In this paper we consider the effect of sinusoidal gravity modulation of size ε on a differentially heated infinite slot in which a vertical temperature stratification is imposed on the walls. The slot problem is characterized by a Rayleigh number, Prandtl number, and the imposed uniform stratification on the walls. When ε is small, we show by regular perturbation expansion in ε that the modulation interacts with the natural mode of the system to produce resonances, confirming the results of Farooq & Homsy (1994). For ε ∼ O(1) we show that the modulation can potentially destabilize the longwave eigenmodes of the slot problem. This is achieved by projecting the governing equations onto the least-damped eigenmode, and investigating the resulting Mathieu equation via Floquet theory. No instability was found at large values of the Prandtl number and also low stratification, when there are no travelling modes present.

To understand the nonlinear saturation mechanisms of this growth, we consider a two-mode model of the slot problem with the primary mode being the least-damped travelling parallel-flow mode as before and a secondary mode of finite wavenumber. By projecting the governing equations onto these two modes we obtained the equations for temporal evolution of the two modes. For modulation amplitudes above critical, the growth of the primary mode is saturated resulting in a stable weak nonlinear synchronous oscillation of the primary mode. An unexpected and intriguing feature of the coupling is that the secondary mode exhibits very high-frequency bursts which appear once every cycle of the forcing frequency.

In a recent work, a simple formula was derived for the ‘wave drift damping’ in a two-dimensional floating body and the obtained expression is exact within the context of the related theory, where only leading-order terms in the forward speed are retained. This formula is now generalized for a three-dimensional problem and the coefficients of the ‘wave drift damping matrix’ are given explicitly in terms of the standard second-order steady forces and moment in the horizontal plane; Munk's yaw moment, related with the steady second-order potential and discussed in Grue & Palm (1993), is not analysed in this paper and the effect of an eventual small angular velocity around the vertical axis is also not considered.

Numerical results agree in general with the proposed formula although in a specific case a consistent disagreement has been observed, as discussed in §5.

The steady, two-dimensional, isothermal flow of an incompressible Newtonian fluid in a semi-infinite channel is modelled using a finite-element method. The flow is driven by injecting two identical jets through symmetrically placed slit-like nozzles into the otherwise closed end of the channel. Multiple steady-state solutions are observed for Reynolds numbers greater than 18.8, where seven solutions have been found. Six of these solutions exist on branches that are not connected to the Stokes flow solution via continuation in the Reynolds number. Further bifurcations of these solutions has led to the discovery of 17 solutions at a Reynolds number of 40. A two-dimensional linear stability analysis of the solution branches shows that for Reynolds numbers in the range of 18.8 to 26.8 there are three stable solutions. One solution is symmetric about the channel centreline while the other two stable solutions are a pair of mirror-image asymmetric flows. For Reynolds numbers in the range 26.8 to 40, there are four known stable solutions consisting of two asymmetric solutions and their mirror-images.

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin–Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.

The interaction of a planar shock wave with a square cavity is studied experimentally and numerically. It is shown that such a complex, time-dependent, process can be modelled in a relatively simple manner. The proposed physical model is the Euler equations which are solved numerically, using the second-order-accurate high-resolution GRP scheme, resulting in very good agreement with experimentally obtained findings. Specifically, the wave pattern is numerically simulated throughout the entire interaction process. Excellent agreement is found between the experimentally obtained shadowgraphs and numerical simulations of the various flow discontinuities inside and around the cavity at all times. As could be expected, it is confirmed that the highest pressure acts on the cavity wall which experiences a head-on collision with the incident shock wave while the lowest pressures are encountered on the wall along which the incident shock wave diffracts. The proposed physical model and the numerical simulation used in the present work can be employed in solving shock wave interactions with other complex boundaries.

The properties of a relatively uncommon regime of fluid dynamics, low Mach number compressible flow are investigated. This regime, which is characterized by an exceptionally large channel aspect ratio L/d ∼ 106 leads to highly subsonic flows in which friction dominates inertia. Even so, because of the large aspect ratio, finite pressure, temperature, and density gradients are required, implying that compressibility effects are also important. Analytical results are presented which show, somewhat unexpectedly, that for forced channel flow, steady-state solutions exist only below a critical value of heat input. Above this value the flow reverses against the direction of the applied pressure gradient causing fluid to leave both the inlet and outlet implying that the related concepts of a steady-state friction factor and heat transfer coefficient have no validity.

Busse's annulus is considered as a model of thermal convection inside the Earth's liquid core. The conventional tilted base and top are modified by azimuthal sinusoidal corrugations so that the effects of surface topography can be investigated. The annulus rotates rapidly about its axis of symmetry with gravity directed radially inwards towards the rotation axis. An unstable radial temperature gradient is maintained and the resulting Boussinesq convection is considered at small Ekman number. Since the corrugations on the boundaries cause the geostrophic contours to be no longer circular, strong geostrophic flows may be driven by buoyancy forces and damped by Ekman suction. When the bumps are sufficiently large, instability of the static state is dominated by steady geostrophic flow with the convection pattern locked to the bumps. As the bump size is decreased, oscillatory geostrophic flow is possible but the preferred mode is modulated on a long azimuthal length scale and propagates as a wave eastwards. This mode only exists in the presence of bumps and is not to be confused with the thermal Rossby waves which are eventually preferred as the bump height tends to zero. Like thermal Rossby waves, the new modes prefer to occupy the longest available radial length scale. In this long-length-scale limit, two finite-amplitude states characterized by uniform geostrophic flows can be determined. The small-amplitude state resembles Or & Busse's (1987) mean flow instability. On losing stability the solution jumps to the more robust large-amplitude state. Eventually, for sufficiently large Rayleigh number and bump height, it becomes unstable to a long-azimuthal-length-scale travelling wave. The ensuing finite-amplitude wave and the mean flow, upon which it rides, are characterized by a geostrophic flow, which is everywhere westward.

The non-equilibrium behaviour of concentrated colloidal dispersions is studied by Stokesian Dynamics, a general molecular-dynamics-like technique for simulating particles suspended in a viscous fluid. The simulations are of a suspension of monodisperse Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of shear and Brownian forces. Three clearly defined regions of behaviour are revealed. There is first a Brownian-motion-dominated regime (Pe ≤ 1) where departures from equilibrium in structure and diffusion are small, but the suspension viscosity shear thins dramatically. When the Brownian and hydrodynamic forces balance (Pe ≈ 10), the dispersion forms a new ‘phase’ with the particles aligned in ‘strings’ along the flow direction and the strings are arranged hexagonally. This flow-induced ordering persists over a range of Pe and, while the structure and diffusivity now vary considerably, the rheology remains unchanged. Finally, there is a hydrodynamically dominated regime (Pe > 200) with a dramatic change in the long-time self-diffusivity and the rheology. Here, as the Péclet number increases the suspension shear thickens owing to the formation of large clusters. The simulation results are shown to agree well with experiment.

The structure of a viscous two-dimensional vortex core in a imposed weak strain is analysed, in the same spirit as a similar analysis of strained columnar vortices (Moffatt, Kida & Ohkitani 1994). The analysis is recast in terms of a coordinate deformation, ensuring the uniform validity of the perturbation expansion up to the neighbourhood of a dividing streamline, beyond which it is not expected to work, and from where the exponentially weak vorticity is expected to be stripped to infinity. The orientation and ellipticity of the vorticity distribution of the cores is compared with the results of a numerical experiment in two-dimensional turbulence, and shown to agree. This is interpreted both as a confirmation of the theory and as an indication that the vortices of two-dimensional turbulence are sufficiently long-lived to be controlled by viscous diffusion, even at the relatively large Reynolds numbers of our simulation.

The p.d.f. of the velocity gradients in two-dimensional decaying isotropic turbulence is shown to approach a Cauchy distribution, with algebraic s−2 tails, as the flow becomes dominated by a large number of compact coherent vortices. The statistical argument is independent of the vortex structure, and depends only on general scaling properties. The same argument predicts a Gaussian p.d.f. for the velocity components. The convergence to these limits as a function of the number of vortices is analysed. It is found to be fast in the former case, but slow (logarithmic) in the latter, resulting in residual u−3 tails in all practical cases. The influence of a spread Gaussian vorticity distribution in the cores is estimated, and the relevant dimensionless parameter is identified as the area fraction covered by the cores. A comparison is made with the result of numerical simulations of two-dimensional decaying turbulence. The agreement of the p.d.f.s is excellent in the case of the gradients, and adequate in the case of the velocities. In the latter case the ratio between energy and enstrophy is computed, and agrees with the simulations. All the one-point statistics considered in this paper are consistent with a random arrangement of the vortex cores, with no evidence of energy screening.

Results are presented of numerical simulations of passive scalar mixing in homogeneous, incompressible turbulent flows. These results are generated via the Linear Eddy Model (LEM) and Direct Numerical Simulation (DNS) of turbulent flows under a variety of different conditions. The nature of mixing and its response to the turbulence field is examined and the single-point probability density function (p.d.f.) of the scalar amplitude and the p.d.f.s of the scalar spatial-derivatives are constructed. It is shown that both Gaussian and exponential scalar p.d.f.s emerge depending on the parameters of the simulations and the initial conditions of the scalar field. Aided by the analyses of data, several reasons are identified for the non-Gaussian behaviour of the scalar amplitude. In particular, two mechanisms are identified for causing exponential p.d.f.s: (i) a non-uniform action of advection on the large and the small scalar scales, (ii) the nonlinear interaction of the scalar and the velocity fluctuations at small scales. In the absence of a constant non-zero mean scalar gradient, the behaviour of the scalar p.d.f. is very sensitive to the initial conditions. In the presence of this gradient, an exponential p.d.f. is not sustained regardless of initial conditions. The numerical results pertaining to the small-scale intermittency (non-Gaussian scalar derivatives) are in accord with laboratory experimental results. The statistics of the scalar derivatives and those of the velocity-scalar fluctuations are also in accord with laboratory measured results.

Solid-body collisions between smooth particles in a gas would not occur if the lubrication force for a continuum incompressible fluid were to hold at all particle separations. When the gap between the particles is of the order of the mean free path λ0 of the gas, the discrete molecular nature of the gas becomes important. For particles of radii a smaller than about 50 μm colliding in air at a relative velocity comparable to their terminal velocity, the effects of compressibility of the gas in the gap are not important.

The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness h0 ≡ aε, the mean-free path λ0, and the distance aε1/2 over which the effects of curvature become important. The slip-flow regime, a[Gt ]λ0, was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (aε ∼ O(λ0)), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When aε[Lt ]λ0[Lt ]aε1/2, one might expect the local flow in the gap to be governed by Knudsen diffusion. However, an attempt to calculate the Knudsen diffusivity between parallel plates leads to a logarithmic divergence, which is cut off by intermolecular collisions, and the flux is therefore proportional to h0c log(λ0/h0), where c is the mean molecular speed. The non-continuum lubrication force is shown to have a weak, log - log divergence as the particle separation goes to zero. As a result, the energy dissipated in the collision is finite. In the limit of large particle inertia, the energy dissipated is 6πμU0a2(log h0/λ0 – 1.28), where 2U0 is the relative velocity of the particles.

When λ0[Gt ]aε1/2, we have a free molecular flow in the gap. In this case, owing to the curvature of the particles, the flux versus pressure gradient relation is non-local. We analyse the free molecular flow between two cylinders and obtain scalings for the lubrication force.

We examine the problem of determining the particle-phase velocity variance and rhe-ology of sheared gas-solid suspensions at small Reynolds numbers and finite Stokes numbers. Our numerical simulations take into account the Stokes flow interactions among particles except for pairs of particles with a minimum gap width comparable to or smaller than the mean free path of the gas molecules for which the usual lubrication approximation breaks down and particle collisions occur in a finite time. The simulation results are compared to the predictions of two theories. The first is an asymptotic theory for large Stokes number St and nearly elastic collisions, i.e. St [Gt ] 1 and 0 ≤ 1 - e [Lt ] 1, e being the coefficient of restitution. In this limit, the particle velocity distribution is close to an isotropic Maxwellian and the velocity variance is determined by equating the energy input in shearing the suspension to the energy dissipation by inelastic collisions and viscous effects. The latter are estimated by solving the Stokes equations of motion in suspensions with the hard-sphere equilibrium spatial and velocity distribution while the shear energy input and energy dissipation by inelastic effects are estimated using the standard granular flow theory (i.e. St = ∞). The second is an approximate theory based on Grad's moments method for which St and 1 – e are O(1). The two theories agree well with each other at higher values of volume fraction ϕ of particles over a surprisingly large range of values of St. For smaller ϕ however, the two theories deviate significantly except at sufficiently large St. A detailed comparison shows that the predictions of the approximate theory based on Grad's method are in excellent agreement with the results of numerical simulations.

A class of problems of natural convection in tilted boxes is studied by analytical and numerical methods. The convection is assumed to be driven by uniform fluxes of heat (or mass) at two opposing walls, the remaining walls being perfect insulators. Disregarding end-region effects, an exact analytical solution is derived for the state which occurs after initial transients have decayed. This state is steady except for a spatially uniform, linear growth in the temperature (or the species concentration) which occurs whenever the fluxes are not equal. It is characterized by a uni-directional flow, a linear stratification and wall-to-wall temperature profiles which, except for the difference in absolute values due to the stratification, are the same at each crosssection. The mathematical problem is in essence nonlinear and multiple solutions are found in some parameter regions. The Bénard limit of horizontal orientation and heating from below is found to give a first bifurcation for which the steady states both before and after the bifurcation are obtained analytically. For a tilted Bénard-type problem, a steady state with top-heavy stratification is found to exist and compete with a more natural solution. The analytical solution is verified using numerical simulations and a known approximate solution for a vertical enclosure at high Rayleigh numbers. The presented solution admits arbitrary Rayleigh numbers, inclination angles and heat fluxes. Some restrictions on its validity are discussed in the paper.

This paper analyses the sound radiated from the front face of a hard-walled circular cylindrical duct in a subsonic mean flow when the duct contains acoustic sources typical of those in a ducted-fan aeroengine. Two main results are established for modes of any given frequency and circumferential order. The first result is that in certain easily calculated directions, called here the nil-shielding directions, the sound radiated by ducted sources is the same as the sound radiated by the corresponding open sources, i.e. by unducted sources of the same distribution and strength radiating into free space. Thus in these special directions the duct has no noise-shielding effect. The second result is that, in the Kirchhoff approximation, the sound radiated by the open sources in the nil-shielding directions determines the sound radiated by the ducted sources in all directions; i.e. the sound fields radiated by open and ducted sources are related by an open-to-ducted transfer function. This function is such that the sound radiated by the ducted sources is a linear combination of certain diffraction functions, in which the coefficients are given by the sound radiated by the open sources in the nil-shielding directions. The diffraction functions do not depend on the sources and are here calculated explicitly in terms of Bessel functions. The method used in the paper is Kirchhoff's approximation; within linear theory this gives the nil-shielding directions exactly, i.e. in agreement with the Wiener—Hopf solution, and gives the main beam of the radiated field, including the major side-lobes, to good accuracy. The results are relevant to the sound radiated into the forward arc by a ducted turbofan aeroengine.

Thermal convection is investigated experimentally in a dilute liquid mixture of 3He in 4He at four temperatures between 20 and 100 mK above the superfluid transition temperature, chosen for their proximity to the codimension-two and hydrodynamic tricritical points. Two experimental cells of aspect ratio 2.76 and 1.00 were used. For the cell with the higher aspect ratio, two convective transitions at each of the four temperatures were observed above a critical angular velocity, and only one observed below. At temperatures lower than that of the hydrodynamic tricritical point the transition with the lower critical Rayleigh number is hysteretic for all angular velocities; above this temperature hysteresis is absent. The critical Rayleigh numbers are compared with theoretical predictions that take into account the existence of convection modes with azimuthal angular dependence. In the case of the cell with the smaller aspect ratio thermal relaxation oscillations were observed when heating from below. Convective thresholds were again observed but their critical Rayleigh numbers are almost independent of angular velocity. Some suggestions are advanced for this unexpected behaviour.