We study theoretically and numerically the downstream flow near the corner of a bluff body partially submerged at a deadrise depth ${\rm\Delta}h$ into a uniform stream of velocity $U$, in the presence of gravity, $g$. When the Froude number, $\mathit{Fr}=U/\sqrt{g{\rm\Delta}h}$, is large, a three-dimensional steady plunging wave, which is referred to as a corner wave, forms near the corner, developing downstream in a similar way to a two-dimensional plunging wave evolving in time. We have performed an asymptotic analysis of the flow near this corner to describe the wave’s initial evolution and to clarify the physical mechanism that leads to its formation. Using the two-dimensions-plus-time approximation, the problem reduces to one similar to dam-break flow with a wet bed in front of the dam. The analysis shows that, at leading order, the problem admits a self-similar formulation when the size of the wave is small compared with the height difference ${\rm\Delta}h$. The essential feature of the self-similar solution is the formation of a mushroom-shaped jet from which two smaller lateral jets stem. However, numerical simulations show that this self-similar solution is questionable from the physical point of view, as the two lateral jets plunge onto the free surface, leading to a self-intersecting flow. The physical mechanism leading to the formation of the mushroom-shaped structure is discussed.

The dynamics of a large deformable bubble ($\mathit{Re}\sim \mathit{O}(10^{3})$) rising near a vertical wall in quiescent water is experimentally investigated. The reference (without the wall) rising path of the considered bubble is a two-dimensional zigzag. For a range of wall configurations (i.e. initial wall distance and boundary condition), using high-speed shadowgraphy, various rising behaviours such as periodic bouncing, sliding, migrating away, and non-periodic oscillation without collisions are measured and analysed. Unlike low-Reynolds- and Weber-number bubbles, the contribution of the surface deformation to the transport between the energy components becomes significant during the bubble’s rise. In particular, across the bubble–wall collision, the excess surface energy compensates the deficit of kinetic energy. This enables a large deformable bubble to maintain a relatively constant bouncing kinematics, despite the obvious wall-induced energy dissipation. The wall effect, predominantly appearing as energy loss, is found to decrease as the initial distance from the bubble centre to the wall increases. Compared to the regular (no-slip) wall, a hydrophobic surface enhances or reduces the wall effect depending on the wall distance, whereas a porous surface reduces the energy loss due to the wall, regardless of the initial distance from the wall. Furthermore, the bubble–wall collision behaviour is assessed in terms of a restitution coefficient and modified impact Stokes number (ratio of the inertia to viscous forces), which shows a good correlation, the trends of which agree well with the variations in the energy components. The dependence of near-wall bubble motion on the wall distance and boundary condition may suggest a way of predicting or controlling the near-wall gas void-fraction distribution in gas–liquid flow systems.

The flow of an electrically conducting fluid in an array of square ducts, separated by arbitrary thickness conducting walls, subject to an applied magnetic field is studied. The analytical solution presented here is valid for thick walls and is based on the homogeneous solution obtained by Shercliff (Math. Proc. Camb. Phil. Soc., vol. 49 (01), 1953, pp. 136–144). Arrangements of ducts arise in a number of applications, most notably in fusion blankets, where liquid metal is used both as coolant and for tritium generation purposes. Analytical solutions, such as those presented here, provide insight into the physics and important benchmarking and validation data for computational magnetohydrodynamics (MHD), as well as providing approximate flow parameters for 1D systems codes. It is well known that arrays of such ducts with conducting walls exhibit varying degrees of coupling, significantly affecting the flow. An important practical example is the so-called Madarame problem (Madarame et al., Fusion Technol., vol. 8, 1985, pp. 264–269). In this work analytical results are derived for the relevant hydrodynamic and magnetic parameters for a single duct with thick walls analogous to the Hunt II case. These results are then extended to an array of such ducts stacked in the direction of the applied magnetic field. It is seen that there is a significant coupling affect, resulting in modifications to pressure drop and velocity profile. In certain circumstances, counter-current flow can occur as a result of the MHD effects, even to the point where the mean flow is reversed. Such phenomena are likely to have significant detrimental effects on both heat and mass transfer in fusion applications. The dependence of this coupling on parameters such as conductivities, wall thickness and Hartmann number is studied.

The spatio-temporal pressure–velocity correlation in a turbulent boundary layer is investigated so as to understand the link between pressure fluctuations and turbulent coherent structures. A new experimental set-up is developed to measure the pressure fluctuations at the wall and in the field and, simultaneously, the velocity field by stereoscopic particle image velocimetry. The present measurement area covers the whole boundary layer thickness, and the spatial resolution of the measurement is good enough to assess the representative length scales of the flow. The Reynolds number effect is quantified from the data at $\mathit{Re}_{{\it\theta}}=7300$, 10 000, 18 000. The spatio-temporal three-dimensional structures of the pressure–velocity correlations, $\boldsymbol{R}_{pu}$, $\boldsymbol{R}_{pv}$ and $\boldsymbol{R}_{pw}$, are evaluated. The wall pressure fluctuations are closely coupled with coherent structures which occupy a large region of the boundary layer in the wall-normal and spanwise directions and up to $10{\it\delta}/U_{e}$ in time, where ${\it\delta}$ and $U_{e}$ denote the boundary layer thickness and the free stream velocity. Reynolds number effects are mainly observed on the size and intensity of the pressure–velocity correlations. Conditioning the correlations on the pressure signal sign shows different types of flow phenomena linked to the positive and negative pressure events. For the wall pressure, positive pressure fluctuations appear to be correlated with the leading edge of a large sweeping motion of splatting type followed by a large ejection. The negative pressure fluctuations are linked to a localized ejection upstream, followed by a large sweeping motion downstream. For the pressure fluctuations in the field, in addition to the structures observed with the wall pressure, the pressure–velocity correlations exhibit a significant correlation in a region very extended in time. Such long structures appear to be independent of the one observed at the wall and to grow significantly in time with the Reynolds number when scaling with external variables. When conditioned by the pressure sign, clear ejection and sweeping motions are observed with associated streamwise vortical structures at a scale of the order of $0.2{\it\delta}$. These structures can be linked to the large-scale motion and very-large-scale motion previously observed by different authors and seem to organize in a scheme analogous to the near-wall cycle, but at a much larger scale.

We perform particle image velocimetry (PIV) measurements of various terms of the non-homogeneous Kármán–Howarth–Monin equation in the most inhomogeneous and anisotropic region of grid-generated turbulence, the production region which lies between the grid and the peak of turbulence intensity. We use a well-documented fractal grid which is known to magnify the streamwise extent of the production region and abate its turbulence activity. On the centreline around the centre of that region the two-point advection and transport terms are dominant and the production is significant too. It is therefore impossible to apply usual Kolmogorov arguments based on the Kármán–Howarth–Monin equation and resulting dimensional considerations to deduce interscale flux and spectral properties. The interscale energy transfers at this location turn out to be highly anisotropic and consist of a combined forward and inverse cascade in different directions which, when averaged over directions, gives an interscale energy flux that is negative (hence forward cascade on average) and not too far from linear in $r$, the modulus of the separation vector $\boldsymbol{r}$ between two points. The energy spectrum of the streamwise fluctuating component exhibits a well-defined $-5/3$ power law over one decade, even though the streamwise direction is at a small angle to the inverse cascading direction.

The interaction of two laser-induced bubbles in bulk water is investigated. The strength and direction of the emerging liquid jets can be controlled by adjusting the relative bubble positions, the time difference between bubble generation, and the laser pulse energies determining the bubble sizes. Experimental and numerical studies are performed for millimetre-sized bubble pairs. Taking bubbles of equal energy, a maximum jet velocity is found for close anti-phase bubbles, i.e. when the second bubble is produced at the maximum volume of the first one and the bubble walls are almost touching and not merging. Under these conditions, one bubble produces a fast jet with a peak velocity of about $150~\text{m}~\text{s}^{-1}$ that reaches a distance into the surrounding liquid of at least three times the maximum bubble radius. Collapse of the other bubble results in a slow jet of large mass that rapidly converts into a ring vortex. Correspondingly, the interaction with adjacent structures is dominated either by localized jet impact or by shear stresses extending over a larger area. Furthermore, interactions between micrometre-sized bubble pairs are investigated numerically to understand and predict how the effects of the physical parameters on bubble dynamics would change when the bubbles become smaller. The results are discussed with respect to micropumping and opto-injection.

Direct numerical simulations (DNS) are conducted for turbulent flow through pipes with three-dimensional sinusoidal roughnesses explicitly represented by body-conforming grids. The same viscous-scaled roughness geometry is first simulated at a range of different Reynolds numbers to investigate the effects of low Reynolds numbers and low $R_{0}/h$, where $R_{0}$ is the pipe radius and $h$ is the roughness height. Results for the present class of surfaces show that the Hama roughness function ${\rm\Delta}U^{+}$ is only marginally affected by low Reynolds numbers (or low $R_{0}/h$), and observations of outer-layer similarity (or lack thereof) show no signs of sensitivity to Reynolds number. Then, building on this, a systematic approach is taken to isolate the effects of roughness height $h^{+}$ and wavelength ${\it\lambda}^{+}$ in a turbulent wall-bounded flow in both transitionally rough and fully rough regimes. Current findings show that while the effective slope $\mathit{ES}$ (which for the present sinusoidal surfaces is proportional to $h^{+}/{\it\lambda}^{+}$) is an important roughness parameter, the roughness function ${\rm\Delta}U^{+}$ must also depend on some measure of the viscous roughness height. A simplistic linear–log fit clearly illustrates the strong correlation between ${\rm\Delta}U^{+}$ and both the roughness average height $k_{a}^{+}$ (which is related to $h^{+}$) and $\mathit{ES}$ for the surfaces simulated here, consistent with published literature. Various definitions of the virtual origin for rough-wall turbulent pipe flow are investigated and, for the surfaces simulated here, the hydraulic radius of the pipe appears to be the most suitable parameter, and indeed is the only virtual origin that can ever lead to collapse in the total stress. First- and second-order statistics are also analysed and collapses in the outer layer are observed for all cases, including those where the largest roughness height is a substantial proportion of the reference radius (low $R_{0}/h$). These results provide evidence that turbulent pipe flow over the present sinusoidal surfaces adheres to Townsend’s notion of outer-layer similarity, which pertains to statistics of relative motion.

The physical interpretation of velocity potential is used to propose a model of the mean flow boundary of a row of plumes or jets. Generalised plume equations incorporating the plume area and net entrainment are closed with an entrainment assumption. The resulting model is shown to approach the appropriate limiting similarity solutions above and below the merging height in an unstratified environment. The virtual origin of the far-field flow is hence predicted. An application to plume rise in channels of varying aspect ratio shows that the model may be used to predict the depth of the outflow along the channel.

We report the results of three-dimensional direct numerical simulations for incompressible viscous fluid in a circular pipe flow with a gradual expansion. At the inlet, a parabolic velocity profile is applied together with a constant finite-amplitude perturbation to represent experimental imperfections. Initially, at low Reynolds number, the solution is steady. As the Reynolds number is increased, the length of the recirculation region near the wall grows linearly. Then, at a critical Reynolds number, a symmetry-breaking bifurcation occurs, where linear growth of asymmetry is observed. Near the point of transition to turbulence, the flow experiences oscillations due to a shear layer instability for a narrow range of Reynolds numbers. At higher Reynolds numbers, the recirculation region breaks into a turbulent state which remains spatially localised and unchanged when the perturbation is removed from the flow. Spatial correlation analysis suggests that the localised turbulence in the gradual expansion possesses a different flow structure from the turbulent puff of uniform pipe flow.

Recently published experimental observations of slender bacteria swimming in channel flow demonstrate that the bacteria become trapped in regions of high shear, leading to reduced concentrations near the channel’s centreline. However, the commonly used advection–diffusion equation, formulated in macroscopic space variables and originally derived for unbounded homogeneous shear flow, predicts that the concentration of bacteria is uniform across the channel in the absence of chemotactic bias. In this paper, we instead use a Smoluchowski equation to describe the probability distribution of the bacteria, in macroscopic (physical) and microscopic (orientation) space variables. We demonstrate that the Smoluchowski equation is able to predict the trapping phenomena and compare the full numerical solution of the Smoluchowski equation with experimental results when there is no chemotactic bias and also in the presence of a uniform cross-channel chemotactic gradient. Moreover, a simple analytic approximation for the equilibrium distribution provides an excellent approximate solution for slender bacteria, suggesting that the dominant effect on equilibrium behaviour is flow-induced modification of the bacteria’s swimming direction. A continuum framework is thus provided to explain how the equilibrium distribution of slender chemotactic bacteria is altered in the presence of spatially varying shear flow. In particular, we demonstrate that while advection is an appropriate description of transport due to the mean swimming velocity, the random reorientation mechanism of the bacteria cannot be simply modelled as diffusion in physical space.