Advanced continuum modelling of gas-particle flows beyond the hydrodynamic limit

The accurate prediction of dilute gas-particle flows using Euler–Euler models is challenging because particle–particle collisions are usually not dominant in such flows. In other words, in dilute flows the particle Knudsen number is not small enough to justify a Chapman–Enskog expansion about the collision-dominated near-equilibrium limit. Moreover, due to the fluid drag and inelastic collisions, the granular temperature in gas-particle flows is often small compared to the mean particle kinetic energy, implying that the particle-phase Mach number can be very large. In analogy to rarefied gas flows, it is thus not surprising that two-fluid models fail for gas-particle flows with moderate Knudsen and Mach numbers. In this work, a third-order quadrature-based moment method, valid for arbitrary Knudsen number, coupled with a fluid solver has been applied to simulate dilute gas-particle flow in a vertical channel with particle-phase volume fractions between 0.0001 and 0.01. In order to isolate the instabilities that arise due to fluid-particle coupling, a fluid mass flow rate that ensures that turbulence would not develop in a single phase flow (Re = 1380) is employed. Results are compared with the predictions of a two-fluid model with standard kinetic theory based closures for the particle phase. The effect of the particle-phase volume fraction on flow instabilities leading to particle segregation is investigated, and differences with respect to the two-fluid model predictions are examined. The influence of the discretization on the solution of both models is investigated using three different grid resolutions. Radial profiles of phase velocities and particle concentration are shown for the case with an average particle volume fraction of 0.01, showing the flow is in the core-annular regime.     

Linear and nonlinear instabilities of a granular bed: Determination of the scales of ripples and dunes in rivers

Granular media are frequently found in nature and in industry and their transport by a fluid flow is of great importance to human activities. One case of particular interest is the transport of sand in open-channel and river flows. In many instances, the shear stresses exerted by the fluid flow are bounded to certain limits and some grains are entrained as bed-load: a mobile layer which stays in contact with the fixed part of the granular bed. Under these conditions, an initially flat granular bed may be unstable, generating ripples and dunes such as those observed on the bed of rivers. In free-surface water flows, dunes are bedforms that scale with the flow depth, while ripples do not scale with it. This article presents a model for the formation of ripples and dunes based on the proposition that ripples are primary linear instabilities and that dunes are secondary instabilities formed from the competition between the coalescence of ripples and free surface effects. Although simple, the model is able to explain the growth of ripples, their saturation (not explained in previous models) and the evolution from ripples to dunes, presenting a complete picture for the formation of dunes.     

Features of the Formation of Composite Materials in Anisotropic Liquid Systems with Elongated Suspended Particles

The equations of rotational motion of nondeformable spherical and axisymmetric elongated particles and a rheologic equation for stresses in arbitrary gradient flows of dilute suspensions of such particles in an anisotropic carrying fluid are obtained within the framework of a structural-phenomenological approach. As a rheologic model of the suspension-carrying fluid and a hydrodynamic model of the suspended particles, we use the Ericksen simple anisotropic fluid and a symmetric triaxial dumbbell, respectively. The constitutive equations obtained are used to study the effect of anisotropy of the carrying fluid on the dynamics of suspended particles and on the rheologic properties of suspensions in a simple shear flow. A stationary orientation of the elongated suspended particles under the action of hydrodynamic forces is discovered. The possibility of applying this phenomenon to the formation of composite materials is discussed.     

Simulations of collision of ice particles

The objective of this paper is to develop a realistic model for ice–structure interaction. To this end, the experiments made by Bridges et al. [Bridges FG, Hatzes A, Liu DNC. Structure, stability and evolution of Saturn’s rings. Nature 1984;309:333–5] in order to measure the coefficient of restitution for ice particles are thoroughly analyzed. One particularly troublesome aspect of the aforementioned experiments is fracture of the ice particles during a collision. In the present effort, the collisional properties of the ice particles are investigated using a Finite Element approach. It is found that a major challenge in modeling collision of the ice balls is the prediction of the onset of fracture and crack propagation in them. In simulations of a block of ice collision to a structure, it is crucial that fracture is determined correctly, as it will influence the collisional properties of the ice particles. The results of the simulation, considering fracture criterion implemented into the Finite Element Model [Zamankhan P, Bordbar M-H. Complex flow dynamics in dense granular flows. Part I: experimentation. J Appl Mech (T-ASME) 2006;73:648–57; Zamankhan P, Huang J. Complex flow dynamics in dense granular flows. Part II: simulations. J Appl Mech (T-ASME) 2007;74:691–702] together with a material model for the ice, imply that most of the kinetic energy dissipation occurs as a result of fracturing at the contact surface of the ice particles. The results obtained in the present study suggest that constitutive models such as those proposed by Brilliantov et al. [Brilliantov NV, Spahn F, Hertzsch JM, Poschel T. Model for collisions in granular gases. Phys Rev E;1996;53:5382–92] for collisions of ice particles are highly questionable.     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.     

Jamming,plasticity and diffusion in dense granular materials

In this paper, we present experimental results based on photoelastic techniques that allow us to obtain complete details of processes in dense granular materials. This work focuses in particular on the jamming transition and on the nature of affine and non-affine deformations in the shear band occuring in granular Couette flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fluid mechanical model for granular flow in silos

Granular materials may display both solid and fluid like behaviour. For low densities and high strain rates as in avalanches or during the discharge of silos the behaviour is mainly governed by interparticle collisions. On the other hand, frictional contacts characterise the solid state which is represented within the framework of plasticity theory. A fluid like constitutive model describes granular materials when subjected to large deformations and high strain rates. It bases upon a modified viscoplastic model that is valid for both yielded and unyielded regions. The central idea is the distinction between fluid and solid regions by means of comparing actual shear stress and Coulomb yield stress. The application to the simultion of the discharge of silos shows the feasibility of the chosen method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unsteady flow of viscoelastic fluid with the fractional K-BKZ model between two parallel plates

In the present paper, we investigate the unsteady flow of a viscoelastic fluid between two parallel plates which is generated by the impulsively accelerated motion of the bottom plate. Based on the result of (Jaishankar and McKinley, 2014), the fractional K-BKZ constitutive equation is obtained from the fractional Maxwell model. Using respectively the fractional Maxwell model and fractional K-BKZ model, the unidirectional flows between two plates are simulated and compared. The velocity field and shear stress of the flows are calculated by developing efficient finite difference schemes. The results show that the fluid with the fractional Maxwell model gradually loses the viscoelasticity, but the fluid with the fractional K-BKZ model continues to preserve the viscoelasticity. The dependence of the flow velocity on various parameters of the fractional K-BKZ model is analyzed graphically.     

Magnetic field effects on Newtonian and non-Newtonian ferrofluid flow past a circular cylinder

The changes in the flow properties under the action of electromagnetic body forces are investigated numerically for ferrofluid flow past a circular cylinder. Ferrofluid is modeled as both a Newtonian and a non-Newtonian Power-Law fluid. Magnetic forces are applied by placing magnets at different locations on the surface of the cylinder. The magnetostatic effects on the structure of the wake region, on drag reduction and on vortex formation length and frequency are shown and compared in terms of Reynolds number, interaction parameter, Power-Law index and magnet location. It is shown that the increase in the interaction parameter reduces drag for both Newtonian and non-Newtonian model. This decrease is observed to be higher for shear thinning and lower for shear thickening fluid compared to Newtonian case. It is also shown that vortex street formation in the wake region behind the cylinder may be delayed under high magnetic effects. The Strouhal number is higher for shear thinning case at both low and high Reynolds numbers, and lower for shear thickening case at high Reynolds numbers, compared to Newtonian fluid. The vortex formation frequency also decreases under the action of the magnetic field in all cases, however the vortex formation length increases. Placing the magnet towards the front region of the cylinder increases considerably the drag coefficient for both Newtonian and non-Newtonian model. This increase in drag coefficient is higher in the shear thinning fluid and lower in the shear thickening fluid compared to the Newtonian case.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.     

The Influence of Variable Convective Velocities on Longitudinal Diffusion

Solutions are presented for the impulsively started uniformstream and simple shear flows past a point source of momentum,which can be interpreted to describe the position and the widthof the front which transmits the knowledge of the singularitythrough a slightly viscous fluid. These understandings are thengeneralized to show that the front always moves with velocityslower than that of a (strictly monotonic) convective velocity,and also that its width always grows faster than with simplediffusion. Finally, a remarkably simple, exact expression is given forvorticity due to a simple shear flow past a point vortex.     

A rapid granular chute avalanche impinging on a small fixed obstacle: DEM modeling,experimental validation and exploration of granular stress

Granular flows are constrained by applied stresses. When a granular flow moves rapidly and impinges on an obstacle, the stress is significantly increased along the contact force networks. Granular stresses are still incompletely understood. The aim of this study is to investigate a rapid avalanche of spherical glass beads in an inclined chute with a small fixed semi-cylindrical obstacle by using particle image velocimetry (PIV) technique and discrete element method (DEM). The proposed DEM model produces good agreement with the corresponding avalanche experiment in terms of the velocity profiles. The validated DEM results are then used to explore the internal flow characteristics of a granular avalanche that are not directly observable in experiments, such as the solid fraction, the average coordination number and the granular stress. Rectangular measurement cells, similar to representative volume elements, are developed to determine the spatial variation in stresses for the granular avalanche. The internal flow characteristics of a rapid granular avalanche with and without obstacles are compared. For the unobstructed flow, the normal and shear stresses decrease in the downstream direction because the solid fraction and the average coordination number decrease, resulting from the gravitational acceleration. On the other hand, granular jamming forms in front of the semi-cylindrical obstacle and results in a significant increase in the normal and shear stresses. The unobstructed flow shows slightly anisotropic stress states, giving an earth pressure coefficient of approximately 1.0, whereas the disturbed flow exhibits strongly anisotropic stress states. The simulation results show that the corresponding earth pressure coefficient can be much higher than unity and increases to a maximum value of roughly 5.0. A shear band develops at a distance of roughly twice the particle diameter above the basal surface and a stronger shear band forms in the upstream vicinity of the obstacle.     

Barotropic instability of shear flows

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm-Liouville theory and hypergeometric functions.     

Dense underflow into a lake or reservoir—Sub-critical flow.

Steady two-dimensional flow of a dense stream down a slight embankment into a lake or a reservoir is considered. The inflowing water is separated from the ambient lake water by a density interface. This work follows on from earlier work in which the flows down a steep incline with a relatively high flow rate were considered. Here, the flow is slow and the entry angle is small, resulting in waves on the interface. The fluid is assumed to be of finite depth and the incoming channel makes an angle α to the horizontal. Limiting flows are found when the fluid separates at a stagnation point or alternatively when the waves reach maximum steepness. The regions in parameter space where such solutions are obtained are delineated for different flow conditions.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.Received: December 16, 2003; revised: October 11, 2004     

Elliptical flows perturbed by shear waves

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Resonant Interactions between Vortical Flows and Water Waves. Part I: Deep Water

Any weak, steady vortical flow is a solution to leading order of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic, resonant interactions can occur between the various components of the flow. The periodic vortical component of the flow is proposed as a model for more complicated vortical flows that would affect surface waves in the ocean, such as the turbulence in the wake of a ship. These resonant interactions are studied in two dimensions, both in the limit of deep water (Part I) and shallow water (Part II). For deep water, the resonant set of surface waves is governed by “triad-like” ordinary differential equations for the wave amplitudes, whose coefficients depend on the underlying rotational flow. These coefficients are calculated explicitly and the stability of various configurations of waves is discussed. The effect of three dimensionality is also briefly mentioned.     

On Wave Interactions III. Miscellaneous Results

In this paper, a collection of results pertaining to resonant triads in fluid flow systems is presented. The main topic is the extension of the analysis by Mahoney for wind-driven shear flows to general, layered flows, that is to say, flows which have piecewise continuous velocity and density profiles. It is seen how the results for calculating the triad interaction coefficients for two-layer flows generalize and carry over for multilayer flows.