The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A second-order sensitivity analysis is conducted for the linear temporal stability of parallel flows $U_{0}$ subject to small-amplitude spanwise-periodic modification ${\it\epsilon}U_{1},{\it\epsilon}\ll 1$. It is known that spanwise-periodic flow modifications have a quadratic effect on stability properties, i.e. the first-order eigenvalue variation is zero, hence the need for a second-order analysis. A second-order sensitivity operator is computed from a one-dimensional calculation, which allows one to predict how eigenvalues are affected by any flow modification $U_{1}$, without actually solving for modified eigenvalues and eigenmodes. Comparisons with full two-dimensional stability calculations in a plane channel flow and in a mixing layer show excellent agreement. Next, optimisation is performed on the second-order sensitivity operator: for each eigenmode streamwise wavenumber ${\it\alpha}_{0}$ and base flow modification spanwise wavenumber ${\it\beta}$, the most stabilising/destabilising profiles $U_{1}$ are computed, together with lower/upper bounds for the variation in leading eigenvalue. These bounds increase like ${\it\beta}^{-2}$ as ${\it\beta}$ goes to zero, thus yielding a large stabilising potential. However, three-dimensional modes with wavenumbers ${\it\beta}_{0}=\pm {\it\beta}$, $\pm {\it\beta}/2$ are destabilised, and therefore larger control wavenumbers should be preferred. The most stabilising $U_{1}$ optimised for the most unstable streamwise wavenumber ${\it\alpha}_{0,max}$ has a stabilising effect on modes with other ${\it\alpha}_{0}$ values too. Finally, the potential of transient growth to amplify perturbations and stabilise the flow is assessed with a combined optimisation. Assuming a separation of time scales between the fast unstable mode and the slow transient evolution of the optimal perturbations, combined optimal perturbations that achieve the best balance between transient linear amplification and stabilisation of the nominal shear flow are determined. In the mixing layer with ${\it\beta}\leqslant 1.5$, these combined optimal perturbations appear similar to transient growth-only optimal perturbations, and achieve a more efficient overall stabilisation than optimal spanwise-periodic and spanwise-invariant modifications computed for stabilisation only. These results are consistent with the efficiency of streak-based control strategies.

In this work, we employ a kinetic-theory-based approach to predict the hydrodynamic forces on electromechanical resonators operating in gaseous media. Using the Boltzmann–BGK equation, we investigate the influence of the resonator geometry on the fluid resistance in the entire range of non-dimensional frequency variation 0 ≤ τω ≤ ∞; here the fluid relaxation time τ = μ/p is determined by the gas viscosity μ and pressure p at thermodynamic equilibrium, and ω is the (angular) oscillation frequency. Our results here capture two important aspects of recent experimental measurements that covered a broad range of experimental parameters. First, the experimentally observed transition from viscous to viscoelastic flow in simple gases at τω ≈ 1 emerges naturally in the numerical data. Second, the calculated effects of resonator geometry are in agreement with experimental observations.

Measurements were made of the wave height of periodic, quasi-periodic, and chaotic parametrically forced cross-waves in a long rectangular channel. In general, three frequencies (and their harmonics) may be observed: the subharmonic frequency and two slow temporal modulations — a one-mode instability associated with streamwise variation and a sloshing motion associated with spanwise variation. Their interaction, as forcing frequency, f, and forcing amplitude, a, were varied, produced a pattern of Arnold tongues in which two or three frequencies were locked. The overall picture of frequency-locked and -unlocked regions is explained in terms of the Arnold tongues predicted by the circle-map theory describing weakly coupled oscillators. Some of the observed tongues are apparently folded by a subcritical bifurcation, with the tips of the tongues lying on the unstable manifold folded under the observed stable manifold. Near the intersection of the neutral stability curves for two adjacent modes, a standing wave localized on one side of the tank was observed in agreement with the coupled-mode analysis of Ayanle, Bernoff & Lichter (1990). At large cross-wave amplitudes, the spanwise wave structure apparently breaks up, because of modulational instability, into coherent soliton-like structures that propagate in the spanwise direction and are reflected by the sidewalls.

We present a model describing the evolution of the small-scale Navier–Stokes turbulence due to its stochastic distortion by much larger turbulent scales. This study is motivated by numerical findings (Laval et al. Phys. Fluids vol. 13, 2001, p. 1995) that such interactions of separated scales play an important role in turbulence intermittency. We introduce a description of turbulence in terms of the moments of $k$-space quantities using a method previously developed for the kinematic dynamo problem (Nazarenko et al. Phys. Rev. E vol. 68, 2003, 0266311). Working with the $k$-space moments allows us to introduce new useful measures of intermittency such as the mean polarization and the spectral flatness. Our study of the small-scale two-dimensional turbulence shows that the Fourier moments take their Gaussian values in the energy cascade range whereas the enstrophy cascade is intermittent. In three dimensions, we show that the statistics of turbulence wavepackets deviates from Gaussianity toward dominance of the plane polarizations. Such turbulence is formed by ellipsoids in the $k$-space centred at its origin and having one large, one neutral and one small axis with the velocity field pointing parallel to the smallest axis.

Understanding the small-scale structure of incompressible turbulence and its implications for the non-local pressure field is one of the fundamental challenges in fluid mechanics. Intense velocity gradient structures tend to cluster on a range of scales which affects the pressure through a Poisson equation. Here we present a quantitative investigation of the spatial distribution of these structures conditional on their intensity for Taylor-based Reynolds numbers in the range [160, 380]. We find that the correlation length of the second invariant of the velocity gradient is proportional to the Kolmogorov scale. It is also a good indicator for the spatial localization of intense enstrophy and strain-dominated regions, as well as the separation between them. We describe and quantify the differences in the two-point statistics of these regions and the impact they have on the non-locality of the pressure field as a function of the intensity of the regions. Specifically, across the examined range of Reynolds numbers, the pressure in strong rotation-dominated regions is governed by a dissipation-scale neighbourhood. In strong strain-dominated regions, on the other hand, it is determined primarily by a larger neighbourhood reaching inertial scales.

The effect of a temperature gradient in a gas inclined at an angle to a boundary wall has been investigated. For an infinite half-space of gas it is found that, in addition to the conventional temperature slip problem, the component of the temperature gradient parallel to the wall induces a net mass flow known as thermal creep. We show that the temperature slip and thermal creep effects can be decoupled and treated quite separately.

Expressions are obtained for the creep velocity and heat flux, both far from and at the boundary; it is noted that thermal creep tends to reduce the effective thermal conductivity of the medium.

By assuming that an uncharged drop situated in a uniform electric field E retains a spheroidal shape while oscillating about its equilibrium configuration, two approximate equations of motion are derived for the deformation ratio γ expressed as the ratio a/b of the major and minor axis of the drop. Solutions of these equations of motion indicate that the stability of a drop of undistorted radius R and surface tension T depends upon E(R/T)½ and the initial displacement of γ from its equilibrium value. The predictions of the two equations are compared to assess the accuracy of the spheroidal assumption as applied to such a dynamical situation. The analysis is used to determine the stability criterion of a drop subject to a step function field. Finally, the limit of validity of the spheroidal assumption is discussed in terms of Rayleigh's criterion for the stability of charged spherical drops. By applying Rayleigh's criterion to the poles of a spheroidal drop, the stage at which the drop departs from spheroidal form to form conical jets was approximately determined.

We report on the influence of a quasi-one-dimensional periodic forcing on the pattern selection process in Rayleigh–Bénard convection (RBC). The forcing was introduced by a lithographically fabricated periodic texture on the bottom plate. We study the convection patterns as a function of the Rayleigh number ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ra}$) and the dimensionless forcing wavenumber ($q_f$). For small $\mathit{Ra}$, convection takes the form of straight parallel rolls that are locked to the underlying forcing pattern. With increasing $\mathit{Ra}$, these rolls give way to more complex patterns, due to a secondary instability. The forcing wavenumber $q_f$ was varied in the experiment over the range of $0.6q_c<q_f<1.4q_c$, with $q_c$ being the critical wavenumber of the unforced system. We investigate the stability of straight rolls as a function of $q_f$ and report patterns that arise due to a secondary instability.

We investigate experimentally and theoretically the streamwise transport and dispersion properties of steady quasi-two-dimensional plane turbulent jets discharged vertically from a slot of width into a fluid confined between two relatively close rigid boundaries with gap . We model the evolution in time and space of the concentration of passive tracers released in these jets using a one-dimensional time-dependent effective advection–diffusion equation. We make a mixing length hypothesis to model the streamwise turbulent eddy diffusivity such that it scales like , where is the streamwise coordinate, is the jet width, is the maximum time-averaged vertical velocity. Under these assumptions, the effective advection–diffusion equation for , the horizontal integral of the ensemble-averaged concentration, is of the form , where is time, (the advection parameter) and (the dispersion parameter) are empirical dimensionless parameters which quantify the importance of advection and dispersion, respectively, and is the source momentum flux. We find analytical solutions to this equation for in the cases of a constant-flux release and an instantaneous finite-volume release. We also give an integral formulation for the more general case of a time-dependent release, which we solve analytically when tracers are released at a constant flux over a finite period of time. From our experimental results, whose concentration distributions agree with the model, we find that and , for both finite-volume releases and constant-flux releases using either dye or virtual passive tracers. The experiments also show that streamwise dispersion increases in time as . As a result, in the case of finite-volume releases more than 50 % of the total volume of tracers is transported ahead of the purely advective front (i.e. the front location of the tracer distribution if all dispersion mechanisms are ignored and considering a ‘top-hat’ mean velocity profile in the jet); and in the case of constant-flux releases, at each instant in time, approximately 10 % of the total volume of tracers is transported ahead of the advective front.

The wings of butterflies are relatively heavier than those of other insects, and the inertial force and torque due to the wing mass are likely to have a significant effect on agility and manoeuvrability in the flapping flight of butterflies. In the present study, the effect of wing mass on the free flight of butterflies is investigated by numerical simulations based on an immersed boundary–lattice Boltzmann method. We use a butterfly-like model consisting of two square wings with mass connected by a rod-shaped body. We simulate the free flights of the model by changing the ratio of the wing mass to the total mass of the model and also changing the mass distributions of the wings. As a result, we find that the aerodynamic vertical and horizontal forces decrease as the wing-mass ratio increases, since for a large wing-mass ratio the body has large vertical and horizontal oscillations in each stroke and consequently the speeds of the wing tip and the leading edge relatively decrease. In addition, we find that the wing-mass ratio has a dominant effect on the rotational motion of the model, and a large wing-mass ratio reduces aerodynamic force and intensifies the time variation of the pitching angle. From the results of our free flight simulations, we clarify the critical wing-mass ratio between upward flight and downward flight and find that the critical wing-mass ratio is a function of the non-dimensional total mass and almost independent of the wing length. Then, we evaluate the effect of the wing-mass distribution on the critical wing-mass ratio. Finally, we discuss the limitations of the model.

Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around $47^{\circ }$, for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below $90^{\circ }$ can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

The present work is a numerical study of fluid flow and energy transport in single rock fractures. Each fracture is modelled as a parallel channel with a distribution of contact areas where the fracture aperture is zero. The pressure drop required to sustain a given rate of flow is calculated by solving the Stokes equations along with the incompressibility constraint. The permeability of the fracture is then determined as the ratio of mean flow and pressure drop. With flow distribution known, the advection–diffusion equation is solved for a thermal mixing problem. All differential equations have been solved by the finite-element technique.

Results show that the fracture permeability is affected by the presence of contact areas and the magnitude of h, the mean fracture aperture. The limiting case of zero aperture is realized at h/L [les ] 0.016, where L is the characteristic dimension of the fracture. The other limiting case of a two-dimensional fracture is seen for h/L [ges ] 1. The thermal dispersion profiles at steady state are unaffected by the size of aperture h and the presence of contact areas for Péclet numbers Pe less than 50 and a contact fraction of up to 20%. However, the transient dispersion problem is seen to be influenced by the contact areas for Pe > 10.

By using the methods of the theory of two- and three-dimensional linear absolute and convective instabilities, we examine the nature of the instability at the onset of convection in a model of convection in an extended horizontal layer of a saturated porous medium with either horizontal or vertical salinity and inclined temperature gradients, and horizontal throughflow. First, normal modes are analysed and the critical values of the vertical thermal Rayleigh number, Rv, wavenumber vector, (k, l) and frequency, ω, are obtained for a variety of values of the horizontal thermal and salinity Rayleigh numbers, Rh and Sh, respectively, the vertical salinity Rayleigh number Sv and the horizontal Péclet number, Qh. In the computations, a high-precision pseudo-spectral Chebyshev-collocation method is used. In most of the cases of parameter combinations considered, the onset of convection occurs through a longitudinal mode. Most of the non-longitudinal critical modes are oscillatory. Further, it is revealed that there exists an absolute/convective instability dichotomy at the onset of three-dimensional convection in a set of the base states given by exact analytic solutions of the equations of motion in the model. This echoes the results of Brevdo (vol. 641, 2009, p. 475) for transverse modes in a model with inclined temperature gradient and vertical throughflow, but with no salinity. The dependence of the dichotomy on the inclined thermal gradient, and on the horizontal and the vertical salinity gradients is investigated, for the longitudinal modes treated both as two-dimensional as well as three-dimensional modes, and for the non-longitudinal modes. For a certain set of parameter cases, it was found that the destabilization through longitudinal modes treated as two-dimensional modes has the character of absolute instability whereas a three-dimensional analysis of these modes revealed that the instability is convective, with the group velocity vector of the emerging unstable wavepacket being parallel to the axis of the convection rolls. Since a similar effect was reported by Brevdo (vol. 641, 2009, p. 475) for a model with no salinity, we conclude that this effect is not a separate case. In most of the cases considered in which a marginally unstable base state is absolutely stable, but convectively unstable, the direction of propagation of the emerging unstable wavepacket is either parallel or perpendicular to the axis of the convection rolls. Only in the absolutely stable, but convectively unstable cases in which non-longitudinal modes are favourable, the angle, ϕ, between the group velocity vector of the unstable wavepacket and the axis of the rolls satisfies 0 < ϕ < 90°.

Attractive colloidal dispersions, suspensions of fine particles which aggregate and frequently form a space-spanning elastic gel are ubiquitous materials in society with a wide range of applications. The colloidal networks in these materials can exist in a mode of free settling when the network weight exceeds its compressive yield stress. An equivalent state occurs when the network is held fixed in place and used as a filter through which the suspending fluid is pumped. In either scenario, hydrodynamic instabilities leading to loss of network integrity occur. Experimental observations have shown that the loss of integrity is associated with the formation of eroded channels, so-called streamers, through which the fluid flows rapidly. However, the dynamics of growth and subsequent mechanism of collapse remain poorly understood. Here, a phenomenological model is presented that describes dynamically the radial growth of a streamer due to erosion of the network by rapid fluid back flow. The model exhibits a finite-time blowup – the onset of catastrophic failure in the gel – due to activated breaking of the inter-colloid bonds. Brownian dynamics simulations of hydrodynamically interacting and settling colloids in dilute gels are employed to examine the initiation and propagation of this instability, which are in good agreement with the theory. The model dynamics is also shown to accurately replicate measurements of streamer growth in two different experimental systems. The predictive capabilities and future improvements of the model are discussed and a stability-state diagram is presented providing insight into engineering strategies for avoiding settling instabilities in networks meant to have long shelf lives.

We consider wavenumber selection for the periodic two-dimensional Rayleigh–Bénard problem on a horizontally infinite domain. The temperature difference (and hence the Rayleigh number) is assumed to be a slowly varying function of the horizontal coordinate perpendicular to the convection rolls. Under this condition Kramer et al. have shown that a unique wavenumber of convection is selected by a certain solvability condition if the domain contains a subcritical region. They performed analytic calculations for a model problem and for small-amplitude convection of an infinite-Prandtl-number fluid between stress-free boundaries. Their results are extended here to the realistic case of large-amplitude convection of a finite-Prandtl-number fluid between rigid boundaries. The temperature difference may be ‘ramped’ by changing either the temperature at the lower boundary or at the upper boundary, or both. It is shown that the choice has a significant effect on the ‘mean flow’, but no effect on the selected wavenumber.

Turbulent convection into stratified two-layer fluid systems in the presence of rotation has been investigated in a cylindrical laboratory tank. For a wide range of conditions the vertical stability of the flow depends only on the Richardson number Ri = g′h0/(B0R)2/3 (here, g′ = gδ ρi/ρ0 is the jump in the reduced gravity across the density interface, δ ρi is the jump in density itself, h0 is the depth of the top layer, B0 is the surface buoyancy flux and R is the radius of the source). We have found that for values of Ri greater than a critical value of Ric ≈ 11, the convective flow did not penetrate through the density interface, regardless of the values of the convective Rossby number Ro* = (B0/f3h02)1/2 of the flow (here f is the Coriolis parameter). In this case after the convective layer interacted with the density interface the mixed fluid, of intermediate density, propagated radially along the interface in the form of an intrusion. Later, if Ro* was less than approximately 5, mesoscale vortices with mean diameter D ≈ 8(RoR)2/3 and maximum swirl velocity v ≈ (B0R)1/3 were generated at the edge of this propagating front (here RoR = (B0/f3R2)1/2 is the Rossby number based on R). When Ri was less than 11, the convective flow eventually penetrated through the density interface and into the bottom layer. This occurred through the formation of discrete ‘convective holes’ beneath the source. At large values of Ro*, and at early times, one turbulent plume penetrated into the bottom layer from each hole. Later the initial holes coalesced to create one large lesion beneath the source.