32 research outputs found
Particle-size and -density segregation in granular free-surface flows
No full textWhen a mixture of particles, which differ in both their size and their density, avalanches downslope, the grains can either segregate into layers or remain mixed, dependent on the balance between particle-size and particle-density segregation. In this paper, binary mixture theory is used to generalize models for particle-size segregation to include density differences between the grains. This adds considerable complexity to the theory, since the bulk velocity is compressible and does not uncouple from the evolving concentration fields. For prescribed lateral velocities, a parabolic equation for the segregation is derived which automatically accounts for bulk compressibility. It is similar to theories for particle-size segregation, but has modified segregation and diffusion rates. For zero diffusion, the theory reduces to a quasilinear first-order hyperbolic equation that admits solutions with discontinuous shocks, expansion fans and one-sided semi-shocks. The distance for complete segregation is investigated for different inflow concentrations, particle-size segregation rates and particle-density ratios. There is a significant region of parameter space where the grains do not separate completely, but remain partially mixed at the critical concentration at which size and density segregation are in exact balance. Within this region, a particle may rise or fall dependent on the overall composition. Outside this region of parameter space, either size segregation or density segregation dominates and particles rise or fall dependent on which physical mechanism has the upper hand. Two-dimensional steady-state solutions that include particle diffusion are computed numerically using a standard Galerkin solver. These simulations show that it is possible to define a Péclet number for segregation that accounts for both size and density differences between the grains. When this Péclet number exceeds 10 the simple hyperbolic solutions provide a very useful approximation for the segregation distance and the height of rapid concentration changes in the full diffusive solution. Exact one-dimensional solutions with diffusion are derived for the steady-state far-field concentration.LH
A 2-DIMENSIONAL MODEL FOR THE DYNAMICS OF SEA-ICE
No full textThis paper develops a systematic analysis of a sea ice pack viewed as a thin layer of coherent ice floes and open water regions at the ocean surface. The pack is driven by wind stress and Coriolis force, with responsive water drag on the base of the floes. Integration of the mass and momentum balances through the layer thickness result in a two-dimensional theory for the interface between ocean and atmosphere. The theory is presented for a plane horizontal interface, but the construction is readily extended to a non-planar interface. An interacting continua framework is adopted to describe the layer mixture of ice and water, which introduces the layer thickness h and ice area fraction A as smoothly varying functions of the plane coordinate and time, on a pack length scale and weather system timescale. It is shown how an evolution equation for A which ignores ridging can lead to the area fraction exceeding unity in maintained converging flow, which is physically invalid. This is a feature and weakness of current models, and is eliminated by artificial cut-off in numerical treatments. Here we formulate a description of the ridging process which redistributes smoothly the excess horizontal ice flux into increasing thickness of a ridging zone of area fraction Ar, and a simple postulate for the vertical ridging flux yields an evolution equation for A which shows how A can approach unity asymptotically, but not exceed unity, in a maintained converging flow. This is a significant feature of the new model, and eliminates a serious physical and numerical flaw in existing models. The horizontal momentum balance involves the gradients of the extra stress integrated through the layer thickness, extra to the integrated water pressure over the depth of a local floe edge below sea level. These extra stresses are zero in diverging flow and arise as a result of interactions between floes during converging flow. It is shown precisely how a mean stress in a floe is determined by such edge tractions, and in turn provides an interpretation of the local extra stress in the pack. The interpretation introduces the further model function f(A) which defines the fraction of ice-ice contact length over the boundary of a floe, describing an increase of the contact fraction as A increases. Model interaction mechanisms then suggest a qualitative law for the pack stress in terms of relative motions of the floes which define the pack-scale strain rates. A simple viscous law is presented for illustration, but it is shown that even this simple model can reflect a conventional motion of a failure criterion on the stresses in a ridging zone where the convergence greatly exceeds a threshold value. We have therefore defined precisely the two-dimensional ice pack stress arising in the momentum balance, and determined its relation to the contact forces between adjacent floes. The foregoing analyses hinge on the introduction of dimensionless variables and coordinate scalings which reflect the orders of magnitude of the many physical variables and their gradients in both individual floe and ice pack motions. A variety of small dimensionless parameters arise, which allows the derivation of leading-order equations defining a reduced model which describes the major balances in the motion. The distinct equations for diverging and converging flow regions indicates the existence of moving boundaries (in the two-dimensional pack domain) in the flow, satisfying appropriate matching conditions to be determined as part of the complete evolution. This feature appears to have been ignored in previous treatments. Here we illustrate the evolution of a moving boundary by constructing an exact solution to a one-dimensional pack motion which describes onshore drift due to increasing, then decreasing, wind stress. During the second phase a region of diverging flow expands from the free edge. The solution demonstrates the influence of various parameters, but, importantly, will provide a test solution for numerical algorithms which must be constructed to determine more complex one and two-dimensional motions
A 2-DIMENSIONAL MODEL FOR THE DYNAMICS OF SEA-ICE
No full textThis paper develops a systematic analysis of a sea ice pack viewed as a thin layer of coherent ice floes and open water regions at the ocean surface. The pack is driven by wind stress and Coriolis force, with responsive water drag on the base of the floes. Integration of the mass and momentum balances through the layer thickness result in a two-dimensional theory for the interface between ocean and atmosphere. The theory is presented for a plane horizontal interface, but the construction is readily extended to a non-planar interface. An interacting continua framework is adopted to describe the layer mixture of ice and water, which introduces the layer thickness h and ice area fraction A as smoothly varying functions of the plane coordinate and time, on a pack length scale and weather system timescale. It is shown how an evolution equation for A which ignores ridging can lead to the area fraction exceeding unity in maintained converging flow, which is physically invalid. This is a feature and weakness of current models, and is eliminated by artificial cut-off in numerical treatments. Here we formulate a description of the ridging process which redistributes smoothly the excess horizontal ice flux into increasing thickness of a ridging zone of area fraction Ar, and a simple postulate for the vertical ridging flux yields an evolution equation for A which shows how A can approach unity asymptotically, but not exceed unity, in a maintained converging flow. This is a significant feature of the new model, and eliminates a serious physical and numerical flaw in existing models. The horizontal momentum balance involves the gradients of the extra stress integrated through the layer thickness, extra to the integrated water pressure over the depth of a local floe edge below sea level. These extra stresses are zero in diverging flow and arise as a result of interactions between floes during converging flow. It is shown precisely how a mean stress in a floe is determined by such edge tractions, and in turn provides an interpretation of the local extra stress in the pack. The interpretation introduces the further model function f(A) which defines the fraction of ice-ice contact length over the boundary of a floe, describing an increase of the contact fraction as A increases. Model interaction mechanisms then suggest a qualitative law for the pack stress in terms of relative motions of the floes which define the pack-scale strain rates. A simple viscous law is presented for illustration, but it is shown that even this simple model can reflect a conventional motion of a failure criterion on the stresses in a ridging zone where the convergence greatly exceeds a threshold value. We have therefore defined precisely the two-dimensional ice pack stress arising in the momentum balance, and determined its relation to the contact forces between adjacent floes. The foregoing analyses hinge on the introduction of dimensionless variables and coordinate scalings which reflect the orders of magnitude of the many physical variables and their gradients in both individual floe and ice pack motions. A variety of small dimensionless parameters arise, which allows the derivation of leading-order equations defining a reduced model which describes the major balances in the motion. The distinct equations for diverging and converging flow regions indicates the existence of moving boundaries (in the two-dimensional pack domain) in the flow, satisfying appropriate matching conditions to be determined as part of the complete evolution. This feature appears to have been ignored in previous treatments. Here we illustrate the evolution of a moving boundary by constructing an exact solution to a one-dimensional pack motion which describes onshore drift due to increasing, then decreasing, wind stress. During the second phase a region of diverging flow expands from the free edge. The solution demonstrates the influence of various parameters, but, importantly, will provide a test solution for numerical algorithms which must be constructed to determine more complex one and two-dimensional motions
A two-dimensional depth-averaged μ(I)-rheology for dense granular avalanches
Get PDFSteady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged μ(I)-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional μ(I)-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory
Granular Vacua
No full textA continuum model of a channelized, free-surface granular flow is developed to calculate the rate at which it expands into an initially grain-free region when lateral constraints are removed. The spreading is driven by cross-stream pressure gradients and resisted by basal drag. The boundary between the granular vacuum and the flowing grains is elucidated both in the near and far fields
Granular Vacua
No full textA continuum model of a channelized, free-surface granular flow is developed to calculate the rate at which it expands into an initially grain-free region when lateral constraints are removed. The spreading is driven by cross-stream pressure gradients and resisted by basal drag. The boundary between the granular vacuum and the flowing grains is elucidated both in the near and far fields
