Understanding erosion by flows
Erosion by flows is a commonly seen natural phenomenon. This happens on many different length scales. At large length scales one sees the effects of erosion by rivers. This is an intriguing geological phenomenon. The landscape is eroded by the river, which in turn can change the path of the river itself. This interconnectedness between boundaries and the flow is an essential aspect of erosion.
The problem manifests itself at different scales. If you consider
river banks or even coastlines, you find that they are not simple
shapes. A coastline typically has multifractal shape. That is the
reason why measuring coastlines (or borders between countries) is
nontrivial, see, e.g.,
When faced with such complex problems, physicists start by looking for "simple but not simpler approach". Consider a clay ball dipped in a flow and watch how it erodes. Exactly this experiment was performed by a group of applied mathematicians at the Courant Institute in New York. The first thing they notice is that the boundary of the clay ball changes very slowly compared to the flow velocities, as is observed in geological problems. This separation of time-scales will play a crucial role in our treatment of this problem. A summary of their results is:
- The rate of erosion is proportional to the local wall-stress.
- After a short time, the shape of the eroded body becomes self-similar.
- Sharp corners can develop, particularly at stagnation points of the flow.
Taking this experiment as our inspiration, we would like to start with a numerical experiment. Consider a two dimensional flow problem. A circle (a cylinder in three dimensions) in a cross-flow as shown in the figure below:
We expect that due to the wall-stress on the boundary material in the circle are going to erode away. This erosion will change the boundary of the solid object, the circle will possibly not remain a circle. Hence we need a computational method that can deal with boundaries that are not regular shape. In addition, we need to deal with a dynamical boundary, but we are going to simplify the problem by virtue of the separation of the time scales; as I suggest below:
Let us use a two-dimensional lattice Boltzmann code (LBM) to solve this problem. Let us parametrise the boundary by a arc-length variable $s$. So the closed boundary implies that $s$ is periodic. We assume that the rate of erosion from the surface is proportional to the wall-stress, so the mass loss over a time $\tau$ is given by \begin{equation} \Delta M(s,\tau) \equiv \int_0^{\tau} \sigma_{\rm wall}(s,t) dt \end{equation} We need to assume $\tau$ to be neither to small or too large. Then we run the code for time $\tau$ and calculate the mass-loss. We use this mass-loss to deduce a new boundary. Then run the simulations again with this new boundary. How does various shapes erode under this method ? I expect that the shape is going to be more like a peeble (smooth with sharp corners) than a multifractal surface.
In addition to erosion, we can also use to same idea to deal with deposition. For example, we could introduce heavy inertial particles in the problem. Some of these particles would hit the cylinder and depending on the coefficient of resitution between the cylinder and the particles they can stick. We can again wait for time $\tau$ and calculate $\Delta M(s,\tau)$ with contributions from both deposition and erosion.
The problem is also interesting from astrophysical point of view. The central object could be a kilometer-sized planetesimal in a protoplanetary disk. Would the erosion favours certain shapes and also selects sizes larger than certain sizes (e.g., smaller objects are eroded away) ?
The problem could then continued to 3 dimensions. The possibilities are endless.
Collaborations: The problem has been designed by myself (Dhrubaditya Mitra, NORDITA) and Anders Johansen from the Lund Observatory.