An exact analytical solution is obtained for the dynamical system derived in Part 1 of this series (Moffatt & Kimura, J. Fluid Mech, vol. 861, 2019a, pp. 930–967), which describes the approach of two initially circular vortices of finite but small cross-section symmetrically located on inclined planes. This exact solution, applicable in the inviscid limit, allows determination of the amplification $\mathcal {A}_{\omega }$ of the axial vorticity within the finite time $T$ during which the basic assumptions of the model continue to apply. It is first shown that, for arbitrarily prescribed $\mathcal {A}_{\omega }$, it is possible to specify smooth initial conditions of finite energy such that, in the inviscid limit, this amplification is achieved within the time $T$. When viscosity is included, an estimate is provided for the minimum vortex Reynolds number that is sufficient for the same result to hold. The predictions are broadly compatible with results from direct numerical simulations at moderate Reynolds numbers. Moreover, it is shown that one may come arbitrarily close to a finite-time singularity of the Navier–Stokes equation by appropriate choice of an initial, smooth, finite-energy velocity field; however, this approach to a singularity is ultimately thwarted through breach of the assumptions on which the dynamical system is based. Thus we make no claim here concerning realisation of a Navier–Stokes singularity. Moreover, we find that the conditions required to attain a large amplification $\mathcal {A}_{\omega }\gg 1$ during the time $T$ are far beyond those that can be realised in either experiment or direct numerical simulation.

Boundary-layer flow over a realistic porous wall might contain both the effects of wall-permeability and wall-roughness. These two effects are typically examined in the context of a rough-wall flow, i.e. by defining a ‘roughness’ length or equivalent to capture the effect of the surface on momentum deficit/drag. In this work, we examine the hypothesis of Esteban et al. (Phys. Rev. Fluids, vol. 7, no. 9, 2022, 094603), that a turbulent boundary layer over a porous wall could be modelled as a superposition of the roughness effects on the permeability effects by using independently obtained information on permeability and roughness. We carry out wind tunnel experiments at high Reynolds number ($14\ 400 \leq Re_{\tau } \leq 33\ 100$) on various combinations of porous walls where different roughnesses are overlaid over a given permeable wall. Measurements are also conducted on the permeable wall as well as the rough walls independently to obtain the corresponding length scales. Analysis of mean flow data across all these measurements suggests that an empirical formulation can be obtained where the momentum deficit ($\Delta U^+$) is modelled as a combination of independently obtained roughness and permeability length scales. This formulation assumes the presence of outer-layer similarity across these different surfaces, which is shown to be valid at high Reynolds numbers. Finally, this decoupling approach is equivalent to the area-weighted power-mean of the respective permeability and roughness length scales, consistent with the approach recently suggested by Hutchins et al. (Ocean Engng, vol. 271, 2023, 113454) to capture the effects of heterogeneous rough surfaces.

Compound surfaces, consisting of periodic arrays of solid patches and free surfaces, exhibit hydrodynamic slipperiness which is quantified by their slip length. The limit of small solid fractions, where the slip length diverges, is of fundamental interest. This paper addresses longitudinal liquid flows over a periodic array of grooves which are partially invaded by the liquid. Assuming that the slats separating the grooves are infinitely thin, the solid fraction $\epsilon$ is set by the invasion depth. Inspired by the singular small-solid-fraction limit for non-invaded grooves (Schnitzer, Phys. Rev. Fluids, vol. 1, 2016, 052101R), we consider the idealised geometry of $90^{\circ }$ protrusion angles, where an integral force balance in the limit $\epsilon \to 0$ implies a slip length that scales as $\epsilon ^{-1}$. The problem exhibits a nested structure, where the liquid domain is conceptually decomposed into four distinct regions: a unit-cell region on the scale of the period, where the wetted portion of the slat appears as a point singularity; two regions on the scale of the wetted slat, where the flow essentially varies in one dimension; and a transition region about the tip of the slat. Analysing these regions using matched asymptotic expansions and conformal mappings yields the ratio of slip length to semi-period as $2\epsilon ^{-1} - (10/{\rm \pi} )\ln 2 + \cdots$.

Non-spherical particles transported by turbulent flow have a rich dynamics that combines their translational and rotational motions. Here, the focus is on small, heavy, inertial particles with a spheroidal shape fully prescribed by their aspect ratio. Such particles undergo an anisotropic, orientation-dependent viscous drag with the carrier fluid flow whose associated torque is given by the Jeffery equations. Direct numerical simulations of homogeneous, isotropic turbulence are performed to study systematically how the translational motion of such spheroidal particles depends on their shape and size. It is found that the Lagrangian statistics of both velocity and acceleration can be described in terms of an effective Stokes number obtained as an isotropic average over angles of the particle's orientation. Corrections to the translational motion of particles due to their non-sphericity and rotation can hence be recast as an effective radius obtained from such a mean.

The thermo-osmotic flow (TOF) of an electrolyte solution in a slit channel with a Navier slip condition at the channel walls is studied. An analytical expression for the TOF velocity profile, based on the long-wavelength and Debye–Hückel approximations, is derived and compared to numerical solutions based on the finite-element method. The TOF between graphene surfaces whose charge is created via polarization through an applied electric field is considered as a special case. Using the relationship between the surface charge and the slip length obtained from molecular dynamics simulations, a giant flow amplification is uncovered. Specifically, for such flow in a channel with a width of 10 nm, compared to the flow between no-slip walls, a flow velocity enhancement by a factor of up to 250 is predicted.

The aim of this work is to analyse the formation mechanisms of large-scale coherent structures in the flow around a wall-mounted square cylinder, due to their impact on pollutant transport within cities. To this end, we assess causal relations between the modes of a reduced-order model obtained by applying proper orthogonal decomposition to high-fidelity simulation data of the flow case under study. The causal relations are identified using conditional transfer entropy, which is an information-theoretical quantity that estimates the amount of information contained in the past of one variable about another. This allows for an understanding of the origins and evolution of different phenomena in the flow, with the aim of identifying the modes responsible for the formation of the main vortical structures. Our approach unveils that vortex-breaker modes are the most causal modes, in particular, over higher-order modes, and no significant causal relationships were found for vortex-generator modes. We validate this technique by determining the causal relations present in the nine-equation model of near-wall turbulence developed by Moehlis et al. (New J. Phys., vol. 6, 2004, p. 56), which are in good agreement with literature results for turbulent channel flows.

A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. These functions serve as Galerkin basis functions, capturing correct analyticity and boundary conditions for vortices in an unbounded domain. The incompressible Euler or Navier–Stokes equations linearised on a swirling flow are transformed into a standard matrix eigenvalue problem of toroidal and poloidal streamfunctions, solving perturbation velocity eigenmodes with their complex growth rate as eigenvalues. This reduces the problem size for computation and distributes collocation points adjustably clustered around the vortex core. Based on this method, strong swirling $q$ vortices with linear perturbation wavenumbers of order unity are examined. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. The inviscid critical-layer eigenmodes numerically tend to appear in pairs, implying their singular degeneracy. With viscosity, the spectra pertaining to physical regularisation of critical layers stretch out toward an area, referring to potential eigenmodes with wavepackets found by Mao & Sherwin (J. Fluid Mech., vol. 681, 2011, pp. 1–23). However, the potential eigenmodes exhibit no spatial similarity to the inviscid critical-layer eigenmodes, doubting that they truly represent the viscous remnants of the inviscid critical-layer eigenmodes. Instead, two distinct continuous curves in the numerical spectra are identified for the first time, named the viscous critical-layer spectrum, where the similarity is noticeable. Moreover, the viscous critical-layer eigenmodes are resolved in conformity with the $Re^{-1/3}$ scaling law. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracy.

The structure and the frequency spectra of wall pressure fluctuations beneath a planar turbulent boundary layer interacting with a conical shock wave at Mach number $M_\infty =2.05$ and Reynolds number $\textit {Re}_\theta \approx 630$ (based on the upstream boundary layer momentum thickness) are examined to elucidate the effects of pressure gradient and flow separation on the characteristics of the wall pressure fluctuations, by exploiting a direct numerical simulation database. Upstream of the interaction, in the zero pressure gradient region, wall pressure statistics compare well with canonical compressible boundary layers in terms of fluctuation intensities and frequency spectra. Across the main interaction zone (APG1), the root-mean-square of wall pressure fluctuations becomes very large (corresponding to approximately 173.3 dB), with maximum increase approximately 12.7 dB from the incoming level. In the second adverse pressure gradient zone (APG2), the root-mean-square of wall pressure fluctuations attains a second peak (corresponding to $164.7$ dB), with an increase of 8.4 dB from the upstream level. Both the APG1 and APG2 regions feature a substantial fraction of flow reversal events, which are, however, scattered and interspersed with regions of attached flow. The wall pressure power spectral density exhibits a broadband and energetic low-frequency component associated with the global unsteadiness of the separation bubble/conical shock system. Analysis of the two-point correlations and wavenumber/frequency spectra of wall pressure fluctuations further suggests that the typical eddies become more elongated along the spanwise direction, as the flow in the separated region tends to escape the centreline, and the convection velocity is significantly reduced.

Shock buffet on wings is a phenomenon caused by strong shock-wave/boundary-layer interaction resulting first in self-sustained flow unsteadiness and eventually in a detrimental structural response called buffeting. While it is an important aspect of wing design and aircraft certification, particularly for modern transonic air transport, not all of the underlying multidisciplinary physics is thoroughly understood. Building upon a single-discipline shock-buffet stability study, this work now investigates the impact of an elastic structure in these extreme flow conditions. Specifically, a triglobal stability analysis of a fluid–structure coupled system is presented, utilising the implicitly restarted Arnoldi method with a sparse iterative Krylov solver and novel preconditioner. Asymmetry resulting from a static aeroelastic simulation based on a finite-element model of the underlying geometry in a wind tunnel modifies the global modes of the earlier fluid-only symmetric full-span analysis. A flutter stability analysis at wind-tunnel flow conditions below shock-buffet onset finds no instability in the structural degrees-of-freedom, whereas in shock-buffet flow with globally unstable fluid modes additional marginally unstable structural (and fluid) modes emerge. The developed stability tool for coupled analysis is instrumental in identifying those physically relevant and strongly coupled modes where a standard pk-type (p being eigenvalue and k reduced frequency) flutter analysis fails. With the complementary computation of adjoint eigenmodes, the core of the instability is pinpointed to a relatively small wing area which may help to effect the control and delay of this detrimental transonic unsteadiness. We contribute to the question on how the presence of the elastic wing structure impacts on the otherwise pure aerodynamic three-dimensional shock-buffet dynamics.

The propagation of a finite blister of fluid beneath an elastic sheet is controlled by the local dynamics at the peeling periphery of the current. Previous works have described constant volume elastic blisters by considering the peeling due to curvature at these edges. Here, we show that an along-slope component of gravity fundamentally changes the dynamics by removing the role of curvature at the trailing, upslope edge. The local dynamics of this trailing edge is instead controlled by shear stress in the sheet, as in the elastic Landau–Levich problem, and thereby allows for a receding edge, in contrast to propagation by peeling for which only an advancing contact line is possible. Using an asymptotic analysis, we show that this receding edge condition allows for a new, nearly translating regime in which the body of the blister moves at an approximately constant speed, leaving behind a thin layer of fluid. This prediction is verified by detailed numerical modelling of the two-dimensional downslope spreading. We conclude by discussing the applicability of these results in the rapid spreading of subglacial meltwater.

Power minimisation of fluid transport in branched fluidic networks has become of paramount importance for microfluidics, additive manufacturing and hierarchical functional materials. For fully developed laminar flow of Newtonian fluids, Murray's theory provides a solution for the channel and network dimensions that minimise power consumption. However, design and optimisation of networks that transport complex fluids is still challenging. Here, we generalise Murray's theory towards fluid rheologies, including non-Newtonian (power-law) and yield-stress fluids (Bingham, Herschel–Bulkley, Casson). A straightforward graphical approach is presented that provides the optimal radii in a branching network, and the angles between these branches. The wall shear stress is found to be uniform over the entire network, and the velocity profile is self-similar. Furthermore, the effect of non-optimal channel radii on the power consumption of the network is investigated. Finally, examples illustrate how this approach applies to a wide variety of systems.

The development of novel drug delivery systems, which are revolutionizing modern medicine, is benefiting from studies on microorganisms’ swimming. In this paper we consider a model microorganism (a squirmer) enclosed in a viscous droplet to investigate the effects of medium heterogeneity or geometry on the propulsion speed of the caged squirmer. We first consider the squirmer and droplet to be spherical (no shape effects) and derive exact solutions for the equations governing the problem. For a squirmer with purely tangential surface velocity, the squirmer is always able to move inside the droplet (even when the latter ceases to move as a result of large fluid resistance of the heterogeneous medium). Adding radial modes to the surface velocity, we establish a new condition for the existence of a co-swimming speed (where squirmer and droplet move at the same speed). Next, to probe the effects of geometry on propulsion, we consider the squirmer and droplet to be in Newtonian fluids. For a squirmer with purely tangential surface velocity, numerical simulations reveal a strong dependence of the squirmer's speed on shapes, the size of the droplet and the viscosity contrast. We found that the squirmer speed is largest when the droplet size and squirmer's eccentricity are small, and the viscosity contrast is large. For co-swimming, our results reveal a complex, non-trivial interplay between the various factors that combine to yield the squirmer's propulsion speed. Taken together, our study provides several considerations for the efficient design of future drug delivery systems.

In dispersed two-phase flows, particle electrification is a prevalent phenomenon that plays a crucial role in particle transport. However, the influences of electrostatic forces on particle behaviour in wall-bounded turbulent flows, especially in bidisperse cases, is not well understood. In this study, using direct numerical simulations based on a coupled Eulerian–Lagrangian point-particle approach at friction Reynolds number $Re_\tau =550$, we demonstrate that when the electrostatic Stokes number is of the order of $O(10^{-1})$, electrostatic forces could considerably alter particle behaviour in both monodisperse and bidisperse particle-laden turbulent channel flows. Specifically, the wall-normal profiles of the particle concentration are determined by the competition of turbophoresis, biased sampling and electrostatic effects. The electrostatic forces are found to reduce the concentrations of lighter particles by electrostatic drift directly, whereas they alter those of heavier particles by strengthening turbophoresis indirectly. With increasing electrical charge, the dynamics of the lighter particles remains approximately unchanged, but that of the heavier particles is modulated significantly due to their relatively strong particle–electrostatic interaction. In the near-wall region, electrostatic forces tend to homogenize the distribution of lighter particles in the spanwise direction by inhibiting the formation and destruction of particle clusterings and voids, thereby maintaining the anisotropic streaky clusterings. Furthermore, even though the clustering dynamics remains unchanged, the spatial extents of the clusterings at the channel centreline are suppressed (enhanced) by a factor of two, probably due to the remarkable reduction (increase) of particle concentration in this layer.

Direct numerical simulations of turbulent channel flow subjected to spanwise wall oscillations in the form of streamwise travelling waves (STW) were performed in an effort to elucidate the mechanism responsible for the observed drag reduction. We imposed large amplitudes to identify the proper effects of STW, while keeping the angular frequency and wavenumber fixed at a particular values. We primarily focus on the vorticity transport mechanism, to better understand the influence of STW actuation on the near-wall turbulence. We identify key terms appearing in the turbulent enstrophy transport equations that are directly linked to the STW actuation. The analysis reveals that the primary effect of the STW forcing is to attenuate the spanwise turbulent enstrophy at the wall, which is linked to the fluctuating wall shear stress. The suppression of the wall-normal turbulent enstrophy is deemed to be subordinate. To strengthen this point, we performed numerical experiments, where the streamwise fluctuating velocity, and consequently the spanwise vorticity, is artificially suppressed next to the wall. The anisotropic invariant maps show striking resemblance for large amplitude STW actuation and artificially forced cases. Detailed analysis of various structural features is provided, which includes the response of the near-wall streaks and shear layers of spanwise fluctuating velocity field. The quasistreamwise vortices, which play a key role in the regeneration mechanism, are shown to be pushed away from the wall, resulting in their weakened signature at the wall.

The present paper numerically investigates the fluid dynamics associated with the flow over an elastically mounted circular cylinder at a Reynolds number of 500. A slit normal to the flow direction is placed at the cylinder's centre. The study covers the large parametric set of calculations for the combined mass-damping ratio $m^{*}\zeta = 0.2$, including the slit offset for six different offset angles ($-30^\circ \leq \alpha \leq +30^\circ$) measured from the vertical axis at the centre point of the cylinder and the effect of slit widths ($0.1 \leq s/D \leq 0.3$) on the aerodynamic loading, vibration response and associated flow characteristics. Furthermore, a wide range of reduced velocities ($3\leq U_r \leq 7$) are examined for the complete closure of studying the effect on the vortex-induced vibration response. The results demonstrate that adding the normal slit increases the periodic suction-blowing phenomena, strengthening the vortex shedding. The results suggest that the normal slit can suppress or increase vortex-induced vibration depending on the slit-offset angle. Placing it toward the front stagnation point results in the increased oscillation amplitude (with a wider wake behind the cylinder) while shifting it toward the rear stagnation point diminishes the cylinder's oscillations. The paper reveals that this dual behaviour of the normal slit (based on its offset placement) is closely related to the phase difference between the lift force and the oscillation amplitude. Various vortex-shedding patterns associated with the different slit-offset angles are duly reported in the paper. Furthermore, a magnet is attached to the slit cylinder, and its effect on the energy harvesting capability via a coil-magnet arrangement is explored in detail for a wide range of reduced velocities.

Axisymmetric steady solutions of Taylor–Couette flow at high Taylor numbers are studied numerically and theoretically. As the axial period of the solution shortens from approximately one gap length, the Nusselt number goes through two peaks before returning to laminar flow. In this process, the asymptotic nature of the solution changes in four stages, as revealed by the asymptotic analysis. When the aspect ratio of the roll cell is approximately unity, the solution has the Nusselt number proportional to the quarter power of the Taylor number, and captures quantitatively the characteristics of the classical turbulence regime. By shortening the axial period the Nusselt number can even reach the experimental value around the onset of the ultimate turbulence regime. However, at higher Taylor numbers, the theoretical predictions eventually underestimate the experimental values. An important consequence of the asymptotic analyses is that the mean angular momentum should become uniform in the core region unless the axial wavelength is too short. The theoretical scaling laws deduced for the steady solutions can be carried over to Rayleigh–Bénard convection.

In this paper, we study the linear stability of a two-dimensional shear flow of an air layer overriding a water layer of finite depth. The air layer is considered to be of an infinite extent with an exponential velocity profile. Three different background conditions are considered in the finite-depth water layer: a quiescent background, a linear velocity profile and a quadratic velocity profile. It is known that the cases of the quiescent water layer and the linear velocity profile allow for analytical treatment. We further provide an analytical solution for the case of the quadratic velocity field: we specifically consider a flow-reversal profile, although the result could be generalized to other quadratic profiles as well. The role of water layer depth on the growth rate of the Miles and rippling instabilities is studied in each of the three cases. Using asymptotic analysis, with the air–water density ratio being a small parameter, we obtain an analytical expression for the growth rate of the Miles mode and discuss the condition for the existence of a long-wave cutoff for these profiles. We provide analytical expressions for the stability boundary in the parameter space of inverse squared Froude number and wavenumber. In scenarios where a long-wave cutoff does not exist, we have carried out a long-wave asymptotic study to obtain the growth rate behaviour in that regime.

We demonstrate that the relative motion of horizontal parallel plates can be generated using patterned heating. This movement is driven by nonlinear thermal streaming associated with a pitchfork bifurcation. The propulsive effect is strongest when all the heating energy is concentrated in a single Fourier mode of the spatial heating pattern; it increases with a decrease in the Prandtl number and increases with the addition of a uniform heating component.

The near-surface locomotion of microswimmers under the action of background flows has been studied extensively, whereas the intervening effects of complex surface properties remain hitherto unknown. Intending to delineate the shear-driven dynamics near a planar slippery wall, we adopt the squirmer model of microswimmers and employ a three-dimensional analytical-numerical framework in bispherical coordinates. It is interpreted that both the self-propulsion and the external shear flow are redistributed due to hydrodynamic slippage, followed by modulations in the thrust torque on the microswimmer. Phase portraits of the quasi-steady dynamics indicate that the stable upstream swimming states, known as ‘rheotaxis’, are significantly modulated by the slip length compared with the no-slip case. For puller swimmers, an intricate interplay among the modulated interfacial friction near a slippery surface, velocity gradients of the shear flow and the strength of the squirmer parameter promotes a critical shear rate beyond which a wide range of new rheotactic states exist. Consequently, an escaping microswimmer may exhibit rheotaxis or an existing rheotactic state annihilates due to crashing. Although stable states are absent for pushers without steric interactions, transitions from escaping and undamped oscillations to ‘rheotaxis’ occur in the presence of wall repulsion, but only until the other characteristics are overwhelmed by escape due to enhanced shear. Disclosing the ability of hydrodynamic slippage in broadening the scope of migration against a background flow for a wide range of parameters, the present work paves the way for investigations on the entrapment of microswimmers near complex pathways or sorting using selective rheotaxis.

The log law of the wall, joining the inner, near-wall mean velocity profile (MVP) in wall-bounded turbulent flows to the outer region, has been a permanent fixture of turbulence research for over hundred years, but there is still no general agreement on the value of the prefactor, the inverse of the Kármán ‘constant’ $\kappa$, or on its universality. The choice diagnostic tool to locate logarithmic parts of the MVP is to look for regions where the indicator function $\varXi$ (equal to the wall-normal coordinate $y^+$ times the mean velocity derivative $\mathrm {d} U^+/\mathrm {d} y^+$) is constant. In pressure-driven flows, however, such as channel and pipe flows, $\varXi$ is significantly affected by a term proportional to the wall-normal coordinate, of order $O({Re}_{\tau }^{-1})$ in the inner expansion, but moving up across the overlap to the leading $O(1)$ in the outer expansion. Here we show that, due to this linear overlap term, ${Re}_{\tau }$ values well beyond $10^5$ are required to produce one decade of near constant $\varXi$ in channels and pipes. The problem is resolved by considering the common part of the inner asymptotic expansion carried to $O({Re}_{\tau }^{-1})$, and the leading order of the outer expansion. This common part contains a superposition of the log law and a linear term $S_0 \,y^+{Re}_{\tau }^{-1}$, and corresponds to the linear part of $\varXi$, which, in channel and pipe, is concealed up to $y^+ \approx 500\unicode{x2013}1000$ by terms of the inner expansion. A new and robust method is devised to simultaneously determine $\kappa$ and $S_0$ in pressure-driven flows at currently accessible ${Re}_{\tau }$ values, yielding $\kappa$ values which are consistent with the $\kappa$ values deduced from the Reynolds number dependence of centreline velocities. A comparison with the zero-pressure-gradient turbulent boundary layer, further clarifies the issues and improves our understanding.