Lubrication theory is adapted to incorporate the large normal stresses that occur for order-one Deborah numbers, $De$, the ratio of the relaxation time to the residence time. Comparing with the pressure drop for a Newtonian viscous fluid with a viscosity equal to that of an Oldroyd-B fluid in steady simple shear, we find numerically a reduced pressure drop through a contraction and an increased pressure drop through an expansion, both changing linearly with $De$ at high $De$. For a constriction, there is a smaller pressure drop that plateaus at high $De$. For a contraction, much of the change in pressure drop occurs in the stress relaxation in a long exit channel. An asymptotic analysis for high $De$, based on the idea that normal stresses are stretched by an accelerating flow in proportion to the square of the velocity, reveals that the large linear changes in pressure drop are due to higher normal stresses pulling the fluid through the narrowest gap. A secondary cause of the reduction is that the elastic shear stresses do not have time to build up to their steady-state equilibrium value while they accelerate through a contraction. We find for a contraction or expansion that the high $De$ analysis works well for $De>0.4$.

We present theoretical results related to the experimental findings of Matsubara & Alfredsson (J. Fluid Mech., vol. 430, 2001, pp. 149–168) on the scaling of the energy spectra of the Klebanoff modes, i.e. streamwise-elongated vortical disturbances generated by free-stream turbulence in a flat-plate transitional boundary layer. The scaling is explained by a model that describes the streamwise evolution of the streamwise and spanwise energy spectra. The theoretical framework is based on the quasi-steady asymptotic solution of the boundary-region equations, on an axial-symmetric model of the free-stream spectrum, and on the spectral response of the boundary layer to the external perturbations.

The interaction between a uniform current with a circular cylinder submerged in a fluid covered by a semi-infinite ice sheet is considered analytically. The ice sheet is modelled as an elastic thin plate, and the fluid flow is described by the linearised velocity potential theory. The Green function or the velocity potential due to a source is first obtained. As the water surface is divided into two semi-infinite parts with different boundary conditions, the Wiener–Hopf method (WHM) offers significant advantages over alternative approaches and is consequently adopted. To do that, the distribution of the roots of the dispersion equation for fluid fully covered by an ice sheet in the complex plane is first analysed systematically, which does not seem to have been done before. The variations of these roots with the Froude number are investigated, especially their effects or factorisation and decomposition required in the WHM. The result is verified by comparing with that obtained from the matched eigenfunction expansion method. Through differentiating the Green function with respect to the source position, the potentials due to multipoles are obtained, which are employed to construct the velocity potential for the circular cylinder. Extensive results are provided for hydrodynamic forces on the cylinder and wave profiles, and some unique features are discussed. In particular, it is found that the forces can be highly oscillatory with the Froude number when the body is below the ice sheet, whereas such an oscillation does not exist when the body is below the free surface.

A Marangoni surfer is an object embedded in a gas–liquid interface, propelled by gradients in surface tension. We derive an analytical theorem for the lower bound on the viscous dissipation by a Marangoni surfer in the limit of small Reynolds and capillary numbers. The minimum dissipation can be expressed with the reciprocal difference between drag coefficients of two passive bodies of the same shape as the Marangoni surfer, one in a force-free interface and the other in an interface with surface incompressibility. The distribution of surface tension that gives the optimal propulsion is given by the surface tension of the solution for the incompressible surface and the flow is a superposition of both solutions. For a surfer taking the form of a thin circular disk, the minimum dissipation is $16\mu a V^2$, giving a Lighthill efficiency of $1/3$. This places the Marangoni surfers among the hydrodynamically most efficient microswimmers.

We carry out direct numerical simulation (DNS) of turbulent flow in rough pipes. Two types of irregular roughness are investigated, namely a grit-blasted and a graphite surface. A wide range of Reynolds numbers is tested, from the laminar up to the fully rough regime, attempting to replicate Nikuradse's pioneering study. Despite the large relative roughness, outer-layer similarity is achieved at high Reynolds number as hypothesised by Townsend, with deviations from the smooth wall case of 4 % for the grit-blasted surface and 13 % for the graphite surface. This makes it possible to define a roughness function and the equivalent sand-grain roughness. The results are compared with those obtained in plane channels, with small differences pointing to the residual influence of the duct cross-sectional shape in the presence of relatively large roughness. The computed friction factors behave similar to those Nikuradse's chart, with differences in terms of the friction factor in the laminar region and of the critical Reynolds number, which are partly absorbed by using the hydraulic radius as reference length scale. The distributions of the velocity fluctuations intensities point to a isotropisation of turbulence in the near-wall region resulting from the roughness, with influence of the roughness geometry. Comparison of the computed equivalent sand-grain roughness height suggest that existing correlations suffer from poor predictive power, at least for surfaces with large relative roughness.

A depth-averaged, convection-dispersion equation is derived for the particle volume fraction distribution in the pressure-driven flow of a concentrated suspension of neutrally buoyant, non-colloidal particles between two parallel plates, by implementing a two-time-scale perturbation expansion of the suspension balance model (Nott & Brady, J. Fluid Mech., vol. 275, 1994, pp. 157–199) coupled with the constitutive equations of Zarraga et al. (J. Rheol., vol. 52, issue 2, 2000, pp. 185–220). The Taylor-dispersion coefficient in the macrotransport equation scales as $U_c^{\prime}B^{\prime 3}/a^{\prime 2}$, where $U_c^{\prime}$ is the characteristic depth-averaged velocity, $B^{\prime}$ is the half-depth of the channel and $a^{\prime}$ is the particle radius. Taylor dispersion relaxes gradients in the depth-averaged volume fraction along the local velocity vector. Perpendicular to the flow, however, only shear-induced migration can cause particle redistribution, leading to fluxes down gradients in volume fraction, shear rate and streamline curvature that scale as $U_c^{\prime}a^{\prime 2}/B^{\prime}$. To determine the velocity and particle distributions in Hele-Shaw suspension flows, one only needs to solve two coupled partial differential equations in the pressure and the depth-averaged volume fraction, achievable on commercially available solvers. Analogous to the macrotransport equation for suspension flow through a circular tube (Ramachandran, J. Fluid Mech., vol. 734, 2013, pp. 219–252), the evolution of the particle volume fraction distribution is dependent only on the total strain experienced by the suspension, and is independent of the suspension velocity scale. However, unlike the tube problem, a positive concentration gradient along the flow direction is susceptible to viscous miscible fingering. A linear stability analysis performed for a step increase in the volume fraction in the direction of flow with a velocity $U^{\prime}$ reveals that the growth rate and wavenumber corresponding to fastest growing mode scale as $U^{\prime}a^{\prime 2}/B^{\prime 3}$ and $a^{\prime 2/3}/B^{\prime 5/3}$, respectively.

Experiments are conducted in an open-channel flow where half of the section is smooth and the other half consists of an array of cubes, which are either submerged or emergent. A shear layer featuring large-scale Kelvin–Helmholtz structures develops between the two subsections. The flows are first analysed in the framework of the double-averaging method (averaging of the flow both in time and space). Double averaging could be performed thanks to an experimental set-up (three-dimensional, two-component telecentric scanning particle image velocimetry) that allows to measure the velocity field in a large volume, including the interstices between the cubes. A momentum balance performed on the smooth subsection indicates that the loss of momentum towards the rough subsection has the same order of magnitude than the momentum loss through bed friction. This lateral momentum flux occurs nearly exclusively through turbulent shear stress, whereas secondary currents plays a minor role and dispersive shear stress is negligible. A pattern recognition technique is then applied to investigate statistically the large-scale Kelvin–Helmholtz structures that develop in the shear layer. The structures appear to be coherent over the water depth and to be strongly inclined in the vertical, the top part being ahead. The educed coherent structure is responsible by itself for the shape of the velocity profile across the shear layer and for a large part of the turbulence (up to 60 % for the turbulent shear stress). Finally, a coupling is identified between the passage of the Kelvin–Helmholtz structures and the instantaneous wake flow around the cubes at the interface.

We analyse the results of direct numerical simulations of rotating convection in spherical shell geometries with stress-free boundary conditions, which develop strong zonal flows. Both the Ekman number and the Rayleigh number are varied. We find that the asymptotic theory for rapidly rotating convection can be used to predict the Ekman number dependence of each term in the governing equations, along with the convective flow speeds and the dominant length scales. Using a balance between the Reynolds stress and the viscous stress, together with the asymptotic scaling for the convective velocity, we derive an asymptotic prediction for the scaling behaviour of the zonal flow with respect to the Ekman number, which is supported by the numerical simulations. We do not find evidence of distinct asymptotic scalings for the buoyancy and viscous forces and, in agreement with previous results from asymptotic plane layer models, we find that the ratio of the viscous force to the buoyancy force increases with Rayleigh number. Thus, viscosity remains non-negligible and we do not observe a trend towards a diffusion-free scaling behaviour within the rapidly rotating regime.

Small differences in the densities of a river confluence's tributaries (i.e. 0.5 kg m$^{-3}$) have been proposed to cause coherent streamwise-oriented vortices (SOVs) in its mixing interface. These secondary flow structures are thought to result from density-driven gravity currents being laterally confined between the converging flows. However, empirical evidence for density SOVs and the confined gravity current mechanism is lacking. To this end, experiments are carried out in a laboratory confluence permitting a spectrum of thermal density differences between its tributaries. Particle image velocimetry and laser-induced fluorescence are used simultaneously to study the mixing interface's dynamics. The sensitivity of the mixing interface's secondary flow structure to the confluence's momentum ratio and the magnitude of the density difference is evaluated. Density SOVs are confirmed in the mixing interface and are caused by the gravity currents being confined laterally as the opposing flows merge and accelerate downstream. The SOVs are largest and most coherent when the momentum of the dense channel is greater than that of the light channel. The dynamics of these secondary flow structures is strongly coupled to periodic vertically orientated Kelvin–Helmholtz instabilities. The striking similarities between the empirically reproduced SOVs herein and those recently observed at the Coaticook-Massawippi confluence (Quebec, Canada), despite a two-order magnitude difference in physical scale, suggest density SOVs are a scale-independent flow structure at confluences when specific, yet relatively common, hydraulic and density conditions align.

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

Energy transfer in turbulent fluids is non-Gaussian. We quantify non-Gaussian energy transfer between the atmosphere and bodies of water using a turbulent diffusion operator coupled with temporally self-affine velocity distributions and a recursive integration method that produce multifractal measures. The measures serve as input to a system of moment field equations (derived from Navier–Stokes) that generate and track high-frequency gravity waves that propagate through the water surface (as a result of the air–water interactions). The dimension of the support of the air–water turbulence produced by our methods falls within the range of theory and observation, and correspondingly, hindcast statistical measures of the water-wave surface such as significant water-wave height and wave period are well correlated to observational buoy data. Further, our recursive integration method can be used by spectral resolving phase-averaged models to interpolate temporal wind data to smaller scales to capture the non-Gaussian behaviour of the air–water interaction.

A multiscale asymptotic theory is formulated for surface gravity waves and currents in finite-depth water with a vegetation canopy that provides a drag force on both flows with known drag coefficients. It assumes that the density is uniform and the depth is uniform pro tem and that the wave frequency is fast compared to the current advective rate. It is a quasi-linear theory in which the wave dynamics is independent of current and drag to leading order but provides perturbative corrections, and in which wave nonlinear interactions are neglected while quadratic wave-averaged wave fluxes and quadratic wave-drag effects are retained. The primary surface wave is modified by drag and current interactions, and the wave-averaged current momentum balance includes a wave-augmented drag force and several vortex forces due to Earth's rotation, current vorticity, Stokes drift and drag-induced wave vorticity. The wave-averaged current equations derived here are a suitable basis for future large-eddy simulation and submesoscale circulation computational models.

Shock waves are of great interest in many fields of science and engineering, but the mechanisms of their formation, maintenance and dissipation are still not well understood. While all transport processes existing in a shock wave contribute to its compression and irreversibility, they are not of equal importance. To figure out the roles of viscosity and heat conduction in shock transition, the existence of smooth shock solutions and the counter-intuitive entropy overshoot phenomenon (the specific entropy is not monotonically increasing and exhibits a peak inside the shock front) are theoretically and numerically investigated, with emphasis on the effects of viscosity and heat conduction. Instead of higher-order hydrodynamics, the Navier–Stokes formalism is employed for its stability and simplicity. Supplemented with nonlinear thermodynamically consistent constitutive relations, the Navier–Stokes equations are adequate to demonstrate the general nature of shock profiles. It is found that heat conduction cannot sustain strong shocks without the presence of viscosity, while viscosity can maintain smooth shock transition at all strengths, regardless of heat conduction. Hence, the critical role in shock compression is played by viscosity rather than heat conduction. Nevertheless, the dispensability of heat conduction would not compromise its essential role in the emergence of an entropy peak. It is the entropy flux resulting from heat conduction that neutralises the positive entropy production and thus prevents the decreasing entropy from violating the second law of thermodynamics. This mechanism of entropy overshoot has not been addressed previously in the literature and is revealed using the entropy balance equation.

Steady shock reflection where the incident shock is free of interaction with other waves has been well studied. In this paper, we consider the less studied shock reflection problem where the incident shock interacts with the wedge trailing-edge expansion fan, which occurs when the wedge trailing-edge height surpasses a threshold. The influence of this interaction on the advance of transition from Mach reflection to regular reflection is quantified in terms of the wedge trailing-edge height ratio. The wave pattern, including primary and reflected Mach waves, for Mach reflection with interaction is clarified using computational fluid dynamics (CFD) and the method of characteristics. Those reflected Mach waves having an important effect on Mach reflection are identified. A simplified Mach stem model that accounts for the direct role of the interaction on the incident shock and its indirect role on the reflected shock and slipline is built up on a past model without interaction. Both theory and CFD show that the Mach stem height decreases nonlinearly with increasing trailing-edge height.

We experimentally and numerically study the dynamics of a liquid jet issued from a rotating orifice, whose breakup is regulated by a vibrating piezo element. The helical trajectory of the spiralling jet yields fictitious forces varying along the jet whose longitudinal projections stretch and thin the jet, affecting the growth of perturbations. We show that by quantifying these fictitious forces, one can estimate the jet intact length and size distribution of drops formed at jet breakup. The presence of the locally varying fictitious forces may render high-frequency perturbations, that would otherwise be stable in the abscence of stretching, unstable, as observed similarly in the case of straight jets stretching under gravity. The perturbation amplitude then dictates how strong the perturbation is coupled to the jet compared with random noise that is inherently present in any experimental set-up. In the present study we exploit the slenderness of the jet to separate the calculation of the base flow and the growth of perturbations. The fictitious forces calculated from the base flow trajectory are then used in a nonlinear slender-jet model, which treats the spiralling jet as a quasi-straight jet with locally varying body forces. We show both experimentally and numerically that jet breakup characteristics (e.g. intact length and drop size distribution) can be controlled by finite-amplitude perturbations created by mechanically induced pressure modulations. Finally, we revisit the integrated net gain approach developed for straight jets under gravity and we provide simple analogous relations for spiralling jets.