The problem formulated in Part 1 (Loper, J. Fluid Mech., vol. 892, 2020, A16) for flow in the turbulent boundary layer beneath a vortex is solved for a power-law swirl: $v_{\infty }(r)\sim r^{2\unicode[STIX]{x1D703}-1}$ , where $r$ is cylindrical radius and $\unicode[STIX]{x1D703}$ is a constant parameter, with turbulent diffusivity parameterized as $\unicode[STIX]{x1D708}=v_{\infty }L$ and the diffusivity function $L$ either independent of axial distance $z$ from a stationary plane (model A) or constant within a rough layer of thickness $z_{0}$ adjoining the plane and linear in $z$ outside (model B). Model A is not a useful model of vortical flow, whereas model B produces realistic results. As found in Part 1 for $\unicode[STIX]{x1D703}=1.0$ , radial flow consists of a sequence jets having thicknesses that vary nearly linearly with $r$ . A novel structural feature is the turning point $(r_{t},z_{t})$ , where the primary jet has a minimum height. The radius $r_{t}$ is a proxy for the eye radius of a vortex and $z_{t}$ is a proxy for the size of the corner region. As $r$ decreases from $r_{t}$ , the primary jet thickens, axial outflow from the layer increases and axial oscillations become larger, presaging a breakdown of the boundary layer. For small $\unicode[STIX]{x1D703}$ , $r_{t}\sim z_{0}/\unicode[STIX]{x1D716}\unicode[STIX]{x1D703}$ and $z_{t}\sim z_{0}/\unicode[STIX]{x1D703}^{3/2}$ . The lack of existence of the turning point for $\unicode[STIX]{x1D703}\gtrsim 0.42$ and the acceleration of the turning point away from the origin of the meridional plane as $\unicode[STIX]{x1D703}\rightarrow 0$ provide partial explanations why weakly swirling flows do not have eyes, why strongly swirling flows have eyes and why a boundary layer cannot exist beneath a potential vortex.

The equations governing the mean fluid motions within a turbulent boundary layer adjoining a stationary plane beneath an axisymmetric circumferential flow $v_{\infty }(r)$ , where $r$ is cylindrical radius, are solved by assuming the eddy diffusivity is proportional to $v_{\infty }$ times a diffusivity function $L(r,z)$ , where $z$ is axial distance from the plane. The boundary-layer shape and structure depend on the dimensionless vorticity $\unicode[STIX]{x1D703}=\text{d}(rv_{\infty })/2v_{\infty }\,\text{d}r$ , but are independent of the strength of the circumferential flow. This problem has been solved using a spectral method in the case of rigid-body motion ( $\unicode[STIX]{x1D703}=1$ and $v_{\infty }\sim r$ ) for two models of  $L$ : $L$ constant (model A) and $L$ constant within a rough layer of thickness $z_{0}$ adjoining the boundary and increasing linearly with $z$ outside that layer (model B). The influence of the rough layer is quantified by the dimensionless radial coordinate $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D716}r/z_{0}$ , where $\unicode[STIX]{x1D716}\ll 1$ . The boundary-layer thickness varies parabolically with $r$ for model A and nearly linearly with $r$ for model B. Inertial stability of the outer flow causes the velocity components to decay with axial distance as exponentially damped oscillations, with the radial flow consisting of a sequence of jets. Axial flow is positive (flowing out of the boundary layer). Outflow from the layer, velocity gradients at the bounding plane, meridional-plane circulation and oscillations all increase as radius decreases.

We use matched asymptotics to derive analytical formulae for the acoustic impedance of a subwavelength orifice consisting of a cylindrical perforation in a rigid plate. In the inviscid case, an end correction to the length of the orifice due to Rayleigh is shown to constitute an exponentially accurate approximation in the limit where the aspect ratio of the orifice is large; in the opposite limit, we derive an algebraically accurate correction, depending upon the logarithm of the aspect ratio, to the impedance of a circular aperture in a zero-thickness screen. Viscous effects are considered in the limit of thin Stokes boundary layers, where a boundary-layer analysis in conjunction with a reciprocity argument provides the perturbation to the impedance as a quadrature of the basic inviscid flow. We show that for large aspect ratios the latter perturbation can be captured with exponential accuracy by introducing a second end correction whose value is calculated to be in between two guesses commonly used in the literature; we also derive an algebraically accurate approximation in the small-aspect-ratio limit. The viscous theory reveals that the resistance exhibits a minimum as a function of aspect ratio, with the orifice radius held fixed. It is evident that the resistance grows in the long-aspect-ratio limit; in the opposite limit, resistance is amplified owing to the large velocities close to the sharp edge of the orifice. The latter amplification arrests only when the plate is as thin as the Stokes boundary layer. The analytical approximations derived in this paper could be used to improve circuit modelling of resonating acoustic devices.

Although numerous studies have investigated the individual effects of salinity, irrigation and fertilization on soil microbial communities, relatively less attention has been paid to their combined influences, especially using molecular techniques. Based on the field of orthogonal designed test and deoxyribonucleic acid sequencing technology, the effects of saline water irrigation amount, salinity level of irrigation water and nitrogen (N) fertilizer rate on soil bacterial community structure were investigated. The results showed that the irrigation amount was the most dominant factor in determining the bacterial richness and diversity, followed by the irrigation water salinity and N fertilizer rate. The values of Chao1 estimator, abundance-based coverage estimator and Shannon indices decreased with an increase in irrigation amount while increased and then decreased with an increase in irrigation water salinity and N fertilizer rate. The highest soil bacterial richness and diversity were obtained under the least irrigation amount (25 mm), medium irrigation water salinity (4.75 dS/m) and medium N fertilizer rate (350 kg/ha). However, different bacterial phyla were found to respond distinctively to these three factors: irrigation amount significantly affected the relative abundances of Proteobacteria and Chloroflexi; irrigation water salinity mostly affected the members of Actinobacteria, Gemmatimonadetes and Acidobacteria; and N fertilizer rate mainly influenced the Bacteroidetes' abundance. The results presented here revealed that the assessment of soil microbial processes under combined irrigation and fertilization treatments needed to be more careful as more variable consequences would be established by comparing with the influences based on an individual factor, such as irrigation amount or N fertilizer rate.

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

We present a new theoretical model describing gravity–capillary waves in orbitally shaken cylindrical containers. Our model can account for both one-layer free-surface and two-layer interfacial wave systems. A set of modal equations for irrotational waves is formulated that we complement with viscous damping rates to incorporate energy dissipation. This approach allows us to calculate explicit formulas for the phase shifts between wave and shaker which are practically important for the mixing efficiency in orbitally shaken bioreactors. Resonance dynamics is described using eight dimensionless numbers, revealing a variety of different effects influencing the forced wave amplitudes. As an unexpected result, the model predicts the formation of novel spiral wave patterns resulting from a damping-induced symmetry breaking mechanism. For validation, we compare theoretical amplitudes, fluid velocities and phase shifts with three different and independent experiments and, when using the slightly deviating experimental values of the resonance frequencies, find a good agreement within the theoretical limits.

‘Ground effect’ refers to the enhanced performance enjoyed by fliers or swimmers operating close to the ground. We derive a number of exact solutions for this phenomenon, thereby elucidating the underlying physical mechanisms involved in ground effect. Unlike previous analytic studies, our solutions are not restricted to particular parameter regimes, such as ‘weak’ or ‘extreme’ ground effect, and do not even require thin aerofoil theory. Moreover, the solutions are valid for a hitherto intractable range of flow phenomena, including point vortices, uniform and straining flows, unsteady motions of the wing, and the Kutta condition. We model the ground effect as the potential flow past a wing inclined above a flat wall. The solution of the model requires two steps: firstly, a coordinate transformation between the physical domain and a concentric annulus; and secondly, the solution of the potential flow problem inside the annulus. We show that both steps can be solved by introducing a new special function which is straightforward to compute. Moreover, the ensuing solutions are simple to express and offer new insight into the mathematical structure of ground effect. In order to identify the missing physics in our potential flow model, we compare our solutions against new experimental data. The experiments show that boundary layer separation on the wing and wall occurs at small angles of attack, and we suggest ways in which our model could be extended to account for these effects.

We have performed fully resolved simulations of turbulent flows over various submerged rigid canopies covering the wall of an open channel. All the numerical predictions have been obtained considering the same nominal bulk Reynolds number (i.e. $Re_{b}=U_{b}H/\unicode[STIX]{x1D708}=6000$ , $H$ being the channel depth and $U_{b}$ the bulk velocity). The computations directly tackle the region occupied by the canopy by imposing the zero-velocity condition on every single stem, while the outer flow is dealt with a highly resolved large-eddy simulation. Four canopy configurations have been considered. All of them share the same in-plane solid fraction while the canopy to channel height ratios have been selected to be $h/H=(0.05,0.1,0.25,0.4)$ . The lowest and the highest values lead to flow conditions approaching the two asymptotic states that in the literature are usually termed the sparse and dense regimes (see Nepf (Annu. Rev. Fluid Mech., vol. 44, 2012, pp. 123–142)). The other two $h/H$ selected ratios are representative of transitional regimes, a generic category that incorporates all the non-asymptotic states. While the interaction of a turbulent flow with a filamentous canopy in the two asymptotic conditions is relatively well understood, not much is known on the transitional flows and on the physical mechanisms that are responsible for the variations of flow regimes when the canopy solidity is changed. The effects of the latter on the flow developing in the intra-canopy region, on the outer flow and on their mutual interactions have been numerically explored and are reported in this work. By systematically varying the canopy height, we have unravelled the main character of the different regimes that are generated by the interplay between the outer flow structures, the emerging instabilities driven by the canopy drag and the interstitial flow between the canopy stems. The key role played by the relative positions of the inflection points of the mean velocity profile and the location of the virtual wall origin (as seen from the outer flow) is put forward and used to define a new condition to infer the canopy flow regime when the solidity is changed. Finally, the presence and the effects of an instability occurring close to the bed, nearby the interior inflectional point of the mean velocity profile is highlighted together with its consequences on the flow structure within the canopy region. To the best of our knowledge, this is the first time that the emergence of close-to-the-bed coherent structures induced by the inner inflection point is reported in the literature.