A mixture of a gas and small solid particles is considered which, far upstream, is in a constant equilibrium state, and moves with a constant velocity. The existence of shock waves is investigated in the four possible cases, namely for frozen flow, for two kinds of partly frozen flow, and for equilibrium flow. It is shown that, in all these cases, compressive shocks may exist, if the upstream velocity exceeds the velocity of sound appropriate to the type of flow. Rarefaction shocks are impossible in each case. Moreover, it is shown that the downstream values of the flow parameters are determined uniquely, and the direction of their change is given. Only rather general assumptions concerning the behaviour of the gas are needed. The paper takes into account the influence of the finite particle volume fraction unlike most previous papers on the topic.

We present a theory for Mach reflection of a detonation undergoing glancing incidence reflection off of a rigid wall. Our focus is on condensed-phase explosives, which we describe with a constant adiabatic gamma equation of state and an irreversible and either state-independent or weakly state-dependent reaction rate. We consider two detonation models: (1) the instantaneous reaction heat-release Chapman–Jouguet (CJ) limit and (2) the spatially resolved reaction heat-release Zeldovich–von Neumann–Dø̈ring (ZND) limit, where here we only consider that a small fraction of the detonation energy release is spatially resolved (the SRHR limit). We observe a three-shock reflection in the CJ limit case, with a Mach shock that is curved. We develop an analytical expression for the triple-point track angle as a function of the angle of incidence. For the SRHR model, we observe a smooth lead shock, akin to von Neumann reflection, with no reflected shock in the reaction zone. Only at larger angles of incidence is a three-shock Mach reflection observed.

This work is focused on the modelling of the shear and normal stresses in fibre suspensions that are subjected to a simple shear flow in the presence of short-range lubrication forces, van der Waals and electrostatic forces, as well as solid friction forces between fibres. All of these forces are weighed by the contact probability. The theory is developed for attractive fibres with van der Waals interaction dominating over electrostatic repulsion. The model predicts a simple Bingham law for both the shear stress and the first normal stress difference, with the apparent shear and normal yield stresses proportional to the second and the third power of the particle volume fraction respectively. The model is applied to the experimental data of Rakatekar et al. (Adv. Mater., vol. 21, 2009, pp. 874–878) and Natale et al. (AIChE J., vol. 60, 2014, pp. 1476–1487) on suspensions of carbon nanotubes dispersed in a Newtonian epoxy resin. It reproduces well the quadratic dependence of the apparent yield stress on the particle volume fraction $(\unicode[STIX]{x1D70E}_{Y}\propto \unicode[STIX]{x1D719}^{2})$ for average particle aspect ratios of $r=160$ and 1200, while it underpredicts the power-law exponent for $r=80$ (always predicting $\unicode[STIX]{x1D719}^{2}$ behaviour instead of $\unicode[STIX]{x1D719}^{3.2}$).

A weak, laminar shear flow of a monodisperse suspension of high-Reynolds-number, low-Weber-number bubbles is studied in a novel experimental configuration. Nitrogen bubbles are formed through an array of small capillaries at the base of a tall channel with a small inclination from the vertical. The bubbles generate a unidirectional shear flow, in which the denser suspension near the bottom wall falls and the lighter suspension near the top wall rises. Profiles of the bubble and liquid velocities and the bubble volume fraction are obtained using hot-film and dual impedance probes. To our knowledge, measurements of the laminar shear properties of a nearly homogeneous bubble suspension have not previously been reported.

A steady shear flow is observed in which the bubble velocity variation across the channel is typically less than 20% of the mean bubble velocity. The velocity and volume fraction gradients increase with channel inclination and exhibit little or no dependence on the mean gas volume fraction. To explain the magnitude of the volume fraction gradients, it is necessary to consider the effects of both the lift force and the effective bubble-phase diffusivity in balancing the segregating tendency of the cross-channel component of the buoyancy force. The bubble velocity gradient can be understood in terms of a balance of the component of the buoyancy force parallel to the channel walls and an effective viscosity associated with the Reynolds stresses produced by bubble-induced liquid velocity fluctuations. Theories for bubbles rising with potential-flow hydrodynamic interactions predict an instability of the homogeneous state due to a negative Maxwell pressure. However, the hydrodynamic diffusivity inferred from our experiments is large enough to mitigate the clustering effects of the Maxwell pressure. Consistent with this, a vigorous instability of the homogeneous state of the bubble suspension is only observed at volume fractions larger than 5%–20% with the critical volume fraction depending on the angle of inclination.

A computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio $\varepsilon $ between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit $\varepsilon \ll 1$ all predictions of the theory can be validated. The larger $\varepsilon $ is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to $30\hspace{0.167em} \% $ of the total balance. The practical relevance of this deviation remains to be assessed in future work.

We study mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyse the behaviour of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyse a double-gyre synthetic velocity field. Second, a Rayleigh–Bénard instability (the Horton–Rogers–Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh–Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated and convection shutdown. The interface compression model correctly predicts the behaviour of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behaviour of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behaviour between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns.

The flow of a planar jet (sheet) impinging onto a solid flat plate with slip is examined theoretically. The jet is assumed to spread out in a thin layer bounded by a hydraulic jump, and draining at the edge of the plate. In contrast to an adhering jet, a slipping jet does not admit a similarity solution. Taking advantage of the different scaling in each region, series expansions are used in the developing and fully viscous layers, which are matched at the transition point. We show that a slipping film exhibits a singularity in the normal stress at the leading edge of the boundary layer, as opposed to the singularity in velocity and shear stress for an adhering film. The boundary-layer and film heights are both found to decrease with slip relative to a smooth substrate, roughly like $\sqrt{30x/Re}-2S$, whereas the slip velocity intensifies like $S\sqrt{Re/30x}$ with slip. Here, $x$ is the distance along the plate, $S$ is the slip length and $Re$ is the Reynolds number (in units of the jet width). The transition is delayed by slip. Guided by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, p. 54002) for a circular adhering jet, the hydraulic-jump height and location are determined for a planar jet, and are found to increase with the Froude number (flow rate) like $Fr^{1/4}$ and $Fr^{5/8}$ respectively, essentially independently of slip length.

The finite-amplitude Rossby wave activity introduced recently by Nakamura and co-workers measures disturbances in terms of the areal displacement of potential vorticity (PV) from zonal symmetry and possesses exact Eliassen–Palm and non-acceleration theorems. This article investigates both theoretically and numerically how this wave activity, denoted , relates to previously defined quantities such as the generalized Lagrangian-mean (GLM) pseudomomentum density and the impulse-Casimir (IC) wave activity in the context of barotropic flow on a sphere. It is shown that under the barotropic constraint both the new and GLM formalisms derive the non-acceleration theorem from the conservation of Kelvin’s circulation, but the two differ in the way the circulation is partitioned into a mean flow and wave activity/pseudomomentum density. The new wave activity differs from the (negative of) GLM pseudomomentum density by the Stokes correction to angular momentum density, which is not negligible even in the small-amplitude limit. In contrast, converges to the IC wave activity and the familiar linear pseudomomentum density in the conservative small-amplitude limit, provided that their reference states are identical. Both the GLM pseudomomentum density and the zonal-mean IC wave activity may be cast in a flux conservation form in equivalent latitude, which may then be related to an exact Eliassen–Palm theorem through a gauge transformation. However, of the three wave activity forms, only satisfies an exact non-acceleration theorem for the zonal-mean zonal wind . A simple jet forcing experiment is used to examine the quantitative differences among these diagnostics. In this experiment, and the IC wave activity behave similarly in the domain average; however, they differ substantially in the local profiles, the former being more closely related to the flow modification. Despite their close conceptual relationship, the GLM pseudomomentum fails to capture the meridional structure of because the Stokes correction term dominates the former. This demonstrates various advantages of as a diagnostic of eddy–mean flow interaction.

The concept of diffusion by bulk convection formulated by Bradshaw is applied to the transport equations for the turbulent kinetic energy, turbulent shear stress and an integral length scale. The resulting set of hyperbolic partial differential equations is solved by an explicit finite-difference scheme for the cases of incompressible axisymmetric wakes and jets in a coflowing air stream. It is found that the profiles of mean velocity and shear stress are almost insensitive to the empirical input whereas the profiles of kinetic energy are very sensitive.

The formation of shock waves in. dense argon is studied using a numerical technique related to the molecular-dynamics approach. The kinetic and total pressures, density, temperature and mass velocity are calculated when a simulated tungsten piston is driven into the stationary gas. It is found that the pressure generated is similar to that found using binary collision assumptions, however the temperature is lower and the density much higher than under more rarefied conditions. Results are also given for the same experiments when the generating piston is composed of dense argon atoms. It is shown that the results are almost independent of the type of piston material and that the shock structure is a function of the gas interparticle force law only. The reflexion of a spherically imploding shock wave in dense argon is also examined and it is found that the shock wave reflects before reaching the centre due to high tangential stresses. Some data is also given, upon the velocity distribution and wall pressure fluctuations.

We present a study of bidispersed particulate gravity currents at high Reynolds numbers flowing along a V-shaped valley. The speed and width of the currents, the mass deposited by the currents and the density of the deposits were examined by both a box model and lock-exchange experiments in a 5 m long tank. Silicon carbide and glass beads were used for the bidispersed suspension models. The initial conditions of the currents were similar, except that the grain size of the glass beads was successively chosen to be 2, 2.5 and 4 times that of the silicon carbide. For all experiments a Stokes’ settling velocity model, assuming that both particles are spherical, gives a settling rate of the glass beads that is greater than that of the silicon carbide by a factor ranging from 1.6 to 16.5. When the ratio of the Stokes’ settling velocity of the glass beads to that of the silicon carbide is greater than ∼6, we find a complete agreement between the box model and the experiment. In particular, the deposit shows a substantial decline in the mass of the coarser glass beads in the first metre, so that it only contains the finer silicon carbide further downstream. By contrast, when the Stokes’ settling velocity ratio is less than ∼4, only the speed of the current and the total sedimented mass can be well described by the box model. The experimental deposit is otherwise characterized by a slightly increasing density, which the box model fails to match. There is no difference in the deposit density across the valley. For all experiments in the V-shaped valley, the width of the currents decreases with time $t$ according to ${t}^{\ensuremath{-} 2/ 7} $. Analogue experiments in a flat-bottom tank were also performed to assess the influence of the valley on the sedimentation dynamics described above. A similar behaviour with settling velocity ratios was found. This study eventually shows the need for considering particle interactions in even dilute gravity currents at high Reynolds numbers.

Phase-averaged and directionally triggered digital particle image velocimetry measurements were taken in longitudinal and transverse planes in the near field of the flow emerging from a fluidic precessing jet nozzle. Measurements were performed at nozzle inlet Reynolds and Strouhal numbers of 59000 and 0.0017, respectively. Results indicate that the jet emerging from the nozzle departs with an azimuthal component in a direction opposite to that of the jet precession. In addition, the structure of the ‘flow convergence’ region, reported in an earlier study, is better resolved here. At least three unique vortex-pair regions containing smaller vortical ‘blobs’ are identified for the first time. These include a vortex-pair region originating from the foci on the downstream face of the nozzle centrebody, a vortex-pair region shed from the edge of the centrebody and a vortex-pair region originating from the downstream surface of the nozzle exit lip.

Breaking waves generated by a two-dimensional hydrofoil moving near a free surface at constant speed (U∞), angle of attack and depth of submergence were studied experimentally. The measurements included the mean and fluctuating shape of the breaking wave, the surface ripples downstream of the breaker and the vertical distribution of vertical and horizontal velocity fluctuations at a single station behind the breaking waves. The spectrum of the ripples is highly peaked and shows little variation in both its peak frequency and its shape over the first three wavelengths of the wavetrain following the breaker. For a given speed, as the breaker strength is increased, the high-frequency ends of the spectra are nearly identical but the spectral peaks move to lower frequencies. A numerical instability model, in conjunction with the experimental data, shows that the ripples are generated by the shear flow developed at the breaking region. The spectrum of the vertical velocity fluctuations was also found to be highly peaked with the same peak frequency as the ripples, while the corresponding spectrum of the horizontal velocity fluctuations was found not to be highly peaked. The root-mean-square (r.m.s.) amplitude of the ripples (νrms) increases with increasing speed and with decreasing depth of submergence of the hydrofoil, and decreases as x-1/2 with increasing distance x behind the breaker. The quantity (gνrms)/(U∞Vrms) (where Vrms is the maximum r.m.s. vertical velocity fluctuation and g is the gravitational acceleration) was found to be nearly constant for all of the measurements.

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.