A theory intended for slow, dense flows of cohesionless granular materials is developed for the case of planar deformations. By considering granular flows on very fine scales, one can conveniently split the individual particle velocities into fluctuating and mean transport components, and employ the notion of granular temperature that plays a central role in rapid granular flows. On somewhat larger scales, one can think of analogous fluctuations in strain rates. Both kinds of fluctuations are utilized in the present paper. Following the standard continuum approach, the conservation equations for mass, momentum and particle translational fluctuation energy are presented. The latter two equations involve constitutive coefficients, whose determination is one of the main concerns of the present paper. We begin with an associated flow rule for the case of a compressible, frictional, plastic continuum. The functional dependence of the flow rule is chosen so that the limiting behaviours of the resulting constitutive relations are consistent with the results of the kinetic theories developed for rapid flow regimes. Following Hibler (1977) and assuming that there are fluctuations in the strain rates that have, for example, a Gaussian distribution function, it is possible to obtain a relationship between the mean stress and the mean strain rate. It turns out, perhaps surprisingly, that this relationship has a viscous-like character. For low shear rates, the constitutive behaviour is similar to that of a liquid in the sense that the effective viscosity decreases with increasing granular temperature, whereas for rapid granular flows, the viscosity increases with increasing granular temperature as in a gas. The rate of energy dissipation can be determined in a manner similar to that used to derive the viscosity coefficients. After assuming that the magnitude of the strain-rate fluctuations can be related to the granular temperature, we obtain a closed system of equations that can be used to solve boundary value problems. The theory is used to consider the case of a simple shear flow. The resulting expressions for the stress components are similar to models previously proposed on a more ad hoc basis in which quasi-static stress contributions were directly patched to rate-dependent stresses. The problem of slow granular flow in rough-walled vertical chutes is then considered and the velocity, concentration and granular temperature profiles are determined. Thin boundary layers next to the vertical sidewalls arise with the concentration boundary layer being thicker than the velocity boundary layer. This kind of behaviour is observed in both laboratory experiments and in granular dynamics simulations of vertical chute flows.

A weakly nonlinear analysis is used to study the initial evolution of the Rayleigh–Taylor instability of two superposed miscible layers of viscous fluid between impermeable and traction-free planes in a field of gravity. Analytical solutions are obtained to second order in the small amplitude of the initial perturbation of the interface, which consists of either rolls or squares or hexagons with a horizontal wavenumber k. The solutions are valid for arbitrary values of k, the viscosity ratio (upper/lower) γ, and the depth ratio r, but are presented assuming that k=kmax(γ, r), where kmax is the most unstable wavenumber predicted by the linear theory. For all planforms, the direction of spouting (superexponential growth of interfacial extrema) is determined by the balance between the tendency of the spouts to penetrate the less viscous layer, and a much stronger tendency to penetrate the thicker layer. When these tendencies are opposed (i.e. when γ>1 with r>1), the spouts change direction at a critical value of r=rc(γ). Hexagons with spouts at their centres are the preferred planform for nearly all values of γ and r, followed closely by squares; the most slowly growing planform is hexagons with spouts at corners. Planform selectivity is strongest when γ[ges ]10 and r[ges ]γ1/3. Application of the results to salt domes in Germany and Iran show that these correspond to points (γ, r) below the critical curve r=rc(γ), indicating that the domes developed from interfacial extrema having subexponential growth rates.

Equations of mean and fluctuating velocities in rotation and translation have been derived for rigid thin inertialess fibres moving in a turbulent fluid. The derived equations for mean motion are general to fluid velocities that vary nonlinearly along the length of the fibre. From the equations of fluctuating fibre velocity, rotational and translational dispersion coefficients were derived. The resulting dispersion coefficients were shown to decrease as the ratio of fibre length to Lagrangian integral length scale of the turbulence increased.

The structure and dynamics of vorticity ω and rate of strain S are studied using direct numerical simulations (DNS) of incompressible homogeneous isotropic turbulence. In particular, characteristics of the pressure Hessian Π, which describe non-local interaction of ω and S, are presented. Conditional Lagrangian statistics which distinguish high-amplitude events in both space and time are used to investigate the physical processes associated with their evolution. The dynamics are examined on the principal strain basis which distinguishes vortex stretching and induced rotation of the principal axes of S. The latter mechanism is associated with misaligned ω with respect to S, a condition which predominates in isotropic turbulence and is dynamically significant, particularly in rotation-dominated regions of the flow. Locally-induced rotation of the principal axes acts to orient ω towards the direction of either the intermediate or most compressive principal strain. The tendency towards compressive straining of ω is manifested at the termini of the high-amplitude tube-like structures in the flow. Non-locally-induced rotation, associated with Π, tends to counteract the locally-induced rotation. This is due to the strong alignment between ω and the eigenvector of Π corresponding to its smallest eigenvalue and is indicative of the controlling influence of the proximate structure on the dynamics. High-amplitude rotation-dominated regions deviate from Burgers vortices due to the misalignment of ω. Although high-amplitude strain-dominated regions are promoted primarily by local dynamics, the associated spatial structure is less organized and more discontinuous than that of rotation-dominated regions.

This paper presents a linear stability analysis of plane Couette flow of a granular material using a kinetic-theory-based model for the rheology of the medium. The stability analysis, restricted to two-dimensional disturbances, is carried out for three illustrative sets of grain and wall properties which correspond to the walls being perfectly adiabatic, and sources and sinks of fluctuational energy. When the walls are not adiabatic and the Couette gap H is sufficiently large, the base state of steady fully developed flow consists of a slowly deforming ‘plug’ layer where the bulk density is close to that of maximum packing and a rapidly shearing layer where the bulk density is considerably lower. The plug is adjacent to the wall when the latter acts as a sink of energy and is centred at the symmetry axis when it acts as a source of energy. For each set of properties, stability is determined for a range of H and the mean solids fraction [barvee ]. For a given value of [barvee ], the flow is stable if H is sufficiently small; as H increases it is susceptible to instabilities in the form of cross-stream layering waves with no variation in the flow direction, and stationary and travelling waves with variation in the flow and gradient directions. The layering instability prevails over a substantial range of H and [barvee ] for all sets of wall properties. However, it grows far slower than the strong stationary and travelling wave instabilities which become active at larger H. When the walls act as energy sinks, the strong travelling wave instability is absent altogether, and instead there are relatively slow growing long-wave instabilities. For the case of adiabatic walls there is another stationary instability for dilute flows when the grain collisions are quasi-elastic; these modes become stable when grain collisions are perfectly elastic or very inelastic. Instability of all modes is driven by the inelasticity of grain collisions.

We experimentally study the formation of cusps at the free surfaces of viscous fluids in three simple cases that portray possible natural or industrial processes. Two cases concern levelling driven by gravity: the case (A) of a sinusoidal surface and the case (B) of a single groove. In the third case (C), an initially sinusoidal surface evolves under the action of a fast enough lateral compression that the effects of gravity are negligible. Case (A) shows a critical aspect ratio above which the cusps form. Case (B) allows a more detailed study of the evolution of the cuspidal structure, which does not change in shape but reduces its size according to a simple power law dependence in time. In case (C), cusps form even for small initial aspect ratios.

We analyse the motion of a current of water migrating under gravity into a hot vapour-saturated porous rock accounting for the vaporization which occurs as the water invades the hot rock. We present a series of similarity solutions to describe the rate of advance of both planar and axisymmetric currents when the total mass of water injected after time t is proportional to tγ. Three distinct cases arise. When γ>1/2 (planar) or γ>1 (axisymmetric), the depth of the current increases at all points from the source, and therefore vaporization occurs at all points on its surface. This case is described by a simple extension of the well-known similarity solutions for non-vaporizing currents. When 0<γ<1/2 (planar) or 0<γ<1 (axisymmetric), there is a region near the source where the depth of the current decreases. The depth only increases at the more distant points. Vaporization therefore only occurs in the leading part of the current where it is advancing into the superheated rock. In this case, we develop modified similarity solutions which account for the vaporization in the distal part of the current. The third case involves the finite release of fluid. Owing to the vaporization, the mass of the current decreases with time. Since there is no injection, the rate of advance of the current can no longer be found by comparing the exponents of time in the local and global equations for mass conservation. Instead, the motion is described by a class of similarity solutions of the second kind, analogous to those described by Barenblatt (1997), in which the total mass of the current is proportional tγ, where γ is a function of the mass fraction which vaporizes, [Fscr ], such that γ→0 as [Fscr ]→0 and γ→−1 as [Fscr ]→1. The model is extended to include the effects of capillary retention of fluid in the pore spaces and we discuss the relevance of our results to the process of liquid reinjection in the geothermal power industry.

Experimental results are presented for the influence of Reynolds number on multifractal scale similarity in turbulent flows. These are obtained from single-point measurements of a dynamically passive Sc≈1 conserved scalar quantity ζ(t) in a turbulent shear flow at outer-scale Reynolds numbers of 14000[les ]Reδ[les ]110000. Statistical criteria based on the maximum allowable scale-to-scale variation L1(ε) in multiplier distributions P(Mε) from multifractal gauge sets allow accurate discrimination between multifractal and non-multifractal scaling. Results show that the surrogate scalar energy dissipation rate χs(t)≡(dζ/dt)2is found to display a scale-invariant similarity consistent with a random multiplicative cascade characterized by a bilinear multiplier distribution P(Mε) over a range of scales extending downward from the outer scaleTδ. For a range of scales extending upward from the inner (diffusive) scale TD, the dissipation rate displays a different scale-invariant similarity characterized by a uniform multiplier distribution. The former scale-invariance becomes evident in the present Sc≈1 data only when Reδ is sufficiently large. Comparisons with results from Sc 1 data indicate that this scale-invariant similarity applies when the outer-to-inner scale ratio Tδ/TD≈0.09 Re3/4δSc1/2 is greater than about 400. In contrast to the scalar dissipation rate field, the scalar field is found to lack any multifractal scale similarity.

The yield conditions for the displacement of three-dimensional fluid droplets from solid boundaries are studied through a series of numerical computations. The study considers low-Reynolds-number shear flows over plane boundaries and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio γ, capillary number Ca and advancing and receding contact angles, θA and θR. This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(γ, θA, Δθ) where Δθ=θA−θR is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on solid surfaces. Additional constraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While good agreement is found in the joint asymptotic limits Δθ[Lt ]θA[Lt ]1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets.

We perform laboratory experiments in a recirculating shear flow tank of non-uniform salt-stratified water to examine the excitation of internal gravity waves (IGW) in the wake of a tall, thin vertical barrier. The purpose of this study is to characterize and quantify the coupling between coherent structures shed in the wake and internal waves that radiate from the mixing region into the deep, stationary fluid. In agreement with numerical simulations, large-amplitude internal waves are generated when the mixing region is weakly stratified and the deep fluid is sufficiently strongly stratified. If the mixing region is unstratified, weak but continuous internal wave excitation occurs. In all cases, the tilt of the phase lines of propagating waves lies within a narrow range. Assuming the waves are spanwise uniform, their amplitude in space and time is measured non-intrusively using a recently developed ‘synthetic schlieren’ technique. Using wavelet transforms to measure consistently the width and duration of the observed wavepackets, the Reynolds stress is measured and, in particular, we estimate that when large-amplitude internal wave excitation occurs, approximately 7% of the average momentum across the shear depth and over the extent of the wavepacket is lost due to transport away from the mixing region by the waves.

We propose that internal waves may act back upon the mean flow modifying it so that the excitation of waves of that frequency is enhanced. A narrow frequency spectrum of large-amplitude waves is observed because the feedback is largest for waves with phase tilt in a range near 45°. Numerical simulations and analytic theories are presented to further quantify this theory.

Analytical support is given to Fornberg's numerical evidence that the steady axially symmetric flow of a uniform stream past a bluff body has a wake eddy which tends towards a large Hill's spherical vortex as the Reynolds number tends to infinity. The viscous boundary layer around the eddy resembles that around a liquid drop rising in a liquid, especially if the body is a circular disc, so that the boundary layer on it does not separate. This makes it possible to show that if the first-order perturbation of the eddy shape from a sphere is small then the eddy diameter is of order R1/5 times the disc diameter, where R is the Reynolds number based on the disc diameter. Previous authors had suggested R1/3 and lnR, but they appear to have made unjustified assumptions.

A cylindrical pipe facility with a length of 32 m and a diameter of 40 mm has been designed. The natural transition Reynolds number, i.e. the Reynolds number at which transition occurs as a result of non-forced, natural disturbances, is approximately 60 000. In this facility we have studied the stability of cylindrical pipe flow to imposed disturbances. The disturbance consists of periodic suction and injection of fluid from a slit over the whole circumference in the pipe wall. The injection and suction are equal in magnitude and each distributed over half the circumference so that the disturbance is divergence free. The amplitude and frequency can be varied over a wide range.

First, we consider a Newtonian fluid, water in our case. From the observations we compute the critical disturbance velocity, which is the smallest disturbance at a given Reynolds number for which transition occurs. For large wavenumbers, i.e. large frequencies, the dimensionless critical disturbance velocity scales according to Re−1, while for small wavenumbers, i.e. small frequencies, it scales as Re−2/3. The latter is in agreement with weak nonlinear stability theory. For Reynolds numbers above 30 000 multiple transition points are found which means that increasing the disturbance velocity at constant dimensionless wavenumber leads to the following course of events. First, the flow changes from laminar to turbulent at the critical disturbance velocity; subsequently at a higher value of the disturbance it returns back to laminar and at still larger disturbance velocities the flow again becomes turbulent.

Secondly, we have carried out stability measurements for (non-Newtonian) dilute polymer solutions. The results show that the polymers reduce in general the natural transition Reynolds number. The cause of this reduction remains unclear, but a possible explanation may be related to a destabilizing effect of the elasticity on the developing boundary layers in the entry region of the flow. At the same time the polymers have a stabilizing effect with respect to the forced disturbances, namely the critical disturbance velocity for the polymer solutions is larger than for water. The stabilization is stronger for fresh polymer solutions and it is also larger when the polymers adopt a more extended conformation. A delay in transition has been only found for extended fresh polymers where delay means an increase of the critical Reynolds number, i.e. the number below which the flow remains laminar at any imposed disturbance.

Direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is O(1) and deformations of the bubbles are small. The rise velocity of a regular array of three-dimensional bubbles at different volume fractions agrees relatively well with the prediction of Sangani (1988) for Stokes flow. A regular array of two- and three-dimensional bubbles, however, is an unstable configuration and the breakup, and the subsequent bubble–bubble interactions take place by ‘drafting, kissing, and tumbling’. A comparison between a finite Reynolds number two-dimensional simulation with sixteen bubbles and a Stokes flow simulation shows that the finite Reynolds number array breaks up much faster. It is found that a freely evolving array of two-dimensional bubbles rises faster than a regular array and simulations with different numbers of two-dimensional bubbles (1–49) show that the rise velocity increases slowly with the size of the system. Computations of four and eight three-dimensional bubbles per period also show a slight increase in the average rise velocity compared to a regular array. The difference between two- and three-dimensional bubbles is discussed.

Space–time correlations and frequency spectra of wall-pressure fluctuations, obtained from direct numerical simulation, are examined to reveal the effects of pressure gradient and separation on the characteristics of wall-pressure fluctuations. In the attached boundary layer subjected to adverse pressure gradient, contours of constant two-point spatial correlation of wall-pressure fluctuations are more elongated in the spanwise direction. Convection velocities of wall-pressure fluctuations as a function of spatial and temporal separations are reduced by the adverse pressure gradient. In the separated turbulent boundary layer, wall-pressure fluctuations are reduced inside the separation bubble, and enhanced downstream of the reattachment region where maximum Reynolds stresses occur. Inside the separation bubble, the frequency spectra of wall-pressure fluctuations normalized by the local maximum Reynolds shear stress correlate well compared to those normalized by free-stream dynamic pressure, indicating that local Reynolds shear stress has more direct influence on the wall-pressure spectra. Contour plots of two-point correlation of wall-pressure fluctuations are highly elongated in the spanwise direction inside the separation bubble, implying the presence of large two-dimensional roller-type structures. The convection velocity determined from the space–time correlation of wall-pressure fluctuations is as low as 0.33U0 (U0 is the maximum inlet velocity) in the separated zone, and increases downstream of reattachment.