The resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) is applied to supersonic turbulent boundary layers to study the validity of Morkovin’s hypothesis, which postulates that high-speed turbulence structures in zero-pressure-gradient turbulent boundary layers remain largely the same as their incompressible counterparts. Supersonic zero-pressure-gradient turbulent boundary layers with adiabatic wall boundary conditions at Mach numbers ranging from 2 to 4 are considered. Resolvent analysis highlights two distinct regions of the supersonic turbulent boundary layer in the wave parameter space: the relatively supersonic region and the relatively subsonic region. In the relatively supersonic region, where the flow is supersonic relative to the free-stream, resolvent modes display structures consistent with Mach wave radiation that are absent in the incompressible regime. In the relatively subsonic region, we show that the low-rank approximation of the resolvent operator is an effective approximation of the full system and that the response modes predicted by the model exhibit universal and geometrically self-similar behaviour via a transformation given by the semi-local scaling. Moreover, with the semi-local scaling, we show that the resolvent modes follow the same scaling law as their incompressible counterparts in this region, which has implications for modelling and the prediction of turbulent high-speed wall-bounded flows. We also show that the thermodynamic variables exhibit similar mode shapes to the streamwise velocity modes, supporting the strong Reynolds analogy. Finally, we demonstrate that the principal resolvent modes can be used to capture the energy distribution between momentum and thermodynamic fluctuations.

This work develops a methodology for approximating the shape of leading resolvent modes for incompressible, quasi-parallel, shear-driven turbulent flows using prescribed analytic functions. We demonstrate that these functions, which arise from the consideration of wavepacket pseudoeigenmodes of simplified linear operators (Trefethen, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 461, 2005, pp. 3099–3122. The Royal Society), give an accurate approximation for the energetically dominant wall-normal vorticity component of a class of nominally wall-detached modes that are centred about the critical layer. We validate our method on a model operator related to the Squire equation, and show for this simplified case how wavepacket pseudomodes relate to truncated asymptotic expansions of Airy functions. Following the framework developed in McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), we next apply a sequence of simplifications to the resolvent formulation of the Navier–Stokes equations to arrive at a scalar differential operator that is amenable to such analysis. The first simplification decomposes the resolvent operator into Orr–Sommerfeld and Squire suboperators, following Rosenberg & McKeon (Fluid Dyn. Res., vol. 51, 2019, 011401). The second simplification relates the leading resolvent response modes of the Orr–Sommerfeld suboperator to those of a simplified scalar differential operator – which is the Squire operator equipped with a non-standard inner product. This characterisation provides a mathematical framework for understanding the origin of leading resolvent mode shapes for the incompressible Navier–Stokes resolvent operator, and allows for rapid estimation of dominant resolvent mode characteristics without the need for operator discretisation or large numerical computations. We explore regions of validity for this method, and show that it can predict resolvent response mode shape (though not necessary the corresponding resolvent gain) over a wide range of spatial wavenumbers and temporal frequencies. In particular, we find that our method remains relatively accurate even when the modes have some amount of ‘attachment’ to the wall, and that that the region of validity contains the regions in parameter space where large-scale and very-large-scale motions typically reside. We relate these findings to classical lift-up and Orr amplification mechanisms in shear-driven flows.