River damming produces alterations on the natural river functionality, both in the water discharge, as well as in the sediment transport and connectivity [1
]. In particular, large dams [3
] are estimated to trap more than 99% of the sediments entering the reservoir [4
]. This causes the progressive silting of the reservoir and inhibits the sediment load in the river flow downstream of the dam, thus altering the river morphology [5
] and the aggradation/degradation dynamics that are closely linked to the balance between upstream sediment supply and local transport capacity conditions [6
]. Sediment supply-limited conditions from upstream cause the (selective) erosion of finer particles from the granular bed, until the flow is unable to move the coarser grains and new equilibrium conditions are reached [7
]. During this degradation process, the median size of the bed material progressively coarsens and the sediment transport rate decreases, leading to a process known as bed armoring [8
Occasionally, armored stream beds may be subject to high sediment loads, for example during dam flushing or removal operations, or during flood events associated to large sediment input from lateral inflows. Under these conditions, finer particles infiltrate into the void spaces of the immobile coarse bed grains, according to a selective trapping mechanism that is reciprocal to the selective erosion that caused bed armoring [11
]. If the infiltration of fine particles into the coarse bed interstices is extensive, the volume of voids drops increasing the compactness of the stream bed texture, thus decreasing its hydraulic conductivity and increasing its resistance to flow (e.g., [12
]). This process is known as colmation or clogging (see, for example, [14
] for a review), and has significant impacts on stream ecology (e.g., [15
]), exchanges of water, dissolved substances and heat with the underlying hyporheic zone and groundwater [19
], and flow and turbulence structure [20
]. Under high flow conditions, the armour layer can break up and the entire river bed becomes mobilized, hence resetting the bed morphology and grain size distribution. However, if the erosion capacity of the flow is not sufficient to remove the coarse grains, as soon as the sediment load from upstream declines, the reestablishment of sediment supply-limited conditions reactivates selective erosion of finer particles, sustaining the armoring of the stream bed.
The cleaning dynamics controlling the erosion of finer particles from coarse granular beds is inherently different from that typical of uniform sized beds. In fact, in armored beds, the presence of macro-roughness due to the coarser particles alters the flow structure and, consequently, the distribution of the stress components below the gravel crest level. In these beds, besides turbulent stresses, form-induced stresses and form drag also contribute to the total shear stress distribution [23
]. In addition, the vertical component of the stress responsible for lifting and transporting fine material was found to be decreasing below gravel crest level [22
]. This alteration is mirrored in the reduced sediment entrainment and transport capacity of the flow, which is affected also by the smaller fine sediment-water active interface with respect to that of a uniform bed.
It is therefore clear that the traditional formulae derived for uniform bed cases fail to describe erosion and sediment transport processes over immobile gravel beds. In fact, these formulae do not account for the reduction in the effective part of the shear stress, nor the reduction in the fine sediment-water active interface. In this respect, performing laboratory experiments is a common methodological practice for investigating selective transport dynamics in armored beds and gaining useful elements to derive empirical formulas of fine sediment transport (e.g., [26
]). Experimental research is typically carried out in laboratory flumes using laser-scanner (e.g., [29
]) or digital photogrammetry (e.g., [30
]) to measure the changes in the topography. The track of these changes in the fine sediment level inside the gravel matrix is usually coupled with measures of transport rate (trough sieves or density cells (e.g., [27
]) or concentration (e.g., [26
]) to quantify the fine sediment transport and/or erosion rate between the gravel. Based on these experimental approaches, useful fine sediment transport formulae has been proposed in previous literature, as for instance in [26
Despite the above cited empirical formulae and despite the examples of direct measurements of the flow field in the roughness layer [23
] (e.g., [22
]), a comprehensive framework on the inter-grain flow and sediments dynamics in gravel bed rivers still poses some scientific challenges. These challenges are primarily due to the operational difficulties to perform velocity measurements far below the gravel crest level and to quantify the relative contribution of the form drag to the total shear stress [25
]. In addition, a fair comparison among existing studies is not obvious due to the differences in the bed topographies, which chiefly controls the distribution of the shear stress components [25
], thus hampering the derivation of general considerations. In this context, fine-scale numerical models can offer a valid alternative to overcome the inherent difficulties of fine-resolution, inter-grain experimental measurements. At the same time, they can easy the investigation of the role of the geometry in affecting the stresses distribution, provided that the setup and repetition of laboratory experiments with different configurations is not a minor matter. While examples of Direct Numerical Simulation (DNS) over rough bed configurations do exist (e.g., [33
]), the inclusion of sediment active layers in fine-resolution hydrodynamics model is a relatively unexplored area of research.
In this study, we present and test a new semi-implicit numerical scheme for the solution of the two-dimensional Navier-Stokes equations, in which we included the possibility to easily simulate sediment entertainment and transport processes. The scheme, based on the method proposed by [36
], is mass-conservative, computationally efficient, and able to solve the small-scale structures that characterize inter-grain flow field. In this study, we present proof-of-concept and preliminary results of this model as a first step towards its extension to a complete three-dimensional model coupled with a turbulence closure scheme. To this end, we focus on the validation of the proposed model against numerical tests and on showing its potential for applications in the context of fine sediment transport dynamics in gravel bed rivers.
A second order semi-implicit numerical scheme on staggered Cartesian meshes for the incompressible Navier-Stokes equations, based on the method proposed by [36
] in presence of a time dependent sedimentation/erosion process, was derived. In the scheme, we defined the hydraulic head in the cells centers and the velocity at the cells interfaces. By formally substituting the discrete momentum equations into the discrete continuity equation, we obtained a symmetric semi-positive definite linear system where the only unknown is the hydraulic head at the new time step. The system is then solved using a fast iterative linear solver such as the conjugate gradient algorithm [49
]. We note that the method is built in such a way that the computation in each cell involves only its direct neighbors. This makes the algorithm particularly suitable to parallelization, since the data that need to be synchronized are limited to the single layer of cells surrounding each parallel region.
For the entrainment and deposition of the sediment we used an explicit finite volume scheme in combination with a general mass flux between suspended and deposited sediment. The deposition of the suspended sediment changes the effective domain sizes in terms of volume and edges length in the cells. The method is mass-conservative and limited in the time discretization by a classical CFL-type time restriction based on the local fluid velocity. However, if the convective-viscous terms are solved by an Eulerian-Lagrangian method combined with a local time stepping/subcycling approach for the sediment dynamics, the method becomes unconditionally stable. Furthermore, compared to [36
], the pressurized system allows for avoiding the solution of the mildly nonlinear contribution through the Nested-Newton approach [50
], which ultimately reduces the computational time thus allowing for a fine resolution in the mesh.
The method was validated against some classical benchmarks, i.e., the Blasius boundary layer and the lid-driven cavity test. Moreover, a modified version of the lid-driven cavity test was run, to verify the conservation of sediment and water mass in presence of erodible sediment, and the robustness of the model in presence of time-varying boundaries of the fluid domain.
Once validated, the model was used to simulate three simple cases representative of typical experiments for gravel bed rivers at different filling rates. The numerical results for the inter-grain flow field show the formation of main and secondary circulation cells, forced by the presence of the macro-roughness elements, which generates a double inflection in the time-averaged velocity profile for the lower filling rates. This information would probably be lost if performing only experimental measurements of the flow dynamics, which are possible just at discrete points, especially below the gravel crest. Furthermore, the use of a numerical model simplifies the repetition of the experiments considering different topographies, which contributes at improving the understanding of the geometry role in the stresses distribution.
We note that the model is two-dimensional, therefore it does not account for the three-dimensional effects and its results are not immediately representative of the real case. While a full description of the inter-grain flow field and turbulence structure would require the use of a three-dimensional model coupled with a proper turbulence closure scheme, even in this form the proposed model provides useful clues on the approximation effects introduced when using simplified geometries to represent real topographies.
Future work will concern the extension of the present approach to the complete VOF (Volume Of Fluid) method such as proposed in [36
], and its extension to the three-dimensions in the presence of erodible sediment, together with the inclusion of a proper turbulence closure and high-performance parallelization standards.