Numerical Experiments

Gravel bed rivers are often represented in laboratory experiments by covering the flume bed with spheres or hemispheres (e.g., [26,28,46,47]). We therefore assumed two simplified topographic configurations with the gravel represented by homogeneous spheres with different spacing, according to the most common simplified configurations used in laboratory flumes: in-line arrangement (Figure 2a) and closest-packing arrangement (Figure 2b). In particular, in the first case, we examined the section where two consecutive spheres touch each other (Figure 2d), while in the second case we examined the section with the maximum inter-sphere spacing (Figure 2e).

To investigate the differences between a simplified and a real bed topography, we considered also a real gravel topography obtained from point-laser-scanning a longitudinal section in a laboratory flume covered by gravel (Figures 2c). In this case, we selected a 20 cm long section (Figure 2f), whose granulometric distribution was characterized by *D*<sup>50</sup> = 26.5 mm and *D*<sup>90</sup> = 29.5 mm. In order to set up equivalent geometry configurations, the two simplified cases described above were drawn considering spheres of 30.0 mm diameter.

In addition to the bed topography of the gravel matrix, in the numerical experiments we considered also the presence of inter-grain inerodible fine sediments at fixed level, to see the variations in the flow field depending on the depth. The fine sediment interface was schematized as a horizontal line, whose level was assigned according to four different filling rates: *Z* = 0, 0.25, 0.50, and 0.75, where *<sup>Z</sup>* <sup>=</sup> *<sup>D</sup>*/2 <sup>+</sup> *zb <sup>D</sup>*/2 and *<sup>D</sup>* is either the diameter of the spheres in the simplified case, or *<sup>D</sup>*<sup>90</sup> in the real topography. The vertical coordinate *z* was defined positive upwards, with *z* = 0 at the gravel crest.

All the simulations were performed considering a 0.2 m long domain with a rigid top boundary at 0.03 m above the gravel crest. The domain was discretized according to a 400 × 600 grid for a total of 240,000 elements having horizontal and vertical dimension of *dx* = <sup>5</sup> × <sup>10</sup>−<sup>4</sup> m and *dz* = <sup>10</sup>−<sup>4</sup> m, respectively. We set *θ* = 1, and *g* = 9.81 m/s2, and *ν* = 10−<sup>6</sup> m2/s. The upstream and downstream boundary conditions were as follows: a uniform horizontal inflow velocity equal to 0.3 m/s and

transmissive conditions downstream, with an assigned upstream/downstream pressure gradient equal to 2.5 × <sup>10</sup>−<sup>3</sup> m/m. No slip boundary conditions were assigned at the bottom, and free slip conditions were assigned at the top boundary. We notice that solving the incompressible Navier-Stokes equations has the undeniable advantage of reducing the computational burden, while the simulation setup is equivalent to considering a free-surface flow with a bed slope equal to the pressure gradient.

**Figure 2.** Schematic of the simulated gravel bed topographies: (**a**) in-line arrangement, view from above; (**b**) closest-packing arrangement, view from above; (**c**) real gravel bed, 3D view; (**d**) in-line arrangement, longitudinal section; (**e**) closest-packing arrangement, longitudinal section; (**f**) real gravel bed, longitudinal section. In panels (**d**–**f**), the horizontal dashed lines indicate the fine sediment filling rate, while the vertical lines indicate the gravel crest (dotted) and inter-grain cavity (continuous) sections.
