A Roe-type scheme for two-phase shallow granular flows over variable topography

Abstract

Abstract.We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Wellbalancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions. Mathematics Subject Classification. 65M99, 76T25.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

“…Pitman & Le [4] developed a depth-averaged version of Jackson's [5] three-dimensional, two-phase granular-fluid equations, and various extensions of their model have been proposed (e.g. [6][7][8]). These models are expressed as hyperbolic systems of partial differential equations, yet they possess an unstable elliptic degeneracy associated with loss of hyperbolicity in particular flow regimes [6,9].…”

“…[6][7][8]). These models are expressed as hyperbolic systems of partial differential equations, yet they possess an unstable elliptic degeneracy associated with loss of hyperbolicity in particular flow regimes [6,9].…”

An exponential time-differencing method for monotonic relaxation systems

“…Granular gases have lately been the subject of considerable theoretical, numerical and experimental studies [9,30,29,33,31]. In this work we consider a continuum model for granular-gas flow, in which the dynamics are accounted for by a hyperbolic conservation law with relaxation.…”

Modeling Shallow Over-Saturated Mixtures on Arbitrary Rigid Topography

“…In the model by Berzi et al [8][9][10] for a twolayer flow with a sediment-water mixture in the lower layer and clean water in the upper layer, σ s n 1 = 0 is presumed at the layer interface and, as in Fraccarollo and Capart [3], a uniform υ along the avalanche depth is considered -this is different from (24). Here we use (24), which is based on (23), and deal with a solid volume fraction which slightly deviates from uniformity along the depth, except, eventually, near the layer interface. We introduce Boussinesq coefficients (see below) to account for these deviations, and let υ at the interface assume a value determined from experiments (Egashira et al [30] used υ = 0.05, cf.…”

“…Moreover, noting that the trend in avalanche modeling is to account for curvature effects (see, e.g., Sivakumaran et al [14], Dewals et al [15], Iverson [16], Pudasaini et al [17,18], Bouchut and Westdickenberg [19], De Toni and Scotton [20], Bouchut et al [21], Tai and Kuo [22], Tai and Lin [23], Pelanti et al [24], Luca et al [25][26][27]5]), emphasis is on the description of the flow on arbitrary terrain, using the approach initiated by Bouchut and Westickenberg [19], and not on the formulation of constitutive laws, even if we indicate such laws as possible choices. In particular, the shallow saturated mixture in the lower layer is treated in Truesdell's sense, in the same manner as it has been done by Luca et al [27].…”