The flow development in a groove-modified channel consisting of flat and grooved walls was investigated by direct numerical simulations based on the Navier–Stokes equations at a Reynolds number of $5\times 10^{3}$ based on the full channel height and the bulk velocity. Simulations were performed for highly disturbed initial flow conditions leading to the almost instantaneous appearance of turbulence in channels with flat walls. The surface morphology was designed in the form of profiled grooves aligned with the flow direction and embedded in the wall. Such grooves are presumed to allow development of only the statistically axisymmetric disturbances. In contrast to the rapid production of turbulence along a flat wall, it was found that such development was suppressed over a grooved wall for a remarkably long period of time. Owing to the difference in the flow structure, friction drag over the grooved wall was more than 60 % lower than that over the flat wall. Anisotropy-invariant mapping supports the conclusion, emerging from analytic considerations, that persistence of the laminar regime is due to statistical axisymmetry in the velocity fluctuations. Complementary investigations of turbulent drag reduction in grooved channels demonstrated that promotion of such a state across the entire wetted surface is required to stabilize flow and prevent transition and breakdown to turbulence. To support the results of numerical investigations, measurements in groove-modified channel flow were performed. Comparisons of the pressure differentials measured along flat and groove-modified channels reveal a skin-friction reduction as large as $\text{DR}\approx 50\,\%$ owing to the extended persistence of the laminar flow compared with flow development in a flat channel. These experiments demonstrate that early stabilization of the laminar boundary layer development with a grooved surface promotes drag reduction in a fully turbulent flow with a preserving magnitude as the Reynolds number increases.

When a torus oscillates horizontally in a linearly stratified fluid, the wave rays form a double cone, one upward and one downward, with two focal points where the wave amplitude has a maximum due to wave focusing. Following a former study on linear aspects of wave focusing (Ermanyuk et al., J. Fluid Mech., vol. 813, 2017, pp. 695–715), we here consider experimental results on the nonlinear aspects that occur in the focal region below the torus for higher-amplitude forcing. A new non-dimensional number that is based on heuristic arguments for the wave amplitude in the focal area is presented. This focusing number is defined as $Fo=(A/a)\unicode[STIX]{x1D716}^{-1/2}f(\unicode[STIX]{x1D703})$, with oscillation amplitude $A$, $f(\unicode[STIX]{x1D703})$ a function for the variation of the wave amplitude with wave angle $\unicode[STIX]{x1D703}$, and $\unicode[STIX]{x1D716}^{1/2}=\sqrt{b/a}$ the increase in amplitude due to the focusing, with $a$ and $b$, respectively, the minor and major radius of the torus. Nonlinear effects occur for $Fo\geqslant 0.1$, with the shear stress giving rise to a mean flow which results in the focal region in a central upward motion partially surrounded by a downward motion. With increasing $Fo$, the Richardson number $Ri$ measured from the wave steepness monotonically decreases. Wave breaking occurs at $Fo\approx 0.23$, corresponding to $Ri=0.25$. In this regime, the focal region is unstable due to triadic wave resonance. For the different tori sizes under consideration, the triadic resonant instability in these three-dimensional flows resembles closely the resonance observed by Bourget et al. (J. Fluid Mech., vol. 723, 2013, pp. 1–20) for a two-dimensional flow, with only minor differences. Application to internal tidal waves in the ocean are discussed.

We consider the two- and three-dimensional spreading of a finite volume of viscous power-law fluid released over a denser inviscid fluid and subject to gravitational and capillary forces. In the case of gravity-driven spreading, with a power-law fluid having strain rate proportional to stress to the power , there are similarity solutions with the extent of the current being proportional to in the two-dimensional case and in the three-dimensional case. Perturbations from these asymptotic states are shown to retain their initial shape but to decay relatively as in the two-dimensional case and in the three-dimensional case. The former is independent of , whereas the latter gives a slower rate of relative decay for fluids that are more shear-thinning. In cases where the layer is subject to a constraining surface tension, we determine the evolution of the layer towards a static state of uniform thickness in which the gravitational and capillary forces balance. The asymptotic form of this convergence is shown to depend strongly on , with rapid finite-time algebraic decay in shear-thickening cases, large-time exponential decay in the Newtonian case and slow large-time algebraic decay in shear-thinning cases.

The two-dimensional steady flow of a non-Newtonian fluid (a dilute polymer solution) is examined. The flow domain is composed of a parallel-walled inflow region, a contraction region in which the walls are rectangular hyperbolae, and a parallel-walled outflow region. The problem is formulated in terms of the vorticity, stream function and appropriate rheological equation of state, i.e. an Oldroyd-type constitutive equation (with no shear-thinning) for the total shear and normal-stress components. Computational results from the numerical solution of the equations are presented. In particular, the molecular extension and pressure distribution along the centre-line are presented as well as contour plots of the different flow variables. The alignment of the molecules with the principal axes of strain rate is shown by a qualitative comparison of the streamwise normal-stress contours with contours of the eigenvalues of the strain-rate matrix.

Langmuir circulation is a convective motion commonly observed in the oceanic mixed layer. Internal waves are a prominent feature of stratified regions, particularly the thermocline bounding the mixed layer. Here, the potential for Langmuir-circulation–internal-wave coupling is investigated using a two-layer ocean model. The density jump across the sharp thermocline confines all rotational motions, including the wind-aligned Langmuir vortices, to the upper (‘mixed’) layer. Linear analysis indicates: (i) that thermocline compliance enhances the onset of Langmuir circulation, and (ii) that the ‘vortex force’ arising from the interaction of surface waves with the wind-driven shear modifies the dynamics of cross-wind propagating internal waves. Weakly nonlinear analysis reveals that resonant cross-wind propagating internal waves can be nonlinearly reflected from stationary Langmuir circulation, a dynamic reminiscent of the ‘Bragg reflection’ of surface waves propagating over sand bars. A key feature of the reflection mechanism is the modification of the linear internal-wave dynamics by the vortex force.

This paper uses measurements of surface heat transfer to study roughness-induced turbulent wedges in a hypersonic boundary layer on a blunt cylinder. A family of wedges was produced by changing the height of an isolated roughness element, providing conditions in the following range: fully effective tripping, for the largest element, with a turbulent wedge forming immediately downstream of the element; a long wake, in length several hundred times the boundary layer thickness, leading ultimately to transition; and retention of laminar flow, for the smallest element. With appropriate element size, a fully intermittent wedge formed, comprising a clear train of turbulent spots.

The equation usually used to describe the conservation of volume flux in free turbulence (such as mixing layers, jets, etc.) is discussed.

We numerically assess the influence of non-uniform magnetic flux density and connected load resistance on turbulent duct flows in a liquid metal magnetohydrodynamic (MHD) electrical power generator. When increasing the magnetic flux density (or Hartmann number), an M-shaped velocity profile develops in the plane perpendicular to the magnetic field; the maximum velocity in the sidewall layer of the M-shaped profile increases to maintain the flow rate. Under the conditions of a relaminarized flow, the turbulence structures align along the magnetic field and flow repeatedly like a von Kármán vortex sheet. At higher Hartmann numbers, the wall-shear stress in the sidewall layer increases and the sidewall jets transit to turbulence. The sidewall jets in the MHD turbulent duct flows have profiles similar to the non-MHD wall jets, i.e. a mean velocity profile with outer scaling, Reynolds shear stress with the opposite sign in a sidewall jet, and two maxima for the turbulent intensities in a sidewall jet. The Lorentz force suppresses the vortices of the secondary mean flow near the Hartmann layer for low Hartmann numbers, whereas the secondary vortices remain near the Hartmann layer for high Hartmann numbers. An optimal load resistance (or load factor) to obtain a maximum electrical efficiency exists, because the strong Lorentz force for a low load factor and unextracted eddy currents for a high load factor reduce efficiency. When the value of the load factor is changed, the profiles of mean velocity and r.m.s. for the optimal load factor produce almost the same profiles as the high load factor near the open-circuit condition.

The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as ‘vorticity waves’, which enables us to treat the wave–flow resonance as the resonant wave–wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the ‘fast’ solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; ‘subcritical’ localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking.

The properties of a relatively uncommon regime of fluid dynamics, low Mach number compressible flow are investigated. This regime, which is characterized by an exceptionally large channel aspect ratio L/d ∼ 106 leads to highly subsonic flows in which friction dominates inertia. Even so, because of the large aspect ratio, finite pressure, temperature, and density gradients are required, implying that compressibility effects are also important. Analytical results are presented which show, somewhat unexpectedly, that for forced channel flow, steady-state solutions exist only below a critical value of heat input. Above this value the flow reverses against the direction of the applied pressure gradient causing fluid to leave both the inlet and outlet implying that the related concepts of a steady-state friction factor and heat transfer coefficient have no validity.

Wake visualization experiments were conducted on a finite curved cylinder whose plane of curvature is aligned with the free stream. The stagnation face of the cylinder is oriented concave or convex to the flow at $230\leqslant Re_{D}\leqslant 916$, where $Re_{D}$ is the cylinder Reynolds number and the curvature is constant and ranges from a straight cylinder to a quarter-ring. While the magnitude of the local angle of incidence to the flow is the same for both orientations, the contrast in their wakes demonstrates a violation of a common approximation known as the ‘independence principle’ for curved cylinders. Vortex shedding always occurred for the convex-oriented cylinder for the Reynolds-number range investigated, along most of the cylinder span, at a constant vortex shedding angle. In contrast, a concave-oriented cylinder could exhibit multiple concurrent wake regimes along its span: two shedding regimes (oblique, normal) and two non-shedding regimes. The occurrence of these wake regimes depended on the curvature, aspect ratio and Reynolds number. In some cases, vortex shedding was entirely suppressed, particularly at higher curvatures. In the laminar wake regime, increasing the curvature or decreasing the aspect ratio restricts vortex shedding to smaller regions along the span of the cylinder. Furthermore, the local angle of incidence where vortex shedding occurs is self-similar across cylinders of the same aspect ratio and varying curvature. After the wake transitions to turbulence, the vortex shedding extends along most of the cylinder span. The difference in the wakes between the concave and convex orientations is attributed to the spanwise flow induced by the finite end conditions, which reduces the generation of spanwise vorticity and increases the incidence of non-shedding and obliquely shedding wakes for the concave cylinder.

Scale analysis of unstable density-driven miscible convection in porous media is performed. The main conclusions for instabilities in the developed (long time scales) regime are that (i) large-scale structures are responsible for the bulk of the production of concentration variance, (ii) variance dissipation is dominated by the small (diffusive) scales and that (iii) both the production and dissipation rates are independent of the Rayleigh number. These findings provide a strong basis for a new modelling approach, namely, large-mode simulation (LMS), for which closure is achieved by replacing the actual diffusivity with an effective one. For validation, LMS results for vertical flow in a homogeneous rectangular domain are compared with direct numerical simulations (DNS). Some of the analysis is based on the derivation and closure of the concentration mean and variance equations, whereby averaging over the ensemble of all possible initial perturbations is considered. While self-similar solutions are obtained for vertical, statistically one-dimensional fingering, triple correlation of concentration and scalar dissipation rate (rate at which the concentration variance decays due to diffusion) have to be modelled in the general case. For this purpose, an ensemble-averaged Darcy modelling (EADM) approach is proposed.

The relative velocity and extra pressure drop of a single vesicle flowing through a square microchannel are quantified via boundary element simulations, lubrication theory and microfluidic experiments. The vesicle is modelled as a fluid sac enclosed by an inextensible, fluidic membrane with a negligible bending stiffness. All results are parametrized in terms of the vesicle sphericity (i.e. the reduced volume) and flow confinement (i.e. the ratio of the vesicle radius to the channel hydraulic radius). Direct comparison is made to previous studies of vesicle flow through circular tubes, revealing several distinct features of the square-channel geometry. Firstly, fluid in the suspending medium bypasses the vesicle through the corners of the channel, which in turn reduces the dissipation created by the vesicle. Secondly, the absence of rotational symmetry about the channel axis permits surface circulation in the membrane (tank treading), which in turn reduces the vesicle’s speed. At very high confinement, both theory and experiment indicate that the vesicle’s speed can be reduced below the mean speed of the suspending fluid through this mechanism. Finally, the contact area for lubrication is greatly reduced in the square-duct geometry, which in turn weakens the stress singularity predicted by lubrication theory. This fact directly leads to a breakdown of the lubrication approximation at low flow confinement, as verified by comparison to boundary element simulations. Since the only distinct property assumed of the membrane is its ability to preserve surface area locally, it is expected that the results of this study are applicable to other types of soft particles with immobilized surfaces (e.g. Pickering droplets, gel beads and biological cells).

The dynamics of a spherical elastic capsule, containing a Newtonian fluid bounded by an elastic membrane and immersed in another Newtonian fluid, in a uniform DC electric field is investigated. Discontinuity of electrical properties, such as the conductivities of the internal and external fluid media as well as the capacitance and conductance of the membrane, leads to a net interfacial Maxwell stress which can cause the deformation of such an elastic capsule. We investigate this problem considering well-established membrane laws for a thin elastic membrane, with fully resolved hydrodynamics in the Stokes flow limit, and describe the electrostatics using the capacitor model. In the limit of small deformation, the analytical theory predicts the dynamics fairly satisfactorily. Large deformations at high capillary number, though, necessitate a numerical approach (axisymmetric boundary element method in the present case) to solve this highly nonlinear problem. Akin to vesicles, at intermediate times, highly nonlinear biconcave shapes along with squaring and hexagon-like shapes are observed when the outer medium is more conducting. The study identifies the essentiality of parameters such as high membrane capacitance, low membrane conductance, low hydrodynamic time scales and high capillary number (the ratio of the destabilizing electric force to the stabilizing elastic force) for observation of these shape transitions. The transition is due to large compressive Maxwell stress at the poles at intermediate times. Thus such shape transition can be seen in spherical globules admitting electrical capacitance, possibly irrespective of the nature of the interfacial restoring force.

Several recent studies have reported that there exists a self-similar form of invariant solutions down to the Kolmogorov microscale in the bulk region of turbulent Couette flow. While their role in a fully developed turbulent flow is yet to be identified, here we report that there exists a related mechanism of turbulence production at the Kolmogorov microscale in the bulk region of turbulent Couette flow by performing a set of minimal-span direct numerical simulations up to friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\simeq 800$. This mechanism is found to essentially originate from the non-zero mean shear in the bulk region of the Couette flow, and involves eddy turn-over dynamics remarkably similar to the so-called self-sustaining process (SSP) and/or vortex–wave interaction (VWI). A numerical experiment that removes all other motions except in the core region is also performed, which demonstrates that the eddies at a given wall-normal location in the bulk region are sustained in the absence of other motions at different wall-normal locations. It is proposed that the self-sustaining eddies at the Kolmogorov microscale correspond to those in uniform shear turbulence at transitional Reynolds numbers, and a quantitative comparison between the eddies in uniform shear and near-wall turbulence is subsequently made. Finally, it is shown that turbulence production by the self-sustaining eddies at the Kolmogorov microscale is much smaller than that of full-scale simulations, and that the difference between the two increases with Reynolds number.

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destabilising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical electric-field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

Slender liquid jets that have a curved trajectory have been examined in a range of papers using a method introduced in Wallwork et al. (Proc. IUTAM Symp. on Free-Surface Flows, 2000, Kluwer; J. Fluid Mech., vol. 459, 2002, pp. 43–65) and Decent et al. (J. Engng Maths, vol. 42, 2002, pp. 265–282), for jets that emerge from an orifice on the surface of a rotating cylindrical container, successfully comparing computational results to measurements arising from laboratory experiments. Wallwork et al. (2000, 2002) and Decent et al. (2002) based their analyses on the slenderness of the jet, and neglected the torsion of the centreline of the jet, which is valid since in most situations examined the torsion is zero or small. Shikhmurzaev & Sisoev (J. Fluid Mech., vol. 819, 2017, pp. 352–400) used differential geometry and incorporated the torsion. This paper shows that these two methods produce identical results at leading order when the torsion is zero or when the torsion is $O(1)$, in an asymptotic framework based upon the slenderness of the jet, and shows that the method of Wallwork et al. (2000, 2002) and Decent et al. (2002) is accurate for parameters corresponding to scenarios previously examined and also when the torsion is $O(1)$. It is shown that the method of Shikhmurzaev & Sisoev (2017) should be used when the torsion is asymptotically large or when the jet is not slender.