Present work is aimed at the derivation of a simply usable equation for the total flow resistance associated with river bedforms, by a unifying approach allowing for bypassing some of the limiting restrictions usually adopted in similar types of studies. Specifically, we focused on the effect induced by the out-of-phase free surface undulations appearing in presence of sand dunes. The proposed expression, obtained by combining the balance of momentum referred to the control volume whose longitudinal dimension coincides with the dune wavelength and the energy balance integrated between its extreme sections, was tested by comparison with some laboratory experimental measurements available in the literature and referred to steady flow past fixed, variably rough bedforms. In terms of shear stress or friction factor, the proposed theory provides estimates in good agreement with the real data, especially if evaluated against the performances provided by other classical similar approaches. Moreover, when analyzed in terms of hydrodynamic dispersive properties as a function of the skin roughness on the basis of a previously derived analytical solution, the dune-covered beds seem to behave like meandering channels, responsible for a globally enhanced fluid particles longitudinal spreading, with a relatively reduced effect in the presence of less pronounced riverbed modelling.

Stream flow along undulated beds has been widely studied in the past, both analytically and experimentally, in terms of resistance factors [

Indeed, for a long time it has been commonly accepted that large-scale bedforms like dunes and bars are generated by large-scale turbulence, vertical or horizontal, respectively [

The periodic (or, more realistically, pseudoperiodic) distortion of the velocity field, in its turn, triggers off the deformation of the mobile bed and, therefore, the creation of typical bedforms. From turbulence measurements, it has been found that the fully-developed dune wavelength

The aim of present work is represented by the analytical derivation of a closed-form solution for the estimation of the total resistance in presence of river dunes (a typically two-dimensional sand bed pattern) in perfect equilibrium conditions, that is, neglecting the effects of their downstream migration, based on the elaboration of the global momentum and energy balances. The final expression accounts for skin and shape roughness in a nontrivially additive way, and for the free surface undulation as a function of Froude number; the intrinsically nonstationary phenomenon of the flow separation near the dune crest is evaluated in terms of resistance resorting to an equivalent steady flow configuration.

The proposed approach is then tested by comparing the analytical estimates of the total friction factor with the laboratory measurements provided by Engel and Lau [

Consider steady and periodically uniform two-dimensional streams. For weak curvatures of the fluid particles trajectory determined by bottom and free surface undulations, the pressure can be assumed as hydrostatically distributed along all the verticals. Note that, in the separated flow zone downstream the dune crest, the validity of the assumption is guaranteed by Helmholtz theory and by the substantial immobility of the fluid subject to the geopotential. In the same zone, the shear stresses are usually neglected due to their very small negative values. Finally, in the low Froude number regime flows, water and bed sediment waves are commonly out of phase. Specifically, it is reasonable to expect the maximum drop of the free surface near the dune peak, due to the acceleration induced by its upstream positive slope, and the maximum rise in the separated and highly turbulent zone where the flow, after having achieved the minimum velocity, realigns and accelerates again (see Figure

Sketch of the physical model.

with

It is worth noting that in (

In order to test the proposed formulation, we employed the experimental data provided by Engel and Lau [

Comparison between measured [

Comparison between measured [

As Figures

Histograms of percentage errors corresponding to the use of (

Figure

Behaviour of the calculated friction factor as a function of Froude number.

Effect of Froude number on the free surface relative drop.

In order to double-check the soundness of the proposed approach, (

Comparison between measured [

Finally, for the sake of completeness, Figure

Behaviour of the dimensionless asymptotic dispersion coefficient as a function of the skin friction factor for different dune slopes.

As Figure

Present work has proposed the analytical derivation of a simple and easily usable closed-form solution for the estimation of the total flow resistance in presence of river dunes, based on the application of the global equilibrium equations. The final expression accounts for skin and shape roughness in a non trivially additive way and for the free surface undulation as a function of the Froude number; the intrinsically non stationary phenomenon of flow separation at the dune crest is handled in terms of resistance based an equivalent steady flow configuration.

That expression is tested by comparing the analytical estimates of the total friction factor with the laboratory measurements provided by Engel and Lau [

In order to double-check the soundness of the proposed approach, (

Finally, when analyzed in terms of hydrodynamic dispersive properties as a function of the skin roughness on the basis of a previously derived analytical solution, the dune-covered beds seem to behave like meandering channels, responsible for a globally enhanced hydrodynamic solute particles spreading, though with a relatively reduced effect in presence of a less pronounced riverbed modelling.