Movable bed roughness in unsteady oscillatory flow

A model to predict the roughness in unsteady oscillatory flows over movable, noncohesive beds is presented. The roughness over movable beds is shown to be a function of the boundary shear stress, rather than a fixed geometrical scale as is the case for fully rough turbulent boundary shear flows over immobile beds. The model partitions the roughness into two distinct contributions. These two contributions are due to the form drag around individual bed forms and to the near‐bed sediment transport. The form drag over the bed forms is treated explicitly as a function of the boundary geometry and shear stress. The ripples are predicted as a function of the local skin friction, and a semiempirical expression is derived using standard law‐of‐the‐wall arguments, which gives the ripple or form roughness as a function of the boundary geometry. The ripple roughness is found to be proportional to the product of the ripple steepness and height. Favorable comparison of the form drag model with the results of Bagnold's (1946) fixed ripple study is found. The value of z 0 associated with intense sediment transport in oscillatory flow over a flat bed is determined from Carstens et al.'s (1969) experiments. This value is found to be 7 or 8 grain diameters. An expression is derived for the roughness associated with the maximum thickness of a near‐bottom sediment‐transporting layer consistent with Owen's (1964) roughness hypothesis for saltation of uniform grains in air. At large values of the boundary shear stress relative to the critical value for initial sediment motion, the derived expression is similar to the results of Smith and McLean's (1977) unidirectional flow approach modified for oscillatory flow. The total roughness model is found to compare favorably with Carstens et al.'s (1969) data. In contrast to Smith and McLean's (1977) steady flow findings, the results here show that when ripples are present, they account for a significant portion of the boundary roughness.