Rheology Of Dense Granular Flows Research Articles

Dynamical yield criterion for granular matter from first principles.

We investigate, using a recently developed model of liquid state theory describing the rheology of dense granular flows, how a yield stress appears in granular matter at the yielding transition. Our model allows us to predict an analytical equationof the corresponding dynamical yield surface, which is compared to the usual models of solid fracture. In particular, this yield surface interpolates between the typical failure behaviors of soft and hard materials. This work also underlines the central role played by the effective friction coefficient at the yielding transition.

Coupled continuum modelling of size segregation driven by shear-strain-rate gradients and flow in dense, bidisperse granular media

Dense granular systems that consist of particles of disparate sizes segregate based on size during flow, resulting in complex, coupled segregation and flow patterns. The ability to predict how granular mixtures segregate is important in the design of industrial processes and the understanding of geophysical phenomena. The two primary drivers of size segregation are pressure gradients and shear-strain-rate gradients. In this work, we isolate size segregation driven by shear-strain-rate gradients by studying two dense granular flow geometries with constant pressure fields: gravity-driven flow down a long vertical chute with rough parallel walls and annular shear flow with rough inner and outer walls. We perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in both flow geometries, while varying system parameters, such as the flow rate, flow configuration size, fraction of large/small grains and grain-size ratio, and use DEM data to inform continuum constitutive equations for the relative flux of large and small particles. When the resulting continuum model for the dynamics of size segregation is coupled with the non-local granular fluidity model – a non-local continuum model for dense granular flow rheology – we show that both flow fields and segregation dynamics may be simultaneously captured using this coupled, continuum system of equations.

Solid Concentration as a Main Proxy for Basal Force Fluctuations Generated by Highly Concentrated Sediment Flows

AbstractSediment flows generate ground vibrations by exerting force fluctuations on the riverbed. Linking force fluctuations to properties of highly concentrated sediment flows, however, remains particularly challenging due to complexities arising from grain‐to‐grain interactions. Here, we conduct downscaled flume experiments in which we specifically measure force fluctuations and local seismic vibrations together with flow properties of highly concentrated sediment flows at high spatial and temporal resolution. We observe hysteresis behaviors between force fluctuations amplitude and flow surface elevation and mass that occur during complex changes in internal flow dynamics. By contrast, force fluctuations amplitude exhibits a unique negative relationship with solid concentration. We suggest this is due to the rheology of dense granular flows, where solid concentration is a proxy for particle agitation. We therefore advance that solid concentration should be incorporated in seismic models of such sediment flows as a key parameter describing inter‐particle collisions and impacts to the bed.

Granular flows in drums of non-uniform widths

We study how channel width variations influence the dynamics of free-surface granular flows. For this purpose, we extend a continuum model framework to granular flows passing through channels that narrow or widen. Our theory uses a linearized approximation to an established dense granular flow rheology and a Coulomb friction law to model interaction between flow and sidewalls. We test the theoretical predictions using two novel 40 cm-diameter drums (convex and concave) filled halfway with 2 mm diameter particles rotated at rates in which the shear layer remains shallow and dense. We apply particle tracking velocimetry to enable quantitative comparisons between experimental data and theoretical predictions. We find that our experimental kinematics and energy profiles largely agree with the theoretical predictions. In general, flows through narrowing channels are faster and deeper than flows through widening channels. The influence of width variations grows with increasing flow speed, and the form of the rate dependence changes fundamentally as the regime changes from one in which kinetic energy is dissipated locally to one in which it is advected downstream. For both regimes, theoretical scaling analysis leads us to experimentally validated power laws, in which the exponent depends on the flow regime, and the multiplicative coefficient depends on channel geometry alone. Finally, we discuss how the differences between theoretical predictions and experimental data may be useful for improving our understanding of flows through non-uniform channels.

A multi-dimensional two-phase mixture model for intense sediment transport in sheet flow and around pipeline

A two-phase mixture model is developed to simulate intense sediment transport covering the bed-load layer and suspended load layer. The proposed model maintains high accuracy as an Eulerian two-phase model but requires low computational cost. The proposed model applies an analytical formula for relative velocity between phases. The dense granular flow rheology is employed to close particle stress economically. The closure of Reynolds stress considers turbulence damping and small-scale fluctuation of fluid–particle interaction and particle collision. A damping function is adopted in eddy viscosity for extra turbulence damping from inter-particle interaction. The optimal exponent of the damping function refers to sediment shape and size. The sediment diffusion includes turbulence diffusion and shear-induced self-diffusion originating from dense sediment. The proposed model is validated by several sets of sheet flow cases (Shields number Θ = 0.44–2.20 and particle Reynolds number Res = 1.6–603.0) and shows a wide applicable range and good accuracy. The small-scale fluctuation and shear-induced self-diffusion improve the computation in the lower sheet flow layer where volumetric sediment concentration is larger than 0.2. Furthermore, the proposed model shows reasonable applicability on the multi-dimensional pipeline scour development. The scour profiles are well predicted and the Brier Skill Score = 0.809. However, the proposed model does not perform the wake characteristic around the pipeline sufficiently, and slight scour difference exists between the simulation and experiment.

The effect of grain shape and material on the nonlocal rheology of dense granular flows.

Nonlocal rheologies allow for the modeling of granular flows from the creeping to intermediate flow regimes, using a small number of parameters. In this paper, we report on experiments testing how particle properties affect the model parameters used in the Kamrin & Koval cooperative nonlocal model, using particles of three different shapes (circles, ellipses, and pentagons) and three different materials, including one which allows for the measurement of stresses via photoelasticity. Our experiments are performed on a quasi-2D annular shear cell with a rotating inner wall and a fixed outer wall. Each type of particle is found to exhibit flows which are well-fit by nonlocal rheology, with each particle having a distinct triad of the local, nonlocal, and frictional parameters. While the local parameter b is always approximately unity, the nonlocal parameter A depends sensitively on both the particle shape and material. The critical stress ratio μs, above which Coulomb failure occurs, varies for particles with the same material but different shape, indicating that geometric friction can dominate over material friction.

Rheology of two-dimensional granular chute flows at high inertial numbers

Contrary to the popular inertial number-based rheology of dense granular flows, recent studies suggest a non-monotonic variation of the effective friction coefficient μ(I) with the inertial number I in plane shear flows. While the popular rheology assuming monotonic variation of μ(I) with I suggests existence of an upper limit of inclination angle for steady chute flows, the non-monotonic variation suggests the possibility of two different flow states for chute flows at a given inclination angle. In this work, we perform DEM simulations of chute flow of frictional inelastic disks and show that steady, fully developed flows are possible at inclinations much higher than those predicted from the monotonic μ−I rheology. We observe steady flows up to inertial number I≈2 and find non-monotonic variation of the effective friction at high inertial numbers for chute flow of disks. The flows at high inertial numbers exhibit a constant density bulk region supported on top of a very dilute energetic basal layer of particles. We show that, in addition to a modified effective friction law that accounts for the non-monotonic variation of μ(I) and the dilatancy law relating the solids fraction ϕ with I, the rheological description also needs to account for the stress anisotropy by means of a normal stress difference law. By accounting for the presence of the normal stress difference, we also establish that only a single flow state is possible at any given inclination angle despite the non-monotonic variation of the effective friction coefficient.

Explosively driven dynamic compaction of granular media

This paper reports experimental investigations into the dynamic compaction of particle rings subjected to moderate explosions confined in a radial Hele-Shaw cell. The findings reveal marked transitions in the flow regimes corresponding to the evolution of the transient pressure fields inside the granular medium induced by unsteady gas infiltration. As the pressure fields evolve from being localized to diffusive with a substantial reduction in intensity, three sequent flow regimes with distinct rheologies are identified. Specifically, these flow regimes are found to be governed by the localized strong pressure field, then the competition between the diffusive pressure field and wall friction, and finally, solid stresses in the presence of rarefaction waves. A Bingham-type rheology can adequately describe the granular compaction when the pressure gradients remain the dominant driving forces, whereas the frictional nature of the granular flows becomes increasingly significant as the solid stresses set in. As the pressure gradients phase out, rarefaction decompaction commences. However, this only manages to relax the innermost layers of the compacted particles due to a distinctive compressive deformation pattern, giving rise to a discontinuous flow field. These findings shed light on the rheology of dense granular flows subjected to unsteady pressure loadings involving diverse flow–particle and particle–particle interactions.

Modelling of scour formation using SedFoam, continuum approach

SedFOAM is a two-phase flow solver built upon consecutive laws, based on the CFD toolbox OpenFOAM. The sediment body is considered as a continuum phase. The intergranular and fluid stresses are modeled with the dense granular flow rheology and the k–ϵ turbulent model, respectively. The model setup will be based on an experimental study on the scour due to a water jet subjected to lateral confinement. A comparison study will be made based on precise experimental data by Martino et al. (2019) that will show the advantages and defaults of SedFoam. The objective of this work is to reproduce the digging and filling cycle of the scour formation due to the water jet in a confined canal. The first numerical results show that it needs to use 3D numerical simulations because of the fluctuation of the jet direction induced by the presence of a driven flow cavity.

DEM simulation of dense granular flows in a vane shear cell: Kinematics and rheological laws

The rheology of dense granular flows is investigated through discrete element method (DEM) simulation of a vane shear cell. From the simulation, profiles of shear stress, shear rate, and velocity are obtained, which demonstrates that the flow features in the vane shear cell are equivalent to those in the classic annular Couette cell. A novel correlation for the apparent viscosity is formulated and leads to a new expression for μKT in the kinetic theory analysis. The μKT formulation is able to qualitatively capture the μ-I relation in the shear cell. A correlation length is added in the energy dissipation term to account for the effects of the particle motion correlation. A simplified correlation length model is derived based on DEM results and is compared with the literature. The modified granular kinetic energy equation is able to correctly predict the granular temperature profiles in the shear cell.

A two-phase SPH model for massive sediment motion in free surface flows

Massive sediment motion in water with a free surface is an important kind of geophysical flows such as hyper-concentrated sediment laden river flows discharging into estuarine delta and turbidity currents generated by subaqueous landslides. One of the key and common characteristics of such flows is that interactions between water and sediment as well as those among sediment particles are equally important in affecting the sediment motion and the fluid flow. This paper presents a numerical model that builds on and extends an earlier two-phase SPH model based on a continuum formulation of solid-liquid mixtures (Shi et al., 2017) to provide a unified description of massive sediment motion in free surface flows. In the model, a constitutive law based on the rheology of dense granular flow is introduced to express the intergranular stresses while the interphase drag force is determined by combining the Ergun equation for dense solid-fluid mixtures and the power law for dilute suspensions. The proposed model is firstly applied to the study of collapse of loosely or densely packed granular columns submerged in water. The computed surface profiles of the granular column are found to be in good agreement with the experimental data. It shows that the loosely packed and the densely packed columns behave rather differently due to the differences in water-sediment interaction processes. The model is then used to simulate a dam-break flow over a mobile sediment bed. The computed configurations of the flow and the movable bed also agree well with the measured data. The predicted position on the leading edge of the flow has a mean error of 0.8% while the mean error for the maximum bed height is 12.9%. To further identify the dynamic processes involved, effects of water-sediment interactions on the motion of bed materials are investigated by examining the spatial and temporal variations of pressure and flow velocity. As shown in the applications, the proposed two-phase SPH model can successfully represent both the gravity-driven underwater granular flows and the shear flow driven intense sediment transport, implying its potential use in practical scenarios in which the two kinds of flows exist simultaneously, such as landslides triggered by storm in shallow sea and flows resulted in barrier or dam breaks.

Development of a new multiphase sediment transport model for free surface flows

Modeling of sediment transport in estuaries and coastal areas requires a lot of compromises to keep the computational costs within acceptable limits. Due to that, existing sediment transport models do not account for particle-scale physics, e.g. particle-particle interaction and turbulence modulation by sediment, which play a significant role, especially in the non-dilute regime. In the current study, a newly developed physics-based sediment transport model for free surface flows and its numerical implementation within the OpenFOAM framework is introduced. The new model is based on the multiphase mixture theory to account for interactions between sediment and water while tracking the free surface at the same time. A modified VOF equation for sediment-laden free surface flow was derived and implemented. The interphase momentum transfer is considered by solving an additional closure for the slip velocity which includes the effects of drag force, turbulent dispersion, and shear-induced diffusion. Dense granular flow rheology is used to supply the required closures for particle stresses. Additionally, suitable closures for the mixture and turbulent viscosities are introduced. The model was validated using experimental data and analytical solutions of five test cases of variable complexity. This includes pure sedimentation, laminar bedload transport, turbulent sheet flow, local scour due to a submerged jet, and wave-induced scour under a submarine pipeline.

Quantitative comparison of hydrodynamic and elastoplastic approaches for modeling granular flow in silo

Hydrodynamic and elastoplastic theories are two commonly used continuum approaches for modeling granular materials. Here, the elastoplastic approach is extended to incorporate the rheology of dense granular flow, which therefore allows a quantitative comparison with the hydrodynamic approach under the same setting of rheological laws, material parameters, and numerical method. The flow patterns yielded by two approaches are apparently similar and the discharge rates are close. Yet the elastoplastic approach creates a narrow dome‐like flow zone in contrast to the wide cone‐like flow zone generated by the hydrodynamic approach. The shear localization is also less prominent in elastoplastic modeling owing to the existence of elastic deformation. The stresses predicted by two approaches match well in flow zones but show significant differences in stagnant zones. The proposed elastoplastic approach incorporating flow rheology can be used generally in both solid and fluid states of granular materials. © 2019 American Institute of Chemical Engineers AIChE J, 65: e16533, 2019

Regularization by compressibility of the μ(I) model of dense granular flow

The following article deals with the role of compressibility in regularizing the well-known μ(I) model, i.e., eliminating the short-wavelength (Hadamard) instability revealed by Barker et al. [“Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow,” J. Fluid Mech. 779, 794–818 (2015)]. In particular, we discuss the compressible-flow models proposed in the recent papers by Heyman et al. [“Compressibility regularizes the μ(I)-rheology for dense granular flows,” J. Fluid Mech. 830, 553–568 (2017)] and Barker et al. [“Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology,” Proc. R. Soc. A 473(2201), 20160846 (2017)]. In addition to a critique of certain aspects of their proposed constitutive models, we show that the main effect of their regularizations is to add viscous effects to the shear response in a way that appears unfortunately to eliminate quasi-static yield stress. Another goal of the present work is to show how the development and analysis of visco-plastic constitutive relations are facilitated by dissipation potentials and the dissipative analog of elastic potentials. We illustrate their utility in Sec. IV of this article, where it is shown that a constant non-zero yield stress leads to loss of convexity that can only be restored by substituting viscous effects or else by adding spatial-gradient effects proposed previously by the present authors [Goddard, J. and Lee, J., “On the stability of the μ(I) rheology for granular flow,” J. Fluid Mech. 833, 302–331 (2017)].

Rheology of dense granular flows in two dimensions: Comparison of fully two-dimensional flows to unidirectional shear flow

This work utilizes soft-particle discrete element simulations to examine the rheology of steady two-dimensional granular flows with reference to a unidirectional shear flow, which has been extensively employed for validating the local visco-plastic model of Jop et al. [Nature 441, 727--730 (2006)]. The $\mu$-$I$ scaling proposed by Jop et al. is found to be valid in both two-dimensional and unidirectional flows, as observed in previous studies, however, each flow type results in a different curve. Here $\mu$, ratio of the shear stress magnitude to the pressure, is the friction coefficient and $I$ is the dimensionless inertial number, which is proportional to the ratio of the magnitude of the rate of strain tensor, $\dot{\gamma}$, to the square root of the pressure. The friction coefficient is found not to scale in a simple way with the flow classification parameter $\psi$, which characterizes the local flow type. All the data collapse to a single curve using the scaling proposed by Zhang and Kamrin [Phys. Rev. Lett. 118, 058001 (2017)], in which the scaled granular fluidity ($f=1/(\mu T)$, where $T \propto u/\dot{\gamma}$ and $u$ is the fluctuation velocity) is found to depend only on the solid fraction $\phi$. The data for variation of $\phi$ with inertial number $I$ collapse to a single curve for all the flows.

Nonlocal rheology of dense granular flow in annular shear experiments.

The flow of dense granular materials at low inertial numbers cannot be fully characterized by local rheological models; several nonlocal rheologies have recently been developed to address these shortcomings. To test the efficacy of these models across different packing fractions and shear rates, we perform experiments in a quasi-2D annular shear cell with a fixed outer wall and a rotating inner wall, using photoelastic particles. The apparatus is designed to measure both the stress ratio μ (the ratio of shear to normal stress) and the inertial number I through the use of a torque sensor, laser-cut leaf springs, and particle-tracking. We obtain μ(I) curves for several different packing fractions and rotation rates, and successfully find that a single set of model parameters is able to capture the full range of data collected once we account for frictional drag with the bottom plate. Our measurements confirm the prediction that there is a growing lengthscale at a finite value μs, associated with a frictional yield criterion. Finally, we newly identify the physical mechanism behind this transition at μs by observing that it corresponds to a drop in the susceptibility to force chain fluctuations.

Rheology of dense granular flows for elongated particles.

We study the rheology of dense granular flows for frictionless spherocylinders by means of 3D numerical simulations. As in the case of spherical particles, the effective friction μ is an increasing function of the inertial number I, and we systematically investigate the dependence of μ on the particle aspect ratio Q, as well as that of the normal stress differences, the volume fraction, and the coordination number. We show in particular that the quasistatic friction coefficient is nonmonotonic with Q: from the spherical case Q=1, it first sharply increases, reaches a maximum around Q≃1.05, and then gently decreases until Q=3, passing its initial value at Q≃2. We provide a microscopic interpretation for this unexpected behavior through the analysis of the distribution of dissipative contacts around the particles: as compared to spheres, slightly elongated grains enhance contacts in their central cylindrical band, whereas at larger aspect ratios particles tend to align and dissipate by preferential contacts at their hemispherical caps.

SedFoam-2.0: a 3-D two-phase flow numerical model for sediment transport

Abstract. In this paper, a three-dimensional two-phase flow solver, SedFoam-2.0, is presented for sediment transport applications. The solver is extended from twoPhaseEulerFoam available in the 2.1.0 release of the open-source CFD (computational fluid dynamics) toolbox OpenFOAM. In this approach the sediment phase is modeled as a continuum, and constitutive laws have to be prescribed for the sediment stresses. In the proposed solver, two different intergranular stress models are implemented: the kinetic theory of granular flows and the dense granular flow rheology μ(I). For the fluid stress, laminar or turbulent flow regimes can be simulated and three different turbulence models are available for sediment transport: a simple mixing length model (one-dimensional configuration only), a k − ε, and a k − ω model. The numerical implementation is demonstrated on four test cases: sedimentation of suspended particles, laminar bed load, sheet flow, and scour at an apron. These test cases illustrate the capabilities of SedFoam-2.0 to deal with complex turbulent sediment transport problems with different combinations of intergranular stress and turbulence models.

Rheology of granular flows across the transition from soft to rigid particles

The rheology of dense granular flows is often seen as dependent on the nature of the energy landscape defining the modes of energy relaxation under shear. We investigate numerically the transition from soft to rigid particles, varying $S$, their stiffness compared to the confining pressure over three decades and the inertial number $I$ of the shear flow over five decades. We show that the rheological constitutive relation, characterized by a dynamical friction coefficient of the form $\mu(I)=\mu_c + a I^{\alpha}$ is marginally affected by the particle stiffness, with constitutive parameters being essentially dependent on the inter-particle friction. Similarly, the distribution of local shear rate mostly depends on the inertial number $I$, which shows that the characteristic timescale of plastic events is primarily controlled by the confining pressure and is insensitive to $S$. By contrast, the form under which energy is stored between these events, but also the contact network properties such as the coordination number and the distance to isostaticity, are strongly affected by stiffness, allowing us to discuss the different regimes in the $(S,I)$ phase space.

A comprehensive two-phase flow model for unidirectional sheet-flows

ABSTRACTIn this paper a comprehensive one-dimensional two-phase flow model for sediment transport under sheet-flow conditions is presented. The model is based on the dense granular flow rheology for the intergranular stresses and on a mixing length model for the Reynolds shear stress. A drift velocity model is introduced to represent the vertical dispersion effects. Recent experimental data are used to improve the mixing length model and the numerical model is validated for four configurations. The model is further assessed by comparing the dependence of the sediment transport rate and the sheet layer thickness on the Shields number. The good agreement with experimental data confirms the capability of the model formulation to reproduce unidirectional sheet-flows. The agreement with the latest kinetic theory of granular flows based model demonstrate that the dense granular flow rheology can be used as an alternative approach to kinetic theory in two-phase flow models.