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.

In this work, we report the excitation of inertial waves in a librating sphere even for libration frequencies where these waves are not directly forced. This spontaneous generation comes from the localized turbulence induced by the centrifugal instabilities in the Ekman boundary layer near the equator and does not depend on the libration frequency. We characterize the key features of these inertial waves in analogy with previous studies of the generation of internal waves in stratified flows from localized turbulent patterns. In particular, the temporal spectrum exhibits preferred values of excited frequency. This first-order phenomenon is generic to any rotating flow in the presence of localized turbulence and is fully relevant for planetary applications.

Resonance characteristics of tidal waves in a network are investigated with the linearized, one-dimensional shallow water equations. The network comprises a semi-enclosed main channel with an adjacent secondary channel at an arbitrary position. Water motion is forced by a prescribed incoming wave at the entrance of the main channel. The model is used to compute the ratio of sea surface height amplitude in the presence and absence of the secondary channel. Relevance lies in the possible construction of secondary channels to reduce tidal range in the main channel. When $\mu = 2\mathrm{\pi} { l}_{b}^{\ast } / { l}_{t}^{\ast } \gtrsim \mathrm{\pi} $ ( ${ l}_{b}^{\ast } $ being the length of the main channel, ${ l}_{t}^{\ast } $ the tidal wavelength) and friction is weak, it is found that reduction (amplification) of the tidal range occurs for secondary channels located less (more) than a quarter wavelength from the closed end of the main channel. Furthermore, a minimum is observed halfway between the closed end and the nodal point, and a maximum a quarter wavelength further seaward. With increasing friction and/or decreasing length of the main channel, amplitude ratios become less pronounced and depend weakly on the location of the secondary channel. The mechanism responsible for amplification or reduction of the sea surface height in the main channel is identified and explained in terms of the forced wave and waves radiating away from the secondary channel.

Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a ${k}^{- 3} \log k$ spectrum in the transient stage, before the Kolmogorov ${k}^{- 5/ 3} $ asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of transient decaying turbulence.

The simple pendulum remains one of the most fundamental systems studied in physics. It is commonly used as a model to illustrate a broad variety of mechanisms in a wide range of areas. However, in spite of this popularity, subtle behaviours still remain to be discovered and to be explored when the pendulum is strongly coupled to fluid mechanics. This is for instance illustrated in recent studies by Neill, Livelybrooks & Donnelly (Am. J. Phys., vol. 75, 2007, pp. 226–229) and Bolster, Hershberger & Donnelly (Phys. Rev. E, vol. 81, 2010, pp. 1–6) which highlight the impact on a simple spherical pendulum of vortex shedding and added mass effects. In the present work we show that the equilibrium of a pendular disk facing a flow exhibits bi-stability and hysteresis. We give a simple interpretation of this behaviour in terms of a two-potential-well description, only requiring to know the angular dependence of the normal drag coefficient of an inclined plate. We investigate the influence of turbulence on the equilibrium of the pendulum in general and on the observed bi-stability in particular. Our results have potentially important fundamental and practical consequences: (i) they extend the attractiveness of the pendulum as a model to investigate generic questions related to bi-stable stochastic processes, (ii) they highlight important fluid dynamic mechanisms, including turbulent drag enhancement and fluid–structure interaction issues.

We present an experimental investigation of the dynamics and flow coupling of convective turbulent flows in a cylindrical Rayleigh–Bénard convection (RBC) cell with two immiscible fluids, water and Fluorinert FC-77 electronic liquid (FC77). With the lighter water above FC77, the latter is under the condition of constant heat flux at its top and bottom boundaries. It is found that one large-scale circulation (LSC) roll exists in each of the fluid layers, and that their circulation planes have two preferred azimuthal orientations separated by ${\sim }\mathrm{\pi} $ . A surprising finding of the study is that cessations/reversals of the LSC in FC77 of the two-layer system occur much more frequently than they do in single-layer turbulent RBC, and that a cessation is most likely to result in a flow reversal of the LSC, which is in sharp contrast with the uniform distribution of the orientational angular change of the LSC before and after cessations in single-layer turbulent RBC. This implies that the dynamics governing cessations and reversals in the two systems are very different. Two coupling modes, thermal coupling (the flow directions of the two LSCs are opposite to each other at the fluid–fluid interface) and viscous coupling (the flow directions of the two LSCs are the same at the fluid–fluid interface), are identified, with the former as the predominant mode. That most cessations (in the FC77 layer) end up as reversals can be understood as a symmetry breaking imposed by the orientation of the LSC in the water layer, which remains unchanged most of the time. Furthermore, the frequently occurring cessations and reversals are caused by the system switching between its two metastable states, i.e. thermal and viscous coupling modes. It is also observed that the strength of the LSC in water becomes weaker when the LSC in FC77 rotates faster azimuthally and that the flow strength in FC77 becomes stronger when the LSC in water rotates faster azimuthally, i.e. the influence of the LSC in one fluid layer on the other is not symmetric.

This study refers to the article of Chicón, Castellanos & Martion (J. Fluid Mech., vol. 344, 1997, pp. 43–66), who presented a numerical study of electroconvection in a layer of dielectric liquid induced by unipolar injection. An important characteristic of the numerical strategy proposed by Chicón et al. lies in the fact that the Navier–Stokes equations are never solved to obtain the velocity field, which is subsequently needed in the charge density transport equation. Instead, the velocity field is explicitly provided by an expression obtained with some assumptions about the flow structure and related to the electric field (the imposed velocity field approach; IVF). The validity of the above simplification is examined through a direct comparison of the solutions obtained by solving the Navier–Stokes equations (the Navier–Stokes computation approach; NSC). It is clearly demonstrated that, even in the strong injection regime ($C= 10$), the results look very similar for a given range of the mobility parameter $M$; however, in the weak injection regime ($C= 0. 1$), significant discrepancies are observed. The rich flow structures obtained with the NSC approach invalidate the use of the IVF approach in the weak injection regime.

Flow instabilities encountered in the homogeneous cooling of a gas–solid system are considered via lattice-Boltzmann simulations. Unlike previous efforts, the relative contribution of the two mechanisms leading to instabilities is explored: viscous dissipation (fluid-phase effects) and collisional dissipation (particle-phase effects). The results indicate that the instabilities encountered in the gas–solid system mimic those previously observed in their granular (no fluid) counterparts, namely a velocity vortex instability that precedes in time a clustering instability. We further observe that the onset of the instabilities is quicker in more dissipative systems, regardless of the source of the dissipation. Somewhat surprisingly however, a cross-over of the kinetic energy levels is observed during the evolution of the instability. Specifically, the kinetic energy of the gas–solid system is seen to become greater than that of its granular counterpart (i.e. same restitution coefficient) after the vortex instability sets in. This cross-over of kinetic energy levels between a more dissipative system (gas–solid) and a less dissipative system (granular) can be explained by the alignment of particle motion found in a vortex. Such alignment leads to a reduction in both collisional and viscous energy dissipation due to the more glancing nature of collisions.

Exact one-dimensional solutions to the equations of fluid dynamics are derived in the $\mathit{Pr}\rightarrow \infty $ and $\mathit{Pr}\rightarrow 0$ limits (where $\mathit{Pr}$ is the Prandtl number). The solutions are analogous to the $\mathit{Pr}= 3/ 4$ solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large- $\mathit{Pr}$ solution is very similar to Becker’s solution, differing only by a scale factor. The small- $\mathit{Pr}$ solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite- $\mathit{Pr}$ equations is $\mathit{O}({\mathit{Pr}}^{- 1} )$ as $\mathit{Pr}\rightarrow \infty $ and $\mathit{O}(\mathit{Pr})$ as $\mathit{Pr}\rightarrow 0$ .

We note that the mirror-symmetric solution in plane Couette flow, found recently by Gibson, Halcrow & Cvitanović (J. Fluid Mech., vol. 611, 2009, pp. 107–130) and Itano & Generalis (Phys. Rev. Lett., vol. 102, 2009, p. 114501), belongs to the solution group classified as ‘ribbon’ in rotating-plane Couette flow (RPCF). It represents a subcritical (in terms of the system rotation) solution at zero rotation rate on the three-dimensional tertiary flow branch which bifurcates from the second streamwise-independent flow in RPCF. The way of its appearance is similar to that of the Nagata solution (J. Fluid Mech., vol. 217, 1990, pp. 519–527), which lies on the subcritical three-dimensional tertiary flow branch bifurcating from the first streamwise-independent flow in RPCF.

We study drop impact on a deep pool of the same fluid, with an emphasis on the air layer trapped under the droplets from its formation to its rupture. The penetration velocity of the air layer at a very short time scale prior to its rupture is shown, using an energy argument and experimental verification, to be one-half of the impact velocity. We then deduce the dependence of the rupture position on the liquid viscosity and the impact velocity. We show that the volume of the resulting air bubbles can be related to both those resulting from droplets impacting on solid surfaces and those resulting from rigid spheres impacting on liquid surfaces.

We consider the development of nonlinear three-dimensional vortex–wave interaction equilibria of laminar plane Couette flow for a range of spanwise wavenumbers. The results are computed using a hybrid approach that captures the required asymptotic structure while at the same time providing a direct link with full numerical calculations of equilibrium states. Each equilibrium state consists of a streak flow, a roll flow and a wave propagating on the streak. Direct numerical simulations at finite Reynolds numbers using initial conditions constructed from these parts confirm that the scheme generates equilibrium solutions of the Navier–Stokes equations. Consideration of the form of the vortex–wave interaction equations in the high-spanwise-wavenumber limit predicts that for small wavelengths the equilibria take on a self-similar structure confined near the centre of the flow. These states feel no influence from the walls, and lead to a class of canonical states relevant to arbitrary shear flows. These predictions are supported by an analysis of computational results at increasing values of the spanwise wavenumber. For the wave part of these new canonical states, it is shown that the mass-specific kinetic energy density per unit wavenumber scales with the $- 5/ 3$ power of the wavenumber.

Existing control-volume models for predicting the front velocity of internal bores enforce the conservation of mass and streamwise momentum, but not vertical momentum. Instead, they usually invoke an empirical assumption relating the up- and downstream energy fluxes to obtain an additional equation required for determining the pressure jump across a bore. The present investigation develops a control-volume model for internal bores on the basis of mass and momentum conservation alone, without the need for considering energy. This is accomplished by combining the streamwise and vertical momentum equations to obtain a vorticity relation that no longer involves pressure. Hence, this vorticity equation, in combination with the conservation of mass, is sufficient for evaluating the bore velocity. The energy loss across the bore can then be predicted by the streamwise energy equation and compared to the assumptions underlying earlier models. The flux of vorticity across the internal bore predicted by the new model is seen to be in close agreement with direct numerical simulation results. Any discrepancies with experimentally measured bore velocities are shown to be due to the effects of downstream mixing.

Buoyancy-driven convection is modelled using the Navier–Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, ${c}_{p} $ , must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of $- \mathrm{D} P/ \mathrm{D} t$ and not that of $- \alpha T(\mathrm{D} P/ \mathrm{D} t)$ . Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, $\gamma - 1\ll 1$ , independent of the dissipation number.

The dynamics of moving contact lines in a two-phase Couette flow is investigated by using a matched asymptotic procedure. The walls are assumed to be partially wetting, and the microscopic contact angle is finite but sufficiently small so that the lubrication approach can be used. Explicit formulas are derived to characterize the shear-induced interface deformation and the critical capillary number for the onset of wetting transition. It is found that the apparent contact angle vanishes for liquid–air systems and remains finite for liquid–liquid systems when the wetting transition occurs.