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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...

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.

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

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 stability of ripples and dunes in flows of finite depth are investigated for the linear, small-amplitude case and also for the more realistic finite-amplitude situation. For the former perturbation approach is taken, leading to the general result that, without the inclusion of some lag between the sediment transport rate and the boundary shear stress, all wavelengths are unstable. This results from an upstream shift in the stress relative to the bedform, leading to deposition at the crest. By assuming a realistic lag based on the mechanics of bed load transport, a fastest growing wave at ripple-scales results: similarly, if suspended sediment were included, longer lags result and a dune-scale instability could result. Some investigators have invoked a gravitational effect, arguing that sediment transport is enhanced downslope and retarded upslope. This creates an effective downstream shift in the transport rate and some argue that both ripple and dune instabilities result, however, a realistic evaluation of this effect shows that it is much too small to be effective. Naturally occurring bedforms are of finite-amplitude; typically they are asymmetrical, often causing flow separation. This highly non-linear process creates a vastly different flow environment from that assumed in the initial stability case. A wake-like flow structure develops downstream of the separation zone and an internal boundary layer develops beneath. The complex interaction of these two different flow regimes as well as the effect induced by the topography of the bedform itself, produces a shear stress field that dictates the stability of the feature itself.

[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.

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.