The lift and drag forces acting on a small spherical particle moving with a finite slip in single-wall-bounded flows are investigated via direct numerical simulations. This study is an extension of our previous work that considered the lift and drag forces acting on a sphere moving near a wall in the presence shear, but in the absence of slip (Ekanayake et al., J. Fluid Mech., vol. 904, 2020, A6). The effect of slip velocity on the particle force is analysed as a function of separation distance for low slip and shear Reynolds numbers ($10^{-3} \leq Re_{{slip}} \leq 10^{-1}$ and $10^{-3} \leq Re_{\gamma } \leq 10^{-1}$) in both quiescent and linear shear flows. A generalised lift model valid for arbitrary particle–wall separation distances and $Re_{\gamma }, Re_{{slip}} \leq 10^{-1}$ is developed based on the results of the simulations. The proposed model can now predict the lift forces in linear shear flows in the presence or absence of slip, and in quiescent flows when slip is present. Existing drag models are also compared with numerical results for both quiescent and linear shear flows to determine which models capture near-wall slip velocities most accurately for low particle Reynolds numbers. Finally, we compare the results of the proposed lift model to previous experimental results of negatively buoyant particles and to numerical results of neutrally buoyant (force-free) particles moving near a wall in quiescent and linear shear flows. The generalised lift model presented can be used to predict the behaviour of particle suspensions in biological and industrial flows where the particle Reynolds numbers based on slip and shear are ${O}(10^{-1})$ and below.

We present results of $\approx$300 miscible Newtonian displacement flow experiments carried out in a dimensionally scaled laboratory set-up. Annulus eccentricity, density difference and viscosity of the fluids are varied, over a wide range of laminar Reynolds numbers. Comparisons with predictions from the two-dimensional gap-averaged (2DGA) model of Carrasco-Teja et al. (J. Fluid Mech., vol. 605, 2008, pp. 293–327) show excellent agreement in predicting the underlying competition between buoyancy and eccentricity, which results in either top side or slumping flows. Other features of the experiments are not predicted as well. The main discrepancy results from a variety of dispersive effects that are not present in the 2DGA model, e.g. dispersion within the annular gap and due to azimuthal secondary flows. We find that dispersive effects dominate to the extent that the slumping flows are best described by bulk diffusive spreading of the height-averaged concentrations, relative to the mean flow. A variety of flow structures and wave-like instabilities are discussed. The study is relevant to the oilfield process of primary cementing of horizontal wells.

In the flamelet regime of turbulent premixed combustion the enhancement in the burning rates originates primarily from surface wrinkling. In this work we investigate the Reynolds number dependence of burning rates of spherical turbulent premixed methane/air flames in decaying isotropic turbulence with direct numerical simulations. Several simulations are performed by varying the Reynolds number, while keeping the Karlovitz number the same, and the temporal evolution of the flame surface is compared across cases by combining the probability density function of the radial distance of the flame surface from the origin with the surface density function formalism. Because the mean area of the wrinkled flame surface normalized by the area of a sphere with radius equal to the mean flame radius is proportional to the product of the turbulent flame brush thickness and peak surface density within the brush, the temporal evolution of the brush and peak surface density are investigated separately. The brush thickness is shown to scale with the integral scale of the flow, evolving due to decaying velocity fluctuations and stretch. When normalized by the integral scale, the wrinkling scale defined as the inverse of the peak surface density is shown to scale with Reynolds number across simulations and as turbulence decays. As a result, the area ratio and the burning rate are found to increase as ${Re}_{\lambda }^{1.13}$, in agreement with recent experiments on spherical turbulent premixed flames. We observe that the area ratio does not vary with turbulent intensity when holding the Reynolds number constant.

The linear stability of thermal buoyant flow in a fluid-saturated vertical porous slab is studied under the assumption of weak and strong horizontal heterogeneities of the permeability. The two end vertical isothermal boundaries are impermeable and some paradigmatic cases of linear, quadratic and exponential heterogeneity models are deliberated. The stability/instability of the basic flow is examined by carrying out a numerical solution of the governing equations for the disturbances as Gill's proof (A.E. Gill, J. Fluid Mech, vol. 35, 1969, pp. 545–547) of linear stability is found to be ineffective. The possibilities of base flow becoming unstable due to heterogeneity in permeability are recognized, in contrast to Gill's stability problem. The neutral stability curves are presented and the critical Darcy–Rayleigh number for the onset of convective instability is computed for different values of the variable permeability constant. The similarities and differences between different heterogeneity models on the stability of fluid flow are clearly discerned.

The development of the governing equations for fluid flow in a surface-following coordinate system is essential to investigate the fluid flow near an interface deformed by propagating waves. In this paper, the governing equations of fluid flow, including conservation of mass, momentum and energy balance, are derived in an orthogonal curvilinear coordinate system relevant to surface water waves. All equations are further decomposed to extract mean, wave-induced and turbulent components. The complete transformed equations include explicit extra geometric terms. For example, turbulent stress and production terms include the effects of coordinate curvature on the structure of fluid flow. Furthermore, the governing equations of motion were further simplified by considering the flow over periodic quasi-linear surface waves wherein the wavelength of the disturbance is large compared to the wave amplitude. The quasi-linear analysis is employed to express the boundary layer equations in the orthogonal wave-following curvilinear coordinates with the corresponding decomposed equations for the mean, wave and turbulent fields. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocity–vorticity formulations. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over wind-generated surface waves. Experimental results are presented in a companion paper.

Lee waves generated by stratified flow over rough bottom topography in the ocean extract momentum and energy from the geostrophic flow, causing drag and enhancing turbulence and mixing in the interior ocean when they break. Inviscid linear theory is generally used to predict the generation rate of lee waves, but the location and mechanism of wave breaking leading to eventual dissipation of energy and irreversible mixing are poorly constrained. In this study, a linear model with viscosity, diffusivity and an upper boundary is used to demonstrate the potential importance of the surface in reflecting lee wave energy back into the interior, making the case for treating lee waves as a full water-column process. In the absence of critical levels, it is shown that lee waves can be expected to interact with the upper ocean, resulting in enhanced vertical velocities and dissipation and mixing near the surface. The impact of the typical oceanic conditions of increasing background velocity and stratification with height above bottom are investigated and shown to contribute to enhanced upper ocean vertical velocities and mixing.

We study the dynamics of a cavitation bubble near beds of sand of different grain sizes. We use high-speed imaging to observe the motion of the bubble and the sand for different values of the stand-off parameter $\gamma$ (dimensionless bubble-boundary distance) between 0.3 and 5.3. Compared with a rigid boundary, we find that a granular boundary leads to bubbles with shorter lifetimes and reduced centroid displacements. Above $\gamma \approx 1.3$, the behaviour of the bubble is almost independent of the granularity of the sand. When the stand-off parameter lies between $0.6$ and $1.3$, a mound of sand develops beneath the bubble, which can force the latter to assume a conical shape as it collapses. For $\gamma \lesssim 0.6$, the bubble develops a bell-shaped form, leading to the formation of thin and surprisingly fast micro-jets ($v_{jet} > 1000\,{\rm m}\,{\rm s}^{-1}$). Moreover, between $\gamma \approx 1.3$ and $\gamma \approx 0.3$, granular jets erupt from the sand surface following the bubble collapse. We additionally develop a simple numerical model, based on the boundary integral method, to predict the dynamics of a cavitation bubble near a bed of sand, which we replace by an equivalent liquid. The simulations are remarkably consistent with experimental observations for values of $\gamma$ down to $1.3$. We also show how the anisotropy parameter $\zeta$, a dimensionless version of the Kelvin impulse, can be adapted for the case of a nearby bed of sand and predict the displacement of the bubble centroid for $\zeta \lesssim 0.08$.

We present an exploratory study of the hydrodynamical interaction between two bubbles rising at high Reynolds numbers in a thin-gap cell. When they are isolated, the bubbles exhibit oscillatory motions and develop an unsteady wake with periodic release of vortices. Experiments combine bubble tracking and measurements of the liquid velocity field through volumetric time-resolved particle image velocimetry. This enabled us to analyse the kinematics of the bubbles during their interaction in relationship with the liquid flow field induced by their motion and governing their behaviour. We first investigate how the kinematics of a bubble, already submitted to the intrinsic instability of its path and wake, is modified by the interaction, i.e. by the presence of a liquid flow field generated by the companion bubble. Two main effects are highlighted in association with (i) the role of the ascending flow generated by the leading bubble, and of its spatial evolution, leading to a slowly varying vertical entrainment of the trailing bubble, and (ii) the role of the vortices released by the leading bubble inducing strong localized horizontal deviations on a bubble in line or in oblique positioning. In the latter case, two major scenarios are identified: deviations of the trailing bubble towards the wake centre line (centring in the wake) or away from it (ejection from the wake). We also show that a regular succession of ejections and re-alignments events may take place (cyclic alternation of ejections and centrings). The analysis is built on the knowledge of the behaviour of isolated bubbles, which is used as the basis for comparison to characterize the effect of the interaction, for the modelling of the vertical entrainment, and for the definition of a criteria on a dimensionless parameter characterizing the ability of a vortex to drive the bubble motion. In turn, we investigate the effect of a bubble passage in the liquid flow field generated by the companion bubble, highlighting the destruction or reinforcement of vortices. We show in particular that both effects can occur without a significant impact on the bubble kinematics.

This study investigates deterministic wave forecasting from the perspective of the Zakharov equation. Forecasts based on linear dispersion, weakly nonlinear amplitude dispersion and the Zakharov equation are compared for reference ocean surfaces generated from the Joint North Sea Wave Observation Project and Pierson–Moskowitz spectra. This approach allows for the role of nonlinearity to be isolated, and demonstrates the success of simple frequency corrections in forecasting wave fields up to moderate steepness. The role of second-order bound waves is investigated by means of analytical formulae, and their impact on forecast accuracy is illustrated for a range of forecast times.

The transient dynamics of accelerating turbulent pipe flow has been examined using direct numerical simulation (DNS) data sets with a high spatiotemporal resolution, starting from low and moderate Reynolds numbers. The time-dependent evolution of the mean flow dynamics reveals that internal flows, during and after a rapid increase in the flow rate, experience four unambiguous transient stages: inertial, pre-transition, transition and core relaxation before they reach their final steady-state. The first stage is characterised by a rapid and substantial increment in the viscous forces within the viscous sublayer, together with the frozen behaviour of the existing turbulent eddies. The pre-transitional stage reveals a weak response of the turbulent inertia within the near-wall region, together with a rapid reduction in the viscous forces. At the third stage, termed transition, balanced growth in the magnitude of both the turbulent and the viscous forces within $y^{+0} \lesssim 50$ is observed. The final stage, referred to as core relaxation, shows a quasi-steady behaviour at $y^{+0} < 50$ and reveals the slow propagation of turbulence from the near-wall region into the wake region. A decomposition of the skin friction coefficient ($C_f$), using an FIK identity suitable for unsteady pipe flow, shows a progressive increment in the turbulent contribution during the core-relaxation period. Simultaneously, the unsteady contribution decreases proportionally, maintaining a plateau in $C_f$. The principal mechanism responsible for this slow regeneration in the wake is a temporal turbulence stratification at the inner region of the flow, together with a quiescent core, which maintains geometric coherence across extensive periods.

Characteristics of particle clustering in a particle-laden turbulent boundary layer at a moderate frictional Reynolds number ($Re_\tau =5500$) are experimentally investigated based on Voronoï analysis. High-inertia sand grains with a large viscous-time-based Stokes number, i.e. $St_p=10^2 - 10^3$, are used as the dispersed phase. The bulk particle volume fraction is $\varPhi _V\sim O(10^{-5})$. Two-dimensional velocity fields of the fluid and particle phases in the streamwise–wall-normal plane are simultaneously measured via particle image/tracking velocimetry. Under the net sedimentation condition, a self-similarity of the geometries of particle clusters is clearly seen in the log layer. This can be characterized by the $-5/3$ power law of the probability density functions of the particle cluster areas and the $y$-scaling of their first- and second-order moments. Considering the self-similarity of wall-attached structures in the attached eddy hypothesis, we propose a conceptual model in which a large proportion of particle clusters prefer to reside on the back ridges of low-momentum attached $u$-structures due to the high-strain effect caused by converging sweep–ejection events. Such a scenario can be partly evidenced by the conditional streamwise spectra of the streamwise and wall-normal fluid-phase velocity components being conditioned inside cluster-occupied regions, which present a distinct $y$-scaling in the log layer. Furthermore, the fluid-phase sweep and ejection events inside particle clusters are observed to be weaker than the cases outside cluster-occupied regions, and the local rate of strain inside cluster-occupied regions is relatively higher than in all unconditional statistics. Finally, conditional statistics reveal that those sand grains with relatively small slip velocities are prone to be captured by low-momentum turbulent motions to form particle clusters.

Fish schools and bird flocks, which involve a variety of orderly formations, have fascinated people for a long time. The collective performance of two flapping foils in parallel, which can self-propel in both lateral and longitudinal directions, is numerically studied through two-dimensional simulations in this paper. For the first time it is numerically confirmed that two flapping swimmers can simultaneously converge to equilibrium distances in both lateral and longitudinal directions. Two types of stable formations have been observed and which type occurs depends on the phase difference between two foils. The staggered formation appears in the in-phase scenario, and the side-by-side formation occurs in the anti-phase scenario. Moreover, both types of the stable formations strongly depend on the flapping frequency and amplitude, but are independent of the perturbation of the initial distance between the two foils when the perturbation is small. In addition, considerable velocity enhancement can be achieved by two foils in all of the stable formations, as compared with that of a single foil. Moreover, the velocity of the two-foil system can be couched as a specific scaling law, which is determined by the type of the stable formations. Finally, the mechanism behind the stable formations has been analysed. The results obtained here may shed some light on the understanding of collective behaviour of fish schools and bird flocks.