The mean skin-friction drag in a wall-bounded turbulent flow can be decomposed into different physics-informed contributions based on the mean and statistical turbulence quantities across the wall layer. Following Renard & Deck’s study (J. Fluid Mech., vol. 790, 2016, pp. 339–367) on the skin-friction drag decomposition of incompressible wall-bounded turbulence, we extend their method to a compressible form and use it to investigate the effect of density and viscosity variations on skin-friction drag generation, using direct numerical simulation data of compressible turbulent channel flows. We use this novel decomposition to study the skin-friction contributions associated with the molecular viscous dissipation and the turbulent kinetic energy production and we investigate their dependence on Reynolds and Mach number. We show that, upon application of the compressibility transformation of Trettel & Larsson (Phys. Fluids, vol. 28, 2016, 026102), the skin-friction drag contributions can be only partially transformed into the equivalent incompressible ones, as additional terms appear representing deviations from the incompressible counterpart. Nevertheless, these additional contributions are found to be negligible at sufficiently large equivalent Reynolds number and low Mach number. Moreover, we derive an exact relationship between the wall heat flux coefficient and the skin-friction drag coefficient, which allows us to relate the wall heat flux to the skin-friction generation process.

Bidimensional empirical mode decomposition (BEMD) is used to identify attached eddies in turbulent channel flows and quantify their relationship with the mean skin-friction drag generation. BEMD is an adaptive, non-intrusive, data-driven method for mode decomposition of multiscale signals especially suitable for non-stationary and nonlinear processes such as those encountered in turbulent flows. In the present study, we decompose the velocity fluctuations obtained by direct numerical simulation of channel flows into BEMD modes characterized by specific length scales. Unlike previous works (e.g. Flores & Jiménez, Phys. Fluids, vol. 22(7), 2010, 071704; Hwang, J. Fluid Mech., vol. 767, 2015, pp. 254–289), the current approach employs naturally evolving wall-bounded turbulence without modifications of the Navier–Stokes equations to maintain the inherent turbulent dynamics, and minimize artificial numerical enforcement or truncation. We show that modes identified by BEMD exhibit a self-similar behaviour, and that single attached eddies are mainly composed of streaky structures carrying intense streamwise velocity fluctuations and vortex packets permeating in all velocity components. Our findings are consistent with the existence of attached eddies in actual wall-bounded flows, and show that BEMD modes are tenable candidates to represent Townsend attached eddies. Finally, we evaluate the turbulent-drag generation from the perspective of attached eddies with the aid of the Fukagata–Iwamoto–Kasagi identity (Fukagata et al., Phys. Fluids, vol. 14(11), 2002, pp. L73–L76) by splitting the Reynolds shear stress into four different terms related to the length scale of the attached eddies.

We develop a new methodology to assess the streamwise inclination angles (SIAs) of the wall-attached eddies populating the logarithmic region with a given wall-normal height. To remove the influences originating from other scales on the SIA estimated via two-point correlation, the footprints of the targeted eddies in the vicinity of the wall and the corresponding streamwise velocity fluctuations carried by them are isolated simultaneously, by coupling the spectral stochastic estimation with the attached-eddy hypothesis. Datasets produced with direct numerical simulations spanning $Re_{\tau } \sim O(10^2)\unicode{x2013}O(10^3)$ are dissected to study the Reynolds number effect. The present results show, for the first time, that the SIAs of attached eddies are Reynolds-number-dependent in low and medium Reynolds numbers, and tend to saturate at $45^{\circ }$ as the Reynolds number increases. The mean SIA reported by vast previous experimental studies are demonstrated to be the outcomes of the additive effect contributed by multi-scale attached eddies. These findings clarify the long-term debate and perfect the picture of the attached-eddy model.

No abstract