Abstract
In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the $\unicode[STIX]{x1D707}(I)$-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number $I$. This formulation greatly benefits from its similarity to the incompressible Navier–Stokes equations as it allows many existing numerical methods to be used. Unfortunately, this system of equations is ill posed when the inertial number is too high or too low. The consequence of ill posedness is that the growth rate of small perturbations tends to infinity in the high wavenumber limit. Due to this, numerical solutions are grid dependent and cannot be taken as being physically realistic. In this paper changes to the functional form of the $\unicode[STIX]{x1D707}(I)$ curve are considered, in order to maximise the range of well-posed inertial numbers, while preserving the overall structure of the equations. It is found that when the inertial number is low there exist curves for which the equations are guaranteed to be well posed. However when the inertial number is very large the equations are found to be ill posed regardless of the functional dependence of $\unicode[STIX]{x1D707}$ on $I$. A new $\unicode[STIX]{x1D707}(I)$ curve, which is inspired by the analysis of the governing equations and by experimental data, is proposed here. In order to test this regularised rheology, transient granular flows on inclined planes are studied. It is found that simulations of flows, which show signs of ill posedness with unregularised models, are numerically stable and match key experimental observations when the regularised model is used. This paper details two-dimensional transient computations of decelerating flows where the inertial number tends to zero, high-speed flows that have large inertial numbers, and flows which develop into granular rollwaves. This is the first time that granular rollwaves have been simulated in two dimensions, which represents a major step towards the simulation of other complex granular flows.
Highlights
Reliable and accurate constitutive modelling of granular flows is vital in both industrial and natural settings
The significance of this number was demonstrated by GDR MiDi (2004) who found that the ratio μ of the shear stress τ and pressure p collapsed onto a single μ(I) curve for flows in a range of geometries
For flows with low inertial numbers the analysis demonstrates that values of μ must be smaller in order to remain well posed whereas for high inertial numbers the values of μ must be larger than in the rheology of Jop et al (2005)
Summary
Reliable and accurate constitutive modelling of granular flows is vital in both industrial and natural settings. Trulsson et al (2013) showed that acoustic waves could be transmitted through shear flows when compressibility was taken into account and Pouliquen & Forterre (2009) and Kamrin & Henann (2015) were able to model the point of transition between static and moving material on a frictional inclined plane These effects introduce additional complexity into the governing equations and require new numerical schemes to be developed in order to solve them. It is conceivable that in the regions of the flow with ill-posed inertial numbers, the code reverted to solving the classical Navier–Stokes equations with a constant Newtonian viscosity, which are well posed. It is the case that any μ(I) curve is guaranteed to lead to ill-posed equations for very large inertial numbers Given these limitations it is useful to consider the regions for which well-posed rheologies can be constructed. There are additional restrictions on the choice of rheology, such as the condition that the μ(I) curve should be monotonically increasing
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