We study entrainment in lock-release gravity currents using highly spatially resolved optical transmission experiments and quantitative analysis of the available potential energy of the flow. The principal results provide a resolution to the debate regarding the mechanism and degree of mixing in the head of a gravity current during the slumping phase. The nature of the complex internal mixing structure changes as a bore propagates from the tail to the head of the current during the slumping phase and overtakes its leading edge. We use quantitative methods to identify the connection between dynamics and entrainment and show that its manifestation as examined using different methodologies is the cause of previous contradictory experimental findings. Therefore, we conclude that the two main perspectives previously considered at odds are in accord.

Direct and non-invasive measurement of the pressure distribution in test sections of a micro-channel is a challenging, if not an impossible, task. Here, we present an analytical method for extracting the pressure distribution in a deformable micro-channel under flow. Our method is based on a measurement of the channel deflection profile as a function of applied hydrostatic pressure; this initial measurement generates ‘constitutive curves’ for the deformable channel. The deflection profile under flow is then matched to the constitutive curves, providing the hydrodynamic pressure distribution. The method is validated by measurements on planar microfluidic channels against analytic and numerical models. The accuracy here is independent of the nature of the wall deformations and is not degraded even in the limit of large deflections, ${\zeta }_{max} / 2{h}_{0} = O(1)$, with ${\zeta }_{max} $ and $2{h}_{0} $ being the maximum deflection and the unperturbed height of the channel, respectively. We discuss possible applications of the method in characterizing micro-flows, including those in biological systems.

We study theoretically the dynamics of porous ellipsoids rotating in simple shear flows. We use the Brinkman–Debye–Bueche (BDB) model to simulate flow within and through particles and solve the coupled Stokes–BDB equations to calculate the overall flow field and the rotation rate of porous ellipsoids. Our results show that the permeability has little effect on the rotational behaviour of particles, and that Jeffery’s prediction of the angular velocity of impermeable ellipsoids in simple shear flows (Proc. R. Soc. Lond. A, vol. 102, 1922, pp. 161–179) remains an excellent approximation, if not an exact one, for porous ellipsoids. Employing an appropriate scaling, we also present approximate expressions for the torque exerted on ellipses and spheroids rotating in a quiescent fluid. Our findings can serve as the basis for developing a suspension theory for non-spherical porous particles, or for understanding the orientational diffusion of permeable ellipses and spheroids.

Direct force measurements and qualitative flow visualization were used to compare flow field evolution versus lift and drag for a nominally two-dimensional rigid flat plate executing smoothed linear pitch ramp manoeuvres in a water tunnel. Non-dimensional pitch rate was varied from 0.01 to 0.5, incidence angle from 0 to 90°, and pitch pivot point from the leading to the trailing edge. For low pitch rates, the main unsteady effect is delay of stall beyond the steady incidence angle. Shifting the time base to account for different pivot points leads to collapse of both lift/drag history and flow field history. For higher rates, a leading edge vortex forms; its history also depends on pitch pivot point, but linear shift in time base is not successful in collapsing lift/drag history. Instead, a phenomenological algebraic relation, valid at the higher pitch rates, accounts for lift and drag for different rates and pivot points, through at least 45° incidence angle.

We investigate the levitation of a drop gently deposited onto the inner wall of a rotating hollow cylinder. For a sufficiently large velocity of the wall, the drop steadily levitates over a thin air film and reaches a stable angular position in the cylinder, where the drag and lift balance the weight of the drop. Interferometric measurements yield the three-dimensional (3D) air film thickness under the drop and reveal the asymmetry of the profile along the direction of the wall motion. A two-dimensional (2D) model is presented which explains the levitation mechanism, captures the main characteristics of the air film shape and predicts two asymptotic regimes for the film thickness ${h}_{0} $: for large drops ${h}_{0} \sim {\mathit{Ca}}^{2/ 3} { \kappa }_{b}^{- 1} $, as in the Bretherton problem, where $\mathit{Ca}$ is the capillary number based on the air viscosity and ${\kappa }_{b} $ is the curvature at the bottom of the drop; for small drops ${h}_{0} \sim {\mathit{Ca}}^{4/ 5} {(a{\kappa }_{b} )}^{4/ 5} { \kappa }_{b}^{- 1} $, where $a$ is the capillary length.

This paper considers the effect of flow advection on entropy waves. The interest is in whether entropy waves persist in gas turbine combustors, between the flame, where they are generated, and the combustor exit, where their acceleration generates acoustic waves (known as ‘entropy noise’ or ‘indirect combustion noise’). Entropy wave advection within a simplified fully developed turbulent channel-flow simulation is investigated. Entropy wave dissipation is found to be negligible, with loss of entropy wave strength caused predominantly by mean flow shear dispersion. The impulse response of entropy perturbations downstream of where they are generated is shown to be well modelled by a Gaussian profile in time. This yields a (different) Gaussian form for the inlet–outlet transfer function of entropy perturbations. For representative gas turbine flows, the magnitude of this transfer function is such that significant entropy wave strength will remain at the combustor exit, confirming that entropy-generated acoustic waves are likely to be important.

Laser-induced optical breakdown at two spatial locations in ultrapure water saturated with ambient gas is used to nucleate microscopic bubble clouds with lifetimes of tens of nanoseconds. The liquid is ruptured via the interaction of a pair of laser-induced shocks and bubbles. We find that the acoustically nucleated micro-bubbles appear in a localized region defined by the plane that bisects the pair of foci, where rarefaction waves (reflected from the laser-induced bubbles) merge. We measure the probability for acoustic nucleation as a function of the separation between the foci, and the minimum pressures for each separation are calculated with Euler flow simulations. The simulations show that the liquid is exposed to negative pressures for 3–17 ns. A statistical threshold pressure for cavitation inception (0.5 probability) of $- 20. 1\pm 3. 4~\mathrm{MPa} $ is extracted from the measured probabilities and the calculated minimum pressures.

Spatial dynamical systems are obtained for two-dimensional steady gravity waves with vorticity on water of finite depth. These systems have Hamiltonian structure and Hamiltonian is essentially the flow–force invariant.

A recently proposed stabilization mechanism for the Rayleigh–Taylor instability, using magnetic fluids and azimuthally rotating magnetic fields, is experimentally investigated in a cylindrical geometry and compared with the theoretical model. This approach allows the imperfection of the experimental setup to be exploited for measuring the critical field strength of the instability without ever reaching the supercritical state. Furthermore, we use a fast increase in the magnetic field strength to prevent an already occurring instability and force the system back to its initial state. In this way we measure the growth dynamics repeatedly and acquire the characteristic time scale ${\tau }_{0} $ of the instability.

Ordinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

We introduce a simple variational principle for coherent material vortices in two-dimensional turbulence. Vortex boundaries are sought as closed stationary curves of the averaged Lagrangian strain. Solutions to this problem turn out to be mathematically equivalent to photon spheres around black holes in cosmology. The fluidic photon spheres satisfy explicit differential equations whose outermost limit cycles are optimal Lagrangian vortex boundaries. As an application, we uncover super-coherent material eddies in the South Atlantic, which yield specific Lagrangian transport estimates for Agulhas rings.

The non-stationary transition from Mach to regular reflection followed by a reverse transition from regular to Mach reflection is investigated experimentally. A new experimental setup in which an incident shock wave reflects from a cylindrical concave surface followed by a cylindrical convex surface of the same radius is introduced. Unlike other studies that indicate problems in identifying the triple point, an in-house image processing program, which enables automatic detection of the triple point, is developed and presented. The experiments are performed in air having a specific heats ratio 1.4 at three different incident-shock-wave Mach numbers: 1.2, 1.3 and 1.4. The data are extracted from high-resolution schlieren images obtained by means of a fully automatically operated shock-tube system. Each experiment produces a single image. However, the high accuracy and repeatability of the control system together with the fast opening valve enables us to monitor the dynamic evolution of the shock reflections. Consequently, high-resolution results both in space and time are obtained. The credibility of the present analysis is demonstrated by comparing the first transition from Mach to regular reflection ($\mathrm{MR} \rightarrow \mathrm{RR} $) with previous single cylindrical concave surface experiments. It is found that the second transition, back to Mach reflection ($\mathrm{RR} \rightarrow \mathrm{MR} $), occurs earlier than one would expect when the shock reflects from a single cylindrical convex surface. Furthermore, the hysteresis is observed at incident-shock-wave Mach numbers smaller than those at which the dual-solution domain starts, which is the minimal value for obtaining hysteresis in steady and pseudo-steady flows. The existence of a non-stationary hysteresis phenomenon, which is different from the steady-state hysteresis phenomenon, is discovered.

The Rotne–Prager–Yamakawa approximation is one of the most commonly used methods of including hydrodynamic interactions in modelling of colloidal suspensions and polymer solutions. The two main merits of this approximation are that it includes all long-range terms (i.e. decaying as ${R}^{- 3} $ or slower in interparticle distances) and that the diffusion matrix is positive definite, which is essential for Brownian dynamics modelling. Here, we extend the Rotne–Prager–Yamakawa approach to include both translational and rotational degrees of freedom, and derive the regularizing corrections to account for overlapping particles. Additionally, we show how the Rotne–Prager–Yamakawa approximation can be generalized for other geometries and boundary conditions.

We present a theoretical and experimental analysis of axisymmetric gravity currents of power-law fluids in homogeneous porous media. The non-Newtonian shear-thinning fluid is a mixture of water, glycerol and Xanthan gum ($n= 0. 33{\unicode{x2013}} 0. 53$), and it is injected into a porous layer of glass beads ($d= 1{\unicode{x2013}} 3~\mathrm{mm} $). We compare experiments conducted with constant ($\alpha = 1$) and time-increasing ($\alpha = 1. 5$ and $2. 0$) influxes to theoretical self-similar solutions obtained by the numerical integration of the nonlinear ordinary differential equation that describes one-dimensional transient motion. The theoretical analysis is confirmed by experimental data. In addition, the selection of the most appropriate expression for the tortuosity factor and the choice of the correct range of shear stress for the determination of the rheological parameters are shown to be crucial to obtaining a good fit between the theory and experiments.

Using a large-time-resolved particle image velocimetry field of view, a developing turbulent spot is followed in space and time in a rectangular channel flow for more than 100 advective time units. We show that the flow can be decomposed into a large-scale motion consisting of an asymmetric quadrupole centred on the spot and a small-scale part consisting of streamwise streaks. From the temporal evolution of the energy of the streamwise and spanwise velocity perturbations, it is suggested that a self-sustaining process can occur in a turbulent spot above a given Reynolds number.

The mechanical energy budget for thermally equilibrated Rayleigh–Bénard convection is developed theoretically, with explicit consideration of the role of available potential energy, this being the form in which all the mechanical energy for the flow is supplied. The analysis allows derivation for the first time of a closed analytical expression relating the rate of mixing in symmetric fully developed convection to the rate at which available potential energy is supplied by the thermal forcing. Only about half this supplied energy is dissipated viscously. The remainder is consumed by mixing acting to homogenize the density field. This finding is expected to apply over a wide range of Rayleigh and Prandtl numbers for which the Nusselt number is significantly greater than unity. Thus convection at large Rayleigh number involves energetically efficient mixing of density variations. In contrast to conventional approaches to Rayleigh–Bénard convection, the dissipation of temperature or density variance is shown not to be of direct relevance to the mechanical energy budget. Thus, explicit recognition of available potential energy as the source of mechanical energy for convection, and of both mixing and viscous dissipation as the sinks of this energy, could be of further use in understanding the physics.

Three-dimensional convection in a binary mixture in a porous medium heated from below is studied. For negative separation ratios steady spatially localized convection patterns are expected. Such patterns, spatially localized in two dimensions, are computed and numerical continuation is used to examine their growth and proliferation as parameters are varied. The patterns studied have the form of a core region with four extended side-branches and can be stable. A physical mechanism behind the formation of these unusual structures is suggested.

Many eukaryotic cells use the active waving motion of flexible flagella to self-propel in viscous fluids. However, the criterion governing the selection of particular flagellar waveforms among all possible shapes has proved elusive so far. To address this question, we derive computationally the optimal shape of an internally forced periodic planar flagellum deforming as a travelling wave. The optimum is here defined as the shape leading to a given swimming speed with minimum energetic cost. To calculate the energetic cost, we consider the irreversible internal power expended by the molecular motors forcing the flagellum, only a portion of which is dissipated in the fluid. This optimization approach allows us to derive a family of shapes depending on a single dimensionless number quantifying the relative importance of elastic to viscous effects: the Sperm number. The computed optimal shapes are found to agree with the waveforms observed on spermatozoon of marine organisms, thus suggesting that these eukaryotic flagella might have evolved to be mechanically optimal.

The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments obey ${D}_{m} \geq {D}_{m+ 1} $, the reverse of the usual ${\Omega }_{m+ 1} \geq {\Omega }_{m} $ Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the $1\lt m\lt \infty $ vorticity moments are ordered in a manner consistent with the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with ${ D}_{m}^{2} \rightarrow \sup \vert \boldsymbol{\omega} \vert \sim A_{m}{({T}_{c} - t)}^{- 1} $ where the ${A}_{m} $ are nearly independent of $m$. In the second phase, the new ${D}_{m} $ ordering breaks down as the ${\Omega }_{m} $ converge towards the same super-exponential growth for all $m$. The transition is identified using new inequalities for the upper bounds for the $- \mathrm{d} { D}_{m}^{- 2} / \mathrm{d} t$ that are based solely upon the ratios ${D}_{m+ 1} / {D}_{m} $, and the convergent super-exponential growth is shown by plotting $\log (\mathrm{d} \log {\Omega }_{m} / \mathrm{d} t)$. Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth.

Supersonic turbulence plays an important role in a number of extreme astrophysical and terrestrial environments, yet its understanding remains rudimentary. We use data from a three-dimensional simulation of supersonic isothermal turbulence to reconstruct an exact fourth-order relation derived analytically from the Navier–Stokes equations (Galtier & Banerjee, Phys. Rev. Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like inertial energy cascade in supersonic turbulence previously discussed on a phenomenological level. We show that two compressible analogues of the four-fifths law exist describing fifth- and fourth-order correlations, but only the fourth-order relation remains ‘universal’ in a wide range of Mach numbers from incompressible to highly compressible regimes. A new approximate relation valid in the strongly supersonic regime is derived and verified. We also briefly discuss the origin of bottleneck bumps in simulations of compressible turbulence.