Transient growth in diabatic boundary layers with fluids at supercritical pressure

In the thermodynamic region close to the vapour-liquid critical point and in the proximity of the pseudo-boiling (Widom) line, strong property variations extensively alter the growth of modal instabilities, as revealed in Ren et al. (J. Fluid Mech., vol. 871, 2019, pp. 831-864). Here, we study non-modal disturbances in the spatial framework using an eigenvector decomposition of the linearized Navier-Stokes equations for locally parallel flow. To account for non-ideality, a new energy norm is derived. At a constant free-stream Mach number of 10⁻³ along an isobar, three different regimes are considered by regulating the diabatic wall temperature: subcritical (liquid-like), supercritical (gas-like), and transcritical (across the Widom line). In the non-transcritical regimes, the resulting streamwise-independent streaks originate from the lift-up effect. Wall cooling is found to enhance the energy amplification for both subcritical and supercritical regimes. However, when the temperature profile is increased beyond the Widom line, a strong sub-optimal growth is observed over very short streamwise distances due to the Orr mechanism. The optimal energy growth at large distances is found to arise from an interplay between lift-up and Orr mechanism due to the additional presence of the transcritical Mode II. As a result, optimal disturbances are streamwise-modulated streaks with strong thermal components and with a propagation angle inversely proportional to the local Reynolds number. An N-factor comparison with modal growth reveals that transient growth is scarcely affected by a wall-temperature variation in the non-transcritical regimes. In contrast, in the transcritical regime, transient growth is greatly enhanced, especially when cooling beyond the Widom line. Here, non-modal N-factors are found to resemble the imposition of an adverse pressure gradient in the ideal-gas regime.