*2.2. Validation Tests*

The numerical model was first validated against some standard numerical tests, for which reference solutions are available. In the first validation test, we considered the Blasius analytical solution for the laminar velocity profile in the boundary layer above a semi-infinite plate [43]. This validation test was obtained by simulating a plate 0.2 m long, assuming a uniform upstream velocity parallel to it and equal to *u*<sup>∞</sup> = 0.3 m/s. The two-dimensional x-z domain Ω = [−0.01, 0.2] × [0.0, 0.031] m was discretized according to a 600 × 300 grid, for a total of 180, 000 elements having horizontal and vertical dimension of <sup>Δ</sup>*<sup>x</sup>* = 3.5 × <sup>10</sup>−<sup>4</sup> m and <sup>Δ</sup>*<sup>z</sup>* = 1.03 × <sup>10</sup>−<sup>4</sup> m, respectively. We set *θ* = 0.51, *g* = 1 m/s2, and *ν* = 10−<sup>6</sup> m2/s. As for the boundary conditions (BCs), we assumed the velocity *u*∞ as the left BC, transmissive BCs at the right and at the top edges, and no-slip BC at the bottom plate, beginning from *x* = 0 in order to trigger the boundary layer.

The second validation test was performed considering the so called lid-driven cavity problem, which is another classical benchmark test [44]. The problem consists in a cavity Ω = [−0.5, 0.5] <sup>2</sup> m where the initial velocity field is (*u*, *w*) = 0.0 m. We set *θ* = 1, and *g* = 1 m/s2. We imposed (*u*, *w*)=(1, 0) m/s as the top BC and no-slip BCs at the other three boundaries. We repeated the test for two different values of the Reynolds number, *Re* = 400 and *Re* = 1000, assuming the sediment level *S* ≡ 0. For both tests we discretized the domain according to a square grid 400 × 400, for a total of 160, 000 elements having horizontal and vertical dimensions of <sup>Δ</sup>*<sup>x</sup>* = 2.5 × <sup>10</sup>−<sup>3</sup> m and <sup>Δ</sup>*<sup>z</sup>* = 2.5 × <sup>10</sup>−<sup>3</sup> m, respectively. This benchmark test was chosen due to its analogy to the inter-grain regions, where circulation cells develop bounded laterally and at the bottom by the grains and by the fine sediment, respectively.

The same lid-driven cavity test was performed introducing an erodible sediment at the bottom of the cavity, characterized by density *ρ<sup>s</sup>* = 1553 kg/m3 and porosity *φ* = 0.46. This resulted in considering mobile boundaries of the fluid domain due to the variation in time of the bed level *zb*, according to the following equation:

$$\frac{\partial z\_b}{\partial t} = \frac{D - E}{(1 - \phi)}.\tag{23}$$

where *E* and *D* are erosion and deposition rates, respectively. In Equation (23) we assumed *D* = 0, which corresponds to simulating a non-equilibrium erosive process. We considered the same domain Ω, grid discretization, BCs, and *g* as in the original lid-driven cavity test, but we filled the cavity with

sediment up to 1/3 of the cavity height, set *θ* = 0.51, and performed the test for just one value of the Reynolds number, i.e., *Re* = 1000. We defined the lowering of the fine sediment bed through the van Rijn erosion rate formula [45], which, expressed in [*m*/*s*], reads as follows:

$$E = 0.00033\sqrt{g\Delta d\_s}d\_\*^{0.3}T^{1.5},\tag{24}$$

where <sup>Δ</sup> is the fine sediment relative density <sup>Δ</sup> <sup>=</sup> *<sup>ρ</sup><sup>s</sup> <sup>ρ</sup>* <sup>−</sup> 1, *ds* is the fine sediment median diameter, *<sup>d</sup>*<sup>∗</sup> is the dimensionless grain size *d*<sup>∗</sup> = *ds* Δ*g ν*2 1/3 , and *T* is the dimensionless excess of shear stress *<sup>T</sup>* <sup>=</sup> <sup>Θ</sup> <sup>−</sup> <sup>Θ</sup>*cr* Θ*cr* . In the above formulas, *ρ<sup>s</sup>* is the fine sediment density, Θ is the Shields parameter, and Θ*cr* its critical value. The Shields parameter Θ is defined as:

$$
\Theta = \left(\frac{\tau\_b}{\Delta \rho \lg d\_s}\right) \tag{25}
$$

where *τ<sup>b</sup>* is the shear stress at the bottom, here defined as:

$$
\pi\_b = \mu \left. \frac{\partial u(z)}{\partial z} \right|\_{z=z\_b}. \tag{26}
$$

To fasten the simulation, the erosion rate *E* from Equation (24) was multiplied by a factor 100. This test was introduced to validate the water and sediment mass conservation properties of the model, and to verify its robustness when dealing with time-varying changes in the boundaries of the fluid domain.
