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The goal of this paper is to discuss the spatial averaging concept in environmental hydraulics and develop it further by considering transport equations for fluid momentum, passive substances, and suspended sediments. The averaging theorems, the double-averaged (in time and in space) fluid momentum equation, and advection-diffusion equations for a passive substance and suspended sediments are introduced and their limitations and applications for modeling rough-bed flows, experimental design, and data interpretation are discussed. The suggested equations differ from those considered in terrestrial canopy aerodynamics and porous media hydrodynamics by accounting for roughness mobility, change in roughness density in space and time, and particle settling effects for the case of suspended sediments. We show that the form of the double-averaged equations may depend on the type of decomposition of flow variables and that this difference may have important implications for modeling. We also show that the suggested methodology offers better definitions for hydraulic characteristics, variables, and parameters such as flow uniformity, flow two dimensionality, and bed shear stress.

Vertical profiles of streamwise velocity measured over bed forms are commonly used to deduce boundary shear stress for the purpose of estimating sediment transport. These profiles may be derived locally or from some sort of spatial average. Arguments for using the latter procedure are based on the assumption that spatial averaging of the momentum equation effectively removes local accelerations from the problem. Using analogies based on steady, uniform flows, it has been argued that the spatially averaged velocity profiles are approximately logarithmic and can be used to infer values of boundary shear stress. This technique of using logarithmic profiles is investigated using detailed laboratory measurements of flow structure and boundary shear stress over fixed two-dimensional bed forms. Spatial averages over the length of the bed form of mean velocity measurements at constant distances from the mean bed elevation yield vertical profiles that are highly logarithmic even though the effect of the bottom topography is observed throughout the water column. However, logarithmic fits of these averaged profiles do not yield accurate estimates of the measured total boundary shear stress.

To estimate bed-load sediment transport rates in flows over bed forms such as ripples and dunes, spatially averaged velocity profiles are frequently used to predict mean boundary shear stress. However, such averaging obscures the complex, nonlinear interaction of wake decay, boundary-layer development, and topographically induced acceleration downstream of flow separation and often leads to inaccurate estimates of boundary stress, particularly skin friction, which is critically important in predicting skin friction over 2D bed forms. The approach is based on combining the equations describing the mechanics of the internal boundary layer with semi-empirical structure functions to predict the velocity at the crest of a bed form, where the flow is most similar to a uniform boundary layer. Significantly, the methodology is directed toward making specific predictions only at the bed-form crest, and as a result it avoids the difficulty and questionable validity of spatial averaging. The model provides an accurate estimate of the skin friction at the crest where transport rates are highest. Simple geometric constraints can be used to derive the mean transport rates as long as bedload is dominant.

Detailed laser-Doppler velocity and Reynolds stress measurements over fixed two-dimensional bed forms are used to investigate the coupling between the mean flow and turbulence and to examine effects that play a role in producing the bed form instability and finite amplitude stability. The coupling between the mean flow and the turbulence is explored in both a spatially averaged sense, by determining the structure of spatially averaged velocity and Reynolds stress profiles, and a local sense, through computation of eddy viscosities and length scales. The measurements show that there is significant interaction between the internal boundary layer and the overlying wake turbulence produced by separation at the bed form crest. The interaction produces relatively low correlation coefficients in the internal boundary layer, which suggests that using local bottom stress to predict bed load flux may not only be erroneous, it may also disregard the essence of the bed form instability mechanism. The measurements also indicate that topographically induced acceleration over the bed form stoss slope has a more significant effect in damping the turbulence over bed forms than was previously supposed, which is hypothesized to play a role in the stabilization of fully developed bed forms.

The net transport of sediment by suspension is calculated by integrating the product of the sediment concentration and velocity over the water depth. For many flows, accurate calculations must incl...

[1] We address some unsolved methodological issues in modeling of natural rough-bed flows by critically examining existing approaches that parameterize rough-bed flows. These often utilize loosely defined variables. Here we suggest that double-averaging (in time and in a volume occupying a thin slab in a plane parallel to the mean bed) provides a rigorous, straightforward alternative that can aid in the parameterization process. We further present two examples: two-dimensional bed form and gravel bed flows. We argue that the double-averaging approach based on momentum equations should serve as a better methodological basis for modeling, phenomenological developments, and parameterizations. These equations explicitly include drag terms and form-induced momentum fluxes due to spatial heterogeneity of the time-averaged flow in the near-bed region. They also provide a solid basis for better definitions of basic flow variables including the shear stress partitioning into turbulent and form-induced momentum fluxes, skin friction, and pressure (form) drag. We show that for a range of rough-bed flows the vertical distribution of the double-averaged velocity consists of two distinct regions: (1) a linear region below roughness tops, and (2) a logarithmic region above them.

[1] Dunes formed in response to fluid flow exert a total boundary stress on the fluid that is made up of form drag and skin friction, the latter of which is generally considered important for predicting sediment transport and dune evolution. Previous research has used various methods to estimate the total stress and its subcomponents, with recent work suggesting the application of a spatial average to the equations of motion. A complete, three-dimensional (3-D) form of this analysis is derived and applied to recent measurements of turbulent open-channel flow over 3-D dunes. Spatial averages of Reynolds shear stresses over 3-D dunes cannot be used to predict total boundary shear stress. However, estimations of total boundary shear stress from the spatially averaged equations of motion agree with not only the sum of form drag and skin friction but also the direct measurements of average free surface slope. Form-induced stresses from secondary circulations augment the relatively low Reynolds shear stresses over the 3-D dunes. These low turbulence levels have ramifications for sediment transport predictions, in that use of the total boundary shear stress in prediction of sediment transport over 3-D dunes must account for these low turbulence levels so as not to overpredict transport rates.

Turbulence measurements from the flow over two-dimensional fixed dune shapes are presented along with analysis and discussion of the ramifications of the observations for transport of sediment as bed load over bed forms. The spatial structure of the local transport rate determines the shape and stability of bed forms such as ripples and dunes, and the transport of sediment is a highly nonlinear process that is profoundly affected by the statistics of the temporal fluctuations in the near-bed flow field. The measurements presented herein show strong spatial evolution of the joint probability distribution of the streamwise and bednormal fluctuating velocity components. Unlike measurements in uniform boundary layers, these distributions and the higher moments of the velocity do not scale with the local shear velocity, indicating that it is probably inappropriate to use the shear stress to characterize the sediment flux. This conclusion is supported by observations of sediment flux over a dune.

Calculation of suspended sediment transport rates requires integration of the product of sediment concentration times velocity over the depth of the flow; hence accurate profiles of these quantities are required especially near the sediment bed. Although this would appear to be a straightforward task, many effects can cause significant deviation of these quantities from those values expected from simple known relationships. For example, density stratification by suspended particulates can dampen turbulence fluctuations and significantly reduce the effective viscosity and diffusivity of the flow. The momentum and suspended sediment equations are solved herein, including the effects of stratification by suspended sediment and the effects due to size distribution and bedforms. The results show that stratification and bedforms generally reduce the transport capacity of the flow, whereas treating the sediment as an assemblage of different sizes rather than a single size (D50) increases the expected transport.

Sand ribbons typically are shallow-water sedimentary features characterized as thin, elongate bands of sand overlying a coarser lag sediment which is exposed between them. The ribbons are oriented along the direction of the dominant current, and are formed by a secondary circulation pattern that includes a small component of the boundary shear stress directed toward the center of the sand ribbons. A perturbation expansion, coupled with flume experiments, indicates that this circulation pattern is linked to differences in the bottom roughness between the sand ribbons and the adjacent coarser strips. The non-uniform turbulence field arising from this roughness distribution drives a secondary helical flow. The expansion indicates that there is a preferred ratio of sand ribbon spacing to water depth (∼4:1) for unstratified channel flow conditions. The secondary flow pattern forces the downstream flow to readjust so that the boundary shear stress over the smooth sand strips is considerably smaller than the mean stress. This serves to retard the movement of sand through such a system.