High-speed confined granular flows down smooth inclines: scaling and wall friction laws
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ORIGINAL PAPER
High‑speed confined granular flows down smooth inclines: scaling and wall friction laws Yajuan Zhu1 · Renaud Delannay1 · Alexandre Valance1 Received: 15 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Recent numerical work has shown that high-speed confined granular flows down smooth inclines exhibit a rich variety of flow patterns, including dense unidirectional flows, flows with longitudinal vortices and supported flows characterized by a dense core surrounded by a dilute hot granular gas [1]. Here, we further analyzed the results obtained in [1]. More precisely, we characterize carefully the transition between the different flow regimes, including unidirectional, roll and supported flow regimes and propose for each transition an appropriate order parameter. Importantly, we also uncover that the effective friction at the basal and side√ walls can be described as a unique function of a dimensionless number which is the analog of a Froude number: Fr = V∕ gH cos 𝜃 where V is the particle velocity at the walls, 𝜃 is the inclination angle and H the particle holdup (defined as the depth-integrated particle volume fraction). This universal function provides a boundary condition for granular flows running on smooth boundaries. Additionally, we show that there exists a similar universal law relating the √ local friction to a local Froude number Frloc = V loc ∕ Ploc ∕𝜌 (where V loc and Ploc are the local velocity and pressure at the boundary, respectively, and 𝜌 the particle density) and that the latter holds for unsteady flows. Keywords High-speed granular flows · Longitudinal vortices · Supported flows · Effective friction
1 Introduction The scientific community has paid particular attention to gravitational granular flows over the past 20 years. These flows are ubiquitous in natural and industrial processes [2, 3]. However, their modeling and understanding still leave us with open issues. The complexity arises from grain-grain interactions, and also from grain-boundary interactions which may induce correlations over distances much greater than a grain diameter. The inclined plane geometry was the most employed configuration to study gravity-driven granular flows [4, 5]. It is simple and relevant for many practical situations, but it can be also seen as a rheological test with constant friction. This article is part of the Topical Collection: Flow regimes and phase transitions in granular matter: multiscale modeling from micromechanics to continuum. * Alexandre Valance Alexandre.Valance@univ‑rennes1.fr 1
Institut de Physique de Rennes, CNRS UMR 6251, Univ Rennes, 35042 Rennes CEDEX, France
To date, experiments [4] and simulations [6] have focused mainly on mildly sloping and bumpy planes, leading to slow and dense flows which are now fairly well understood [2, 7]. More complex flows, including span-wise vortices [8–10], were obtained at slightly higher angles suggesting that upon further steepening, granular flows may reveal original features. Obtaining steady and
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