Surface reactions such as the adsorption and desorption at boundaries are very common for solute dispersion in many applications of chemistry, biology, hydraulics, etc. To study how reversible adsorption affects the transient dispersion, Zhang, Hesse & Wang (J. Fluid Mech., vol. 828, 2017, pp. 733–752) have investigated the temporal evolution of moments using the Laplace transform method. Owing to difficulties introduced by the adsorption–desorption boundary condition, great challenges arise from the inverse Laplace transform: dealing with the singularities by the residue theorem can tremendously increase complexities. This work provides a much simpler analytical method to derive solutions in a more compact form that is valid for the entire range of the reactive transport process. Such a progress demonstrates that the classic framework of separation of variables can be extended and applied to this more general adsorption–desorption condition, based on which higher-order statistics including skewness and kurtosis can be explicitly explored in practice. Also extended is Gill's generalised dispersion model for solute concentration distributions, which can now address the entire transient dispersion characteristics, instead of just applied for the long-time asymptotic reactive process as done previously. Regarding the most classic Taylor dispersion problem, we investigate the influence of the reversible adsorption–desorption on the solute cloud in a tube flow. Not only the transient dispersion characteristics of transverse-average concentration distribution but also those of the bulk, surface and total-average distributions are discussed. We further investigate the influence of initial conditions on the non-uniformity of the transient dispersion over the cross-section.

Flexible vegetation plays key roles in the natural channel and riparian ecology. This paper investigates the drag coefficient C_df of an emergent flexible vegetation array in a steady nonuniform flow. Laboratory flume tests of five vegetation densities exposed to a constant flow rate were conducted, and the variation in the water surface was accurately measured. The Saint‐Venant equation was applied to explore vegetation hydrodynamics under nonuniform conditions, and a more widely applicable formula for the vegetated drag of nonuniform flow was proposed. Furthermore, the flexible drag coefficient factor α_b was used to represent the flexibility and drag reconfiguration, which were explored from the perspective of material mechanics. The results reveal that the calculated values of C_df exhibit nonmonotonic variation with increasing Reynolds number along the streamwise direction due to the flow nonuniformity. There are two effects of blockage and sheltering in the emergent vegetated patch: The blockage effect increases the drag coefficient, while the sheltering effect—especially at the leading edge and tail of the vegetated patch—reduces the drag of the flexible canopy due to bending deformation. These two effects achieve equilibration in dense flexible canopies. Finally, by relating α_b to flexibility through the slenderness Cauchy number C_YS, a nonmonotonic pattern can be obtained for each test. A fitting formula was proposed based on α_b as a function of C_YS. The results presented in this study can potentially support applications related to the plant flexible dams and vegetated filter strips for river ecosystems.

Understanding the statistics of bedload particle motions is of great importance. To model the hop events which are defined as trajectories of particles moving successively from the start to the end of their motions, recently, Wu et al. (Water Resour. Res., vol. 56, 2020, p. e2019WR025116) have successfully performed individual-based simulations according to the Fokker–Planck equation for particle velocities. However, analytical solutions are still not available due to (i) difficulties in treating the velocity-dependent diffusivity, and (ii) a knowledge gap in incorporating the termination of particle motions for the equation. To tackle the above-mentioned challenges, we first specify a Robin boundary condition representing the deposition of particles. Second, for analytical solutions of hop statistics, a variable transformation is devised to deal with the velocity-dependent diffusivity. The original bedload transport problem is thus found to be governed by the classic equation for the solute transport in tube flows with a constant diffusivity after the transformation. Finally, through solving the spatial and temporal moments of the governing equation, we investigate the influence of the deposition rate on three key characteristics of particle hops. Importantly, we have related the deposition rate to the mean travel times and hop distances, enabling a direct determination of this physical parameter based on measured particle motion statistics. The analytical solutions are validated by experimental observations with different bedload particle diameters and transport conditions. Based on the limited experimental datasets, the deposition frequency is shown to decrease as the shear stress increases when the flow rate is not small.

The accumulation of swimming microorganisms at surfaces is an essential feature of various physical, chemical, and biological processes in confined spaces. To date, this accumulation is mainly assumed to depend on the change of swimming speed and angular velocity caused by cell-wall contact and hydrodynamic interaction. Here, we measured the swimming trajectories of Heterosigma akashiwo (a biflagellate marine alga) near vertical and horizontal rigid boundaries. We observed that the probability of sharp turns is greatly increased near a vertical wall, resulting in significant changes in the distributions of average swimming speed, angular velocity, and rotational diffusivity near the wall (a quantity that has not previously been investigated) as functions of both distance from the wall and swimming orientation. These cannot be satisfactorily explained by standard hydrodynamic models. Detailed examination of an individual cell trajectory shows that wall contact by the leading flagellum triggers complex changes in the behavior of both flagella that cannot be incorporated in a mechanistic model. Our individual-based model for predicting cell concentration using the measured distributions of swimming speed, angular velocity, and rotational diffusivity agrees well with the experiment. The experiments and model are repeated for a cell suspension in a vertical plane, bounded above by a horizontal wall. The cell accumulation beneath the wall, expected from gyrotaxis, is considerably amplified by cell-wall interaction. These findings may shed light on the prediction and control of cell distribution mediated by gyrotaxis and cell-wall contact.

When motile algal cells are exposed to gyrotactic torques, their swimming directions are guided to form radial accumulation, well known as hydrodynamic focusing. The origin of hydrodynamic focusing from the effects of active swimming, ambient flow and particle anisotropy is elucidated in the present study on the pre-asymptotic dispersion of active particles through a vertical pipe. With an extension of the Galerkin method to pipe flows, time-dependent solutions directly from the Smoluchowski equation in the position and orientation space are derived by series expansions of spherical harmonics and Bessel functions. Ballistic and diffusive scaling laws are examined with the predominance of self-propelled swimming, and computation is validated against an explicit benchmark solution and Lagrangian particle simulation. In the limit of extreme shear, the competitive roles of shear dispersion and Brownian rotation are reflected concretely in the pre-asymptotic phase of hydrodynamic focusing. For flows with various shear strengths, a concentration peak in near-wall regions with a smooth transition to hydrodynamic focusing is illustrated with richer phenomena in upwelling and downwelling flows. A newly observed regime through a vertical pipe, named transient effective trapping, is revealed as a transitional mode towards hydrodynamic focusing. The pre-asymptotic approach to hydrodynamic focusing is elaborated intensively through extensive solutions of concentration moments and macroscopic transport coefficients characterised by swimming and flow Péclet numbers. The unique findings for the origin of hydrodynamic focusing could provide insight into related micro-algae reactor technology and contribute to flow control and biomass transfer in confined environments.

Understanding the swimming characteristics of micro-organisms is significant for modelling the migration of motile cells and corresponding ecological risk assessments associated with harmful algae in oceans and estuaries. Presented in this paper is an experimental and numerical investigation of swimming characteristics of a typical gyrotactic micro-organism, Heterosigma akashiwo (H. akashiwo) in water, based on the technology of planar laser-induced fluorescence and the finite volume method. Two-dimensional swimming velocity of algal cells are obtained by analyzing cells’ trajectories in the vertical plane, and three-dimensional swimming velocity is reconstructed based on the assumption that cells’ swimming is isotropic in the horizontal plane. Four important parameters are given to reflect the swimming characteristics of gyrotactic cells in still water, including the mean swimming speed (V_s = 146 μm/s), the relative strength of reorientation by gravitational torque to rotational diffusion (λ = 1.96), the time scale of reorientation (B = 5.6 s), and rotational diffusivity (D_r = 0.046 rad²/s). A database of the ambient vorticity, mean swimming velocity and diffusivity tensor is established, by solving Fokker-Planck equation for the probability density function of cells’ swimming under the combined action of gravity, rotational diffusion, and the ambient vorticity. The mean swimming velocity and translational diffusion tensor of H. akashiwo are found to change with the horizontal and vertical vorticity. It is also shown that gyrotactic cells swim in a given direction for a weak horizontal vorticity, in contrast to cells’ tumbling and being trapped for a strong horizontal vorticity.