Three-dimensional bifurcating internal flow is studied for a single mother tube branching into two equal but diverging daughter tubes. The mother tube is straight and of circular cross-section, containing a fully developed incident motion, while the diverging daughters are straight and of semi-circular cross-section. This basic configuration is treated first by direct numerical simulation and secondly by slender-flow modelling, for a variety of Reynolds numbers and angles of divergence. The direct simulations and modelling highlight different forms of three-dimensional separation or flow reversal as well as enhanced upstream and downstream influence and pressure loss induced by the bifurcations especially at increased divergence angles. Comparisons between the results from the simulations and those from the slender-flow modelling show relatively close agreement at medium values of Reynolds number. In particular, as the angle of divergence increases for a given Reynolds number, there is generally first an increase in flow attachment on to the inner divider wall(s) and then, at higher angles, an increasing trend to flow reversal at the corners formed by the junctions of the outer wall with the divider; longitudinal vortex motion is also enhanced then. The agreement persists over a surprisingly wide range of divergence angles.

The stability of a viscoplastic fluid film falling down an inclined plane is explored, with the aim of determining the critical Reynolds number for the onset of roll waves. The Herschel–Bulkley constitutive law is adopted and the fluid is assumed two-dimensional and incompressible. The linear stability problem is described for an equilibrium in the form of a uniform sheet flow, when perturbed by introducing an infinitesimal stress perturbation. This flow is stable for very high Reynolds numbers because the rigid plug riding atop the fluid layer cannot be deformed and the free surface remains flat. If the flow is perturbed by allowing arbitrarily small strain rates, on the other hand, the plug is immediately replaced by a weakly yielded ‘pseudo-plug’ that can deform and reshape the free surface. This situation is modelled by lubrication theory at zero Reynolds number, and it is shown how the fluid exhibits free-surface instabilities at order-one Reynolds numbers. Simpler models based on vertical averages of the fluid equations are evaluated, and one particular model is identified that correctly predicts the onset of instability. That model is used to describe nonlinear roll waves.

Experiments are reported on the dynamics of a bed of particles sheared by a viscous Couette flow in an annular channel, with emphasis on the distributions of particle velocities, durations and lengths of the small saltation flights, and surface density of the moving particles. The velocity distributions are shown to decay approximately exponentially, with mean value, $U_{p}$, equal to $0.1\,\gamma d$, where $\gamma$ is the shear rate and $d$ is the particle diameter. The duration of the flights does not depend on the shear rate, and is equal to 15 times the settling time $d/V_{S}$, where $V_{S}$ is the Stokes settling velocity. Starting from an initially loosely packed bed, the surface density of the moving particles, $N_{p}$, was observed to decrease slowly over several days, unlike their velocity which remains constant with time. This decay is related to the increase of the threshold shear rate for particle motion, and corresponds to rearrangement of the particles near the bed surface (armouring). When the stationary state is reached, $N_{p}$ depends linearly on the shear rate, so that the particle flow rate, $Q_{p} \,{=}\, N_{p}U_{p}$, is a quadratic function of the shear rate. Two theoretical models are proposed to account for these observations. In the first one, the erosion and deposition rates are modelled using the two hydrodynamic time scales: the inverse shear rate $\gamma^{-1}$ for the erosion rate, and the settling time $d/V_{S}$ for the deposition rate. This model accounts for the linear dependence of $N_{p}$ on the shear rate. The second model was developed to capture the slow decrease of $N_{p}$, by considering the trapping of moving particles into troughs of the bed. This trapping model does recover the main features observed experimentally, although the characteristic time for the decrease of $N_{p}$ still remains too short. Our observations are, finally, compared to existing numerical and experimental studies on turbulent flows.

A computational study of the penetration of a blade into the core of an initially columnar vortex in an incompressible viscous fluid and the subsequent cutting of the vortex is reported for the case where the blade axis is initially orthogonal to the vortex axis. The vortex is advected toward the fixed blade by a free-stream velocity oriented tangent to the blade chord, where the free-stream speed is sufficiently large that the vortex does not induce ejection of vorticity from the blade boundary layer prior to impact of the vortex core with the blade leading edge. A range of computations are performed for cases both with and without ambient axial flow in the vortex core. As the blade leading edge penetrates into the vortex core, cross-diffusion between the columnar vortex and the blade boundary layer causes vortex lines originating in the columnar vortex to rapidly reconnect to those in the blade boundary layer. This cutting process is found to be always incomplete however, due to a change in sign of the spanwise vortex-induced velocity along the leading edge as the vortex is cut, leaving a thin vortex sheet that wraps around the blade leading edge. Cutting of a vortex with non-zero axial flow causes an asymmetry that results in an impulsive lift force on the blade. This lift force has maximum magnitude during the time period where the blade leading edge penetrates into the vortex core. Both the vortex cutting process and the unsteady lift force on the blade are found to be approximately independent of Reynolds number for the various cases examined.

We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as ‘three-fluid’ flow in the same geometry. Assuming symmetry across the gap and based on the lubrication (‘equilibrium’) approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations. The hyperbolic-to-elliptic change-of-type reflects the failing of the lubrication approximation, underlying the quasi-linear hyperbolic formalism, to describe the problem uniformly. The obtained solution is analogous to non-classical shocks recently suggested in problems with change of type.

We demonstrate that the dynamics of a rigid body falling in an infinite viscous fluid can, in the Stokes limit, be reduced to the study of a three-dimensional system of ordinary differential equations ${\dot\Grav} \,{=}\, \Grav\times M_2\Grav$ where $M_2\,{\in}\,{{\mathbb R}^{3\times 3}}$ is a generally non-symmetric matrix containing certain hydrodynamic mobility coefficients. We further show that all steady states and their stability properties can be classified in terms of the Schur form of $M_2$. Steady states correspond to screw motions (or limits thereof) in which the centre of mass traces a helical path, while the body spins uniformly about the vertical. All rigid bodies have at least one such stable screw motion. Bodies for which $M_2$ has exactly one real eigenvalue have a unique globally attracting asymptotically stable screw motion, while other bodies can have multiple, stable and unstable steady motions. One application of our theory is to the case of rigid filaments, which in turn is a first step in modelling the sedimentation rate of flexible polymers such as DNA. For rigid filaments the matrix $M_2$ can be approximated using the Rotne–Prager theory, and we present various examples corresponding to certain ideal shapes of knots which illustrate the various possible multiplicities of steady states. Our simulations of rigid ideal knots in a Stokes fluid predict an approximate linear relation between sedimentation speed and average crossing number, as has been observed experimentally for the much more complicated system of real DNA knots in gel electrophoresis.

The idea of vortex formation time, originally developed for vortex ring formation, is extended to bluff-body flows. Effects related to characteristic vortex formation time are shown for both the cylinder starting from rest and the cylinder undergoing forced oscillations in a steady flow. By looking at how wake vortices are formed when the cylinder is accelerated from rest, it is found that similarities exist between the formation process for wakes and for vortex rings. This formation process is then observed for forced oscillating cylinders, where the characteristic formation time interacts with the oscillation period. Frequently observed bluff-body phenomena will be recast in light of the vortex formation process.

An axisymmetric tube with a variable cross-sectional area, closed at both ends, containing a polytropic gas is oscillated parallel to its axis at or near a resonant frequency. The resonant gas oscillations in an equivalent tube of constant cross-section contain shocks. We show how cone, horn and bulb resonators produce shockless periodic outputs. The output consists of a dominant fundamental mode, where its amplitude and detuning are connected by a cubic equation – the amplitude–frequency relation. For the same gas, a cone resonator exhibits a hardening behaviour, while a bulb resonator may exhibit a hardening or softening behaviour. These theoretical results agree qualitatively with available experimental results and are the basis for resonant macrosonic synthesis (RMS).

A theoretical analysis of the distortion of unsteady three-dimensional disturbances in a Hiemenz boundary layer and its effect on the heat transfer enhancement is presented. It is shown that the disturbance length scale is a critical parameter in determining the amplification ratio of the incoming vorticity. For large disturbance length scales, the amplification ratio increases when the length scale decreases, and a maximum value occurs at a length scale close to five times the boundary-layer thickness. The unsteadiness of the disturbances is found to reduce the vorticity amplification, but the effect is second order when the frequency is low compared to the mean flow strain rate. The impinging disturbances induce large-amplitude vorticity of opposite sign at the wall whose magnitude controls the heat transfer enhancement. As an application of the present analysis, a new scaling correlation is derived for stagnation-point heat transfer in the presence of free-stream turbulence. The theoretical correlation, expressed in terms of turbulence intensity, integral length scale and mean flow Reynolds number, agrees reasonably well with recent experimental data.

Fully resolved simulations of particles suspended in a sustained turbulent flow field are presented. To solve the Navier–Stokes equations a lattice-Boltzmann scheme was used. A spectral forcing scheme is applied to maintain turbulent conditions at a Taylor microscale Reynolds number of 61. The simulations contained between 2 and 10 vol % particles with a solid to fluid density ratio between 1.15 and 1.73. A lubrication force is used to account for subgrid hydrodynamic interaction between approaching particles. Results are presented on the influence of the particle phase on the turbulence spectrum and on particle collisions. Energy spectra of the simulations show that the particles generate fluid motion at length scales of the order of the particle size. This results in a strong increase in the rate of energy dissipation at these length scales and a decrease of kinetic energy at larger length scales.

Collisions due to uncorrelated particle motion are observed (primary collisions), and collision frequencies are in agreement with theory on inertial particle collisions. In addition to this, a large number of collisions at high frequencies is encountered. These secondary collisions are due to the correlated motion of particles resulting from short-range hydrodynamic interactions and spatial correlation of the turbulent velocity field at short distances. This view is supported by the distribution of relative particle velocities, the particle velocity correlation functions and the particle radial distribution function.

In this paper a systematic numerical study of flow past an impulsively started circular cylinder at low and moderate Reynolds numbers using a lattice-Boltzmann algorithmic approach is presented together with an extended volumetric boundary scheme. Results agree well with some well-known previous works. It is demonstrated that in the nearly incompressible limit, this approach is able to provide accurate direct numerical simulations of unsteady flows with curved geometry.

We demonstrate that the effects of turbulent fluctuations have a striking resemblance to those of microscale (thermal) fluctuations in laminar flows, even to higher order in the Knudsen number. This suggests that there may be a good basis for understanding turbulence in terms of Boltzmann kinetic theory. If so, turbulence may be better described in terms of ‘mixing times’ rather than the more classical ‘mixing lengths’. Comparisons are made to Reynolds-stress turbulence models.

We investigate the possibility of using proper orthogonal decomposition (POD) in reconstructing complete flow fields from gappy data. The incomplete fields are created from DNS snapshots of flow past a circular cylinder by randomly ommiting data points. We first examine the effectiveness of an existing method and subsequently introduce modifications that make the method robust and lead to the maximum possible resolution at a certain level of spatio-temporal gappiness. We simulate three levels of gappiness at approximately 20%, 50% and 80% in order to investigate the limits of applicability of the new procedure. We find that for the two lower levels of gappiness both the temporal and spatial POD modes can be recovered accurately leading to a very accurate representation of the velocity field. The resulting resolution is improved by more than five times compared to the existing method. However, for 80% gappiness only a few temporal modes are captured accurately while the corresponding spatial modes are noisy. We explain this breakdown of the method in terms of a simple perturbation analysis. This new methodology can be a building block in an effort to develop effective data assimilation techniques in fluid mechanics applications.

We investigate the slow spreading of fluid mud over a gently sloped conical surface which may simulate a shallow basin or a hill. The mud is assumed to behave as a Bingham plastic possessing a finite yield stress and the lubrication approximation is used. Because of the finite yield stress, a variety of non-trivial equilibrium profiles can exist, corresponding to the state of deposit at the end of upward or downward motion. Analytical solutions are derived for axially symmetric deposits. It is shown that the front of the final profile in axisymmetric spreading is of a common form in dimensionless variables, independent of the total mud volume. Transient evolutions are then studied numerically by employing a finite-volume scheme for both axially symmetric and asymmetric spreading from a localized source. The characteristic features of the mud pile at different stages of spreading are examined. The final shape in asymmetric spreading is strongly affected by the total volume released and by the rate of discharge.

We first study the rising velocity $U_b$ of long bubbles in vertical tubes of different cross-sections, under the acceleration due to gravity $g$. The vessel being initially filled with a liquid of kinematic viscosity $\nu$, it is known that for cylindrical tubes of radius $R$, high-Reynolds-number bubbles ($\hbox{\it Re}\,{\equiv}\,U_{b}R/\nu \,{\gg}\, 1$) are characterized by {Newton's law} $U_{b}\,{\propto}\,\sqrt{gR}$ and low-Reynolds-number bubbles by {Stokes' law} $U_{b}\,{\propto}\, gR^2/\nu$. We show experimentally that these results can be generalized for vessels of ‘arbitrary’ cross-section (rectangles, regular polygons, toroidal tubes). The high-Reynolds-number domain is shown to be characterized by $U_{b}\,{=}\,(8\pi)^{-1/2} \sqrt{gP}$, and the low-Reynolds-number range by $U_{b}\,{\approx}\, 0.012 gS/\nu$, where $P$ and $S$ respectively stand for the wetted perimeter and the area of the normal cross-section of the tube. We derive an analytical justification of these results, using the rectangular geometry. Finally, the problem of long bubble propagation in an unsteady acceleration field is analysed. The theory is compared to existing data.