Accurate prediction of the global properties of wall-bounded turbulence holds significant importance for both fundamental research and engineering applications. In atmospheric boundary layers, the relationship between friction drag and geostrophic wind is described by the geostrophic drag law (GDL). We use carefully designed large-eddy simulations to study nocturnal stable atmospheric boundary layers (NSBLs), which are characterized by a negative potential temperature flux at the surface and neutral stratification higher up. Our simulations explore a wider range of the Kazanski–Monin parameter, $\mu = L_f / L_s = [16.7, 193.3]$, with $L_f$ the Ekman length scale and $L_s$ the surface Obukhov length. We show collapse of the GDL coefficients onto single curves as functions of $\mu$, thereby validating the GDL's applicability to NSBLs over a very wide $\mu$ range. We show that the boundary-layer height $h$ scales with $\sqrt {L_f L_s}$, while both the streamwise and spanwise wind gradients scale with $u_*^2 / (h^2 f)$, where $u_*$ represents the friction velocity and $f$ the Coriolis parameter. Leveraging these insights, we developed new analytical expressions for the GDL coefficients, significantly enhancing our understanding of the GDL for turbulent boundary layers. These formulations facilitate the analytical prediction of the geostrophic drag coefficient and cross-isobaric angle.

Three-dimensional numerical simulations of hypersonic boundary layer transition delay due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) were carried out on a 7$^{\circ {}}$ half-angle cone for unit Reynolds numbers $Re_m=2.43 \times 10^6$–$6.40\times 10^6\ \text {m}^{-1}$, at the free-stream Mach number $M_\infty =7.4$, for both sharp and 2.5 mm nose tip radii. A broadband time-domain impedance boundary condition was used to model the acoustic effects of the porous surface on the flow field. A quasi-spectral sub-filter-scale dynamic closure was adopted to stabilize the computations upon turbulent breakdown under extreme cooling conditions, with wall-to-adiabatic temperature ratio of $T_{w}/ T_{ad} \simeq 0.08 $, while accurately recovering the growth rates of the unstable modes present in the early transition stages. Good agreement is observed with the reference experimental data, both in terms of the predicted extent of the transition delay and the measured second-mode frequency spectrum. The latter is strongly modulated by the formation of near-wall low-temperature three-dimensional streaks. Pressure disturbances concentrate in corridors of locally thickened boundary layer, with frequencies lower than what predicted by linear theory. Here, trapped wavetrains are formed, which can persist long into the turbulent region. Finally, it is shown that the presence of a porous wall simply shifts the onset of turbulence downstream, without affecting its structure.

The confined Stokesian dynamics (CSD) algorithm recently reported equilibrium properties but was missing hydrodynamic functions required for suspension stress and non-equilibrium properties. In this first of a two-part series, we expand the CSD algorithm to model the traceless part of the stress tensor. To obtain quantities needed to solve the integral expressions for the stress, we developed a general method to solve Stokes’ equations in bispherical coordinates. We calculate the traceless stress tensor for arbitrary particle-to-enclosure size ratio. We next compute rheology of a confined suspension by implementing the stresslet hydrodynamic coefficients into CSD, yielding the deviatoric part of the many-body hydrodynamic stresslet. We employed energy methods to relate this stresslet to the high-frequency viscosity of the confined suspension, finding an increase in viscous dissipation with crowding and confinement well beyond the unconfined value. We show that confinement effects on viscosity are dominated by near-field interactions between the particles that reside very near the cavity wall (rather than particle–wall interactions). Surprisingly, this near-field effect is stronger than the viscosity of an unconfined suspension, showing that entropic exclusion driven by the wall sets up many lubrication interactions that then generate strong viscous dissipation. The limiting case of a particle near a flat wall reveals a correction to prior literature. The theory presented in this work can be expanded to study the Brownian contribution to the viscosity of confined suspensions in and away from equilibrium. In part 2, we report the osmotic pressure, via the trace of the stress tensor.