On the scaling of the instability of a flat sediment bed with respect to ripple-like patterns

We investigate the formation of subaqueous transverse bedforms in turbulent open channel flow by means of direct numerical simulations with fully resolved particles. The main goal of the present analysis is to address the question whether the initial pattern wavelength scales with the particle diameter or with the mean fluid height. A previous study (Kidanemariam & Uhlmann, J. Fluid Mech., vol. 818, 2017, pp. 716–743) has observed a lower bound for the most unstable pattern wavelength in the range 75–100 times the particle diameter, which was equivalent to 3–4 times the mean fluid height. In the current paper, we vary the streamwise box length in terms of the particle diameter and of the mean fluid height independently in order to distinguish between the two possible scaling relations. For the chosen parameter range, the obtained results clearly exhibit a scaling of the initial pattern wavelength with the particle diameter, with a lower bound around a streamwise extent of approximately $80$ particle diameters. In longer domains, on the other hand, patterns are observed at initial wavelengths in the range 150–180 times the particle diameter, which is in good agreement with experimental measurements. Variations of the mean fluid height, on the other hand, seem to have no significant influence on the most unstable initial pattern wavelength. We argue that the observed scaling with the particle diameter is due to the wake effect induced by the seeds which are formed by initially dislodged lumps of particles, in accordance with the ideas of Coleman & Nikora (Water Resour. Res., vol. 45, 2009, W04402). Finally, for the cases with the largest relative submergence, we observe spanwise and streamwise sediment waves of similar amplitude to evolve and superimpose, leading to three-dimensional sediment patterns.