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.
A two-dimensional stability analysis of the flow in a straight alluvial channel has been carried out, using the vorticity transport equation. In the analysis an attempt has been made to account for the influence of gravity on bed-load transport, and this turned out to change the stability quite significantly.In the case of instability, the further growth of the dunes has been investigated using a second-order approximation, This nonlinear theory explains the experimental fact that the dunes very soon become asymmetric.
The numerical model is developed consisting of a 1D flow model and the morphological model to simulate the erosion due to the water overtopping. The step method is applied to solve the water surface on the slope and the finite difference method of the modified Lax Scheme is applied for bed change equation. The Meyer-Peter and Muller formulae is used to determine the bed load transport rate. The model is calibrated and verified based on the data in experiment. It is found that the computed results and experiment data are good agreement.
Linear stability analysis of a boundary layer with rotating wall-normal cylindrical roughness elements