios, $\varTheta =T_w/T_{\textit{ad}}=0.25{-}1.75$ . Boundary layer-resolved axisymmetric direct numerical simulations with companion Laguerre polynomials-based linear stability theory provide the supporting numerical datasets. It was found that second-mode instabilities comprise two decks, separated by the pressure node location $(y=y_\pi )$ . The upper deck ( $y\gt y_\pi$ ) is characterised by temperature ( $T^{\prime}$ ) and density ( $\rho'$ ) fluctuations working with in-phase wall-normal velocity fluctuations ( $v'$ ) to sustain the total disturbance energy production term, $-(\rho _0 v'T'\partial T_0/\partial y+\rho ' u_0 v' \partial u_0/\partial y$ ), which peaks at the generalised inflection point $y=y_i$ . The downward-oriented energy flux peaks below the critical layer, $y\lt y_c$ , and sustains acoustic energy accumulation in the lower deck. Effective energy transfer requires the streamwise and wall-normal fluxes to maintain a $90^\circ$ phase difference. This is satisfied especially for colder walls, whereas heated walls yield out-of-phase $v'$ – $T'$ and in-phase pressure ( $p'$ ) – streamwise velocity ( $u'$ ) fluctuations, reducing the disturbance energy production and discouraging the coupling between the two decks. Ray tracing reveals the trajectory of purely acoustic wave paths emanating from the wall, as trapping occurs below the generalised inflection line $(y_i)$ , governed by the mean flow velocity gradients $(\partial u_0/\partial y)$ .