On the formation of ripples on an erodible bed

Abstract

A linear stability analysis is presented of both hydraulically smooth and transitional flows over an erodible bed. The present theory is developed to account for the formation of ripples. It is essentially an extension of the theory of Richards (1980) to include the effect of viscosity upon the bed wave stability. The theory takes into consideration that the formation of ripples does not depend on flow depths, and that only the bed-load transport is involved in the formation of ripples. The effect of gravity is included in the analysis through the local inclination of the wavy bed surface. The results show that the bed is unstable (i.e. ripples exist) when the grain Reynolds number is less than a certain value. The limiting values of the grain Reynolds number for ripple existence obtained through present analysis are found to be in good agreement with observations.

The different ripple formation mechanism

“…For his stability calculation Richards assumed hydraulically rough wall conditions, in contradiction to equation (1). Sumer and Bakioglu [5] removed this difficulty by introducing viscosity and gravity effects into the stability calculations of Richards. The stability concept is discussed at length in the review article of Engelund and Fredsoe [6].…”

“…The main advantage of a structural concept is that the initial wavelength does not have to be evaluated through a stability analysis but can be estimated by equation (5). The mechanism is understood and therefore the effect of manipulations of the flow or bed conditions can be estimated.…”

Ripple formation on a particle bed sheared by a viscous liquid. Part 2. Oscillating flow

“…However, the potential flow theory used by Kennedy was unable to determine the phase lag, which was a free parameter. The hydrodynamic instability mechanism, which is described in more detail in the following section, holds for any steady flow, either turbulent (Richard 1980;Sumer & Bakioglu 1984;Colombini 2004) or viscous (Charru & Mouilleron-Arnould 2002). All wavenumbers are unstable, but the stabilizing effect of gravity dominates for high wavenumbers, hence the occurrence of a long-wave instability with a cut-off wavenumber and a most amplified wavenumber which can be compared to observations.…”

“…Very few growth-rate measurements are available (Betat, Frette & Rehberg 1999), and the observed ripple lengths display large scatter (Yalin 1985), which can be partly attributed to ripple coalescence which occurs at the early stages of the growth. Moreover, several experiments indicate that wavelengths mainly depend on the particle diameter, with only slight dependence on the fluid flow (Coleman & Eling 2000), whereas stability theories predict that the wavelength should scale on a viscous length (Sumer & Bakioglu 1984;Charru & Mouilleron-Arnould 2002). From the modelling point of view, the main reason for the failure of theoretical predictions may be that, apart from the turbulence problem, the dynamics of the particle is poorly accounted for through an algebraic law relating the particle flux to the bottom shear stress.…”

Coupling or decoupling bed and flow dynamics: Fast and slow sediment waves at high Froude numbers

“…[2][3][4][5][6][7][8] In a linear context, information is gathered about the existence of unstable regions in the parameter space and on the wavelength of the most unstable mode. A satisfactory agreement of the theoretical predictions with the experimental observations is found, thus confirming that the basic mechanism of bedform formation is indeed an instability mechanism, which is necessarily associated with the presence of a phase lag between sediment transport and bed topography.…”