Causality in Energy Containing Eddies in Compressible Channels

Compressible wall bounded turbulent flows are ubiquitous in high-speed aerodynamic applications and are characterized by thermodynamic fluctuations and temperature gradients. Morkovin's 1962 hypothesis posits that ``the essential dynamics of these shear flows will follow the incompressible pattern'' for sufficiently low Mach number, provided that the mean property variations are accounted for. This hypothesis has primarily been used to develop velocity transformations for the mean flow field, not necessarily the dynamics of the flow. To measure the dynamics in compressible flows, we use information theoretic approaches to compare causal mechanisms in the energetic turbulent scales in subsonic and supersonic flows. Using minimal flow units at $Re_{\tau} = 220$ of subsonic and supersonic flow, we capture the relevant near-wall turbulent, thermal, and dilatational fluctuations from full channel simulations over the hundreds of eddy-turnover times needed to converge the probability density functions. The Synergistic Unique Redundant Decomposition of causality (SURD) (Martinez-Sanchez, A., et~al.(2024)) is used to quantify the causal links between the compressible signals, here the most energetic POD mode coefficients. Similarities in the temporal evolution of the causal measures between velocity signals provide direct evidence of Morkovin's hypothesis regarding the ``essential dynamics''. Differences emerge when considering the interplay between velocity signals and thermal fluctuations. The active role of the temperature, for instance, is reflected in the direct causation from temperature signals to velocity signals present in the supersonic flow and absent in the subsonic flow. The study also highlights a one-way causal pathway from density fluctuations to the dilatational velocity field only present in the supersonic flow.