Spatial averaging of the Reynolds-averaged Navier-Stokes equations gives the double-averaged Navier-Stokes equations, for which boundary drag appears naturally and explicitly in momentum conservation equations. Increasing use of the double-averaged equations, e.g., for relating flows to three-dimensional bed roughness, for evaluation of profiles of flow stresses and velocities in ecologically significant regions below roughness tops, and for modeling purposes, requires parameterization of boundary drag at subele-ment scales. Based on seven flows over repeated square-rib roughness and ten flows over repeated fixed simulated sand waves, with measured velocities and bed pressures, expressions for form-drag coefficient C_D=f (elevation below roughness top, relative roughness submergence, roughness steepness) are obtained for each of the two-dimensional roughness types. Using these equations, form drag variation with elevation below roughness tops can be calculated using either the double average of the square of local velocity (preferred based on conceptual considerations, trends in coefficient prediction, and also overall drag prediction) or the squared local double-averaged velocity, the roughness area being normal to the flow in each case. Integration of subelement drag given by these expressions is shown to give form-drag coefficient magnitudes and trends for complete individual elements comparable to those obtained by other authors based on measurements or numerical simulations. The ranges of roughness steepness and relative roughness submergence upon which the present equations have been derived need to be noted in consideration of application of the equations. In addition, effective application of the expressions is limited in regions of strongly negative double-averaged velocity. Further work remains to determine drag parameterization for alternative roughness geometries.
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