Particles in fluids

Physics of Fluids, 2014

We develop a simple model for steady, uniform transport in aeolian saltation over a horizontal bed that is based on the computation of periodic particle trajectories in a turbulent shearing flow. The wind and the particles interact through drag, and the particles collide with the bed. We consider collisions with both rigid and erodible beds. The impact velocity in a periodic trajectory over a rigid bed is unconstrained, while that over an erodible bed must have a value that produces a single rebounding particle. The difference in the nature of the collisions results in qualitative differences in the nature of the solutions for the periodic trajectories and, in particular, to differences in the dependence of the particle flow rate on the strength of the turbulent shearing.

Physical Review E

A parallel direct-forcing fictitious domain method is employed to perform fully-resolved numerical simulations of turbulent channel flow laden with finite-size particles. The effects of the particle-fluid density ratio on the turbulence modulation in the channel flow are investigated at the friction Reynolds number of 180, the particle volume fraction of 0.84%, and the particle-fluid density ratio ranging from 1 to 104.2. The results show that the variation of the flow drag with the particle-fluid density ratio is not monotonic, with a larger flow drag for the density ratio of 10.42, compared to those of unity and 104.2. A significant drag reduction by the particles is observed for large particle-fluid density ratios during the transient stage, but not at the statistically stationary stage. The intensity of particle velocity fluctuations generally decreases with increasing particle inertia, except that the particle streamwise root-mean-square velocity and streamwise-transverse velocity correlation in the near-wall region are largest at the density ratio of the order of 10. The averaged momentum equations are derived with the spatial averaging theorem and are used to analyze the mechanisms for the effects of the particles on the flow drag. The results indicate that the drag-reduction effect due to the decrease in the fluid Reynolds shear stress is counteracted by the drag-enhancement effect due to the increase in the total particle stress or the inter-phase drag force for the large particle inertia case. The sum of the total Reynolds stress and particle inner stress contributions to the flow drag is largest at the density ratio of the order of 10, which is the reason for the largest flow drag at this density ratio. The inter-phase drag force obtained from the averaged momentum equation (the balance theory) is significantly smaller than (but agrees qualitatively with) that from the empirical drag formula based on the phase-averaged slip velocity for large density ratios. For the neutrally buoyant case, the balance theory predicts a positive inter-phase force on the particles arising from the negative gradient of the particle inner stress, which cannot be predicted by the drag formula based on the phase-averaged slip velocity. In addition, our results show that both particle collision and particle-turbulence interaction play roles in the formation of the inhomogeneous distribution of the particles at the density ratio of the order of 10.

Earth Surface Processes and Landforms, 2016

The ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker-Planck equation. The diffusive term approximates nonlocal transport. It 'looks' upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High-resolution measurements of bedload particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land surface slope. Particle diffusion during both bedload and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds-averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle-borne substances.

Industrial & Engineering Chemistry Fundamentals, 1969

The random pore model seems to work fairly well on porous solids made by compressing porous alumina powders, which is often done commercially. This model should not be extrapolated and used as yet on other radically different types of porous solids.

The Canadian Journal of Chemical Engineering, 2014

Prediction of particle velocity in a dilute turbulent gas-particle flow nearby a flat solid boundary is investigated numerically using a fully Eulerian two-fluid model. The particulate phase model is based on the kinetic theory of granular flow. The turbulence in both phases is modelled. Interparticle and particle-wall inelastic collisions are considered. Effects of presence/absence of particulate phase turbulence modelling on velocity of particles are investigated. Results indicate that turbulence modelling is necessary for particulate phase, but conventional models should be modified to consider particulate flow nature. Effects of free stream velocity and particle diameter on turbulence modelling results have been discussed.

Coherent structures in …, 2001

The study of trajectories of particles suspended in a fluid in motion is of relevance to many problems; as much those in applied technology as those of theoretical interest. Properties of emulsions, dispersion of contaminants in the atmosphere or the oceans, sedimentation, and mixing, are just a few applications. From the point of view of dynamical systems, this type of problem motivates the study of models with a dimension that is relatively low, but sufficient to encounter theoretical aspects that are as yet little understood.

Journal of Fluid Mechanics, 2004

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 γ d, where γ 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 γ −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.

Journal of Arid Environments, 1998

Experiments conducted in a straight-line suction wind tunnel confirm numerous field observations of sand particles from an upwind source saltating over wet surfaces. Comparison of the mass transport rate on a wet surface (q w) with its dry surface equivalent (q d) yields a strong linear relationship (r 2 > 0•99) over a broad range in freestream velocity. Slope coefficients vary between 0•75 and 1•1. These results indicate that over short distances in the order of 5 m, the mass transport rate is minimally affected by the presence of pore water. However, the boundary-layer structure is substantially modified by grain saltation on a wet surface as compared to the dry condition. Inner region friction velocities associated with dry sand transport (u *d) generally exceed those for clean air (u *o), while u *w for mobile wet surfaces drops below u *o. A model is proposed which relates the ratio q w /q d to several other dimensionless composite variables which address the aerodynamic drag on a wet surface relative to (1) that for an untreated surface and (2) capillary force magnitude. The product of these emulates the variation observed in the slope coefficients from the empirical regressions, thereby quantifying several of the underlying physical processes which govern transport in wet conditions.

Physical Review Letters, 2006

We investigate the airborne transport of particles on a granular surface by the saltation mechanism through numerical simulation of particle motion coupled with turbulent flow. We determine the saturated flux qs and show that its behavior is consistent with a classical empirical relation obtained from wind tunnel measurements. Our results also allow to propose a new relation valid for small fluxes, namely, qs = a(u * − ut) α , where u * and ut are the shear and threshold velocities of the wind, respectively, and the scaling exponent is α ≈ 2. We obtain an expression for the velocity profile of the wind distorted by the particle motion and present a dynamical scaling relation. We also find a novel expression for the dependence of the height of the saltation layer as function of the wind velocity.

Journal of Turbulence, 2007

The motion of small absorbing surfaces is modeled by numerical simulations of turbulent flows with realistic Reynolds numbers. We obtain empirical models for the encounter rate of particles, and in particular the turbulent particle fluxes to given surfaces following the flow. Particular attention is given to the limit, where the particle separations are in the viscous limit, where the scale size of the absorbing surfaces are comparable to or smaller the Kolmogorov length scale. Spherical surfaces represent a special limiting case, but more general shapes are considered as well. The relevance of the problem to the feeding rate of small planktonic organisms in turbulent environments is pointed out in particular, but the results have wider applications. Based on analytical results, we obtain first an empirical expression for the particle flux to a reference spherical surface of interception. This result covers ranges in the universal as well as the viscous subrange. The analysis is then generalized to conical surfaces of interception. We demonstrate the applicability of the results for a wide range of radii, and opening angles of the cones, as well as for two values of turbulence Reynolds numbers. We present also analytical models that account for finite size of particles being transported by a turbulent flow. These latter effects are important for cases where the particles are comparable to the Kolmogorov length in size.