The process which leads to the formation of three-dimensional sand waves is investigated by means of a stability analysis which considers the time development of a small-amplitude bottom perturbation of a shallow tidal sea. The weakly nonlinear interaction of a triad of resonant harmonic components of the bottom perturbation is considered. The results show that the investigated resonance mechanism can trigger the formation of a three-dimensional bottom pattern similar to that observed in the field.

A novel method is developed to simulate suspensions of deformable particles by coupling the lattice-Boltzmann method (LBM) for the fluid phase to a linear finite-element analysis (FEA) describing particle deformation. The methodology addresses the need for an efficient method to simulate large numbers of three-dimensional and deformable particles at high volume fraction in order to capture suspension rheology, microstructure, and self-diffusion in a variety of applications. The robustness and accuracy of the LBM–FEA method is demonstrated by simulating an inflating thin-walled sphere, a deformable spherical capsule in shear flow, a settling sphere in a confined channel, two approaching spheres, spheres in shear flow, and red blood cell deformation in flow chambers. Additionally, simulations of suspensions of hundreds of biconcave red blood cells at 40% volume fraction produce continuum-scale physics and accurately predict suspension viscosity and the shear-thinning behaviour of blood. Simulations of fluid-filled spherical capsules which have red-blood-cell membrane properties also display deformation-induced shear-thinning behaviour at 40% volume fraction, although the suspension viscosity is significantly lower than blood.

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometry.

We present a study of hydraulic jumps with flow predominantly in one direction, created either by confining the flow to a narrow channel with parallel walls or by providing an inflow in the form of a narrow sheet. In the channel flow, we find a linear height profile upstream of the jump as expected for a supercritical one-dimensional boundary layer flow, but we find that the surface slope is up to an order of magnitude larger than expected and independent of flow rate. We explain this as an effect of turbulent fluctuations creating an enhanced eddy viscosity, and we model the results in terms of Prandtl's mixing-length theory with a mixing length that is proportional to the height of the fluid layer. Using averaged boundary-layer equations, taking into account the friction with the channel walls and the eddy viscosity, the flow both upstream and downstream of the jump can be understood. For the downstream subcritical flow, we assume that the critical height is attained close to the channel outlet. We use mass and momentum conservation to determine the position of the jump and obtain an estimate which is in rough agreement with our experiment. We show that the averaging method with a varying velocity profile allows for computation of the flow-structure through the jump and predicts a separation vortex behind the jump, something which is not clearly seen experimentally, probably owing to turbulence. In the sheet flow, we find that the jump has the shape of a rhombus with sharply defined oblique shocks. The experiment shows that the variation of the opening angle of the rhombus with flow rate is determined by the condition that the normal velocity at the jump is constant.

In this study, direct numerical simulations and high-resolved large eddy simulations of turbulent Rayleigh–Bénard convection were conducted with a fluid of Prandtl number Pr = 0.7 in a long rectangular cell of aspect ratio unity in the cross-section and periodic boundaries in a horizontal longitudinal direction. The analysis of the thermal and kinetic energy spectra suggests that temperature and velocity fields are correlated within the thermal boundary layers and tend to be uncorrelated in the core region of the flow. A tendency of decorrelation of the temperature and velocity fields is also observed for increasing Ra when the flow has become fully turbulent, which is thought to characterize this regime. This argument is also supported by the analysis of the correlation of the turbulent fluctuations |u|′ and θ′. The plume and mixing layer dominated region is found to be separated from the thermal dissipation rates of the bulk and conductive sublayer by the inflection points of the probability density function (PDF). In order to analyse the contributions of bulk, boundary layers and plumes to the mean heat transfer, the thermal dissipation rate PDFs of four different Ra are integrated over these three regions. Hence, it is shown that the core region is dominated by the turbulent fluctuations of the thermal dissipation rate throughout the range of simulated Ra, whereas the contributions from the conductive sublayer due to turbulent fluctuations increase rapidly with Ra. The latter contradicts results by He, Tong & Xia (Phys. Rev. Lett., vol. 98, 2007). The results also show that the plumes and mixing layers are increasingly dominated by the mean gradient contributions. The PDFs of the core region are compared to an analytical scaling law for passive scalar turbulence which is found to be in good agreement with the results of the present study. It is noted that the core region scaling seems to approach the behaviour of a passive scalar as Ra increases, i.e. it changes from pure exponential to a stretched exponential scaling.

We develop low-dimensional models for the evolution of a free shear layer in a periodic domain. The goal is to obtain models simple enough to be analysed using standard tools from dynamical systems theory, yet including enough of the physics to model nonlinear saturation and energy transfer between modes (e.g. pairing). In the present paper, two-dimensional direct numerical simulations of a spatially periodic, temporally developing shear layer are performed. Low-dimensional models for the dynamics are obtained using a modified version of proper orthogonal decomposition (POD)/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. Equations are obtained for the rate of change of the shear-layer thickness. A model with two complex modes can describe certain single-wavenumber features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. A model with four complex modes can describe interactions between two wavenumbers (vortex pairing) as well. At least two POD modes are required for each wavenumber in space to sufficiently describe the dynamics, though, for each wavenumber, more than 90% energy is captured by the first POD mode in the scaled space. The comparison of POD modes to stability eigenfunction modes seems to give a plausible explanation. We have also observed a relation between the phase difference of the first and second POD modes of the same wavenumber and the sudden turning point for shear-layer dynamics in both direct numerical simulations and model computations.

We study a one-dimensional model describing buoyancy-driven laminar steady flow of a glass melt in a closed loop under the influence of a localized electromagnetic (Lorentz) force. The loop is a highly simplified representation of a closed streamline in glass melt flow in a real furnace under the influence of an artificially produced Lorentz force. The model is based on the energy equation for the temperature and the Stokes equation for the velocity distribution inside the loop. We take into account the full nonlinear temperature dependence of the viscosity and the electrical conductivity of the melt. The three-dimensional problem is then reduced to a single nonlinear equation for the cross-section averaged velocity from which the one-dimensional temperature distribution along the loop can be readily obtained. We show that the two-way interaction between the velocity and temperature resulting from the temperature-dependent material properties and Lorentz force leads to the result that the mean velocity as a function of the control parameters is non-unique and involves bifurcations. For some parameters we even observe freezing, which refers to a regime in which the fluid is almost at rest. Our model reveals the role of temperature-dependent viscosity and conductivity in glass melt flows in a pure form that is not visible in full numerical simulations.

In this paper, we investigate the dynamics of muddy-seabed motions induced by a surface solitary wave. The muddy seabed is characterized as Bingham-plastic mud. We focus our attention on the situations where the horizontal scale of the wave-induced mud flow is much larger than the vertical scale. The thickness of the mud layer is also assumed to be much smaller than the water depth above. With these simplifications, the dynamic pressure in the mud column remains a constant and the vertical displacement of the water–mud interface is negligible. The horizontal gradient of the wave-induced dynamic pressure along the water–seabed interface drives the motions in the mud bed. For a Bingham-plastic muddy seafloor, the mud moves either like a solid (plug flow) or like a viscous fluid (shear flow) depending on whether the magnitude of shear stress is in excess of the yield stress. Velocities inside these two different flow regimes and the location(s) of the yield surface(s) vary in time as functions of water–mud interfacial pressure gradient and the properties of the Bingham-plastic mud. A semi-analytical approach is developed in this paper to analyse the motions inside the mud bed under a surface solitary wave loading. Three possible scenarios are discussed to illustrate the complexity of the seafloor responses. The formula for the damping rate caused by the energy dissipation inside the muddy seabed is also derived. Using realistic values of the physical parameters, the present results for damping rate agree qualitatively with the available field observations.

We consider moderately dense bounded shear flows of agitated homogeneous inelastic frictionless solid spheres colliding in a gas between two parallel bumpy walls at finite particle Reynolds numbers, volume fractions between 0.1 and 0.4, and Stokes numbers large enough for collisions to determine the velocity distribution of the spheres. We adopt a continuum model in which constitutive relations and boundary conditions for the granular phase are derived from kinetic theory, and in which the gas contributes a viscous dissipation term to the fluctuation energy of the grains. We compare its predictions to recent lattice-Boltzmann (LB) simulations. The theory underscores the role played by the walls in the balances of momentum and fluctuation energy. When particle inertia is large, the solid volume fraction is nearly uniform, thus allowing us to treat the rheology using unbounded flow theory with an effective shear rate, while predicting slip velocities at the walls. When particle inertia decreases or fluid inertia increases, the solid volume fraction becomes increasingly heterogeneous. In this case, the theory captures the profiles of volume fraction, mean and fluctuation velocities between the walls. Comparisons with LB simulations allow us to delimit the range of parameters within which the theory is applicable.

Receptivity in three-dimensional boundary-layer flow to localized surface roughness and free-stream vorticity is studied. A boundary layer of Falkner–Skan–Cooke type with favourable pressure gradient is considered to model the flow slightly downstream of a swept-wing leading edge. In this region, stationary and travelling crossflow instability dominates over other instability types. Three scenarios are investigated: the presence of low-amplitude chordwise localized, spanwise periodic roughness elements on the plate, the impingement of a weak vortical free-stream mode on the boundary layer and the combination of both disturbance sources. Three receptivity mechanisms are identified: steady receptivity to roughness, unsteady receptivity to free-stream vorticity and unsteady receptivity to vortical modes scattered at the roughness. Both roughness and vortical modes provide efficient direct receptivity mechanisms for stationary and travelling crossflow instabilities. We find that stationary crossflow modes dominate for free-stream turbulence below a level of about 0.5%, whereas higher turbulence levels will promote the unsteady receptivity mechanism. Under the assumption of small amplitudes of the roughness and the free-stream disturbance, the unsteady receptivity process due to scattering of free-stream vorticity at the roughness has been found to give small initial disturbance amplitudes in comparison to the direct mechanism for free-stream modes. However, in many environments free-stream vorticity and roughness may excite interacting unstable stationary and travelling crossflow waves. This nonlinear process may rapidly lead to large disturbance amplitudes and promote transition to turbulence.

Optical microscopy and multi-particle tracking are used to study hydrodynamic interactions of monodisperse polymethylmethacrylate (PMMA) spheres at a decalin–water interface. The short-time self-diffusion coefficient measured at low surface coverage has the form DSS(n) = αD0(1 − βn), where n is the area fraction occupied by the particles, and D0 is the Stokes–Einstein diffusion coefficient in the bulk suspension of PMMA spheres in decalin. The measured values of α are found to be in good agreement with the numerical calculation for the drag coefficient of interfacial particles. The measured values of β differ from that obtained for bulk suspensions, indicating that hydrodynamic interactions between the particles have interesting new features at the interface.

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

This investigation focuses on the characteristics of near-bed turbulence in fully rough gravel-bed open-channel flows. The analysis combines results obtained with the double-averaging methodology and local flow characterization, using velocity measurements provided by a high-resolution three-axis Acoustic Doppler Velocity Profiler (ADVP). As a result of the flow heterogeneity induced by the bed topography, the flow is not locally uniform in the near-bed region, and a double-averaging methodology is applied over a length scale much greater than the gravel size. In smooth- and rough-bed flow conditions, without macro-roughness bed elements, maximum turbulent kinetic energy (TKE) production occurs very close to z = 0, while in our case with fully rough flows with macro-roughness elements, maximum turbulence activity is found to occur at gravel crest levels zc (zc/h = 0.1). Turbulent diffusion also reaches a maximum at this elevation. The characteristics of the spatially averaged TKE budget are in good agreement with those obtained in flows over canopies. The hydrodynamic double-averaged properties have strong similarities with mixing layers and reattached mixing layers in flows over backward facing steps. Local time-averaged velocity profiles can be split into three typical classes, namely log, S-shaped and accelerated. It appears that the S-shaped class profiles, located in the wakes of the macro-roughness elements, exhibit an inflectional profile typical of mixing layers. They are of major importance in the double-averaged TKE budget, as they provide a local high contribution to the double-averaged TKE flux, TKE production and dissipation compared to the log class profiles. Consequently, double-averaged TKE production is roughly 75% greater than the dissipation rate at the point of maximal TKE production. Moreover the macro-roughness bed elements imply mixing-layer-type hydrodynamics that play a dominant role in the overall structure of mean near-bed turbulence of gravel-bed channel flows.

We address the issue of the convective or absolute nature of the instability of core-annular pipe flows, in experiments using two miscible fluids of equal density but different viscosities, the core fluid being much less viscous than the wall one. We use a concentric co-current injection of the two fluids. An axisymmetric parallel base state is obtained downstream the injector. The core radius RI and the Reynolds number Re of the so-obtained base state are varied independently due to the control of the flow rate of each fluid. However, a downstream destabilization of this base state was observed within the explored range of the two control parameters RI and Re. Moreover, the fixed location of this destabilization, observed for some particular parameters, suggests an absolute nature of the instability. We present a tentative delineation of the nature (convective or absolute) of the instability and discuss the accessible measurements to experimentally address this issue.

The convective/absolute nature of the instability of miscible core-annular flow with variable viscosity is investigated via linear stability analysis and nonlinear simulations. From linear analysis, it is found that miscible core-annular flows with the more viscous fluid in the core are at most convectively unstable. On the other hand, flows with the less viscous fluid in the core exhibit absolute instability at high viscosity ratios, over a limited range of core radii. Nonlinear direct numerical simulations in a semi-infinite domain display self-excited intrinsic oscillations if and only if the underlying base flow exhibits absolute instability. This oscillator-type flow behaviour is demonstrated to be associated with the presence of a nonlinear global mode. Both the parameter range of global instability and the intrinsically selected frequency of nonlinear oscillations, as observed in the simulation, are accurately predicted from linear criteria. In convectively unstable situations, the flow is shown to respond to external forcing over an unstable range of frequencies, in quantitative agreement with linear theory. As discussed in part 1 of this study (d'Olce, Martin, Rakotomalala, Salin and Talon, J. Fluid Mech., vol. 618, 2008, pp. 305–322), self-excited synchronized oscillations were also observed experimentally. An interpretation of these experiments is attempted on the basis of the numerical results presented here.

Buoyant plumes of CO2 spreading through water-saturated permeable rock, bounded by layers of lower permeability, tend to spread laterally. As they advance, they may gradually leak through fractures or discontinuities in the lower permeability boundary, leading to a gradual waning of the original plume and dispersal of CO2 higher in the formation. With a finite release of CO2, the trailing edge of the plume recedes with time, and capillary forces tend to trap a fraction of this CO2 within the pore space. This also leads to a gradual waning of the plume with time and limits the mass of CO2 which can leak through the boundary and rise higher into the formation. We explore the balance between these two effects and calculate some of the controls on the fraction of the CO2 plume which becomes trapped within the original layer of rock.

We examine the motion of a buoyant fluid injected into a water-saturated porous rock as it spreads along a thin inclined low-permeability barrier. We account for leakage of the fluid across the barrier once the current is sufficiently deep so that the pressure exceeds the capillary threshold. We show that at some distance from the source, the pressure decreases below this threshold, and all the remaining flux spreads laterally along the barrier. We examine the controls on the partitioning of the flow between the draining flux and the laterally spreading flux and also the controls on the lateral extent of the draining region for the case of an instantaneous release and a maintained release of fluid. We consider the implications of our work for the dispersal of CO2 plumes which may be sequestered in deep saline aquifers.