# Formation of sediment patterns in channel flow: minimal unstable systems and their temporal evolution

## TL;DR

This study numerically investigates sediment pattern formation in channel flow, identifying minimal unstable system sizes and analyzing the evolution and characteristics of ripples over different domain lengths.

## Contribution

It determines the minimal domain size for unstable sediment patterns and explores the nonlinear evolution of ripples in large-scale simulations.

## Key findings

- The cutoff length for pattern formation is 75-100 particle diameters.
- Ripple amplitude growth has exponential and nonlinear regimes.
- Ripple shape exhibits a power-law spectrum decay.

## Abstract

The phenomenon of sediment pattern formation in a channel flow is numerically investigated by performing simulations which resolve all the relevant scales of the problem. The numerical approach employed and the flow configuration considered is identical to our previous study (Kidanemariam and Uhlmann J. Fluid Mech., vol. 750, 2014, R2), the only difference being the length of the computational domain. By successively reducing the streamwise length, the minimum box dimension which accommodates an unstable sediment bed is revealed, thus determining the lower threshold of the unstable modes. For the considered parameter point, the cutoff length for pattern formation lies in the range 75-100 particle diameters (3-4 times the clear fluid height). We also simulate the flow in a very long streamwise box with a size of 48 times the clear fluid height (featuring over one million particles), accommodating approximately 11 initial ripple units with a wavelength of 100-110 particle diameters. The amplitude of the patterns exhibits two regimes of growth: an initial exponential regime, with a growth rate independent of the chosen domain size, and a subsequent non-linear regime which is strongly constrained by the domain length. In the small domain cases, after the initial exponential regime, the ripples evolve steadily maintaining their shape and migration velocity, at a mean wavelength equal to the length of the domain. The asymmetric ripple shape is characterized by a spectrum which exhibits a power-law decay over the first few dominant non-dispersive modes propagating at the mean dune migration velocity. The particle flowrate and the mean interface shear stress exhibited an increase with increasing ripple dimensions. Nevertheless, the relationship between the two was observed to be approximately described by the empirical power law formula for sediment transport by Wong & Parker (2006).