The mean velocity profile in the convective atmospheric boundary layer (CBL) is derived analytically. The shear-stress budget equations and the mean momentum equations are employed in the derivation. The multi-point Monin–Obukhov similarity (MMO) recently proposed and analytically derived by Tong & Nguyen (J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) and Tong & Ding (J. Fluid Mech., vol. 864, 2019, pp. 640–669) provides the scaling properties of the statistics in the shear-stress budget equations. Our previous and present studies have shown that the CBL is mathematically a singular perturbation problem. Therefore, we obtain the mean velocity profile using the method of matched asymptotic expansions. Three scaling layers are identified: the outer layer, which includes the mixed layer, the inner-outer layer and the inner-inner layer, which includes the roughness layer. There are two overlapping layers, the local-free-convection layer and the log layer, respectively. Two new velocity-defect laws are discovered: the mixed-layer velocity-defect law and the surface-layer velocity-defect law. The local-free-convection mean profile is obtained by asymptotically matching the expansions in the first two layers. The log law is obtained by matching the expansions in the last two layers. The von Kármán constant is obtained using velocity and length scales, and therefore has a physical interpretation. A new friction law, the convective logarithmic friction law, is obtained. The present work provides an analytical derivation of the mean velocity profile hypothesized in the Monin–Obukhov similarity theory, and is part of a comprehensive derivation of the MMO scaling from first principles.

In this study, the effects of permeable porous walls on momentum and heat transfer in a rectangular duct were studied by means of direct numerical simulation of the turbulent conjugate heat transfer. For this purpose, airflow through a rectangular duct, partially filled with a porous medium consisting of aluminium square bars, was simulated at a bulk mean Reynolds number of 3500, where the geometry of the duct used was identical to that employed in the experimental study of Suga et al. (J. Fluid Mech., vol. 884, 2020, A7). It was found that the large-scale perturbations arising from the Kelvin–Helmholtz type of instability developed over the porous medium wall, and the turbulence intensity, particularly in the porous wall-normal component, was enhanced significantly. The secondary flow was enhanced by a factor of three compared to that in a smooth-walled square duct flow, and could be characterized by a strong upward flow along the lateral walls and downward flow in the symmetry plane. The convection by the secondary flow considerably contributed to the momentum and heat transfer in the top half of the clear flow region, whereas the enhanced turbulence over the porous wall largely affected the momentum and heat transfer just above the porous medium wall, as seen in the case of a porous-walled channel flow. It should be noted that in the porous medium region, the mean temperature at the surface of the porous medium is non-uniform, with the solid- and fluid-phase temperatures reaching the equilibrium state. This could be correctly reproduced only with the conjugate heat transfer. It was found that the mean velocity dispersion as well as the turbulent velocity fluctuation contributed significantly to the energy transfer below the porous wall, which demonstrated the importance of the dispersion heat flux for heat transfer modelling of porous medium flows. Furthermore, it was observed that the secondary flow penetrated the porous medium region resulting in large-scale mean flow currents, which enhanced heat transfer inside the porous medium region.

Large-eddy simulation (LES) is performed to investigate the dynamics of flow and canopy motions and the energy transfer in turbulent canopy flows. Different from the traditional approach that models the canopy as a continuous medium with a drag coefficient prescribed a priori, an immersed boundary method together with a beam model is employed to explicitly capture the dynamics of individual stems and resolve monami. The simulation cases cover a broad range of stem flexibilities from rigid stems to oscillatory stems to stems yielding to the flow. For highly flexible canopies, the stem fluctuation is small such that the canopy behaves like a rigid canopy, which is used to explain the similarities of the flow features between rigid and highly flexible canopies. Analyses of the turbulent kinetic energy (TKE) budget show that, in the flexible canopy cases, the waving term associated with the canopy drag–flow velocity correlation can be as large as one-half of the shear production term near the canopy top. Spectral TKE budget analyses further reveal dominant effects at two characteristic scales: the monami scale associated with the coherent structures in the mixing layer and the wake scale associated with the interval between adjacent stems. For the TKE in flexible canopies, the waving term is found to play an important role in the interscale and wall-normal transport terms. Our LES data show that the spectral shortcut mechanism proposed by previous studies is caused by the waving term.

The processing of dense suspensions is a crucial step in many industries including mining; the production of ceramics; and the manufacture of pharmaceuticals. It is widely reported that these suspensions exhibit nonlinear behaviours such as shear thinning and thickening, with particle surface contacts recently being accepted as a primary culprit in the latter. In light of this, the modelling community have started to explore the role of particle surface tribology, predominantly by incorporating Coulombic friction models borrowed from the field of dry granular matter. Full details of the interactions between particle surfaces remain unclear, however, and it is suggested that physical interlocking of particle asperities may be key. Here, we use particle-based simulations to explore explicitly the effect of interlocking on the rheology of dense suspensions of micron-sized solids in a Newtonian fluid. Our simplified model recovers shear thinning, thickening and jamming phenomena commonly seen in experiments.

We outline a methodology for the simulation of two-way coupled particle-laden flows. The drag force that couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. In addition to hydrodynamics, the method can be extended to other physics present in particle-laden flows such as heat transfer and electromagnetism. The discrete Green's functions for the Navier–Stokes equations are obtained at low particle Reynolds number in a two-plane channel geometry. We perform verification at different Reynolds numbers for a particle settling under gravity parallel to a plane wall, for different wall-normal separations. Compared with other point-particle schemes, the Stokesian discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations. To account for degradation in accuracy away from the wall at finite Reynolds number, we extend the present methodology to an Oseen-like discrete Green's function. The extended discrete Green's function method is found to be accurate within $6\,\%$ at all wall-normal separations for particle Reynolds numbers up to 24. The discrete Green's function approach is well suited to dilute systems with significant mass loading and this is highlighted by comparison against other Euler–Lagrange as well as particle-resolved simulations of gas–solid turbulent channel flow. Strong particle–turbulence coupling is observed in the form of turbulence modification and turbophoresis suppression, and these observations are placed in context of the different methods.

A smooth body, adverse pressure gradient (APG), turbulent boundary layer (TBL) separation is experimentally studied. The geometry features canonical TBL development prior to encountering a smooth, two-dimensional convex ramp geometry of finite span onto which a streamwise APG that is fully adjustable is imposed. Both large- and small-scale separations are studied, and all data are archived on the NASA Turbulence Modeling Resource website. This paper describes the large-scale separation case with focus on the surface topography and topology of both separation and reattachment. Despite the spanwise uniform approach TBL and ramp geometry, the separation is highly three-dimensional but the reattachment is spanwise uniform. The surface flow topology is characterized by the ‘owl-face pattern of the fourth kind’ – found to be highly repeatable over multiple experiments. This ubiquitous topology has been reported for a variety of flows including inclined bodies of revolution. It is demonstrated that the APG and the secondary flow associated with the sidewall–ramp juncture is responsible for the formation of the surface separation patterns.

A rigorous derivation and validation for linear fluid–structure-interaction (FSI) equations for a rigid-body motion problem is performed in an Eulerian framework. We show that the ‘added stiffness’ terms arising in the formulation of Fanion et al. (Revue Européenne des Éléments Finis, vol. 9, issue 6–7, 2000, pp. 681–708) vanish at the FSI interface in a first-order approximation and can be neglected when considering the growth of infinitesimal disturbances. Several numerical tests with rigid-body motion are performed to show the validity of the derived formulation by comparing the time evolution between the linear and nonlinear equations when the base flow is perturbed by identical small-amplitude perturbations. In all cases both the growth rate and angular frequency of the instability matches within $0.1\,\%$ accuracy. The derived formulation is used to investigate the phenomenon of symmetry breaking for a rotating cylinder with an attached splitter plate. The results show that the onset of symmetry breaking can be explained by the existence of a zero frequency linearly unstable mode of the coupled FSI system. Finally, the structural sensitivity of the least stable eigenvalue is studied for an oscillating cylinder, which is found to change significantly when the fluid and structural frequencies are close to resonance.

We investigate drop break-up morphology, occurrence, time and size distribution, through large ensembles of high-fidelity direct-numerical simulations of drops in homogeneous isotropic turbulence, spanning a wide range of parameters in terms of the Weber number $We$, viscosity ratio between the drop and the carrier flow $\mu _r=\mu _d/\mu _l$, where d is the drop diameter, and Reynolds ($Re$) number. For $\mu _r \leq 20$, we find a nearly constant critical $We$, while it increases with $\mu _r$ (and $Re$) when $\mu _r > 20$, and the transition can be described in terms of a drop Reynolds number. The break-up time is delayed when $\mu _r$ increases and is a function of distance to criticality. The first break-up child-size distributions for $\mu _r \leq 20$ transition from M to U shape when the distance to criticality is increased. At high $\mu _r$, the shape of the distribution is modified. The first break-up child-size distribution gives only limited information on the fragmentation dynamics, as the subsequent break-up sequence is controlled by the drop geometry and viscosity. At high $We$, a $d^{-3/2}$ size distribution is observed for $\mu _r \leq 20$, which can be explained by capillary-driven processes, while for $\mu _r > 20$, almost all drops formed by the fragmentation process are at the smallest scale, controlled by the diameter of the very extended filament, which exhibits a snake-like shape prior to break-up.

We perform large-eddy simulations of turbulent flow in a channel constricted by streamwise periodically distributed hill-shaped protrusions. Two Reynolds number cases, i.e. $Re_h=10\,595$ (Fröhlich et al., J. Fluid Mech., vol. 526, 2005, pp. 19–66) and $Re_h=33\,000$ (Kähler et al., J. Fluid Mech., vol. 796, 2016, pp. 257–284), are repeated and utilized to verify and validate our numerical results, including the pressure and skin friction coefficients on bottom and top walls of the channel, mean velocity profiles and Reynolds stresses. All comparisons show reasonable agreement, providing a measure of validity that enables us to further probe simulation results at higher Reynolds number ($Re_h=10^5$) into aspects of flow physics that are not available from experiments. Effects of variation of Reynolds number are studied, with emphasis on the mean skin friction coefficients, separation bubble size and pressure fluctuations that are related to separation and reattachment. In addition, the main large-scale features of the separation behind the hill, including the scaling of the mean velocity profiles, are discussed. Furthermore, the instantaneous near-wall flow field is analysed in terms of skin friction portraits, and we confirm the existence of the local very small separation bubble on the hill crest as observed in experimental and numerical investigations. The flow field at the top wall, which is generally not given sufficient attention, is evaluated with the empirical friction law and universal logarithmic law as in planar channel flows. It is found that these empirical laws compare well with the large-eddy simulation results, although the hill constrictions behave as a perturbation source and the developed shear layer has some effects on the flow field near the top wall.

Falling paper flutters and tumbles through air, whereas a paper airplane glides smoothly if its leading edge is appropriately weighted. We investigate this transformation from ‘plain paper’ to ‘paper plane’ through experiments, aerodynamic modelling and free flight simulations of thin plates with differing centre of mass (CoM) locations. Periodic modes such as fluttering, tumbling and bounding give way to steady gliding and then downward diving as the CoM is increasingly displaced towards one edge. To explain these observations, we formulate a quasi-steady aerodynamic model whose force and torque coefficients are informed by experimental measurements. The dependencies on angle of attack reflect the transition from attached to separated flow and a dynamic centre of pressure, effects that prove critical to reproducing the observed motions of paper planes in air and plates in water. Because the model successfully accounts for unsteady and steady flight modes, it may be usefully applied to further problems involving actuated motions, feedback control and interactions with ambient flows.

The present work investigates the excitation process by which free-stream disturbances are transformed into vortex-core perturbations. This problem of receptivity is modelled in terms of the resolvent in frequency space as the linear response to forcing. This formulation of receptivity suggests that non-normality of the resolvent is necessary to allow free-stream disturbances to excite the vortex core. Considering a local (in frequency) measure of non-normality, we show that vortices are frequency-selectively non-normal in a narrow frequency band of retrograde perturbations while the rest of the range is governed by an effectively normal operator, thus not contributing to receptivity. Canonical decomposition of the resolvent reveals that vortices are most susceptible to coiled filaments localised about the critical layer that induce bending waves on the core. Considering Lamb–Oseen, Batchelor and Moore–Saffman vortices as reference-flow models, we find free-stream receptivity to be essentially generic and independent of the axial wavelength on the considered range. A stochastic interpretation of the results could be a model for trailing-vortex meandering.

We present results of three-dimensional computational simulations of horizontal eccentric annular displacement flows as studied experimentally by Renteria & Frigaard (J. Fluid Mech., vol. 905, 2020, A7). We show the simulations are able to effectively capture detailed dispersive features of the flows, not resolved in simpler models. Top side flows, slumping flows and narrow side residual layers are predicted reliably and we are able to understand the main mechanisms that drive the flow into one of these classifications. Both experiments and simulations agree in their classification of steady and unsteady displacements. However, when the dispersion is properly resolved the concept of a steady state displacement becomes questionable. We study the axial spreading of the displacement front via one-dimensional simplifications, both advective and diffusive. While these do not fully describe the flows, they give some estimates that are useful for practical application.