# Granular rheology and topography

J. M. F. Tsang, S. B. Dalziel, N. M. Vriend
University of Cambridge

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) = kxa, 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.

Figure 1: Results from a DPM simulation in which a current travels downstream with a basal roughness suddenly beginning after a transition point. The colour indicates the speed of each particle. We see the no-slip basal boundary condition being turned on, and this leads to the development of a boundary layer that grows and eventually spreads throughout the current. The Froude number is about 5. (full size)

Figure 2: Internal velocity profiles from a selection of simulations at different Froude numbers. In each box, each curve shows the velocity profile at a different streamwise location, going from black (just after the transition point x = 0) to red (downstream). The profiles are plotted in terms of a rescaled similarity variable. In terms of this coordinate, the curves at the bottom collapse well onto each other, showing that the boundary layer develops in a self-similar way. (The top parts of the curves do not collapse, because the outer flow continues to be accelerated by gravity.)

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)