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The paper briefly outlines the double-averaging methodology for studying environmental rough-bed flows. It focuses on the applications of this methodology in environmental hydraulics by providing several examples illustrating advantages of this methodology over conventional approaches. Examples include: (1) identification of specific flow layers and flow types; (2) vertical distribution of the double-averaged velocity between the roughness tops and troughs; (3) vertical distribution of momentum fluxes and sinks for typical roughness types due to turbulence, mean flow heterogeneity, secondary currents, form drag, and viscous drag; (4) estimates of form-induced (dispersive) stresses and evaluation of their structure using quadrant analysis; and (5) closure development for mass-transfer-uptake processes for stream periphyton. These examples illustrate the advantages of the double-averaging methodology over conventional approaches as well as highlight its potential for studying flows over very rough beds, hig...

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 subelement 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 CD =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 ...

A series of experiments was undertaken to assess fully rough turbulent subcritical flow over two-dimensional transverse repeated-rib roughness of varying spacing λ∕h=1–16 (roughness spacing/height ratio), with h∕H=0.09 (roughness height/flow depth). Each of the 11 experiments involved centerline measurements of three-dimensional velocity vectors, water surface profiles, and bed pressures and forces. The structure of flow over rib roughness consists of a pair of principal vortices of opposite sign set up by the ribs. These vortices are superimposed on an overall double- (time and space)-averaged velocity profile that is (quasi-) logarithmic above roughness tops, and that below roughness tops changes with increasing rib spacing from exponential (λ∕h

Several models for the vertical distribution of the double-averaged (in time and in the plane parallel to the mean bed) longitudinal velocity in the flow region between roughness troughs and roughness tops are suggested. We found that the same model for velocity distribution may be applicable to a range of flow conditions and roughness types, which share some common features. The suggested models for velocity distribution in the near-bed region are: (1) Constant velocity; (2) exponential velocity distribution; and (3) linear velocity distribution. The measured velocity distributions may be approximated by a single model or by a combination of models depending on roughness geometry and flow conditions. The validity of these models for velocity distribution is supported by laboratory data.

The interactions between turbulence events and sediment motions during bed load transport were studied by means of laser-Doppler velocimetry and high-speed cinematography. Sweeps (u′ > 0, w′ 0 w′ > 0) which contribute negatively to the bed shear stress and are relatively rare, individually move as much sediment as sweeps of comparable magnitude and duration, however, and much more than bursts (u′ 0) and inward interactions (u′ < 0, w′ < 0). When the magnitude of the outward interactions increases relative to the other events, therefore, the sediment flux increases even though the bed shear stress decreases. Thus, although bed shear stress can be used to estimate bed load transport by flows with well-developed boundary layers, in which the flow is steady and uniform and the turbulence statistics all scale with the shear velocity, it is not accurate for flows with developing boundary layers, such as those over sufficiently nonuniform topography or roughness, in which significant spatial variations in the magnitudes and durations of the sweeps, bursts, outward interactions, and inward interactions occur. These variations produce significant peaks in bed load transport downstream of separation points, thus supporting the hypothesis that flow separation plays a significant role in the development of bed forms.

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.

[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.

[1] A series of detailed measurements were made of unidirectional turbulent open-channel flow over fixed, artificial, sinuous-crested three-dimensional (3-D) dune shapes. The response of the mean free surface was two-dimensional (2-D). Mean streamwise velocities were largest over the node of the sinuous dune crest line rather than over the maxima or minima of the crest line. Friction coefficients of the 3-D dunes were 50% higher on average in comparison to those of their 2-D counterparts when subjected to flows of similar depths and discharges. Despite the larger friction of the 3-D dunes, the turbulence generated by them was much weaker than that generated by their 2-D counterparts. The spatial structure of turbulence caused by the sinuous crest line of the 3-D dunes was correlated with the presence of secondary currents. These secondary current cells directed flow upward in the regions where the streamwise velocities were low and downward in the regions where streamwise velocities were high, suggesting that they could be responsible for a significant amount of momentum flux not being carried by the low turbulent stresses over the 3-D dunes.

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.

In order to predict the response of deep ocean sediments to the near-bottom currents, accurate estimates of the near-bottom velocity and the boundary shear stress are necessary. Because of the unsteady nature of deep ocean currents, dynamic models are required. This can be accomplished either by time stepping methods or by the use of Fourier transform techniques. Here the latter is applied to the momentum equation and the results are compared to data from the HEBBLE region in the North Atlantic Ocean. An eddy viscosity is employed for closure; however, because the unsteady nature of the problem requires it to vary in time, the uniform one-dimensional Ekman layer equations become nonlinear. Results show that if the Fourier transform method includes the turbulence adjustments in this way, it predicts very accurately both stress and velocity fields.

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.

A knowledge of the transport of sediment in a shallow, high energy environment such as the North Sea is important to the general understanding of such a system. The shape of the basin itself is largely a product of the interaction between the currents and the sediments. The engineers are interested in maintaining harbors and predicting morphological adjustments caused by man-made alterations of currents. The environmentally conscious are concerned about pollution and water quality which both are often strongly linked to the sediments. Sediment transport involves a complex interaction between the flow and the bottom. Ultimately an accurate knowledge of the forces acting on the bottom (i.e., the boundary shear stress) is necessary in order to predict the initiation of sediment movement and subsequent transport rates. In the North Sea this is complicated by the unsteady nature of the flow. This arises from both storms and tides. In order to better understand such a system, it was decided to begin by investigating the boundary layer in regions of active transport where the flow was dominated by tidal effects.