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.

In this paper, we study the effect of lateral wall vibrations on the excitation and evolution of non-modal perturbations in hypersonic boundary layers subject to low-frequency freestream vortical disturbances (FSVDs). A novel, high-efficiency numerical approach, combining the harmonic weakly nonlinear Navier–Stokes and nonlinear parabolised stability equation approaches, is developed, which is sufficient to accommodate both the rapid distortion of the perturbation in the leading-edge vicinity and the nonlinear development of finite-amplitude high-order harmonics in the downstream region. The boundary-layer response to low-frequency FSVDs shows a longitudinal streaky structure, for which the temperature perturbation shows much greater magnitude than the streamwise velocity perturbation. The lateral vibration induces a Stokes layer solution for the spanwise velocity perturbation, which interacts with the FSVD-induced perturbations and leads to a suppression of the non-modal perturbation and an enhancement of the downstream modal perturbation. The new perturbations excited by the FSVD–vibration interaction strengthen as the vibration intensifies, and they could become comparable with the FSVD-induced perturbations in downstream locations at a high vibration intensity, indicating a remarkable modification of the streaky structure and its instability property. Secondary instability (SI) analyses based on the streaky base flow indicate that the vibration could enhance or suppress the SI modes, depending on their initial phases over the vibration period. Overall, the average effect is that the low-frequency and high-frequency SI modes are stabilised and destabilised by the vibration, respectively. Since the high-frequency SI modes undergo higher amplifications, the subsequent bypass transition is likely to be promoted by relatively strong vibrations.

We perform direct numerical simulations of soluble bubbles dissolving in a Taylor–Couette (TC) flow reactor with a radius ratio of $\eta =0.5$ and Reynolds number in the range $0 \leq Re \leq 5000$, which covers the main regimes of this flow configuration, up to fully turbulent Taylor vortex flow. The numerical method is based on a geometric volume-of-fluid framework for incompressible flows coupled with a phase-change solver that ensures mass conservation of the soluble species, whilst boundary conditions on solid walls are enforced through an embedded boundary approach. The numerical framework is validated extensively against single-phase TC flows and competing mass transfer in multicomponent mixtures for an idealised infinite cylinder and for a bubble rising in a quiescent liquid. Our results show that when bubbles in a TC flow are mainly driven by buoyancy, theoretical formulae derived for spherical interfaces on a vertical trajectory still provide the right fundamental relationship between the bubble Reynolds and Sherwood numbers, which reduces to $Sh \propto \sqrt {Pe}$ for large Péclet values. For bubbles mainly transported by TC flows, the dissolution of bubbles depend on the TC Reynolds number and, for the turbulent configurations, we show that the smallest characteristic turbulent scales control mass transfer, in agreement with the small-eddy model of Lamont & Scott (AIChE J., vol. 16, 1970, pp. 513–519). Finally, the interaction between two aligned bubbles is investigated and we show that a significant increase in mass transfer can be obtained when the rotor of the apparatus is operated at larger speeds.

This work introduces a formulation of resolvent analysis that uses wavelet transforms rather than Fourier transforms in time. Under this formulation, resolvent analysis may extend to turbulent flows with non-stationary mean states. The optimal resolvent modes are augmented with a temporal dimension and are able to encode the time-transient trajectories that are most amplified by the linearised Navier–Stokes equations. We first show that the wavelet- and Fourier-based resolvent analyses give equivalent results for statistically stationary flow by applying them to turbulent channel flow. We then use wavelet-based resolvent analysis to study the transient growth mechanism in the near-wall region of a turbulent channel flow by windowing the resolvent operator in time and frequency. The computed principal resolvent response mode, i.e. the velocity field optimally amplified by the linearised dynamics of the flow, exhibits characteristics of the Orr mechanism, which supports the claim that this mechanism is key to linear transient energy growth. We also apply this method to non-stationary parallel shear flows such as an oscillating boundary layer, and three-dimensional channel flow in which a sudden spanwise pressure gradient perturbs a fully developed turbulent channel flow. In both cases, wavelet-based resolvent analysis yields modes that are sensitive to the changing mean profile of the flow. For the oscillating boundary layer, wavelet-based resolvent analysis produces oscillating principal forcing and response modes that peak at times and wall-normal locations associated with high turbulent activity. For the turbulent channel flow under a sudden spanwise pressure gradient, the resolvent modes gradually realign themselves with the mean flow as the latter deviates. Wavelet-based resolvent analysis thus captures the changes in the transient linear growth mechanisms caused by a time-varying turbulent mean profile.

Stochastically generated instantaneous velocity profiles are used to reproduce the outer region of rough-wall turbulent boundary layers in a range of Reynolds numbers extending from the wind tunnel to field conditions. Each profile consists in a sequence of steps, defined by the modal velocities and representing uniform momentum zones (UMZs), separated by velocity jumps representing the internal shear layers. Height-dependent UMZ is described by a minimal set of attributes: thickness, mid-height elevation, and streamwise (modal) and vertical velocities. These are informed by experimental observations and reproducing the statistical behaviour of rough-wall turbulence and attached eddy scaling, consistent with the corresponding experimental datasets. Sets of independently generated profiles are reorganized in the streamwise direction to form a spatially consistent modal velocity field, starting from any randomly selected profile. The operation allows one to stretch or compress the velocity field in space, increases the size of the domain and adjusts the size of the largest emerging structures to the Reynolds number of the simulated flow. By imposing the autocorrelation function of the modal velocity field to be anchored on the experimental measurements, we obtain a physically based spatial resolution, which is employed in the computation of the velocity spectrum, and second-order structure functions. The results reproduce the Kolmogorov inertial range extending from the UMZ and their attached-eddy vertical organization to the very-large-scale motions (VLSMs) introduced with the reordering process. The dynamic role of VLSM is confirmed in the $-u^{\prime }w^{\prime }$ co-spectra and in their vertical derivative, representing a scale-dependent pressure gradient contribution.

A stream of free-falling particles from a rectangular hopper, hereafter called a ‘curtain’, was characterised systematically using a well-resolved, non-intrusive optical shadowgraphic method, to reveal both an additional axial region and an additional dilute region distributed laterally on either side of the curtain, relative to those identified previously. The effects of particle size and hopper outlet thickness on the evolution of the particle curtain were separately isolated, whilst measuring particle mass flow rate. The curtains were characterised into four distinct axial regions, namely a near-field expansion region near to the hopper exit, a neck zone where the curtain contracts, a region of intermediate-field expansion and a far field with particles reaching terminal velocity. The initial expansion half-angle, $2.3^{\circ } \le \alpha \le 4.8^{\circ }$, was found to be insensitive to particle size, but to increase with hopper outlet thickness. The ‘trough’ in the neck zone was deduced to be caused by a pressure gradient driven by particle acceleration. The curtain expansion rate at the intermediate field was found to increase with a decrease in particle size and hopper outlet thickness. The outermost dilute-particle region was deduced to be caused by collisions, induced by gradients in the velocity profile near to the hopper exit. New dimensionless analysis reveals that the dynamics of curtains can be characterised broadly into two regimes, one in which the aerodynamics is dominant and the other where it is weak. Curtain transmittance was found to scale with the Froude number, highlighting the importance of particle momentum.