On the Geometry of Current Ripples in Relation to Stability of Fluid Flow

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Abstract
Very commonly small-scale current ripples display wave-like changes of elevation in a direction transverse as well as parallel to current flow. It is shown theoretically. and confirmed experimentally, that such changes of elevation of a rippled bed record an adjustment of the bed to a three-dimensional fluid motion. Combined with the basic flow, which is characterised by transverse regions of flow separation, are corkscrew vortices parallel to flow that resemble Taylor-Görtler vortices. The instability leading to these vortices in the flow over a rippled bed very probably is largely of centrifugal origin, since there are regions in the flow where the circulation decreases away from the local centres of streamline curvature. Experiments made with beds of a fine-grained quartz sand reveal that the characteristic wavelength of features due to the vortices diminishes with increasing relative roughness and Froude number of flow. But as the stream-wise crest-to-crest distance of ripples changes but little with either relative roughness or Froude number, over the greater part of the practical range of these parameters, the result of an increase of flow roughness or speed is to make ripples more irregular, as is shown quantitatively from the experiments. We find empirically that the mean wavelengths of the spanwise and streamwise features of a rippled bed are related to relative roughness and Froude number by the expression in which is the mean ripple height, is the mean flow depth, and Fr is the Froude number based on mean flow depth and velocity. This expression appears applicable to the much larger dunes formed in water, which also commonly display wave-like ridges and furrows parallel to flow. In having lee projections, certain transverse desert dunes resemble ripples and dunes found in water, and would appear affected by centrifugally unstable wind flows. Wavelength ratios for these dunes overlap in value with ratios measured from subaqueous bed forms.