The granular monoclinal wave: a dynamical systems survey

Kanellopoulos, Giorgos

doi:10.1017/jfm.2021.491

Cited by 9 publications

(12 citation statements)

References 28 publications

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“…Furthermore, we treat the granular material as an incompressible fluid (Savage & Hutter 1989), assuming constant density throughout the flowing sheet. The height h(x, t) and the depth-averaged velocity ū(x, t) are governed by a system of two coupled, nonlinear PDEs Razis et al 2014;Edwards & Gray 2015;Razis et al 2018;Edwards et al 2019;Razis et al 2019;Edwards et al 2021;Kanellopoulos 2021), namely, the mass conservation…”

Section: The Granular Saint-venant Equations and The Associated Const...mentioning

confidence: 99%

“…The fixed point (1, 0) represents the familiar base flow near the inlet, whereas the second fixed point ( h+ , 0) represents a thinner uniform flow. Both these uniform flows can occur simultaneously on the chute, linked by a travelling shock front, forming a 'monoclinal' flood wave or an undular variation of this waveform (Razis et al 2018(Razis et al , 2019Kanellopoulos 2021).…”

Section: Travelling Waves: Reducing the Problem To An Odementioning

confidence: 99%

“…The critical Froude number is given by the expression and depends only on the Froude offset , which plays a role in the friction between the chute and the granular sheet (Forterre & Pouliquen 2003; Edwards et al. 2017; Kanellopoulos 2021). This parameter is determined by the specifics of the experimental set-up and the granular material, see § 2.…”

Section: Introductionmentioning

confidence: 99%

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On the shape and size of granular roll waves

Kanellopoulos

Razis²,

Weele³

2022

J. Fluid Mech.

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This paper describes, from a theoretical point of view, the appearance and characteristics of granular roll waves in chute flow, and the maximal size these waves can attain for a given influx of material into the system. Granular roll waves are steady travelling wave solutions of the generalized Saint-Venant equations for flowing granular matter, appearing when the Froude number $Fr$ of the incoming flow exceeds a critical value, $Fr>Fr_{cr}$ . We focus upon the phase space of the corresponding dynamical system, where the roll waves take the form of a stable limit cycle around an unstable fixed point; this limit cycle gives precise information on the size and periodicity of the roll wave. It is found that, for any given value of $Fr$ , the limit cycle cannot become arbitrarily large because it is constrained by a homoclinic loop in phase space. Roll waves of larger amplitude can be generated by increasing the Froude number $Fr$ .

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Section: The Granular Saint-venant Equations and The Associated Const...mentioning

confidence: 99%

Section: Travelling Waves: Reducing the Problem To An Odementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the shape and size of granular roll waves

Kanellopoulos

Razis²,

Weele³

2022

J. Fluid Mech.

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“…Such states are experienced in nature as a surge between two shallow flowing regions of different height. They have been studied widely in the context of turbulent open water [4,[18][19][20][21] and more recently, in granular flows [22][23][24]. The most mathematically extensive results on their stability are available in the former case, where nonlinear stability theorems for monoclinal profiles have been achieved [25,26].…”

Section: Introductionmentioning

confidence: 99%

“…The most mathematically extensive results on their stability are available in the former case, where nonlinear stability theorems for monoclinal profiles have been achieved [25,26]. In the granular setting, existence and stability of monoclinal traveling waves was explored for a popular model that includes a small diffusive regularization [22][23][24]. In addition to these cases, as shall be demonstrated below, monoclinal waves are available as solutions to Eqs.…”

Section: Introductionmentioning

confidence: 99%

General linear stability properties of monoclinal shallow waves

Langham¹,

Hogg²

2022

Preprint

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We analyze the linear stability of monoclinal traveling waves on a constant incline, which connect uniform flowing regions of differing depths. The classical shallow-water equations are employed, subject to a general resistive drag term. This approach incorporates many flow rheologies into a single setting and enables us to investigate the features that set different systems apart. We derive simple formulae for the onset of linear instability, the corresponding linear growth rates and related properties including the existence of monoclinal waves, development of shocks and whether instability is initially triggered up-or downstream of the wave front. Also included within our framework is the presence of shear in the flow velocity profile, which is often neglected in depth-averaged studies. We find that it can significantly modify the threshold for instability. Constant corrections to the governing equations to account for sheared profiles via a 'momentum shape factor' act to stabilize traveling waves. More general correction terms are found to have a nontrivial and potentially important quantitative effect on the properties explored. Finally, we have investigated the spatial properties of the dominant (fastest growing) linear modes. We derive equations for their amplitude and frequency and find that both features can become severely amplified near the front of the traveling wave. For flood waves that propagate into a dry downstream region, this amplification is unbounded in the limit of high disturbance frequency. We show that the rate of divergence is a function of the spatial dependence of the wave depth profile at the front, which may be determined straightforwardly from the drag law.

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