Depth-Averaged Free-Surface Granular Flows
Free-surface granular ﬂows, such as debris ﬂows and snow avalanches, can spontaneously develop large-amplitude surge waves that are interspersed by periods in which grains are completely stationary. These waves are important in natural hazard mitigation because each individual surge is much more destructive than a steady uniform ﬂow with the same mass ﬂux. A key feature of these ﬂows is the interaction between static and ﬂowing material, which occurs both when the travelling waves erode the static layer of particles in front of them, and when they deposit grains behind them. Although erosion and deposition problems are notoriously difﬁcult, we ﬁnd that a simple model that uses a depth-averaged version of the \(\mu(I)\)-rheology, and Pouliquen and Forterres extended friction law, captures the underlying physics and is in good agreement with experiments. Importantly, this depth-averaged rheology includes not only the leading-order effective basal friction, but also incorporates a depth-averaged viscous term, which is essential for predicting the onset of roll-wave instabilities. Numerical simulations, obtained with a shock-capturing scheme for viscous problems, show that the depth-averaged model quantitatively predicts the wave amplitude, wavespeed and coarsening dynamics of these erosion-deposition waves. This agreement suggests that depth-averaged \(\mu(I)\)-rheology may be more widely useful for prediction of other erosion and deposition phenomena.
Three-dimensional boundary layer states forced by transpiration over short spanwise scales
We consider the three dimensional laminar boundary layer on a semi-infinite plate in a uniform stream. The three dimensionality is induced by a wall transpiration that exists over a spanwise scale that is comparable to the boundary layer thickness. This short-scale transpiration requires the inclusion of spanwise and transverse diffusion in the boundary layer, therefore building upon previous analyses of the corner boundary layer equations. We examine the case in which the spanwise extent of the transpiration region maintains a constant ratio with the boundary layer thickness, allowing similarity solutions to be developed in any cross sectional plane a fixed distance from the leading edge of the plate. Numerical computation is shown to require careful consideration of the far field flow to ensure a global mass balance consistent with the inviscid outer flow. For a blowing transpiration, numerical solutions highlight three regimes of attached flow depending on the magnitude of the mass flux through the boundary. These regimes are differentiated by the (transverse and spanwise) spatial extent of a low-speed streamwise streak. Asymptotic descriptions are presented in the limits of large and small mass flux through the plate, which are seen to be in agreement with the full numerical solutions.