# Granular rheology and topography

Last updated 5 July 2018.

My main project is modelling the way that dry granular currents respond to topographical features that they flow over. A granular current is a continuous release of grains down an inclined slope or chute. The flow is driven by gravity and resisted by basal and internal friction.

A popular way of modelling granular currents is to use depth-averaged or 'shallow water' models, similar to those used in hydrology. These are very useful in practical applications as they are computationally efficient. However, existing depth-averaged models do not accurately describe flows over topographical features, such as changes in basal roughness, or basal curvature. These features change the internal velocity profile of a flow, and depth-averaged models are not able to describe these changes, since they make assumptions about the internal structure of a flow.

To study the changes in the internal structure, we study granular currents
using *rheological* models, such as the
*μ*(*I*) rheology (Jop *et al.* 2006). The
predictions of these models can be compared against results from *discrete
particle model* (DPM) simulations. My work therefore also involves
developing DPM that try to be both physically relevant and computationally
efficient.

We are particularly interested in modelling fast flows, where the
inertial number *I* and the Froude number Fr are
high. These are common in industrial and geophysical contexts. Rheological
models such as the *μ*(*I*) rheology haven't been tested
at high speeds, as it is difficult to create sustained fast flows in the lab. In
our DPM simulations, we take Froude numbers going up to 6.

## The granular Blasius problem

The granular Blasius problem considers what happens with a granular current flows over an inclined surface with a sudden transition in basal roughness.

The surface is smooth upstream but rough downstream. The roughness imposes a no-slip condition on the bottom of the flow (but see Note 1). The flow at the bottom is therefore slowed down, and this is transmitted upwards by internal friction.

Just downstream of the transition point, the main body of the flow is not
affected by the friction, as the flow has inertia. The effects of friction are
localised to a boundary layer, which is reminiscent of the classical Blasius
boundary layer problem When the *μ*(*I*) rheology is
suitably generalised (see Note 2), it can
describe the shape and spreading of this boundary layer using a similarity
solution analogous to Prandtl's solution for the classical problem.

The formation and growth of the boundary layer for one simulation is shown in Figure 1. In Figure 2, we plot the internal velocity profile for simulations at different Froude numbers, suitably scaled to illustrate the self-similarity.

The thickness of the boundary layer develops as
*Λ*(*x*) = *kx*^{a},
where the constant of proportionality *k* and the exponent *a*
depend on the asymptotic properties of *μ*(*I*) that we assume.
The boundary layer thickness also depends on the grain size and the Froude
number of the incident flow. Grain size plays a role similar to viscosity for classical fluids.

We studied a preliminary problem involving a Newtonian fluid instead of a granular flow, and our paper on this problem can be found at doi:10.1017/jfm.2018.2.

**Note 1:** The interplay between
basal roughness and no-slip is very interesting, because basal roughness is a
grain-scale property, but no-slip is a property of the bulk flow. These two
conditions are not equivalent, and basal roughness does not *always* impose
a no-slip condition. This topic is discussed in detail by Bharathraj & Kumaran
2017 and Jing
*et al.* 2016, but it remains an active area of research. (back)

**Note 2:** The generalisation of the
*μ*(*I*) rheology is to choose a new form for the function
*μ*(*I*) in the limit *I* → ∞. This is because
the boundary layer is a region of very high shear rate, so the inertial number
is also very high. It has been shown that the standard form for the function
*μ*(*I*) is ill-posed at high *I* (Barker *et
al.* 2015). The granular Blasius problem
demonstrates this result. It turns out that the standard function
does not admit a solution with a boundary layer (indeed, it predicts
a boundary layer thickness of zero). (back)