An observed, spectacular and puzzling distinctive feature of wall-bounded flows transitioning to/from turbulence is the large-scale ordered coexistence of laminar and turbulent flow over a wide range of Reynolds numbers. In their article, Chantry et al. (J. Fluid Mech., vol. 791, 2016, R8) circumvent the problem of turbulence modulation in genuine plane Couette flow by replacing no-slip boundary conditions at the walls with effective stress-free conditions constraining the interior part of the flow only. In so doing, they are able to collect relevant information on the laminar–turbulent patterning. The approach is then adapted to plane Poiseuille and pipe flow, giving to it a more universal value. Taking advantage of the stress-free conditions, they derive reduced models efficient at capturing the laminar–turbulent coexistence. Such models would be useful to interpret recent findings in the large-aspect-ratio limit.

The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ($Ma_{r}$) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to $Ma_{r}=2.0$) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same $Ma_{r}$ range) with increasing $Ma$. The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local $Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative $Ma$, pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at $Ma=0.8$ and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.

A statistically stationary homogeneous isotropic turbulent flow modified by 64 small fixed non-Stokesian spherical particles is considered. The particle diameter is approximately twice the Kolmogorov length scale, while the particle volume fraction is 0.001. The Taylor Reynolds number of the corresponding unladen flow is 32. The particle-laden flow has been obtained by a direct numerical simulation based on a discretization of the incompressible Navier–Stokes equations on 64 spherical grids overset on a Cartesian grid. The global (space- and time-averaged) turbulence kinetic energy is attenuated by approximately 9 %, which is less than expected. The turbulence dissipation rate on the surfaces of the particles is enhanced by two orders of magnitude. More than 5 % of the total dissipation occurs in only 0.1 % of the flow domain. The budget of the turbulence kinetic energy has been computed, as a function of the distance to the nearest particle centre. The budget illustrates how energy relatively far away from particles is transported towards the surfaces of the particles, where it is dissipated by the (locally enhanced) turbulence dissipation rate. The energy flux towards the particles is dominated by turbulent transport relatively far away from particles, by viscous diffusion very close to the particles, and by pressure diffusion in a significant region in between. The skewness and flatness factors of the pressure, velocity and velocity gradient have also been computed. The global flatness factor of the longitudinal velocity gradient, which characterizes the intermittency of small scales, is enhanced by a factor of six. In addition, several point-particle simulations based on the Schiller–Naumann drag correlation have been performed. A posteriori tests of the point-particle simulations, comparisons in which the particle-resolved results are regarded as the standard, show that, in this case, the point-particle model captures both the turbulence attenuation and the fraction of the turbulence dissipation rate due to particles reasonably well, provided the (arbitrary) size of the fluid volume over which each particle force is distributed is suitably chosen.

Although wind and tidal turbines operate in turbulent shear flow, most theoretical results concerning turbine performance, such as the well-known Betz limit, assume the upstream velocity profile is uniform. To improve on these existing results we extend the classical actuator disc model in this paper to investigate the performance of an ideal turbine in steady, inviscid shear flow. The model is developed on the assumption that there is negligible lateral interaction in the flow passing through the disc and that the actuator applies a uniform resistance across its area. With these assumptions, solution of the model leads to two key results. First, for laterally unbounded shear flow, it is shown that the normalised power extracted is the same as that for an ideal turbine in uniform flow, if the average of the cube of the upstream velocity of the fluid passing through the turbine is used in the normalisation. Second, for a laterally bounded shear flow, it is shown that the same normalisation can be applied, but allowance must also be made for the fact that non-uniform flow bypassing the turbine alters the background pressure gradient and, in turn, the turbines ‘effective blockage’ (so that it may be greater or less than the geometric blockage, defined as the ratio of turbine disc area to cross-sectional area of the flow). Predictions based on the extended model agree well with numerical simulations approximating the incompressible Euler equations. The model may be used to improve interpretation of model-scale results for wind and tidal turbines in tunnels/flumes, to investigate the variation in force across a turbine and to update existing theoretical models of arrays of tidal turbines.

High-speed turbulent flows with shock waves are characterized by high localized surface heat transfer rates. Computational predictions are often inaccurate due to the limitations in modelling of the unclosed turbulent energy flux in the highly non-equilibrium regions of shock interaction. In this paper, we investigate the turbulent energy flux generated when homogeneous isotropic turbulence passes through a nominally normal shock wave. We use linear interaction analysis where the incoming turbulence is idealized as being composed of a collection of two-dimensional planar vorticity waves, and the shock wave is taken to be a discontinuity. The nature of the postshock turbulent energy flux is predicted to be strongly dependent on the angle of incidence of the incoming waves. The energy flux correlation is also decomposed into its vortical, entropy and acoustic contributions to understand its rapid non-monotonic variation behind the shock. Three-dimensional statistics, calculated by integrating two-dimensional results over a prescribed upstream energy spectrum, are compared with available data from direct numerical simulations. A detailed budget of the governing equation is also considered in order to gain insight into the underlying physics.

The effects of a spanwise row of finite-size cylindrical roughness elements in a laminar, compressible, three-dimensional boundary layer on a wing profile are investigated by direct numerical simulations (DNS). Large elements are capable of immediately tripping turbulent flow by either a strong, purely convective or an absolute/global instability in the near wake. First we focus on an understanding of the steady near-field past a finite-size roughness element in the swept-wing flow, comparing it to a respective case in unswept flow. Then, the mechanisms leading to immediate turbulence tripping are elaborated by gradually increasing the roughness height and varying the disturbance background level. The quasi-critical roughness Reynolds number above which turbulence sets in rapidly is found to be $Re_{kk,qcrit}\approx 560$ and global instability is found only for values well above 600 using nonlinear DNS; therefore the values do not differ significantly from two-dimensional boundary layers if the full velocity vector at the roughness height is taken to build $Re_{kk}$. A detailed simulation study of elements in the critical range indicates a changeover from a purely convective to a global instability near the critical height. Finally, we perform a three-dimensional global stability analysis of the flow field to gain insight into the early stages of the temporal disturbance growth in the quasi-critical and over-critical cases, starting from a steady state enforced by damping of unsteady disturbances.

In theoretical treatments of turbulent fountains, the entrainment of ambient fluid into the top of the fountain, hereinafter fountain-top entrainment $Q_{top}$ ($\text{m}^{3}~\text{s}^{-1}$), has been neglected until now. This neglect, which modifies the energetic balance in a fountain, compromises the predictive ability of existing models. Our aim is to quantify $Q_{top}$ by shedding light on the physical processes that are responsible for fountain-top entrainment. First, estimates for $Q_{top}$ are obtained by applying, in turn, an entrainment closure in the vein of Morton et al. (Proc. R. Soc. Lond., vol. 234, 1956, pp. 1–23) and then of Shrinivas & Hunt (J. Fluid Mech., vol. 757, 2014, pp. 573–598) to the time-averaged fountain top. Unravelling the assumptions that underlie these approaches, we argue that neither capture the dynamical behaviour of the flow observed at the fountain top; the top being characterised by quasi-periodic fluctuations, during which large-scale eddies reverse and engulf parcels of ambient fluid into the fountain. Therefore, shifting our mindset to a periodical framework, we develop a new phenomenological model in which we emphasise the role of the fluctuations in entraining external fluid. Our model suggests that $Q_{top}$ is similar in magnitude to the volume flux supplied to the fountain top by the upflow ($Q_{u}$), i.e. $Q_{top}\sim Q_{u}$, in agreement with experimental evidence. We conclude by providing guidance on how to implement fountain-top entrainment in existing models of turbulent fountains.

The movements of fluid–fluid interfaces and the common curve are an important aspect of two-fluid-phase flow through porous media. The focus of this work is to develop, apply and evaluate methods to simulate two-fluid-phase flow in porous medium systems at the microscale and to demonstrate how these results can be used to support evolving macroscale models. Of particular concern is the problem of spurious velocities that confound the accurate representation of interfacial dynamics in such systems. To circumvent this problem, a combined level-set and lattice-Boltzmann method is advanced to simulate and track the dynamics of the fluid–fluid interface and of the common curve during simulations of two-fluid-phase flow in porous media. We demonstrate that the interface and common curve velocities can be determined accurately, even when spurious currents are generated in the vicinity of interfaces. Static and dynamic contact angles are computed and shown to agree with existing slip models. A resolution study is presented for dynamic drainage and imbibition in a sphere pack, demonstrating the sensitivity of averaged quantities to resolution.

The self-propulsion of a spherical squirmer – a model swimming organism that achieves locomotion via steady tangential movement of its surface – is quantified across the transition from viscously to inertially dominated flow. Specifically, the flow around a squirmer is computed for Reynolds numbers ($Re$) between 0.01 and 1000 by numerical solution of the Navier–Stokes equations. A squirmer with a fixed swimming stroke and fixed swimming direction is considered. We find that fluid inertia leads to profound differences in the locomotion of pusher (propelled from the rear) versus puller (propelled from the front) squirmers. Specifically, pushers have a swimming speed that increases monotonically with $Re$, and efficient convection of vorticity past their surface leads to steady axisymmetric flow that remains stable up to at least $Re=1000$. In contrast, pullers have a swimming speed that is non-monotonic with $Re$. Moreover, they trap vorticity within their wake, which leads to flow instabilities that cause a decrease in the time-averaged swimming speed at large $Re$. The power expenditure and swimming efficiency are also computed. We show that pushers are more efficient at large $Re$, mainly because the flow around them can remain stable to much greater $Re$ than is the case for pullers. Interestingly, if unstable axisymmetric flows at large $Re$ are considered, pullers are more efficient due to the development of a Hill’s vortex-like wake structure.

The understanding and accurate prediction of turbulent flow separation on smooth surfaces is still a challenging task because the separation and the reattachment locations are not fixed in space and time. Consequently, reliable experimental data are essential for the validation of numerical flow simulations and the characterization and analysis of the complex flow physics. However, the uncertainty of the existing near-wall flow measurements make a precise analysis of the near-wall flow features, such as separation/reattachment locations and other predicted near-wall flow features which are under debate, often impossible. Therefore, the periodic hill experiment at TU Munich (ERCOFTAC test case 81) was repeated using high resolution particle image velocimetry and particle tracking velocimetry. The results confirm the strong effect of the spatial resolution on the near-wall flow statistics. Furthermore, it is shown that statistically stable values of the turbulent flow variables can only be obtained for averaging times which are challenging to realize with highly resolved large eddy simulation and direct numerical simulation techniques. Additionally, the analysis implies that regions of correlated velocity fluctuations with rather uniform streamwise momentum exist in the flow. Their size in the mean flow direction can be larger than the hill spacing. The possible impact of the correlated turbulent motion on the wake region is discussed, as this interaction might be important for the understanding and control of the flow separation dynamics on smooth bodies.

We study theoretically and experimentally how a thin layer of liquid flows along a flexible beam. The flow is modelled using lubrication theory and the substrate is modelled as an elastica which deforms according to the Euler–Bernoulli equation. A constant flux of liquid is supplied at one end of the beam, which is clamped horizontally, while the other end of the beam is free. As the liquid film spreads, its weight causes the beam deflection to increase, which in turn enhances the spreading rate of the liquid. This feedback mechanism causes the front position ${\it\sigma}(t)$ and the deflection angle at the front ${\it\phi}(t)$ to go through a number of different power-law behaviours. For early times, the liquid spreads like a horizontal gravity current, with ${\it\sigma}(t)\propto t^{4/5}$ and ${\it\phi}(t)\propto t^{13/5}$. For intermediate times, the deflection of the beam leads to rapid acceleration of the liquid layer, with ${\it\sigma}(t)\propto t^{4}$ and ${\it\phi}(t)\propto t^{9}$. Finally, when the beam has sagged to become almost vertical, the liquid film flows downward with ${\it\sigma}(t)\propto t$ and ${\it\phi}(t)\sim {\rm\pi}/2$. We demonstrate good agreement between these theoretical predictions and experimental results.

For understanding the dissipation in a rotating flow when resonance occurs, we study the rotating flow driven by the harmonic force in a periodic box. Both the linear and nonlinear regimes are studied. The various parameters such as the force amplitude $a$, the force frequency ${\it\omega}$, the force wavenumber $k$ and the Ekman number $E$ are investigated. In the linear regime, the dissipation at the resonant frequency scales as $E^{-1}k^{-2}$, and it is much stronger than the dissipation at the non-resonant frequencies on large scales and at low Ekman numbers. In the nonlinear regime, at the resonant frequency the effective dissipation (dissipation normalised with the square of the force amplitude) is lower than in the linear regime and it decreases with increasing force amplitude. This nonlinear suppression effect is significant near the resonant frequency but negligible far away from the resonant frequency. Opposite to the linear regime, in the nonlinear regime at the resonant frequency the lower Ekman number leads to lower dissipation because of the stronger nonlinear effect. This work implies that the previous linear calculations overestimated the tidal dissipation, which is important for understanding the tides in stars and giant planets.

A theory is developed that describes microstreaming generated by two interacting gas bubbles in an acoustic field. The theory is used in numerical simulations to compare the characteristics of acoustic microstreaming at different frequencies, separation distances between the bubbles and bubble sizes. It is shown that the interaction of the bubbles leads to a considerable increase in the intensity of the velocity and stress fields of acoustic microstreaming if the bubbles are driven near the resonance frequencies that they have in the presence of each other. Patterns of streamlines for different situations are presented.

Sand transport and morphological change occur in the wave bottom boundary layer due to sand particle interactions with an oscillatory flow and granular interactions between particles. Although these interactions depend strongly on the characteristics of the particle population, i.e. size and shape, little is known about how natural sand particles behave under oscillatory conditions and how variations in particle size influence transport behaviour. To enable this to be studied numerically, an Euler–Lagrange point-particle model is developed which can capture the individual and collective dynamics of subaqueous natural sand grains. Special treatments for particle collision, friction and hydrodynamic interactions are included to take into account the wide size and shape variations in natural sands. The model is used to simulate sand particle dynamics in two asymmetric oscillatory flow conditions corresponding to the vortex ripple experiments of Van der Werf et al. (J. Geophys. Res., vol. 112, 2007, F02012) and the sheet-flow experiments of O’Donoghue & Wright (Coast. Engng, vol. 50, 2004, pp. 117–138). A comparison of the phase resolved velocity and concentration fields shows overall excellent agreement between simulation and experiments. The particle based datasets are used to investigate the spatio-temporal dynamics of the particle-size distribution and the influence of three-dimensional vortical features on particle entrainment and suspension processes. For the first time, it is demonstrated that even for the relatively well-sorted medium-size sands considered here, the characteristics of the local grain size population exhibit significant space–time variation. Both conditions demonstrate a distinct coarse-over-fine armouring at the bed surface during low-velocity phases, which restricts the vertical mobility of finer fractions in the bed, and also results in strong pickup events involving disproportionately coarse fractions. The near-bed layer composition is seen to be very dynamic in the sheet-flow condition, while it remains coarse through most of the cycle in the vortex ripple condition. Particles in suspension spend more time sampling the upward directed parts of these flows, especially the smaller fractions, which delays particle settling and enhances the vertical size sorting of grains in suspension. For the submillimetre grain sizes considered, most particle–particle collisions occur at low impact Stokes numbers and can be expected to have low rebound velocities. The results are considered to be fundamental to the understanding of particle sorting and transport mechanisms, and can help to improve current transport modelling approaches.

The accuracy of existing impedance boundary conditions is investigated, and new impedance boundary conditions are derived, for lined ducts with inviscid shear flow. The accuracy of the Ingard–Myers boundary condition is found to be poor. Matched asymptotic expansions are used to derive a boundary condition accurate to second order in the boundary layer thickness, which shows substantially increased accuracy for thin boundary layers when compared with both the Ingard–Myers boundary condition and its recent first-order correction. Closed-form approximate boundary conditions are also derived using a single Runge–Kutta step to solve an impedance Ricatti equation, leading to a boundary condition that performs reasonably even for thicker boundary layers. Surface modes and temporal stability are also investigated.

The present work describes the multidimensional behaviour of scale-energy production, transfer and dissipation in wall-bounded turbulent flows. This approach allows us to understand the cascade mechanisms by which scale energy is transmitted scale-by-scale among different regions of the flow. Two driving mechanisms are identified. A strong scale-energy source in the buffer layer related to the near-wall cycle and an outer scale-energy source associated with an outer turbulent cycle in the overlap layer. These two sourcing mechanisms lead to a complex redistribution of scale energy where spatially evolving reverse and forward cascades coexist. From a hierarchy of spanwise scales in the near-wall region generated through a reverse cascade and local turbulent generation processes, scale energy is transferred towards the bulk, flowing through the attached scales of motion, while among the detached scales it converges towards small scales, still ascending towards the channel centre. The attached scales of wall-bounded turbulence are then recognized to sustain a spatial reverse cascade process towards the bulk flow. On the other hand, the detached scales are involved in a direct forward cascade process that links the scale-energy excess at large attached scales with dissipation at the smaller scales of motion located further away from the wall. The unexpected behaviour of the fluxes and of the turbulent generation mechanisms may have strong repercussions on both theoretical and modelling approaches to wall turbulence. Indeed, actual turbulent flows are shown here to have a much richer physics with respect to the classical notion of turbulent cascade, where anisotropic production and inhomogeneous fluxes lead to a complex redistribution of energy where a spatial reverse cascade plays a central role.

We perform a direct numerical simulation (DNS) investigation of the incompressible temporally developing turbulent boundary layer. The approach is inspired by temporal simulations of flows which are generally thought of as developing in space, such as wakes and mixing layers. Compressible boundary layers have previously been studied in this manner yet the temporal approach appears to be under-exploited in the literature concerning incompressible boundary layers. The flow is the turbulent counterpart to the laminar Rayleigh problem or Stokes’ first problem, in which a fluid at rest is set into motion by a wall moving at constant velocity. An initial profile that models the effect of a wall-mounted trip wire is implemented and allows the characterisation of initial conditions by a trip Reynolds number. For the current set-up, a trip Reynolds number of 500 based on the trip-wire diameter successfully triggers transition yet only mildly perturbs the flow so it assumes a natural development at the lowest possible Reynolds number based on momentum thickness. A systematic trip study reveals that as the ratio of momentum thickness to trip-wire diameter approaches unity, our flow approaches a state free from the effects of its starting trip Reynolds number. The transport of a passive scalar by this flow is also simulated. The role played by domain size is investigated with two boxes, sized to accommodate two chosen final Reynolds numbers. Comparisons of the skin friction coefficient, velocity and scalar statistics demonstrate that the temporally developing boundary layer is a good model for the spatially developing boundary layer once initial conditions can be neglected. Analysis of similarity solutions suggests such a rapprochement of the spatial and temporal boundary layers may be expected at high Reynolds numbers given that the only terms that asymptotically persist are those common to both cases. If one seeks statistics for the turbulent boundary layer, the temporal boundary layer is therefore a viable method if modest convergence is sufficient. We suggest that such a temporal set-up could prove useful in the study of turbulence dynamics.

We investigate the inception of cavitation and resulting structures when a sphere collides with a solid surface covered with a layer of non-Newtonian liquid having a kinematic viscosity of up to ${\it\nu}_{0}=20\,000\,000$ cSt. We show the existence of shear-stress-induced cavitation during sphere approach towards the base wall (i.e. the pressurization stage) in ultra-viscous films using a synchronized dual-view high-speed imaging system. For the experimental parameters employed, liquids having viscoelastic properties of $De\geqslant O(1)$ are shown to enable sphere rebound without any prior contact with the solid wall. Cavitation by depressurization (i.e. during rebound) in such non-contact cases is observed to onset after a noticeable delay from when the minimum gap distance is reached. Also, the cavities created originate from remnant bubbles, being the remains of the primary bubble entrapment formed by the lubrication pressure of the air during film entry. Cases where physical contact occurs (contact cases) in 10 000 cSt ${\leqslant}{\it\nu}_{0}\leqslant 1000\,000$ cSt films produce cavities attached to the base wall, which extend into an hourglass shape. In contrast, strikingly different structures occur in the most viscous liquids due to the disproportionality in radial expansion and longitudinal extension along the cavity length. Horizontal shear rates calculated using particle image velocimetry (PIV) measurements show the apparent fluid viscosity to vary substantially as the sphere approaches and rebounds away from the base wall. A theoretical model based on the lubrication assumption is solved for the squeeze flow in the regime identified for shear-induced cavity events, to investigate the criterion for cavity inception in further detail.

This paper is concerned with the high Reynolds number flow over a spanwise-periodic array of roughness elements with interelement spacing of the order of the local boundary-layer thickness. While earlier work by Goldstein et al. (J. Fluid Mech., vol. 644, 2010, pp. 123–163) and Goldstein et al. (J. Fluid Mech., vol. 668, 2011, pp. 236–266) was mainly concerned with smaller roughness heights that produced relatively weak distortions of the downstream flow, the focus here is on extending the analysis to larger roughness heights and streamwise elongated planform shapes that together produce a qualitatively different, nonlinear behaviour of the downstream wakes. The roughness scale flow now has a novel triple-deck structure that is somewhat different from related studies that have previously appeared in the literature. The resulting flow is formally nonlinear in the intermediate wake region, where the streamwise distance is large compared to the roughness dimensions but small compared to the downstream distance from the leading edge, as well as in the far wake region where the streamwise length scale is of the order of the downstream distance from the leading edge. In contrast, the flow perturbations in both of these wake regions were strictly linear in the earlier work by Goldstein et al. (2010, 2011). This is an important difference because the nonlinear wake flow in the present case provides an appropriate basic state for studying the secondary instability and eventual breakdown into turbulence.

We report results of an experimental investigation into the effects of small-scale (mm–cm) heterogeneities on solute spreading and mixing in a Berea sandstone core. Pulse-tracer tests have been carried out in the Péclet number regime $Pe=6{-}40$ and are supplemented by a unique combination of two imaging techniques. X-ray computed tomography (CT) is used to quantify subcore-scale heterogeneities in terms of permeability contrasts at a spatial resolution of approximately $10~\text{mm}^{3}$, while [11C] positron emission tomography (PET) is applied to image the spatial and temporal evolution of the full tracer plume non-invasively. To account for both advective spreading and local (Fickian) mixing as driving mechanisms for solute transport, a streamtube model is applied that is based on the one-dimensional advection–dispersion equation. We refer to our modelling approach as semideterministic, because the spatial arrangement of the streamtubes and the corresponding solute travel times are known from the measured rock’s permeability map, which required only small adjustments to match the measured tracer breakthrough curve. The model reproduces the three-dimensional PET measurements accurately by capturing the larger-scale tracer plume deformation as well as subcore-scale mixing, while confirming negligible transverse dispersion over the scale of the experiment. We suggest that the obtained longitudinal dispersivity ($0.10\pm 0.02$  cm) is rock rather than sample specific, because of the ability of the model to decouple subcore-scale permeability heterogeneity effects from those of local dispersion. As such, the approach presented here proves to be very valuable, if not necessary, in the context of reservoir core analyses, because rock samples can rarely be regarded as ‘uniformly heterogeneous’.