A two-layer approach to modelling the transformation of dilute pyroclastic currents into dense pyroclastic flows

Doyle, Emma E.H.; Hogg, Andrew J.; Mader, Heidy M

doi:10.1098/rspa.2010.0402

Cited by 20 publications

(9 citation statements)

References 64 publications

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“…The model has been generalised to two-dimensional flows over complex topography (Gray et al 1999;Mangeney-Castelnau et al 2003;Pudasaini & Hutter 2003;Bouchut & Westdickenberg 2004;Luca et al 2009) and has been widely used in the snow avalanche community for hazard zone mapping (Grigorian, Eglit & Iakimov 1967;Naaim et al 2004;Sampl & Zwinger 2004;Christen, Kowalski & Bartelt 2010;Fischer, Kowalski & Pudasaini 2012). Analogous theories have been developed for debris flows (Iverson 1997;Johnson et al 2012), pyroclastic flows (Pitman et al 2003;Mangeney et al 2007;Doyle, Hogg & Mader 2011) and landslides (Kuo et al 2009;Mangeney et al 2010).…”

Section: Introductionmentioning

confidence: 99%

Retrogressive failure of a static granular layer on an inclined plane

et al. 2019

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When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwardset al.(J. Fluid. Mech., vol. 823, 2017, pp. 278–315,J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.

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Section: Introductionmentioning

confidence: 99%

Retrogressive failure of a static granular layer on an inclined plane

et al. 2019

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“…Iverson 1997;, pyroclastic flows (e.g. Pitman et al 2003;Doyle, Hogg & Mader 2011) and landslides (e.g. Kuo et al 2009;Mangeney et al 2010).…”

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confidence: 99%

A depth-averaged -rheology for shallow granular free-surface flows

2014

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The µ(I)-rheology is a nonlinear viscous law, with a strain-rate invariant and pressure-dependent viscosity, that has proved to be effective at modelling dry granular flows in the intermediate range of the inertial number, I. This paper shows how to incorporate the rheology into depth-averaged granular avalanche models. To leading order, the rheology generates an effective basal friction, which is equivalent to a rough bed friction law. A depth-averaged viscous-like term can be derived by integrating the in-plane deviatoric stress through the avalanche depth, using pressure and velocity profiles from a steady-uniform solution to the full µ(I)-rheology. The resulting viscosity is proportional to the thickness to the three halves power, with a coefficient of proportionality that is angle dependent. When substituted into the depth-averaged momentum balance this term generates second-order derivatives of the depth-averaged velocity, which are multiplied by a small parameter. Its inclusion therefore represents a singular perturbation to the equations. It is shown that a granular front propagating down a rough inclined plane is completely unaffected by the rheology, but, discontinuities, which naturally develop in inviscid roll-wave solutions, are smoothed out. By comparison with existing experimental data, it is shown that the depth-averaged µ(I)-rheology accurately predicts the growth rate of spatial instabilities to steady-uniform flow, as well as the dependence of the cutoff frequency on the Froude number and inclination angle. This provides strong evidence that, in the steady-uniform flow regime, the predicted decrease in the viscosity with increasing slope is correct. Outside the range of angles where steady-uniform flows develop, the viscosity becomes negative, which implies that the equations are ill-posed. This is a signature of the ill-posedness of the full µ(I)-rheology at both high and low inertial numbers. The depth-averaged µ(I)-rheology therefore cannot be used outside the valid range of angles without additional regularization.

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“…Because of its simple coding and numerical stability, the AB model with an arbitrarily small ε is commonly used for simulations of gravity currents in many geophysical situations (e.g., Denlinger and Iverson 2004;Doyle et al 2007Doyle et al , 2008Doyle et al , 2011Larrieu et al 2006). This model would be applicable in simulating gravity currents with high values of ρ c /ρ a , such as debris flows (e.g., Denlinger and Iverson 2004).…”

Section: Applicability Of the Bc And Ab Modelsmentioning

confidence: 99%

“…Here, we follow Doyle et al (2011) for the basic formulation of the two-layer system, but apply the BC model to the dilute layer and the AB model to the dense layer. We also consider the effects of basal shear drag in the dense layer and entrainment of ambient air into the dilute layer (see Appendix for details).…”

Section: Geophysical Application To Pyroclastic Density Currentsmentioning

confidence: 99%

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A numerical shallow-water model for gravity currents for a wide range of density differences

Shimizu

Koyaguchi

Suzuki

2017

Prog. in Earth and Planet. Sci.

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Gravity currents with various contrasting densities play a role in mass transport in a number of geophysical situations. The ratio of the density of the current, ρ c , to the density of the ambient fluid, ρ a , can vary between 10 0 and 10 3 . In this paper, we present a numerical method of simulating gravity currents for a wide range of ρ c /ρ a using a shallow-water model. In the model, the effects of varying ρ c /ρ a are taken into account via the front condition (i.e., factors describing the balance between the driving pressure and the ambient resistance pressure at the flow front). Previously, two types of numerical models have been proposed to solve the front condition. These are referred to here as the Boundary Condition (BC) model and the Artificial Bed (AB) model. The front condition is calculated as a boundary condition at each time step in the BC model, whereas it is calculated by setting a thin artificial bed ahead of the front in the AB model. We assessed the BC and AB models by comparing their numerical results with the analytical results for a simple case of homogeneous currents. The results from the BC model agree well with the analytical results when ρ c /ρ a 10 2 , but the model tends to overestimate the speed of the front position when ρ c /ρ a 10 2 . In contrast, the AB model generates good approximations of the analytical results for ρ c /ρ a 10 2 , given a sufficiently small artificial bed thickness, but fails to reproduce the analytical results when ρ c /ρ a 10 2 . Therefore, we propose a numerical method in which the BC model is used for currents with ρ c /ρ a 10 2 and the AB model is used for currents with ρ c /ρ a 10 2 .

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A two-layer approach to modelling the transformation of dilute pyroclastic currents into dense pyroclastic flows

Cited by 20 publications

References 64 publications

Retrogressive failure of a static granular layer on an inclined plane

Retrogressive failure of a static granular layer on an inclined plane

A depth-averaged -rheology for shallow granular free-surface flows

A numerical shallow-water model for gravity currents for a wide range of density differences

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