Chris Johnson

Publications

My publications are also listed on Google Scholar, ORCID (0000-0003-2192-3616), Publons / ResearcherID (B-3163-2012), and on my University of Manchester Research Page.

2024
William Webb, Barbara Turnbull and Chris Johnson (2024)
Continuum modelling of a just-saturated inertial column collapse: Capturing fluid-particle interaction,
Granular Matter 26(21) doi:10.1007/s10035-023-01391-2
[Show abstract] [PDF ] This work presents a simple two-phase flow model to analyse a series of axisymmetric granular column collapse tests conducted under elevated gravitational accelerations. These columns were prepared with a just-saturated condition, where the granular pores were filled with a Newtonian fluid up to the column’s free surface. In this configuration, unlike the fully submerged case, air-water-grain contact angles may be important to flow dynamics. The interaction between a Newtonian fluid phase and a monodispersed inertial particle phase was captured by an inter-phase interaction term that considers the drag between the two phases as a function of the particle phase porosity. While this experimental setup has broad applications in understanding various industrial processes and natural phenomena, the focus of this study is on its relevance to predicting the motion of debris flows. Debris flows are challenging to model due to their temporally evolving composition, which can lead to the development of complex numerical models that become intractable. The developed numerical scheme in this study reasonably reproduces the particle-size and gravitational acceleration dependencies observed within the experimental runout and basal fluid pressure dissipation data. However, discrepancies between the model and physical experiments primarily arise from the assumption of modelling the granular phase as a continuum, which becomes less appropriate as particle size increases.
X. Meng, A. M. Taylor-Noonan, C. G. Johnson, W. A. Take, E. T. Bowman and J. M. N. T. Gray (2024)
Granular-fluid avalanches: The role of vertical structure and velocity shear,
Journal of Fluid Mechanics 980, A11, doi:10.1017/jfm.2023.1023
[Show abstract] [PDF ] Field observations of debris flows often show that a deep dry granular front is followed by a progressively thinner and increasingly watery tail. These features have been captured in recent laboratory flume experiments (Taylor-Noonan et al., J. Geophys. Res.: Earth Surf., vol. 127, 2022, e2022JF006622). In these experiments different initial release volumes were used to investigate the dynamics of an undersaturated monodisperse grain–water mixture as it flowed downslope onto a horizontal run-out pad. Corresponding dry granular flows, with the same particle release volumes, were also studied to show the effect of the interstitial fluid. The inclusion of water makes debris flows much more mobile than equivalent volumes of dry grains. In the wet flows, the formation of a dry front is crucially dependent on the heterogeneous vertical structure of the flow and the velocity shear. These effects are included in the depth-averaged theory of Meng et al. (J. Fluid Mech., vol. 943, 2022, A19), which is used in this paper to quantitatively simulate both the wet and dry experimental flows using a high-resolution shock-capturing scheme. The results show that velocity shear causes dry grains (located near the free surface) to migrate forwards to create a dry front. The front is more resistant to motion than the more watery material behind, which reduces the overall computed run-out distance compared with debris-flow models that assume plug flow and develop only small dry snouts. Velocity shear also implies that there is a net transport of water to the back of the flow. This creates a thin oversaturated tail that is unstable to roll waves in agreement with experimental observations.
E. Maguire, T. Barker. M. Rauter, C. G. Johnson and J. M. N. T. Gray (2024)
Particle-size segregation patterns in a partially filled triangular rotating drum,
Journal of Fluid Mechanics 979, A40, doi:10.1017/jfm.2023.1022
[Show abstract] [PDF ] In this paper a fully coupled particle-size segregation model for granular flows (Barker et al., J. Fluid Mech., vol. 909, 2021, p. A22) is used to simulate the development of the patterns in a triangular rotating drum. The results are compared with the experimental patterns formed with bidisperse and tridisperse granular mixtures, and with varying compositions and fill heights. In all cases the agreement between the simulations and experiments is remarkably good. The experimental patterns are generated in a narrow gap between transparent front and back sidewalls. These prevent three-dimensional motion, but also impose friction on the flow, making it thinner and faster than it would otherwise be. This promotes segregation, as it simultaneously increases the shear rate and reduces the local pressure. To obtain the correct flow dynamics and segregation, width-averaged sidewall friction is incorporated into the two-dimensional simulations, which are performed in OpenFOAM. The free-surface avalanche forms a boundary layer within which all the segregation occurs. Material in the lower reach of the avalanche is continuously deposited into an underlying solid body of grains, which rotates with the drum, and is eventually re-entrained into the avalanche along its upper reach. The changing geometry of the granular region (as the drum rotates) implies that the avalanche is constantly adjusting its length, position and depth. This generates a complex quasi-periodic flow, which when combined with particle-size segregation generates amazing patterns in the solid rotating granular body after only two drum rotations.
2023
Shresht Jain, Finn Box, Chris Johnson and Draga Pihler-Puzović (2023)
Material nonlinearities yield doubly negative holey metamaterials,
Extreme Mechanics Letters, doi:10.1016/j.eml.2023.102065
[Show abstract] [PDF ] Mechanical metamaterials with negative stiffness and negative Poisson’s ratio are exciting prospects for advanced material design. A plastic column with a series of periodically-spaced holes exhibits both of these properties when buckled under compression. In this paper, the behaviour of such a column under compression is measured experimentally and described with a simple mathematical model. This model predicts the compression, buckling and post-buckling behaviour of the entire column from the mechanical response of the thin ligaments of material that form the column’s microarchitecture to compression, rotation and shear forces, which are characterised experimentally, The softening behaviour in holey columns beyond the critical level of compression for pattern transformation is shown to be due to material constitutive nonlinearities in the rotation and shear response of the microarchitecture. Geometric perturbations to the columns can cause the observed pattern to change, but result in approximately the same force–displacement measurements as for the column with perfect geometry. This approach provides a useful framework to study systems where both geometric and material nonlinearities underpin observed phenomena.
F. J. Millward, H. N. Webster and C. G. Johnson (2023)
Modeling Wind-Blown Umbrella Clouds in Lagrangian Dispersion Models,
Journal of Geophysical Research: Atmospheres 128 (17), doi:10.1029/2023JD039115
[Show abstract] [PDF ] The ash and gas released by large explosive volcanic eruptions rises to its neutral buoyancy level in the atmosphere, then spreads laterally to form an umbrella cloud. Density stratification of the atmosphere generates buoyancy forces in the cloud, which drive the outward spread. Although umbrella clouds are often modeled as circular axisymmetric structures, in practice they are usually influenced quite strongly by the meteorological wind, with spread in the upwind direction halted by the oncoming wind, and different rates of spreading in the downwind and crosswind directions. In this work, we derive a simple parametrization of non-axisymmetric umbrella cloud spreading from a much more complex physically based shallow-layer intrusion model. The new parametrization is quick to evaluate and so is suitable for use in operational Volcanic Ash Transport and Dispersion Models (VATDMs). In contrast to previous parametrizations, in which there is assumed to be no interaction between a circular umbrella cloud and the meteorological wind, here the umbrella cloud is influenced by the wind and adopts a shape determined by the balance of buoyant spreading and downwind drag forces. We apply the new scheme to four historical case studies of eruptions at Puyehue 2011, Pinatubo 1991, Ulawun 2019, and Calbuco 2015. The results are compared with VATDM simulations using a conventional circular umbrella cloud parametrization. Using the new scheme, good descriptions of cloud spread are recovered and the prediction of horizontal ash distribution is improved relative to the axisymmetric parametrization.
Ben Esse, Mike Burton, Catherine Hayer, Rodrigo Contreras-Arratia, Thomas Christopher, Erouscilla P Joseph, Matthew Varnam and Chris Johnson (2023)
SO2 emissions during the 2021 eruption of La Soufrière St. Vincent, revealed with back-trajectory analysis of TROPOMI imagery,
Geological Society of London, Special Publication "The 2020-21 Eruption of La Soufriere Volcano, St. Vincent", doi:10.1144/SP539-2022-77
[Show abstract] [PDF ] Determining SO₂ emission time series from explosive eruptions can provide important insights into the driving magmatic processes, however accurate measurements are difficult to collect. Satellite-based platforms provide SO₂ imagery, however translating this to the altitude- and time-resolved emission history required to unravel volcanic processes is a major challenge. This means SO₂ emission time series are rarely quantified for major eruptions, producing a gap in our understanding of explosive volcanism.

Here, we combine SO₂ imagery collected by the TROPOspheric Monitoring Instrument (TROPOMI) with PlumeTraj, a back-trajectory analysis toolkit, to reconstruct the SO₂ emission prior to, and during, the explosive eruption of La Soufrière volcano, St. Vincent, in April 2021. Precursory SO₂ emissions were quantified the day before the eruption, with emission rates in agreement with ground-based measurements. We estimate initial magma sulphur contents by comparing the measured SO₂ emissions with erupted magma volumes, finding that the initial explosion was sulphur poor (730 ppm S) compared to the main eruption phase (up to 3400 ppm S). This suggests that the initial explosion cleared old, previously degassed magma resident in the shallow plumbing system, followed by the eruption of the main, mostly un-degassed magma source.
A.N. Edwards, F.M. Rocha, B.P. Kokelaar, C.G. Johnson, J.M.N.T. Gray (2023)
Particle-size segregation in self-channelized granular flows,
Journal of Fluid Mechanics 955, A38, doi:10.1017/jfm.2022.1089
[Show abstract] [PDF ] Geophysical mass flows such as debris flows, dense pyroclastic flows and snow avalanches can self-channelize on shallow slopes. The confinement afforded by formed levees helps to maintain the flow depth, and hence mobility, allowing self-channelized flows to run out significantly farther than unconfined, spreading flows. Levee formation and self-channelization are strongly associated with particle-size segregation, but can also occur in monodisperse flows. This paper uses the monodisperse depth-averaged theory of Rocha et al. (Journal of Fluid Mechanics, vol. 876, 2019, pp. 591–641), which incorporates a hysteretic friction law and second-order depth-averaged viscous terms. Both of these are vital for the formation of a travelling wave that progressively deposits a pair of levees just behind the front. The three-dimensional velocity field is reconstructed in a frame moving with the front assuming Bagnold flow. This enables a bidisperse particle-size segregation theory to be used to solve for the large and small particle concentrations and particle paths in three-dimensions, for the first time. The model shows that the large particles tend to segregate to the surface of the flow, forming a carapace that extends over the centre of the channel, as well as along the external sides and base of the levee walls. The small particles segregate downwards, and are concentrated in the main channel and in the inner levee walls. This supports the contention that a low-friction channel lining provides a secondary mechanism for run-out enhancement. It is also shown that the entire theory scales with particle diameter, so experiments with millimetre-sized particles provide important insights into geophysical-scale flows with boulders and smaller rock fragments. The model shows that self-channelization does not need particle-size segregation to occur, but supports the hypothesis that particle-size segregation and the associated frictional feedback can significantly enhance both the flow mobility and the levee strength.
2022
Xiannan Meng, C.G. Johnson and J.M.N.T. Gray (2022)
Formation of dry granular fronts and watery tails in debris flows,
Journal of Fluid Mechanics 943, A19, doi:10.1017/jfm.2022.400
[Show abstract] [PDF ] Debris flows are particle–fluid mixtures that pose a significant hazard to many communities throughout the world. Bouldery debris flows are often characterised by a deep dry granular flow front, which is followed by a progressively thinner and increasingly watery tail. The formation of highly destructive bouldery wave fronts is usually attributed to particle-size segregation. However, the moving-bed flume experiments of Davies (N. Z. J. Hydrol., vol. 29, 1990, pp. 18–46) show that discrete surges with dry fronts and watery tails also form in monodisperse particle–fluid mixtures. These observations motivate the development of a new depth-averaged mixture theory for debris flows, which explicitly takes account of the differing granular and phreatic surfaces, velocity shear, and relative motion between grains and fluid to explain these phenomena. The theory consists of four coupled conservation laws that describe the spatial and temporal evolution of the grain and water thicknesses and depth-averaged velocities. This system enables travelling wave solutions to be constructed that consist of (i) a large amplitude dry flow front that smoothly transitions to (ii) an undersaturated body, (iii) an oversaturated region and then (iv) a pure water tail. It is shown that these solutions are in good quantitative agreement with Davies’ experiments at high bed speeds and slope inclinations. At lower bed speeds and inclinations, the theory produces travelling wave solutions that connect to a steady-uniform upstream flow, and may or may not have a bulbous flow front, consistent with Davies’ observations.
2021
C. Tregaskis, C.G. Johnson, X. Cui and J.M.N.T. Gray (2021)
Subcritical and supercritical flow around an obstacle on a rough inclined plane,
Journal of Fluid Mechanics 933, A25, doi:10.1017/jfm.2021.1074
[Show abstract] [PDF ] A blunt obstacle in the path of a rapid granular avalanche generates a bow shock (a jump in the avalanche thickness and velocity), a region of static grains upstream of the obstacle, and a grain-free region downstream. Here, it is shown that this interaction is qualitatively altered if the incline on which the avalanche is flowing is changed from smooth to rough. On a rough incline, the friction between the grains and the incline depends on the flow thickness and speed, which allows both rapid (supercritical) and slow (subcritical) steady uniform avalanches. For supercritical experimental flows, the material is diverted around a blunt obstacle by the formation of a bow shock and a static dead zone upstream of the obstacle. Downslope, a grain-free vacuum region forms, but, in contrast to flows on smooth beds, static levees form at the boundary between the vacuum region and the flow. In slower, subcritical, flows the flow is diverted smoothly around the dead zone and the obstacle without forming a bow shock. After the avalanche stops, signatures of the dead zone, levees and (in subcritical flows) a deeper region upslope of the obstacle are frozen into the deposit. To capture this behaviour, numerical simulations are performed with a depth-averaged avalanche model that includes frictional hysteresis and depth-averaged viscous terms, which are needed to accurately model the flowing and deposited regions. These results may be directly relevant to geophysical mass flows and snow avalanches, which flow over rough terrain and may impact barriers or other infrastructure.
Robbie S. J. Bancroft and Chris G. Johnson (2021)
Drag, diffusion and segregation in inertial granular flows,
Journal of Fluid Mechanics 924, A3, doi:10.1017/jfm.2021.560
[Show abstract] [PDF ] Inter-species drag forces in granular flows play a central role in setting the speed and extent of segregation, a process that separates grains of different size or density. Here, we study this drag force in detail, using a novel configuration of discrete element simulations that allows us to completely characterise the drag in inertial granular flows by studying it in a uniform environment. By applying opposing forces to grains in monodisperse and size-bidisperse shear flows, we show that the strength of the drag force scales as I^−7/4, where I is the granular inertial number, and propose a model that explains this scaling by relating the strength of drag to grain velocity fluctuations. These findings suggest that much of the previously observed dependence of the segregation rate on the local shear rate and pressure in dense free-surface flows is due to variation in the strength of the inter-species drag, rather than the strength of forces that drive segregation.
P. Gajjar, C. Johnson, J. Carr, K. Chrispeels, J.M.N.T. Gray and P. Withers (2021)
Size segregation of irregular granular materials captured by time-resolved 3D imaging,
Scientific Reports 11 8352, doi:10.1038/s41598-021-87280-1
[Show abstract] [PDF ] When opening a box of mixed nuts, a common experience is to find the largest nuts at the top. This well-known effect is the result of size-segregation where differently sized ‘particles’ sort themselves into distinct layers when shaken, vibrated or sheared. Colloquially this is known as the ‘Brazil-nut effect’. While there have been many studies into the phenomena, difficulties observing granular materials mean that we still know relatively little about the process by which irregular larger particles (the Brazil nuts) reach the top. Here, for the first time, we capture the complex dynamics of Brazil nut motion within a sheared nut mixture through time-lapse X-ray Computed Tomography (CT). We have found that the Brazil nuts do not start to rise until they have first rotated sufficiently towards the vertical axis and then ultimately return to a flat orientation when they reach the surface. We also consider why certain Brazil nuts do not rise through the pack. This study highlights the important role of particle shape and orientation in segregation. Further, this ability to track the motion in 3D will pave the way for new experimental studies of segregating mixtures and will open the door to even more realistic simulations and powerful predictive models. Understanding the effect of size and shape on segregation has implications far beyond food products including various anti-mixing behaviors critical to many industries such as pharmaceuticals and mining.
Andrew Edwards, Sylvain Viroulet, Chris Johnson and Nico Gray (2021)
Erosion-deposition dynamics and long distance propagation of granular avalanches,
Journal of Fluid Mechanics 915 A9, doi:10.1017/jfm.2021.34
[Show abstract] [PDF ] The net erosion-deposition rate of an avalanche is fundamental to its dynamics and in determining its growth or decay. Small-scale experiments are performed by releasing a given volume of yellow sand onto a stationary erodible red sand layer on a rough inclined plane. Depending on the erodible layer depth and the slope angle, the avalanche is found to either decay, grow, propagate steadily or rapidly shed grains to produce secondary avalanches. The use of different coloured sand with identical properties shows that a particle exchange occurs, which eventually results in a flow that is comprised entirely of particles from the stationary layer rather than the initial release. It is notoriously difficult to model the erosion and deposition processes in granular flows, but it is shown that a two-dimensional depth-averaged avalanche model, with a hysteretic basal friction law, can reproduce all of the observed behaviours. The results illustrate how a continuous exchange of particles with the substrate layer is fundamentally important to the propagation of such avalanches. An investigation into long distance propagation behaviour reveals that avalanches can reach a steady state, the size and speed of which are independent of the initially released volume. In certain conditions avalanches can grow to steady states that are significantly more massive than the flows from which they are originally formed. This paper demonstrates the importance of correctly including erosion-deposition in operational forecast models of snow avalanches and other geophysical mass flows.
Sylvain Viroulet, Chris Johnson and Nico Gray (2021)
Modelling erosion and deposition in geophysical granular mass flows,
Europhysics News 52/1, 29–32, doi:10.1051/epn/2021106
(Magazine article)
[Show abstract] [PDF ] During hazardous geophysical mass flows, such as rock or snow avalanches, debris flows and volcanic pyroclastic flows, a continuous exchange of material can occur between the slide and the bed. The net balance between erosion and deposition of particles can drastically influence the behaviour of these flows. Recent advances in describing the non-monotonic effective basal friction and the internal granular rheology in depth averaged theories have enabled small scale laboratory experiments to be quantitatively reproduced and can also be implemented in large scale models to improve hazard mitigation.
Thomas Barker, Matthas Rauter, Eoin Maguire, Chris Johnson and Nico Gray (2021)
Coupling rheology and segregation in granular flows,
Journal of Fluid Mechanics 909 A22, doi:10.1017/jfm.2020.973
[Show abstract] [PDF ] During the last fifteen years there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the 𝜇(𝐼)-rheology (where 𝜇 is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This paper details theoretical and numerical frameworks (based on OpenFOAM) which unify these currently disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible 𝜇(𝐼)-rheology, which is coupled to the gravity-driven segregation theory of Gray & Ancey (Journal of Fluid Mechanics, vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. Further inclusion of an excess air phase, which segregates away from the granular material, then allows the complex evolution of the free surface to be captured simultaneously. Three primary coupling mechanisms are identified: (i) advection of the particle concentrations by the bulk velocity, (ii) feedback of the particle-size and/or frictional properties on the bulk flow field and (iii) influence of the shear rate, pressure, gravity, particle size and particle-size ratio on the locally evolving segregation and diffusion rates. The numerical method is extensively tested in one-way coupled computations, before the fully coupled model is compared with the discrete element method simulations of Tripathi & Khakhar (Phys. Fluids, vol. 23, 2011, 113302) and used to compute the petal-like segregation pattern that spontaneously develops in a square rotating drum.
2020
Finn Box, Chris Johnson, Draga Pihler-Puzovic (2020)
Hard Auxetic Metamaterials,
Extreme Mechanics Letters (40) 100980, doi:10.1016/j.eml.2020.100980
[Show abstract] [PDF ] Auxetics are materials that contract laterally when compressed, rather than expand, in contrast to common experience. Here we show that common metals and plastics can be rendered auxetic through the introduction of a regular array of holes. Under compression, these hard holey materials bypass localized failure modes, such as shear banding, and instead deform via a global pattern transformation previously reported in elastomeric structures. Despite significant variations in internal structure, the pattern transformation responsible for auxetic behaviour in both metals and plastics is governed by the buckling of the slender struts that comprise the microarchitecture. Furthermore, in contrast to elastomeric structures, holey sheets made from hard materials exhibit significant negative post-buckling stiffness. This suggests that, beyond the geometrical nonlinearities associated with topological modifications, material nonlinearities which arise during plastic deformation offer further potential for altering the material properties of the constituent.
Luke N. Hepworth, J. Stephen Daly, Ralf Gertisser, Chris G. Johnson, C. Henry Emeleus & Brian O’Driscoll (2020)
Rapid crystallization of precious-metal-mineralized layers in mafic magmatic systems,
Nature Geoscience 13, 375–381, doi:10.1038/s41561-020-0568-3
[Show abstract] [PDF ][Article in The Times newspaper] The solidified remnants of mafic magmatic systems host the greatest concentrations of platinum-group metals in the Earth’s crust. Our understanding of precious-metal mineralization in these intrusive bodies is underpinned by a traditional view of magma chamber processes and crystal mush solidification. However, considerable uncertainty remains regarding the physical and temporal controls on concentrating these critical metals, despite their importance to modern society. We present high-precision ⁸⁷Sr/⁸⁶Sr analyses of plagioclase and clinopyroxene from within centimetre-thick precious-metal-enriched layers in the Palaeogene open-system Rum layered intrusion (northwest Scotland). Isotopic heterogeneity is present between plagioclase crystals, between clinopyroxene and plagioclase and within plagioclase crystals throughout the studied section. On the basis of these observations, we demonstrate that platinum-group element mineralization formed by repeated small-volume reactive melt percolation events. The preservation of strontium isotope heterogeneities at 10–100 µm length scales implies cooling of the melts that formed the precious-metal-rich layers occurred at rates greater than 1 °C per year, and cooling to diffusive closure within tens to hundreds of years. Our data highlight the importance of cyclic dissolution–recrystallization events within the crystal mush and raise the prospect that precious-metal-bearing mafic intrusions may form by repeated self-intrusion during cooling and solidification.
C. G. Johnson (2020)
Shocking Granular Flows,
Journal of Fluid Mechanics 890, doi:10.1017/jfm.2020.61 F1 (Invited Focus on Fluids review)
[Show abstract] [PDF ] When a lightning bolt darts across the sky, the thunderclap that reaches our ears a few seconds later is an example of a fluid dynamical shock: a wave across which flow properties such as pressure and density change almost discontinuously. In compressible fluids these shocks are associated with high-energy supersonic flows and so require specialist equipment to realise in steady state. But in granular media, shocks occur much more readily and at flow speeds easily obtainable in the laboratory. In the featured article, Khan et al. (Journal of Fluid Mechanics, vol. 884, 2020, R4) exploit this to explore a remarkable range of steady and oscillatory shocks and shock interactions, which demonstrate many of the unique rheological complexities of granular flow.
2019
F. M. Rocha, C. G. Johnson and J. M. N. T. Gray (2019)
Self-channelisation and levee formation in monodisperse granular flows,
Journal of Fluid Mechanics 876 591–641, doi:/10.1017/jfm.2019.518
[Show abstract] [PDF ] 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 𝜇(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.
A. N. Edwards, A. S. Russell, C. G. Johnson and J. M. N. T. Gray (2019)
Frictional hysteresis and particle deposition in granular free-surface flows,
Journal of Fluid Mechanics 875 1058–1095, doi:/10.1017/jfm.2019.517
[Show abstract] [PDF ] 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(𝜁) on a rough slope inclined at an angle 𝜁 to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle 𝜁_start , or the thickness is increased to h_start(𝜁) > h_stop(𝜁). 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. (Journal of Fluid Mechanics, 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_stop) 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_* 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 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 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.
S. Viroulet, A. N. Edwards, C. G. Johnson, B. P. Kokelaar and J. M. N. T. Gray (2019)
Shedding dynamics and mass exchange by dry granular waves flowing over erodible beds,
Earth Planet. Sci. Lett. 523, doi:/10.1016/j.epsl.2019.07.003
[Show abstract] [PDF ] A continuous exchange of particles between an erodible substrate and the granular flow above it occurs during almost all geophysical events involving granular material, such as snow avalanches, debris flows and pyroclastic flows. The balance between eroded and deposited material can drastically influence the runout distance and duration of the flow. In certain conditions, a perfect balance between erosion and deposition may occur, leading to the steady propagation of material, in which the flow maintains its shape and velocity throughout. It is shown experimentally how the erosion-deposition process in dense flows of sand (160-200 μm) on an erodible bed of the same material produces steadily propagating avalanches that deposit subtle levees at their lateral extent. Moreover, it is shown in this paper, by using two colours of the same sand, that although the avalanche is propagating at constant velocity and maintaining a constant shape, the grains that are initially released are deposited along the flow path and that the avalanche will eventually be composed entirely of particles that are eroded from the bed. Different steady travelling wave regimes are obtained depending on the slope angle, thickness of the erodible layer and the amount of material released. Outside of the range of parameters where steady travelling waves form, the avalanches loose mass and decay if the initial amount of material released is too small, or, if the initial release is too large, they re-adjust to a steadily propagating regime by shedding material and breaking into smaller avalanches at its rear side. Numerical simulations are performed using a shallow-water-like avalanche model together with a friction law that captures the erosion-deposition process in flowing to static regimes and a transport equation for the interface between layers of the two colours. The characteristic behaviours observed in the experiments are qualitatively reproduced. Specifically, the complex processes such as the exchange of particles leading to a change in colour of the avalanche and the formation of lateral levees are captured by the model. Finally a comparison is made with deposits in lunar craters, which are interpreted as closely analogous to the deposits formed in our laboratory experiments.
A. S. Russell, C. G. Johnson, A. N. Edwards, S. Viroulet, F. M. Rocha and J. M. N. T. Gray (2019)
Retrogressive failure of a static granular layer on an inclined plane,
Journal of Fluid Mechanics 869 313–340, doi:/10.1017/jfm.2019.215
[Show abstract] [PDF ] 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 Edwards et 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.
2018
P. Gajjar, Jakob. S. Jørgenson, Jose R. A. Godinho, Chris G. Johnson, A. Ramsey, P. J. Withers (2018)
New software protocols for enabling laboratory based temporal CT,
Review of Scientific Instruments 89 093702, doi:/10.1063/1.5044393
[Show abstract] [PDF ] Temporal micro computed tomography (CT) allows the non-destructive quantification of processes that are evolving over time in 3D. Despite the increasing popularity of temporal CT the practical implementation and optimisation can be difficult. Here, we present new software protocols that enable temporal CT using commercial laboratory CT systems. The first protocol drastically reduces the need for periodic intervention when making time-lapse experiments, allowing a large number of tomograms to be collected automatically. The automated scanning at regular intervals needed for uninterrupted time-lapse CT is demonstrated by analysing the germination of a mung bean (vigna radiata), whilst the synchronisation with an in-situ rig required for interrupted time-lapse CT is highlighted using a shear cell to observe granular segregation. The second protocol uses golden-ratio angular sampling with an iterative reconstruction scheme and allows the number of projections in a reconstruction to be changed as sample evolution occurs. This overcomes the limitation of the need to know a priori what the best time window for each scan is. The protocol is evaluated by studying barite precipitation within a porous column, allowing a comparison of spatial and temporal resolution of reconstructions with different numbers of projections. Both of the protocols presented here have great potential for wider application, including, but not limited to, in-situ mechanical testing, following battery degradation and chemical reactions.
K. van der Vaart, A. R. Thornton, C. G. Johnson, T. Weinhart, L. Jing, P. Gajjar, J. M. N. T. Gray, C. Ancey (2018)
Breaking size-segregation waves and mobility feedback in dense granular avalanches,
Granular Matter 20:46, doi:10.1007/s10035-018-0818-x
[Show abstract] [PDF ] Through experiments and discrete particle method (DPM) simulations we present evidence for the existence of a recirculating structure, that exists near the front of dense granular avalanches, and is known as a breaking size-segregation (BSS) wave. This is achieved through the study of three-dimensional bidisperse granular flows in a moving-bed channel. Particle-size segregation gives rise to the formation of a large-particle-rich front and a small-particle-rich tail with a BSS wave positioned between the tail and front. We experimentally resolve the structure of the BSS wave using refractive-index matched scanning and find that it is qualitatively similar to the structure observed in DPM simulations. Our analysis demonstrates a relation between the concentration of small particles in the flow and the amount of basal slip, in which the structure of the BSS wave plays a key role. This leads to a feedback between the mean bulk flow velocity and the process of particle-size segregation. Ultimately, these findings shed new light on the recirculation of large and small grains near avalanche fronts and the effects of this behaviour on the mobility of the bulk flow.
S. Viroulet, J. L. Baker, F. Rocha, C. G. Johnson, P. Kokelaar and J. M. N. T. Gray (2018)
The kinematics of bidisperse granular roll waves,
Journal of Fluid Mechanics 848, 836–875, doi:10.1017/jfm.2018.348
[Show abstract] [PDF ] Small perturbations to a steady uniform granular chute flow can grow as the material moves downslope and develop into a series of surface waves that travel faster than the bulk flow. This roll wave instability has important implications for the mitigation of hazards due to geophysical mass flows, such as snow avalanches, debris flows and landslides, because the resulting waves tend to merge and become much deeper and more destructive than the uniform flow from which they form. Natural flows are usually highly polydisperse and their dynamics is significantly complicated by the particle size segregation that occurs within them. This study investigates the kinematics of such flows theoretically and through small-scale experiments that use a mixture of large and small glass spheres. It is shown that large particles, which segregate to the surface of the flow, are always concentrated near the crests of roll waves. There are different mechanisms for this depending on the relative speed of the waves, compared to the speed of particles at the free surface, as well as on the particle concentration. If all particles at the surface travel more slowly than the waves, the large particles become concentrated as the shock-like wavefronts pass them. This is due to a concertina-like effect in the frame of the moving wave, in which large particles move slowly backwards through the crest, but travel quickly in the troughs between the crests. If, instead, some particles on the surface travel more quickly than the wave and some move slower, then, at low concentrations, large particles can move towards the wave crest from both the forward and rearward sides. This results in isolated regions of large particles that are trapped at the crest of each wave, separated by regions where the flow is thinner and free of large particles. There is also a third regime arising when all surface particles travel faster than the waves, which has large particles present everywhere but with a sharp increase in their concentration towards the wave fronts. In all cases, the significantly enhanced large particle concentration at wave crests means that such flows in nature can be especially destructive and thus particularly hazardous.
T. Sauma-Perez, C. G. Johnson, Y. Li, T. Mullin (2018)
An experimental study of the motion of a light sphere in a rotating viscous fluid,
Journal of Fluid Mechanics 847, 119–133, doi:10.1017/jfm.2018.330
[Show abstract] [PDF ] We present the results of an experimental investigation of the motion of a light, solid sphere in a horizontal rotating cylinder filled with viscous fluid. At high rotation rates, the sphere sits near the axis of the cylinder. At lower rotation rates, a set of off-axis fixed points are observed for a range of sphere radii. The locations of these fixed points are in quantitative agreement with the predictions of a model based on available theory. The fixed points are observed to become unstable to periodic orbits below a critical Reynolds number Re_c . The radius of the observed orbits increases with Reynolds number more slowly than a typical Hopf bifurcation, in this case, growing as 1/Re_c².
2017
C. G. Johnson, U. Jain, A. L. Hazel, D. Pihler-Puzovic and T. Mullin (2017)
On the buckling of an elastic holey column,
Proc. Royal Soc. A 473:20170477, doi:10.1098/rspa.2017.0477
[Show abstract] [PDF ][Nature research highlight] We report the results of a numerical and theoretical study of buckling in elastic columns containing a line of holes. Buckling is a common failure mode of elastic columns under compression, found over scales ranging from meters in buildings and aircraft to tens of nanometers in DNA. This failure usually occurs through lateral buckling, described for slender columns by Euler’s theory. When the column is perforated with a regular line of holes, a new buckling mode arises, in which adjacent holes collapse in orthogonal directions. In this paper we firstly elucidate how this alternate hole buckling mode coexists and interacts with classical Euler buckling modes, using finite-element numerical calculations with bifurcation tracking. We show how the preferred buckling mode is selected by the geometry, and discuss the roles of localised (hole- scale) and global (column-scale) buckling. Secondly, we develop a novel predictive model for the buckling of columns perforated with large holes. This model is derived without arbitrary fitting parameters, and quantitatively predicts the critical strain for buckling. We extend the model to sheets perforated with a regular array of circular holes and use it to provide quantitative predictions of their buckling.
S. Viroulet, J. L. Baker, A. N. Edwards, C. G. Johnson, C. Gjaltema, P. Clavel, J. M. N. T. Gray (2017)
Multiple solutions for granular flow over a smooth two-dimensional bump,
Journal of Fluid Mechanics 815, 77–116, doi:10.1017/jfm.2017.41
[Show abstract] [PDF ] Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.
2016
J. L. Baker, C. G. Johnson, J. M. N. T. Gray (2016)
Segregation-induced finger formation in granular free-surface flows,
Journal of Fluid Mechanics 809, 168–212, doi:10.1017/jfm.2016.673
[Show abstract] [PDF ] Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the μ(I)-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.
P. Gajjar, K. van der Vaart, A. R. Thornton, C. G. Johnson, C. Ancey, J. M. N. T. Gray (2016)
Asymmetric breaking size-segregation waves in dense granular free-surface flows,
Journal of Fluid Mechanics 794, 460–505, doi:10.1017/jfm.2016.170
[Show abstract] [PDF ] Debris and pyroclastic flows often have bouldery flow fronts, which act as a natural dam resisting further advance. Counter intuitively, these resistive fronts can lead to enhanced run-out, because they can be shouldered aside to form static levees that self-channelise the flow. At the heart of this behaviour is the inherent process of size segregation, with different sized particles readily separating into distinct vertical layers through a combination of kinetic sieving and squeeze expulsion. The result is an upward coarsening of the size distribution with the largest grains collecting at the top of the flow, where the flow velocity is greatest, allowing them to be preferentially transported to the front. Here, the large grains may be overrun, resegregated towards the surface and recirculated before being shouldered aside into lateral levees. A key element of this recirculation mechanism is the formation of a breaking size-segregation wave, which allows large particles that have been overrun to rise up into the faster moving parts of the flow as small particles are sheared over the top. Observations from experiments and discrete particle simulations in a moving-bed flume indicate that, whilst most large particles recirculate quickly at the front, a few recirculate very slowly through regions of many small particles at the rear. This behaviour is modelled in this paper using asymmetric segregation flux functions. Exact non-diffuse solutions are derived for the steady wave structure using the method of characteristics with a cubic segregation flux. Three different structures emerge, dependent on the degree of asymmetry and the non-convexity of the segregation flux function. In particular, a novel ‘lens-tail’ solution is found for segregation fluxes that have a large amount of non-convexity, with an additional expansion fan and compression wave forming a ‘tail’ upstream of the ‘lens’ region. Analysis of exact solutions for the particle motion shows that the large particle motion through the ‘lens-tail’ is fundamentally different to the classical ‘lens’ solutions. A few large particles starting near the bottom of the breaking wave pass through the ‘tail’, where they travel in a region of many small particles with a very small vertical velocity, and take significantly longer to recirculate.
M. Ungarish, C. G. Johnson and A. J. Hogg (2016)
Sustained axisymmetric intrusions in a rotating system,
Euro. J. Mech. B – Fluids 56, 110–119, doi:10.1016/j.euromechflu.2015.10.008
[Show abstract] [PDF ] We analyse the effects of rotation on the propagation of an axisymmetric intrusion through a linearly stratified ambient fluid, arising from a sustained source at the level of neutral buoyancy. This scenario occurs during the horizontal spreading of a volcanic ash cloud, which occurs after the plume has risen to its neutral buoyancy level. A simple and well-accepted approximation for the flow at late times is that inertial effects are negligible. This leads to a lens-shaped intrusion governed by a balance between Coriolis accelerations and horizontal pressure gradients, with the radius scaling with time as r_N∼t^1/3. However, we show using a shallow-layer model that inertial forces cannot be neglected until significant times after the beginning of the influx. These inertial forces result in the flow forming two distinct domains, separated by a moving hydraulic jump: an outer ‘head’ region in which the radial velocity and thickness vary with time, and a thinner ‘tail’ region in which the flow is steady. Initially, the flow expands rapidly and this tail region occupies most of the flow. After about one half-revolution of the system, Coriolis accelerations halt the advance of the front, and the hydraulic jump separating the two regions propagates back towards the source of the intrusion. Only after approximately one and a half rotations of the system does inertia become insignificant and the Coriolis lens solution, with r_N∼t^1/3, become established. Importantly, this means that neither inertia nor Coriolis accelerations can be neglected when modelling intrusions from volcanic eruptions. We exploit the two-region flow structure to construct a new hybrid model, comprising just two ordinary differential equations for the intrusion radius and location of the hydraulic jump. This hybrid model is much simpler than the shallow-layer model, but nonetheless accurately predicts flow properties such as the intrusion radius at all stages of motion, without requiring fitted or adjustable parameters.
S. Pouget, M. I. Bursik, C. G. Johnson, A. J. Hogg, J. C. Phillips, R. S. J. Sparks (2016)
Interpretation of umbrella cloud growth and morphology: implications for flow regimes of short-lived and long-lived eruptions,
Bulletin of Volcanology 78:1, doi:10.1007/s00445-015-0993-0
[Show abstract] [PDF ] New numerical and analytical modeling shows that the growth of a volcanic umbrella cloud, expressed as the increase of radius with time, proceeds through regimes, dominated by different force balances. Four regimes are identified: Regime Ia is the long-time behavior of continuously-supplied intrusions in the buoyancy-inertial regime; regime IIa is the long-time behavior of continuously-supplied, turbulent drag-dominated intrusions; regime Ib is the long-time behavior of buoyancy-inertial intrusions of constant volume; and regime IIb that of turbulent drag-dominated intrusions of constant volume. Power-law exponents for spreading time in each regime are 3/4 (Ia), 5/9 (IIa), 1/3 (Ib), and 2/9 (IIb). Both numerical modeling and observations indicate that transition periods between the regimes can be long-lasting, and during these transitions, the spreading rate does not follow a simple power law. Predictions of the new model are consistent with satellite data from seven eruptions and, together with observations of umbrella cloud structure and morphological evolution, support the existence of multiple spreading regimes.
2015
C. G. Johnson, A. J. Hogg, H. E. Huppert, R. S. J. Sparks, J. C. Phillips, A. C. Slim and M. J. Woodhouse (2015)
Modelling intrusions through quiescent and moving ambients,
Journal of Fluid Mechanics 771, 370–406, doi:10.1017/jfm.2015.180
[Show abstract] [PDF ] Volcanic eruptions commonly produce buoyant ash-laden plumes that rise through the stratified atmosphere. On reaching their level of neutral buoyancy, these plumes cease rising and transition to horizontally spreading intrusions. Such intrusions occur widely in density-stratified fluid environments, and in this paper we develop a shallow-layer model that governs their motion. We couple this dynamical model to a model for particle transport and sedimentation, to predict both the time-dependent distribution of ash within volcanic intrusions and the flux of ash that falls towards the ground. In an otherwise quiescent atmosphere, the intrusions spread axisymmetrically. We find that the buoyancy-inertial scalings previously identified for continuously supplied axisymmetric intrusions are not realised by solutions of the governing equations. By calculating asymptotic solutions to our model we show that the flow is not self-similar, but is instead time-dependent only in a narrow region at the front of the intrusion. This non-self-similar behaviour results in the radius of the intrusion growing with time t as t^3/4, rather than t^2/3 as suggested previously. We also identify a transition to drag-dominated flow, which is described by a similarity solution with radial growth now proportional to t^5/9. In the presence of an ambient wind, intrusions are not axisymmetric. Instead, they are predominantly advected downstream, while at the same time spreading laterally and thinning vertically due to persistent buoyancy forces. We show that close to the source, this lateral spreading is in a buoyancy-inertial regime, whereas far downwind, the horizontal buoyancy forces that drive the spreading are balanced by drag. Our results emphasise the important role of buoyancy-driven spreading, even at large distances from the source, in the formation of the flowing thin horizontally extensive layers of ash that form in the atmosphere as a result of volcanic eruptions.
M. Ungarish, C. G. Johnson and A. J. Hogg (2015)
A novel hybrid model for the motion of sustained axisymmetric gravity currents and intrusions,
Euro. J. Mech. B – Fluids 49A, 108–120, doi:10.1016/j.euromechflu.2014.07.007
[Show abstract] [PDF ] We consider the sustained propagation of axisymmetric intrusions and gravity currents through linearly stratified or unstratified ambient fluids. Such flow configurations are found in a number of atmospheric and oceanic flows, in particular the predominantly horizontal spreading of a volcanic ash cloud after it has ascended through the atmosphere. There is strong theoretical evidence that these flows consist of two domains: an outer annular ‘head’ at the front of the current in which the motion is unsteady; and an inner, much thinner ‘tail’, which is steady, but spatially varying. The transition between the regions is a moving hydraulic jump. While it is possible to investigate these motions by numerically integrating the governing shallow layer equations, here we develop a much simpler mathematical model, which reproduces the more complicated model accurately and addresses issues such as what determines the position of the front and the moving bore between the two regions; what is the partition of influxed volume between the tail and head; and what is the distribution of suspended particles in the flow if present at the source? In such settings a conventional integral model fails, as does scaling based on dimensional analysis and the anticipation of an underlying self-similar form; the predictions they yield for these flows are incorrect. Instead we present a new hybrid model, which combines exact results of the steady shallow-water equations in the tail with simplifying assumptions in the head. This model predicts the flow properties by the straightforward solution of three ordinary differential equations (for front and bore positions and the volume fraction of particles in the head), without using adjustable constants, and obtains the correct asymptotic behaviour for the radius of the current r_N with respect to time t, namely r_N~t^4/5 for gravity currents and r_N~t^3/4 for intrusions. The predictions are obtained with negligible computational effort and accurately capture results from the more complete shallow water models. The model is also applied with success to gravity currents and intrusions that carry particles. For flows in which it is the presence of the particles alone that drives the motion, we identify length and time scales for the runout in terms of dimensional parameters that characterise the release, thus establishing the hybrid model as a useful tool also for modelling radial runout.
2010–14
C. G. Johnson and A. J. Hogg (2013)
Entraining gravity currents,
Journal of Fluid Mechanics 731, 477–508, doi:10.1017/jfm.2013.329
[Show abstract] [PDF ] Entrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow-water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as t^0.447. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as t^4/5. In both cases, entrainment is most significant close to the current front.
M. J. Woodhouse, A. R. Thornton, C. G. Johnson, B. P. Kokelaar and J. M. N. T. Gray (2012)
Segregation-induced fingering instabilities in granular free-surface flows,
Journal of Fluid Mechanics 709, 543–580, doi:10.1017/jfm.2012.348
[Show abstract] [PDF ] [Movies] Particle-size segregation can have a significant feedback on the bulk motion of granular avalanches when the larger grains experience greater resistance to motion than the fine grains. When such segregation-mobility feedback effects occur the flow may form digitate lobate fingers or spontaneously self-channelize to form lateral levees that enhance run-out distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where run-out distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depth-averaged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentration-dependent friction coefficient, which results in a system of non-strictly hyperbolic equations. We compute numerical solutions to the flow of a bidisperse mixture of small mobile particles and larger more resistive grains down an inclined chute. The numerical results demonstrate that our model is able to describe the formation of a front rich in large particles, the instability of this front and the subsequent evolution of elongated fingers bounded by large-rich lateral levees, as observed in small-scale laboratory experiments. However, our numerical results are grid dependent, with the number of fingers increasing as the numerical resolution is increased. We investigate this pathology by examining the linear stability of a steady uniform flow, which shows that arbitrarily small wavelength perturbations grow exponentially quickly. Furthermore, we find that on a curve in parameter space the growth rate is unbounded above as the wavelength of perturbations is decreased and so the system of equations on this curve is ill-posed. This indicates that the model captures the physical mechanisms that drive the instability, but additional dissipation mechanisms, such as those considered in the realm of flow rheology, are required to set the length scale of the fingers that develop.
C. G. Johnson, B. P. Kokelaar, R. M. Iverson, R. G. LaHusen, M. Logan and J. M. N. T. Gray (2012)
Grain-size segregation and levee formation in geophysical mass flows,
Journal of Geophysical Research 117, F01032, doi:10.1029/2011JF002185
[Show abstract] [PDF ] [Movies] Data from large-scale debris-flow experiments are combined with modeling of particle-size segregation to explain the formation of lateral levees enriched in coarse grains. The experimental flows consisted of 10m³ of water-saturated sand and gravel, which traveled ~80m down a steeply inclined flume before forming an elongated leveed deposit 10m long on a nearly-horizontal runout surface. We measured the surface velocity field and observed the sequence of deposition by seeding tracers onto the flow surface and tracking them in video footage. Levees formed by progressive downslope accretion approximately 3.5m behind the flow front, which advanced steadily at ~2m/s during most of the runout. Segregation was measured by placing ~600 coarse tracer pebbles on the bed, which when entrained into the flow, segregated upwards at 6–7.5cm/s. When excavated from the deposit these were distributed in a horseshoe-shaped pattern that became increasingly elevated closer to the deposit termination. Although there was clear evidence for inverse grading during the flow, transect sampling revealed that the resulting leveed deposit was strongly graded laterally, with only weak vertical grading. We construct an an empirical, three-dimensional velocity field resembling the experimental observations, and use this with a particle-size segregation model to predict the segregation and transport of material through the flow. We infer that coarse material segregates to the flow surface and is transported to the flow front by shear. Within the flow head, coarse material is overridden, then recirculates in spiral trajectories due to size-segregation, before being advected to the flow edges and deposited to form coarse-particle-enriched levees.
C. G. Johnson and J. M. N. T. Gray (2011)
Granular jets and hydraulic jumps on an inclined plane,
Journal of Fluid Mechanics 675, 87–116, doi:10.1017/jfm.2011.2
[Show abstract] [PDF ] [Movies] A jet of granular material impinging on an inclined plane produces a diverse range of flows, from steady hydraulic jumps to periodic avalanches, self-channelised flows and pile collapse behaviour. We describe the various flow regimes and study in detail a steady-state flow, in which the jet generates a closed teardrop-shaped hydraulic jump on the plane, enclosing a region of fast-moving radial flow. On shallower slopes, a second steady regime exists in which the shock is not teardrop-shaped, but exhibits a more complex ‘blunted’ shape with a steadily breaking wave. We explain these regimes by consideration of the supercritical or subcritical nature of the flow surrounding the shock. A model is developed in which the impact of the jet on the inclined plane is treated as an inviscid flow, which is then coupled to a depth-integrated model for the resulting thin granular avalanche on the inclined plane. Numerical simulations produce a flow regime diagram strikingly similar to that obtained in experiments, with the model correctly reproducing the regimes and their dependence on the jet velocity and slope angle. The size and shape of the steady experimental shocks and the location of sub- and supercritical flow regions are also both accurately predicted. We find that the physics underlying the rapid flow inside the shock is dominated by depth-averaged mass and momentum transport, with granular friction, pressure gradients and three-dimensional aspects of the flow having comparatively little effect. Further downstream, the flow is governed by a friction–gravity balance, and some flow features, such as a persistent indentation in the free surface, are not reproduced in the numerical solutions. On planes inclined at a shallow angle, the effect of stationary granular material becomes important in the flow evolution, and oscillatory and more general time-dependent flows are observed. The hysteretic transition between static and dynamic friction leads to two phenomena observed in the flows: unsteady avalanching behaviour, and the feedback from static grains on the flowing region, leading to levéed, self-channelised flows.
2005–09
C. G. Johnson and C. J. Davis (2006)
The location of lightning affecting the ionospheric sporadic-E layer as evidence for multiple enhancement mechanisms,
Geophys. Res. Lett. 33, L07811, doi:10.1029/2005GL025294
[Show abstract] [PDF ] We present a study of the geographic location of lightning affecting the ionospheric sporadic-E (Es) layer over the ionospheric monitoring station at Chilton, UK. Data from the UK Met Office's Arrival Time Difference (ATD) lightning detection system were used to locate lightning strokes in the vicinity of the ionospheric monitoring station. A superposed epoch study of this data has previously revealed an enhancement in the Es layer caused by lightning within 200km of Chilton. In the current paper, we use the same data to investigate the location of the lightning strokes which have the largest effect on the Es layer above Chilton. We find that there are several locations where the effect of lightning on the ionosphere is most significant statistically, each producing different ionospheric responses. We interpret this as evidence that there is more than one mechanism combining to produce the previously observed enhancement in the ionosphere.
C. J. Davis and C. G. Johnson (2005)
Lightning-induced intensification of the ionospheric sporadic E layer,
Nature 435, 799–801, doi:10.1038/nature03638
[Show abstract] [PDF ] A connection between thunderstorms and the ionosphere has been hypothesized since the mid-1920s. Several mechanisms have been proposed to explain this connection and evidence from modelling as well as various types of measurements demonstrate that lightning can interact with the lower ionosphere. It has been proposed, on the basis of a few observed events, that the ionospheric 'sporadic E' layer—transient, localized patches of relatively high electron density in the mid-ionosphere E layer, which significantly affect radio-wave propagation—can be modulated by thunderstorms, but a more formal statistical analysis is still needed. Here we identify a statistically significant intensification and descent in altitude of the mid-latitude sporadic E layer directly above thunderstorms. Because no ionospheric response to low-pressure systems without lightning is detected, we conclude that this localized intensification of the sporadic E layer can be attributed to lightning. We suggest that the co-location of lightning and ionospheric enhancement can be explained by either vertically propagating gravity waves that transfer energy from the site of lightning into the ionosphere, or vertical electrical discharge, or by a combination of these two mechanisms.