DPM simulations of granular fingering

J. M. F. Tsang, B. Jin, M. I. Arran, N. M. Vriend
University of Cambridge

Last updated 9 November 2017.

These animations were produced from DPM simulations of granular fingering. A mixed volume of grains is released from behind x = 0 and flows down a slope, which is made bumpy by lining it with grains (which we do not show here). The sides of the domain are periodic. In the simulations shown below, we vary the width W of the domain. On the other pages, linked to above, we vary other parameters.

The periodic boundary conditions ensure that modal perturbations must have wavelengths λ = W/n for n = 1,2,3,.... We can therefore look at which modes are most unstable simply by counting the number of fingers, as well as the downstream position at which they form. This can be compared against the predictions of a linear stability analysis, which give a dispersion relation for the growth rate of linear perturbations against their wavenumber.

These simulations were done using MercuryDPM, an open-source code developed at the University of Twente. We also employed a number of novel techniques that allowed us to create fingers, as well as greatly speeding up the simulations. Here are sketches of the side view and the top view of the simulation setup.

Note: If the gifs don't all play at the same rate, or are out of sync, try letting the page load, and then refreshing the page.

width = 0.5

Figure 1: In a narrow domain, hardly any fingering occurs until quite late on (around x=8). Although segregation takes place, the grains are mostly mixed since the large grains can't be displaced to the sides. This current is therefore rather frictional and doesn't quite travel as far as the other currents in the alloted time, although it doesn't quite run out.

width = 1

Figure 2: A single finger can form for a domain that is about 5.6cm wide.

width = 2

Figure 3: A second finger can form when the domain width is doubled to about 11cm. This would suggest that the λ=5.6cm perturbations grow more quickly than those with λ=11cm.

width = 3

Figure 4: The current develops still only two fingers when the width is further increased to around 17cm. This indicates that the λ=8.4cm state is more 'preferable' to the three-fingered λ=5.6cm state.

width = 4

Figure 5: If the width is increased to around 22cm, then a third finger finally forms, which shows that a λ=7.5cm state is also 'preferable' to the λ=5.6 state. However, there is some variation in the finger widths, as well as their lengths: these cannot be described by linear theory.

Analysis of results

We can analyse these results by looking at the frontal position as a function of cross-stream position and time. A look at the autocorrelation lets us measure the finger width more quantitatively.

Note the interesting sergeant-striped features on the plot of frontal velocity against y and t, suggesting that the channels move in a 'pulsing' manner.

Figure 6: Analysis of the results from the simulation shown in Figure 5. The autocorrelation in the frontal position (bottom panel) allows us to precisely estimate the finger width. The peak around 7cm is consistent with the fact that the domain is around 22cm wide, and that three fingers have formed in this simulation. (full size)