In this paper, we present a detailed investigation of the size, scale and dynamics of macro-turbulent flow structures in gravel-bed rivers. We used an array of seven electromagnetic current meters with high resolution in both space and time to measure the streamwise velocity fluctuations in a gravel-bed river. The array was deployed successively in various configurations in order to quantify the vertical, lateral and longitudinal extent of the flow structures and to estimate their advecting velocities. To depict the spatial and temporal properties of the flow structures, we used space–time velocity matrices, space–time correlation analysis and coherent-structure detection schemes. The results show that the large-scale turbulent flow structures in a gravel-bed river occupy the entire depth of the flow and that they are elongated and narrow. The length of the structures is 3 to 5 times the flow depth while the width is between 0.5 and 1 times flow depth. In spite of the high roughness of the bed, these values are similar to those reported in the literature for laboratory experiments on large-scale turbulent flow structures. The dynamics of the large-scale turbulent flow structures investigated using flow visualization highlight the interactions between the outer flow region and the near-bed region. Our evidence suggests that large-scale flow incursions trigger ejections in the near-bed region that can develop into megabursts that can reach the water surface.

In this paper, the three-dimensional roll-up of a viscoelastic mixing layer is numerically simulated with the pseudospectral method using the FENE-P model. An artificial diffusion is self-adaptively introduced into the constitutive equation to stabilize the time integration. In the three-dimensional mixing layer and within the parameter ranges studied, the effect of the polymer additives on the formation of the coherent structures such as the ribs and the cups is found to be negligible. The polymer normal stresses develop wherever there exist extensional strains that are produced by the coherent structures and they then hinder the development of these structures. Stretching by the quadrupoles and the ribs together gives rise to an enormous enhancement of the polymer normal stress differences in the symmetrical plane between the quadrupole pair. These normal stress differences directly or indirectly weaken all large-scale structures occurring in the flow including the quadrupoles, the cups, the ribs, the spanwise vortices which rotate in the opposite direction to that of the cups, and the thin spanwise vortical sheets. Attenuation of these large-scale structures leads to a diminution of small-scale structures after their breakdown in the secondary roll-up process of the thin sheets. There is a tendency for the small-scale structures in the core region to merge into a large-scale one in the viscoelastic case. As a result, a flat and inclined vortex forms at the end of the simulation which resembles the type of structure observed in an experimental mixing layer with surfactants injected. In addition, the results show that the extensional viscosity is an important quantity to determine the extent to which the coherent structures in a mixing layer are modified by polymer additives.

A hybrid multiscale method is developed for simulating micro- and nano-scale fluid flows. The continuum Navier–Stokes equation is used in one flow region and atomistic molecular dynamics in another. The spatial coupling between continuum equations and molecular dynamics is achieved through constrained dynamics in an overlap region. The proposed multiscale method is used to simulate sudden-start Couette flow and channel flow with nano-scale rough walls, showing quantitative agreement with results from analytical solutions and full molecular dynamics simulations for different parameter regimes. Potential applications of the proposed multiscale method are discussed.

A different approach to the solution of the singular Rayleigh equation is presented in the context of the water wave growth problem as modelled by wind-induced shear instabilities. The approach is based on the analytical solution of a Bessel equation in the vicinity of the singular point, which is obtained from Rayleigh's equation with an arbitrary wind profile. Wave growth rates are computed using an integral expression derived from the dispersion relation of the air–sea interface. Computations of the present approach agree well with those of Conte & Miles (1959) for the special case of a logarithmic wind profile. Effects of the shape of the wind profile on the wave growth rate are investigated by using the 1/7-power law to represent the wind profile. Comparisons of the growth rates for the logarithmic wind profile and for the 1/7 profile reveal appreciable differences which must be investigated further, possibly using measured wind profiles within 10 m above the sea surface.

A three-dimensional analysis is presented of the Stokes flow, adjacent to a Brinkman half-space, that is induced or altered by the presence of a sphere in the flow field that ($a$) translates uniformly without rotating, ($b$) rotates uniformly without translating, or ($c$) is fixed in a shear flow that is uniform in the far field. The linear superposition of these three flow regimes is also considered for the special case of the free motion of a neutrally buoyant sphere. Exact solutions to the momentum equations are obtained in terms of infinite series expansions in the Stokes-flow region and in terms of integral transforms in the Brinkman medium. Attention is focused on the approach to the asymptotic limit as the ratio of Newtonian- to Darcy-drag forces vanishes. From the leading-order asymptotic approximations, implicit recursion relations are derived to determine the coefficients in the series solutions such that those solutions exactly satisfy the boundary and interfacial conditions as well as the continuity equations in both the Stokes-flow and Brinkman regions. For each of the three flow regimes considered, results are presented in terms of the drag force on the sphere and torque about the sphere centre as a function of the dimensionless separation distance between the sphere and the interfacial plane for several small values of the dimensionless hydraulic permeability of the Brinkman medium. Finally, the free motion of a neutrally buoyant sphere is found by requiring that the net hydrodynamic drag force and torque acting on the sphere vanish. Results for this case are presented in terms of the dimensionless translational and rotational speeds of the sphere as a function of the dimensionless separation distance for several small values of the dimensionless hydraulic permeability. The work is motivated by its potential application as an analytical tool in the study of near-wall microfluidics in the vicinity of the glycocalyx surface layer on vascular endothelium and in microelectromechanical systems devices where charged macromolecules may become adsorbed to microchannel walls.

The evolution of infinitesimal three-dimensional perturbations superimposed on a Burgers vortex is studied. By a sequence of variable transformations and scalings this linear evolution problem is reduced to a time-dependent system which is nearly identical to the stability equations governing a Lamb–Oseen vortex. The maximum amplification reached by perturbations over a finite time interval is computed through an iterative scheme based on the direct and adjoint governing equations, and results on the asymptotic stability of the Burgers vortex are deduced. The Burgers vortex is shown to be asymptotically stable, although significant short-term amplification may occur.

The breakup of a thin non-evaporating liquid film that is either flowing down or climbing on a vertical or inclined surface and subject to cocurrent or countercurrent interfacial shear (or gas flow) is investigated analytically. Analytical expressions for the dimensionless liquid film thickness, $\Delta_{\hbox{\scriptsize\it min}}$, and wetting rate, $\Gamma_{\hbox{\scriptsize\it min}}$, at breakup are derived based on the minimization of the total energy of a stable rivulet, formed following the film breakup. For a downflowing liquid film, increasing the cocurrent interfacial shear (or gas velocity) or decreasing the equilibrium contact angle, $\theta_{o}$, decreases both $\Delta_{\hbox{\scriptsize\it min}}$ and $\Gamma _{\hbox{\scriptsize\it min}}$, below their values with zero interfacial shear. Conversely, increasing the countercurrent interfacial shear or $\theta_{o}$, increases both $\Delta_{\hbox{\scriptsize\it min}}$ and $\Gamma_{\hbox{\scriptsize\it min}}$, above their values with zero interfacial shear. The predictions of $\Delta _{\hbox{\scriptsize\it min}}$ and $\Gamma _{\hbox{\scriptsize\it min}}$ for a climbing water film on a vertical surface are in good agreement with reported experimental data for a wide range of cocurrent gas velocities.

The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Re$_{\tau}\,{=}\,1900$. It is found, and explained, that their scaling is anomalous in several respects, including a square-root behaviour of their width with respect to their length, and a velocity scaling of the largest modes with the centreline velocity $U_c$. It is shown that this implies a logarithmic correction to the $k^{-1}$ energy spectrum, and that it leads to a scaling of the total fluctuation intensities away from the wall which agrees well with the mixed scaling of de Graaff & Eaton (2000) at intermediate Reynolds numbers, but which tends to a pure scaling with $U_c$ at very large ones.

We extend the finite-depth interaction theory of Kagemoto & Yue(1986) to water of infinite depth and bodies of arbitrary geometry. The sum over the discrete roots of the dispersion equation in the finite-depth theory becomes an integral in the infinite-depth theory. This means that the infinite dimensional diffraction transfer matrix in the finite-depth theory must be replaced by an integral operator. In the numerical solution of the equations, this integral operator is approximated by a sum and a linear system of equations is obtained. We also show how the calculations of the diffraction transfer matrix for bodies of arbitrary geometry developed by Goo & Yoshida (1990) can be extended to infinite depth, and how the diffraction transfer matrix for rotated bodies can be calculated easily. This interaction theory is applied to the wave forcing of multiple ice floes and a method to solve the full diffraction problem in this case is presented. Convergence studies comparing the interaction method with the full diffraction calculations and the finite- and infinite-depth interaction methods are carried out.

The stability of circular Poiseuille–Couette flow to axisymmetric disturbances is investigated theoretically. First, the governing circular Orr–Sommerfeld equation for linear perturbations is formulated and analysed asymptotically at large values of the Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and inner cylinder velocity is determined explicitly. These findings are confirmed when the linear problem is solved numerically at finite Reynolds number and multiple neutral curves are found. The relevance of these results to the thread injection of medical implants is discussed, and it is shown how the linear modes are connected to nonlinear amplitude-dependent modes at high Reynolds number that exist for $O(1)$ values of the inner cylinder velocity.

Cold O$_{2}$/NO supersonic expansions through an axisymmetric convergent–divergent nozzle into a low-pressure background gas have been investigated both experimentally and numerically. Temperature and density measurements have been carried out, using tracer NO in an O$_{2}$ main flow. NO two-dimensional rotational temperature and density flow field patterns in the jet behind the nozzle have been measured by laser-induced fluorescence (LIF). The spectroscopic investigations are complemented by static and impact pressure measurements along the jet centreline. Two nozzles with the same inner profile but with a short and long divergent section and small and large lip have been examined. Three experiments performed using these nozzles cover a wide range of regimes of underexpanded free jet flow, from highly oscillating multi-cycle structure to the regime with smooth deceleration of the issuing flow by the background gas. The numerical simulation of the whole flow field from the conditions in the stagnation chamber to those in the flooded space is performed in the framework of the full set of Navier–Stokes equations by employing a recently developed algorithm based on a staggered grid. The flow inside a long nozzle is shown to be strongly affected by the boundary layer, which leads to the degeneration of the isentropic core at the nozzle exit and transforms the jet regime from overexpanded for inviscid flow to underexpanded for a real flow. The model reproduces the experimental structures of the low-density free jet flow fairly well.

This paper deals with the temporal stability of the $q$-vortex trailing line vortex model. We describe a family of viscous instabilities existing in a range of parameters which is usually assumed to be stable, namely large swirl parameters ($q\,{>}\,1.5$) and large Reynolds numbers. These instabilities affect negative azimuthal wavenumbers ($m\,{<}\,0$) and take the form of centre-modes (i.e. with a structure concentrated along the vortex centreline). They are related to a family of viscous modes described by Stewartson, Ng & Brown (1988) in swirling Poiseuille flow, and are the temporal counterparts of weakly amplified spatial modes recently computed by Olendraru & Sellier (2002). These instabilities are studied numerically using an original and highly accurate Chebyshev collocation method, which allows a mapping of the unstable regions up to $\hbox{\it Re}\,{\approx}\,10^6$ and $q\,{\approx}\,7$. Our results indicate that in the limit of very large Reynolds numbers, trailing vortices are affected by this kind of instability whatever the value of the swirl number.

The Darcy–Boussinesq equations at infinite Darcy–Prandtl number are used to study convection and heat transport in a basic model of porous-medium convection over a broad range of Rayleigh number $Ra$. High-resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport, i.e. the Nusselt number Nu, from onset at $Ra \,{=}\, 4\pi^2$ up to $Ra\,{=}\,10^4$. Over an intermediate range of increasing Rayleigh numbers, the simulations display the ‘classical’ heat transport $\hbox{\it Nu} \,{\sim}\, Ra$ scaling. As the Rayleigh number is increased beyond $Ra \,{=}\, 1255$, we observe a sharp crossover to a form fitted by $\hbox{\it Nu} \,{\approx}\, 0.0174 \times Ra^{0.9}$ over nearly a decade up to the highest $Ra$. New rigorous upper bounds on the high-Rayleigh-number heat transport are derived, quantitatively improving the most recent available results. The upper bounds are of the classical scaling form with an explicit prefactor: $\hbox{\it Nu} \,{\le}\, 0.0297 \times Ra$. The bounds are compared directly to the results of the simulations. We also report various dynamical transitions for intermediate values of $Ra$, including hysteretic effects observed in the simulations as the Rayleigh number is decreased from $1255$ back down to onset.

The structures and energetics of nearly symmetric modes and nearly baroclinic modes are analysed in detail to examine their instability mechanisms. It is shown that the nearly symmetric modes have their cross-band circulations slanted mainly between the along-band absolute-momentum surface and buoyancy surface of the basic state. Their growth is thus supported mainly by the symmetric-type energy conversion that transports energy from the basic-state along-band velocity and buoyancy to the perturbation along-band velocity and buoyancy, respectively, and then to the cross-band circulation. However, as the band orientations are tilted slightly away from the basic shear, the growth is also assisted by the baroclinic-type energy conversion that transports energy from the basic-state buoyancy to the perturbation buoyancy via the along-band advection and then to the cross-band circulation. When the band orientation is tilted to the warm (or cold) side of the basic shear, the baroclinic-type energy conversion smooths (or sharpens) the near-boundary structures and thus reduces (enhances) the effect of diffusive damping, especially near the non-slip boundaries. This explains why in the presence of diffusivity the symmetric instability yields to the nearly symmetric instability with the band orientation tilted slightly to the warm side of the basic shear. The nearly baroclinic modes transport warm air northward with rising motion and cold air southward with sinking motion, so their growth is supported mainly by the baroclinic-type energy conversion. Since the band orientations are not exactly perpendicular to the basic shear, the growth is also assisted by two additional energy conversions: (i) from the basic-state buoyancy through the cross-band horizontal advection to the perturbation buoyancy; and (ii) from the basic-state along-band velocity to the perturbation along-band velocity. When the band orientation is tilted, by nearly 90$^\circ$ or less, to the warm (or cold) side of the basic shear, the two additional energy conversions smooth (or sharpen) the near-boundary structures and thus reduce (enhance) the effect of diffusive damping, especially near the non-slip boundaries. This explains why the baroclinic instability yields to the warm-side tilted nearly baroclinic instability in the presence of diffusivity.

The translational velocities of two spherical gas bubbles oscillating in water, which is irradiated by a high-intensity acoustic wave field, are calculated. The two bubbles are assumed to be located far enough apart so that shape oscillations can be neglected. Viscous effects are included owing to the small size of the bubbles. An asymptotic solution is obtained that accounts for the viscous drag on each bubble, for large ${\it Re}$ based on the radial part of the motion, in a form similar to the leading-order prediction by Levich (1962), $C_{D} = 48/{\it Re}_{T}$; ${\it Re}_{T} \to \infty$ based on the translational velocity. In this context the translational velocity of each bubble, which is a direct measure of the secondary Bjerknes force between the two bubbles, is evaluated asymptotically and calculated numerically for sound intensities as large as the Blake threshold. Two cases are examined. First, two bubbles of unequal size with radii on the order of $100\,\umu$m are subjected to a sound wave with amplitude $P_{A} < 1.0$ bar and forcing frequency $\omega_{f} = 0.51\omega_{10}$, so that the second harmonic falls within the range defined by the eigenfrequencies of the two bubbles, $\omega_{10} < 2\omega_{f} < \omega_{20}$. It is shown that their translational velocity changes sign, becoming repulsive as $P_{A}$ increases from 0.05 to 0.1 bar due to the growing second harmonic, $2\omega_{f}$, of the forcing frequency. However, as the amplitude of sound further increases, $P_{A} \approx 0.5$ bar, the two bubbles attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles. Second, the case of much smaller bubbles is examined, radii on the order of $10\,\umu$m, driven well below resonance, $\omega_{f}/2\pi = 20$ kHz, at very large sound intensities, $P_{A} \approx 1$ bar. Numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble size corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted, indicating sign inversion of their mutual force. Extensive numerical simulations indicate the formation of bubble pairs with constant average inter-bubble distance, consisting of bubbles with equilibrium radii determined by the primary and secondary resonance frequencies for small and moderate sound amplitudes or by the Blake threshold for large sound amplitudes. It is conjectured that in experiments where ‘acoustic streamers’ are observed, which are filamentary structures consisting of bubbles that are aligned and move rapidly in a cavitating fluid at nearly constant distances from each other, bubbles with size determined by the Blake threshold are predominant because those with size determined by linear resonance are larger and therefore become unstable due to shape oscillations.