Respuestas al ítem: “He identificado las cosas que hemos aprendido del

The mechanisms underlying the erosion processes of sand and/or silt are relatively well understood and documented. Sediments properties such as particle size, shape and packing density can be reasonably used to predict erodibility of these non-cohesive sediments (Grabowski et al., 2011; Mehta, 2014). In addition to the sediment properties, erosion of this class of sediments depends on the flow-induced forces. Consider the forces acting on a particle at the surface of a horizontal bed (Figure 2.19), being a cohesionless particle, it is assumed to be non-deformable, each with an identity when at rest or in motion (some authors have argued that cohesive floc can be ideally treated as sand grain based on these assumptions (e.g. Winterwerp and van Kesteren, 2004; Mehta, 2014; etc.).

Figure 2.19 Forces on a particle at a horizontal bed surface subject to turbulent flow (Adapted from Mehta, 2014)

Above the particle (Figure 2.19), for curved streamlines, a low pressure is generated which induces a vertical lifting force FL. The drag force is as a result of flow by means of

viscous skin friction and low pressure, FD, acting at the same level with bed surface (plane

of zero-velocity). If the cohesive-adhesive force (represented by FC), which may be

assumed to be equal to inter-particle electromagnetic attraction, is considered, then the total resisting force on the particle will be particle buoyant weight (Fg) and FC. Therefore,

the net normal force on the particle can be given as: ‘Fg + FC - FL’. Angle φ𝑎 is termed

angle of repose, and corresponds to the threshold of movement of the first particle Fg + Fc Fg + Fc - FL FD FL Ub P ϕa Plane of zero velocity

Resultant force at the threshold of movement

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anywhere on the bed surface (Mehta, 2014). FD can be assumed to be = 𝑐∗ 𝑑𝑝τ𝑐 (where,

𝑐∗ is a coefficient that takes into account the geometry and packing of the grains and the

variation of the drag coefficient – it is equally expected to vary with boundary Reynolds number; 𝑑_𝑝 is the particle diameter; and 𝜏_𝑐 is the average boundary shear stress).

Shields (1936) developed a parameter to determine the initiation of motion for cohesionless particle (sand) based on the principle of force balancing described above. The model is generally referred to as Shields’ entrainment parameter 𝜃𝑐𝑟 (Equation 2-

22), but when FC is negligible, θ𝑐𝑟 for coarse particles can be defined as a function of

the particle roughness Reynolds number 𝑅𝑒∗ (Equation 2-23). Examples of some flume

experimental data relating Equations (2-22) and (2-23) are shown in Figure 2.20 (i.e. Shields’ diagram). Generally, if Re∗ is known, θ𝑐𝑟 of the particle can be determined and

by extension the critical shear stress (τ𝑐) of the particle. Under given flow conditions, a

Shields parameter greater than the critical line (Figure 2.20) will result in motion of non- cohesive sediments, i.e. the particles start rolling and sliding as the applied shear stress is reaching critical shear stress.

θ𝑐𝑟 = _(ρ |τc|

𝑠− ρ𝑤)𝑔𝑑𝑝 (2-22)

where 𝜏𝑐 (N/m2) is the critical bed shear stress at the threshold of erosion, 𝑔 (m s-2) is the

acceleration due to gravity, 𝑑𝑝 is the particle diameter, and 𝜌𝑠 and 𝜌𝑤 are the particle

and fluid density respectively.

𝑅_𝑒∗ = 𝑢∗𝑐_𝑣𝑑𝑝 ; 𝑢_∗𝑐 = _√ρτ𝑐

𝑤 (2-23)

where 𝑣 is the kinematic viscosity of water and 𝑢∗𝑐 is the critical value of the friction

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Figure 2.20 Initiation of motion according to Shields (1936) [i.e. Shields’ diagram]

A wide range of flow conditions for which there is weak but noticeable sediment movement is one of reasons why determination of particle movement threshold is so challenging, which leads to the general problem of how to define the condition of incipient movement in the first place. This explains why the Shields diagram is under criticism in its application to define condition of incipient movement; for example, because 𝜏_𝑐 and 𝑑_𝑝 both appear in the axis variables it will be impractical to use it to find the threshold shear stress that corresponds to a given sediment diameter, or to find the largest sediment diameter that is moved by a given shear stress. Van Rijn (1993) suggested that the Shields curve in terms of 𝑅_𝑒∗ and 𝜃_𝑐𝑟 is largely not practical because 𝜏𝑐 value can only be obtained by iteration. Nonetheless, the Shields diagram continues to

be used, because it gives good ballpark results for both engineering and sedimentological purposes. Recently various sand erosion laws (often called pick up functions) have been formulated to address the pitfalls of Shields diagram, although experimental validation of such laws is rare. Most of these published sand erosion laws (e.g. van Rijn, 1985; Beach and Sternberg, 1988; Nielsen, 1992; etc.) are expressed by relating erosion rate as a function of the excess-shear stress to a power, which value varies depending on the sand diameter (e.g. higher power for smaller particles) [Le Hir et al., 2008]:

𝐸𝑠𝑎 = 𝐸0,𝑠𝑎[_𝜏𝜏𝑏

𝑒𝑠− 1]

𝑛𝑠

for τb > τ𝑒𝑠 (2-24)

where τ_𝑒𝑠 is the sand particles critical stress for erosion (e.g. deduced from Shields curve, and parametrically formulated in terms of grain size and density by Soulsby, 1997); 𝑛𝑠

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is generally optimised from calibration and varies according to authors [e.g. 𝑛𝑠 = 1.5 in van Rijn (1985); 𝑛𝑠 =1 in Beach and Sternberg (1988); 𝑛𝑠 =0.5 in Waeles et al. (2007)].

Bed shear stress obtainable from the erosion formulae can be used to characterise the modes of sand particles transport. For example, at relatively low bed shear stress but still sufficient enough to initiate motion but incapable of suspending the particles in the overlying water body, the bed load transport regime dominates, where particles are only experiencing the first three modes of sediment transport (i.e. sliding, rolling and saltation). Bedforms in the scale of ripples can be formed in such a situation. Generally, ripples are formed when shear stresses are about 10-20% larger than critical shear stress (Julien, 1995; Soulsby, 1997). For relatively higher bed shear stress, the predominant mode of transport is suspension, although as recorded by Soulsby (1997) bedload can still occur but the quantity af sand that will be carried in suspension will often be very much greater than that carried by bedload, especially for fine sands. The transport regime is called sheet flow regime (i.e. the flow regime in which the shear stress exerted on the sediment bed by the fluid flow is strong enough to set in motion a thick layer of particles) where due to strong flows ripples are washed out.