Relaxation-type nonlocal inertial-number rheology for dry granular flows

Lee, Keng-Lin; Yang, Feng

doi:10.1103/physreve.96.062909

Cited by 18 publications

(21 citation statements)

References 54 publications

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“…This observation may be important in the development of new constitutive equations for granular flows that go beyond the µ(I)-rheology (see e.g. Kamrin & Koval 2012;Bouzid et al 2013;Henann & Kamrin 2013;Kamrin & Henann 2015;Lee & Yang 2017). For example, second normal stress differences can result in variations in free surface height as a function of the downstream velocity (McElwaine et al 2012).…”

Section: Summary Of Resultsmentioning

confidence: 98%

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Self-channelisation and levee formation in monodisperse granular flows

2019

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Dense granular flows can spontaneously self-channelise by forming a pair of parallel-sided static levees on either side of a central flowing channel. This process prevents lateral spreading and maintains the flow thickness, and hence mobility, enabling the grains to run out considerably further than a spreading flow on shallow slopes. Since levees commonly form in hazardous geophysical mass flows, such as snow avalanches, debris flows, lahars and pyroclastic flows, this has important implications for risk management in mountainous and volcanic regions. In this paper an avalanche model that incorporates frictional hysteresis, as well as depth-averaged viscous terms derived from the $\unicode[STIX]{x1D707}(I)$-rheology, is used to quantitatively model self-channelisation and levee formation. The viscous terms are crucial for determining a smoothly varying steady-state velocity profile across the flowing channel, which has the important property that it does not exert any shear stresses at the levee–channel interfaces. For a fixed mass flux, the resulting boundary value problem for the velocity profile also uniquely determines the width and height of the channel, and the predictions are in very good agreement with existing experimental data for both spherical and angular particles. It is also shown that in the absence of viscous (second-order gradient) terms, the problem degenerates, to produce plug flow in the channel with two frictionless contact discontinuities at the levee–channel margins. Such solutions are not observed in experiments. Moreover, the steady-state inviscid problem lacks a thickness or width selection mechanism and consequently there is no unique solution. The viscous theory is therefore a significant step forward. Fully time-dependent numerical simulations to the viscous model are able to quantitatively capture the process in which the flow self-channelises and show how the levees are initially emplaced behind the flow head. Both experiments and numerical simulations show that the height and width of the channel are not necessarily fixed by these initial values, but respond to changes in the supplied mass flux, allowing narrowing and widening of the channel long after the initial front has passed by. In addition, below a critical mass flux the steady-state solutions become unstable and time-dependent numerical simulations are able to capture the transition to periodic erosion–deposition waves observed in experiments.

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Section: Summary Of Resultsmentioning

confidence: 98%

“…Kamrin & Koval 2012; Bouzid et al. 2013; Henann & Kamrin 2013; Kamrin & Henann 2015; Lee & Yang 2017). For example, second normal stress differences can result in variations in free surface height as a function of the downstream velocity (McElwaine et al.…”

Section: Discussionmentioning

confidence: 99%

Self-channelisation and levee formation in monodisperse granular flows

2019

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“…(2017) or a non-local rheology (Bouzid et al. 2013; Kamrin & Henann 2015; DeGiuli & Wyart 2017; Lee & Yang 2017) is needed to model slow flows close to or below yield.…”

Section: Discussionmentioning

confidence: 99%

“…Indeed the standard monotonically increasing µ(I)-rheology is also ill-posed at small inertial numbers, although it is possible to cure ill posedness by forcing the µ(I)-curve to pass through the origin and thereby include a creep state . This is, however, diametrically opposed to the form of the friction that seems necessary to model erosion-deposition phenomena (Edwards & Gray 2015;Edwards et al 2017) and one might imagine that another form of rheology, such as the compressible I-dependent rheology (CIDR) of or a non-local rheology (Bouzid et al 2013;Kamrin & Henann 2015;DeGiuli & Wyart 2017;Lee & Yang 2017) is needed to model slow flows close to or below yield.…”

Section: Discussionmentioning

confidence: 99%

Frictional hysteresis and particle deposition in granular free-surface flows

et al. 2019

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Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness $h_{stop}(\unicode[STIX]{x1D701})$ on a rough slope inclined at an angle $\unicode[STIX]{x1D701}$ to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle $\unicode[STIX]{x1D701}_{start}$, or the thickness is increased to $h_{start}(\unicode[STIX]{x1D701})>h_{stop}(\unicode[STIX]{x1D701})$. This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to $h_{stop}$. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness $h_{\ast }$) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness $h_{\ast }$ it produces a deposit close to $h_{stop}$ for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to $h_{stop}$ for all initial thicknesses. This insensitivity to the initial conditions implies that $h_{stop}$ is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the $h_{stop}$ and $h_{start}$ curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine $h_{stop}$ is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth $h_{stop}$.

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“…The above does not propose an independent order parameter with its own equation, but rather treats I itself as a fluidity-type field within a gradient expansion of the flow rule. If anİ term were added to account for unsteady cases (like in [64]), the model would claim direct diffusion of the I field. For constant ν ∼ d 2 , the form above has the primary feature that effective friction µ is reduced when neighboring material is flowing faster, and increased when neighboring material flows slower.…”

Section: I-gradient Modelmentioning

confidence: 99%

Non-locality in Granular Flow: Phenomenology and Modeling Approaches

Kamrin

2019

Front. Phys.

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This paper reviews the emergence of non-local flow phenomena in granular materials and discusses a range of models that have been proposed to integrate an intrinsic length-scale into granular rheology. The frameworks discussed include micro-polar modeling, kinetic theory, three particular order-parameter-based models, and strongly non-local integral-based models. An extensive commentary is included discussing the current capabilities of these existing models as well as their implementational ease, physical motivation, and breadth of predictive ability.

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Relaxation-type nonlocal inertial-number rheology for dry granular flows

Cited by 18 publications

References 54 publications

Self-channelisation and levee formation in monodisperse granular flows

Self-channelisation and levee formation in monodisperse granular flows

Frictional hysteresis and particle deposition in granular free-surface flows

Non-locality in Granular Flow: Phenomenology and Modeling Approaches

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