Neutrally buoyant particles in low-Reynolds-number pressure-driven suspension flows migrate from regions of high to low shear, and this migration is a strong function of the local concentration. When the particle density differs from that of the suspending fluid, buoyancy forces also affect particle migration. It is the ratio between the buoyancy and viscous forces, as quantified by a dimensionless buoyancy number, which determines the phase distribution of the suspension once the flow is fully developed. Although several experiments have verified shear-induced particle migration in neutrally buoyant suspensions, data for particle migration when buoyancy effects are important are scarce. Electrical impedance tomography (EIT) is used here to non-invasively measure particle concentration across a pipe arising from the low-Reynolds-number flow of heavy conducting particles and light non-conducting particles in a viscous suspending fluid. A range of buoyancy numbers was investigated by varying the flow rate. In all of the experiments, a significant fraction of the particle phase was observed to migrate towards the top or bottom of the pipe, depending on the relative density of the particles. The amount of migration away from the centre of the pipe increased with increasing magnitude of the buoyancy number. Furthermore, observations of the phase distribution at several positions downstream of the inlet indicate that these suspension flows become fully developed earlier than that observed for neutrally buoyant particles. A scaling analysis for the prediction of the fully developed length is presented, which predicts shorter lengths for higher buoyancy numbers and is consistent with experimental observations. The experimental data were compared to an isotropic suspension balance model, and it was found that the particle phase distributions predicted by this model agree fairly well with the experimental observations.

Large-eddy simulation (LES) is conducted of a three-dimensional temporal mixing layer whose lower stream is initially laden with liquid drops which may evaporate during the simulation. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, species, momentum and energy source terms. The drop evolution is modelled using physical drops, or using computational drops to represent the physical drops. Simulations are performed using various LES models previously assessed on a database obtained from direct numerical simulations (DNS). These LES models are for: (i) the subgrid-scale (SGS) fluxes and (ii) the filtered source terms (FSTs) based on computational drops. The LES, which are compared to filtered-and-coarsened (FC) DNS results at the coarser LES grid, are conducted with 64 times fewer grid points than the DNS, and up to 64 times fewer computational than physical drops. It is found that both constant-coefficient and dynamic Smagorinsky SGS-flux models, though numerically stable, are overly dissipative and damp generated small-resolved-scale (SRS) turbulent structures. Although the global growth and mixing predictions of LES using Smagorinsky models are in good agreement with the FC-DNS, the spatial distributions of the drops differ significantly. In contrast, the constant-coefficient scale-similarity model and the dynamic gradient model perform well in predicting most flow features, with the latter model having the advantage of not requiring a priori calibration of the model coefficient. The ability of the dynamic models to determine the model coefficient during LES is found to be essential since the constant-coefficient gradient model, although more accurate than the Smagorinsky model, is not consistently numerically stable despite using DNS-calibrated coefficients. With accurate SGS-flux models, namely scale-similarity and dynamic gradient, the FST model allows up to a 32-fold reduction in computational drops compared to the number of physical drops, without degradation of accuracy; a 64-fold reduction leads to a slight decrease in accuracy.

A simple theory is developed for the equilibrium height of steps in a thermohaline staircase. The model is based on a linear stability analysis for a series of salt-finger interfaces, which reveals a tendency for the staircase to evolve in time until the characteristic thickness of layers reaches a critical value ($H_0$). Relatively thin layers successively merge as a result of the parametric variation of the heat/salt flux ratio ($\gamma$), but these mergers cease when the thickness of layers exceeds $H_0$. The equilibration of thick steps in our model is caused by the slight inhomogeneity of the convecting layers which has a stabilizing effect on the staircase. The instability theory is successfully tested against fully nonlinear numerical simulations and is qualitatively consistent with oceanic observations.

The effects of a large-scale shear on the energy spectrum of a passively advected scalar field are investigated. The shear is superimposed on a turbulent isotropic flow, yielding an Obukhov–Corrsin $k^{-5/3}$ scalar spectrum at small scales. Shear effects appear at large scales, where a different, anisotropic behaviour is observed. The scalar spectrum is shown to behave as $k^{-4/3}$ for a shear fixed in intensity and direction. For other types of shear characteristics, the slope is generally intermediate between the $-5/3$ Obukhov–Corrsin and the $-1$ Batchelor values. The physical mechanisms at the origin of this behaviour are illustrated in terms of the motion of Lagrangian particles. They provide an explanation for the scalar spectra that are shallow and dependent on the experimental conditions in shear flows at moderate Reynolds numbers.

A horizontally infinite water layer cooled from above and absorbing solar radiation is considered. Optical water types are classified according to Jerlov (Marine Optics, Scientific, 1976). The convection and the heat transfer are simulated numerically based on the linear equation of state and the time-dependent two-dimensional heat-conduction and Navier–Stokes equations. To take into account the effect of solar radiation, the conventional heat-conduction law for water with a free surface is corrected by introducing a particular function of heat and solar-radiation fluxes. Analytical expressions for this function are fitted.

Shear flows at sufficiently high Mach numbers support instability waves which travel supersonically relative to the ambient free stream. Such supersonic modes are known to produce intense far-field sound in the form of Mach wave radiation. In this paper, the nonlinear evolution of supersonic modes and the associated Mach wave radiation are analysed in a self-consistent fashion by using the high-Reynolds-number matched asymptotic expansion in conjunction with the multiple-scale method. Attention is focused on the relatively weak disturbances for which the nonlinear effect is comparable with the non-parallel-flow effect. The nonlinear spatial development of the mode is described by an amplitude equation in which the nonlinear term is contributed by the critical layer. The directivity of the radiated Mach waves in the far field is determined explicitly in terms of the amplitude function. The analysis is applicable to plane mixing layers, boundary layers, and planar and circular jets. In particular, it is shown that for the last two flows, the radiated Mach waves are most intensive in a beam which is perpendicular to the Mach wave front and emanates from the streamwise position at which the instability mode attains its maximum amplitude. The theoretical results are compared with direct numerical simulation and experimental data, and favourable qualitative and quantitative agreement is obtained.

The stability properties of the laminar Kolmogorov flow of a viscoelastic Oldroyd-B fluid are investigated both analytically and numerically. Linear stability with respect to large-scale perturbations is studied by means of multiple-scale analysis. This technique yields an effective diffusion equation for the large-scale perturbation where the effective (eddy) viscosity can be computed analytically. Stability amounts to the positive definiteness of the eddy-viscosity tensor as a function of the Reynolds and the Deborah numbers. Two main results emerge from our analysis: (i) at small fluid elasticity, the flow is more stable than in the Newtonian case; (ii) at high elasticity, the flow is prone to elastic instabilities, occurring even at vanishing Reynolds number. The hypothesis of scale separation is verified up to moderate elasticity, as checked by numerical integration of the exact linearized equations by the Arnoldi method. Finally, it is shown that the addition of a stress diffusivity counteracts the effect of elasticity, in agreement with simple physical arguments.

Shock ray theory (SRT) has been found to be useful and computationally efficient in finding successive positions of a curved weak shock front. In this paper, we solve some piston problems and show that the shock ray theory with two compatibility conditions gives shock positions, which are very close to those obtained by solving the same problems by the numerical solution of Euler's equations (Euler solutions). Comparison of the results obtained by shock ray theory and geometrical shock dynamics (GSD) of Whitham (J. Fluid Mech. vol. 2, 1957, p. 146) with the Euler solution shows that the shock ray theory gives more accurate results for any piston motion. The aim of the work is not just this comparison, but also to investigate the role of the nonlinearity in accelerating the process of the evolution of a shock, produced by an explosion of a non-circular finite charge, into a circular shock front. We find that the nonlinear waves propagating on the shock front appreciably accelerate this process. We also discuss a situation, for shock Mach number very close to 1, when GSD and shock ray theory may fail to give any result.

The paper continues our investigations of three-dimensional nonlinear resonant fluid sloshing in a square-base basin with finite depth (mean depth/tank length ratio $h\,{\ge}\, 0.3$). The sloshing is forced by a combined sway/surge resonant harmonic excitation of the two lowest natural modes. The new studies are strongly motivated by a discrepancy between previous quantitative theoretical results and experimental measurements and consist of a more precise description of the strong nonlinear amplification of higher modes. The latter is justified here by secondary resonance. Effective frequency domains of the secondary resonance are quantified. An adaptive asymptotic modal theory improves agreement with earlier and new experimental data both in the transient and steady-state conditions. Local breaking and overturning near the walls, that may lead to a ‘switch’ between distinct steady regimes, increase both global damping and generate random-like excitation of higher modes, are extensively discussed.

A major component of the noise in modern aeroengines is rotor–stator interaction noise generated when the wake from the rotating fan impinges on a stator row downstream. An analytically based model for the prediction of upstream-radiated rotor–stator interaction noise is described, and includes the important effect of mean swirling flow on both the rotor wake evolution and the acoustic response. The analytic nature of the model allows for the inclusion of all wake harmonics and enables the response at all blade passing frequencies to be determined.

An asymptotic analysis based on large rotor blade number is used to model the evolution of the rotor wake downstream in a cylindrical duct carrying mean swirling flow. The equations governing the axial evolution of the wake simplify to three coupled first-order differential equations in the interior, while close to the duct walls, a boundary-layer correction is required in order to satisfy the impermeability conditions at the boundaries. At the stator location, the wake is used as input into a local linear cascade model at each radius. The interaction of each wake harmonic gives rise to acoustic waves of multiple azimuthal order which contribute to the pressure field radiated back upstream. This enables the total acoustic response to be determined in terms of cylindrical duct modes in mean swirling flow.

The effect of stator blade geometry (thickness, camber, angle of attack) and rotor–stator separation on the total upstream-radiated noise is determined. Blade geometry is shown to have a significant effect on the noise generated, and increasing the rotor–stator gap can lead to large reductions in noise levels. Asymptotic treatment of the acoustic field, based on large azimuthal order, is also considered and used to identify the dominant contributions to the total pressure field resulting from the rotor–stator interaction. The ray structure of the acoustic modes in swirl is shown to be very different in some cases from that in uniform flow.

We present high-precision measurements of the Nusselt number $\cal N$ as a function of the Rayleigh number $R$ for cylindrical samples of water (Prandtl number $\sigma \,{=}\, 4.4$) with a diameter $D$ of 49.7 cm and heights $L \,{=}\, 116.3, 74.6$, and 50.6 cm, as well as for $D \,{=}\, 24.8$ cm and $L \,{=}\, 90.2$ cm. For each aspect ratio $\Gamma \,{\equiv}\, D/L \,{=}\, 0.28, 0.43, 0.67$, and 0.98 the data cover a range of a little over a decade of $R$. The maximum $R \,{\simeq}\, 10^{12}$ and Nusselt number ${\cal N} \,{\simeq}\, 600$ were reached for $\Gamma \,{=}\, 0.43$ and $D \,{=}\, 49.7$. The data were corrected for the influence of the finite conductivity of the top and bottom plates on the heat transport in the fluid to obtain estimates of $\cal N_{\infty}$ for plates with infinite conductivity. The results for ${\cal N}_{\infty}$ and $\Gamma \,{\geq}\, 0.43$ are nearly independent of $\Gamma$. For $\Gamma \,{=}\, 0.275$${\cal N}_{\infty}$ falls about 2.5% below the other data. For $R \,{\lesssim}\,10^{11}$, the effective exponent $\gamma_{\hbox{\scriptsize\it eff}}$ of ${\cal N}_{\infty} \,{=}\, N_0 R^{\gamma_{\hbox{\scriptsize\it eff}}}$ is about 0.32, larger than those of the Grossmann–Lohse model with its current parameters by about 0.01. For the largest Rayleigh numbers covered for $\Gamma \,{=}\, $0.98, 0.67, and 0.43, $\gamma_{\hbox{\scriptsize\it eff}}$ saturates at the asymptotic value $\gamma \,{=}\, 1/3$ of the Grossmann–Lohse model. The data do not reveal any crossover to a Kraichnan regime with $\gamma_{\hbox{\scriptsize\it eff}} \,{>}\, 1/3$.

The influence of an external laminar flow on the spreading of a viscous gravity current moving over a horizontal floor is studied theoretically and experimentally. The viscous stress exerted by the ambient flow drives the viscous gravity current streamwise with a velocity proportional to the local height of the current. The one-way coupling between the ambient flow and the spread of the current is examined. Similarity and numerical solutions are developed to describe viscous gravity currents spreading from line and point sources. An experimental study of the spreading of viscous gravity currents issuing from a point source in a channel flow, for both constant-flux and instantaneous releases, confirms the essential character of this description.

Linear stability theory and bifurcation analysis are used to investigate the role of gravity in shear-band formation in granular Couette flow, considering a kinetic-theory rheological model. We show that the only possible state, at low shear rates, corresponds to a ‘plug’ near the bottom wall, in which the particles are densely packed and the shear rate is close to zero, and a uniformly sheared dilute region above it. The origin of such plugged states is shown to be tied to the spontaneous symmetry-breaking instabilities of the gravity-free uniform shear flow, leading to the formation of ordered bands of alternating dilute and dense regions in the transverse direction, via an infinite hierarchy of pitchfork bifurcations. Gravity plays the role of an ‘imperfection’, thus destroying the ‘perfect’ bifurcation structure of uniform shear. The present bifurcation problem admits universal unfolding of pitchfork bifurcations which subsequently leads to the formation of a sequence of a countably infinite number of ‘isolas’, with the solution structures being a modulated version of their gravity-free counterpart. While the solution with a plug near the bottom wall looks remarkably similar to the shear-banding phenomenon in dense slow granular Couette flows, a ‘floating’ plug near the top wall is also a solution of these equations at high shear rates. A two-dimensional linear stability analysis suggests that these floating plugged states are unstable to long-wave travelling disturbances.The unique solution having a bottom plug can also be unstable to long waves, but remains stable at sufficiently low shear rates. The implications and realizability of the present results are discussed in the light of shear-cell experiments under ‘microgravity’ conditions.

A continous two-parameter family of analytical solutions to the Euler equations are presented representing a class of steadily rotating vortex arrays involving $N\,{+}\,1$ interacting vortex patches where $N\,{\ge}\,3$ is an integer. The solutions consist of a central vortex patch surrounded by an $N$-fold symmetric alternating array of satellite point vortices and vortex patches. One of the parameters governs the size of the central patch, the other governs the size of the $N$ satellite patches. In the limit where the areas of the satellite vortex patches tend to zero, the solutions degenerate to the exact solutions of Crowdy (J. Fluid Mech. vol. 469, 2002, p. 209). Limiting states are found in which cusps form only on the central patch, only on the satellite patches, or simultaneously on both central and satellite patches. Contour dynamics simulations are used to check the mathematical solutions and test their robustness. The linear stability of a class of ‘point-vortex models’ (in which the patches are replaced by point vortices) are also studied in order to examine the stability of the distributed-vorticity configurations to pure-displacement modes. On the other hand, a desingularization of all point vortices to Rankine vortices leads to a class of ‘quasi-equilibria’ consisting purely of interacting vortex patches close to hydrodynamic equilibrium.

Low-dimensional empirical Galerkin models are developed for spatially evolving laminar and transitional shear layers, based on a Karhunen–Loève decomposition of incompressible two- and three-dimensional Navier–Stokes simulations. It is shown that the key to an accurate Galerkin model is a novel analytical pressure-term representation. The effect of the pressure term is elucidated by a modal energy-flow analysis in a mixing layer, which generalizes the framework developed by Rempfer (1991). In convectively unstable shear layers, it is shown in particular that neglecting small energy terms leads to large amplitude errors in the Galerkin model. The effect of the pressure term and small neglected energy flows is very important for a two-dimensional mixing layer, is less pronounced for the three-dimensional analogue, and can be considered as small in an absolutely unstable wake flow.

The temporal evolution of baroclinic basin-scale waves in a rotating circular basin following an initial forcing event is investigated using a laboratory study. Experiments conducted in the circular domain containing a two-layer fluid with a flat bottom and vertical sidewalls demonstrate that the response is essentially linear with frictional effects at the boundaries steadily dissipating wave energy. Experiments conducted in the same configuration but with the addition of simple topographic features, either a radially protruding cape or a bathymetric ridge, exhibit wave/topography interactions that result in the formation of an eddy field and an offshore flow, respectively. The rate of wave decay, as well as the amount of horizontal mixing occurring within the basin, is significantly enhanced by such interactions. The results of this study are then considered in terms of their implications for the baroclinic basin-scale wave energy pathways in large stratified lakes influenced by the Earth's rotation.

Natural convection in a cube of fluid-saturated porous medium heated from below and cooled from the top is studied numerically using a non-Darcy flow model. All vertical sidewalls are considered to be impermeable and adiabatic. The evolution of the various flow patterns is investigated from onset up to a Rayleigh number of 1000 where irregularly fluctuating convection prevails. New flow patterns have been found to exist in addition to those mentioned in the previous studies. In the present study, a total of ten steady flow patterns have been identified, of which five show oscillatory behaviour in some Rayleigh-number range. The results are presented in terms of average Nusselt number curves consisting of the solution branches of the convective patterns. The convective patterns are classified in terms of their symmetry properties, and the symmetries broken or gained during bifurcations from one flow structure to the other are identified.