*2.4. Computational Setup*

The Tonstad power plant sand trap is 184 m-long, and the main tunnel has a crosssectional area of around 9.9 m × 11 m (around 119 m2). A picture of the prototype sand trap is shown in Figure 3. A 3D model of the sand trap is developed using engineering CAD software, see Figure 4. It was created from engineering drawings provided by the Sira-Kvina power company—the owner of the plant—and consists of a rectangular inlet section, a diffuser, a long tunnel section with a gentle slope, and a weir in the invert where the tunnel transitions into the penstock. This model is used as a base to add upgrades to in two other models. The additional models were created to test the effects of the upgrades, where one model includes only ribs and another includes both ribs and v-shaped rakes.

**Figure 3.** Picture of a prototype sand trap tunnel at hydropower plant; courtesy of Sira-Kvina kraftselskap.

**Figure 4.** (**a**) Sand trap without upgrades. The distance from the end of the diffuser to the weir is 184 m. The diffuser is 16 m-long. The sand trap tunnel has a cross-sectional area of 119 m2. (**b**) Ribs. The ribs are placed just upstream of the penstock, and are 1 m-wide and spaced 1 m-apart. The ramp has an 8% inclination. (**c**) Top view of the v-shaped rakes in the diffuser. The v-shaped rakes measure 6 m in height. Distances tip-to-tip between rakes are 1 m and 0.8 m for the upstream and downstream row, respectively. Zoomed part shows rake dimensions.

The ribs are placed just upstream of the penstock, in combination with the weir. This setup was tested in experiments [4,5]. The purpose of the ribs is to allow bed load sediments to fall between the ribs, while water is minimally affected. There are five ribs in total. The length of the ribs and the gap between ribs both measure 1 m. A ramp is placed upstream of the ribs to raise them above the bed and create space for sediments to accumulate. Further, in this way, excavating into the tunnel floor is not necessary. In

the work by Richter et al., the ramp was also found to protect sediments that had fallen between the ribs from being resuspended and flushed into the penstock.

The rake structure is made up of two rows of v-shaped rakes, with the tip of the rakes facing downstream. The rakes measure 6 m in height. Distances between rakes are 1 m and 0.8 m for the upstream and downstream row, respectively. The purpose of the rakes is to even the flow of the jet from the inlet and enhance the diffuser effect of slowing down the flow. The distance from the inlet to the rakes is 25 m, which is approximately 3 times the diameter. To ensure that stable and developed flow reaches the rakes, a distance of 5–10 times the inlet diameter is necessary. The fact that the flow reaching the rakes may not be fully developed in the simulations should be taken into consideration when analysing the results.

The three models of the sand trap are meshed similarly. In all three models, the diffuser and the invert are both given tetrahedral mesh structures, while the simpler geometries of the inlet section, tunnel, and penstock have structured hexahedral meshes. The number of cells in each mesh is >23 × <sup>10</sup>6. Inflation layers are added along the tunnel walls so that global *y*<sup>+</sup> < 30. This ensures accurate representation of flow conditions in the boundary layers. The mesh surrounding the ribs and the rakes are refined further to a maximum cell size of 0.01 m. Average element sizes in the diffuser, tunnel, and rib section are 1.05 mm, 1.39 mm, and 0.43 mm, respectively.

The chosen advection scheme is the "High Resolution" scheme available in CFX. This scheme is second-order accurate in smooth regions and reduces its order of accuracy in regions of high gradients, where unboundedness is a factor. A first-order-accurate scheme is generally more robust and reaches convergence criteria faster than a second-orderaccurate scheme. However, this comes at the cost of higher numerical diffusion, resulting in a less-accurate solution. The transient scheme used is the Second-Order Backward Euler scheme.

Steady state multiphase simulations are run to create initial conditions for the transient multiphase simulations. The total simulated time for the transient simulations is set to allow sediments enough time to either reach the bed or exit through the outlet. The simulations use a discharge of 80 m3/s, which is the discharge when the power plant is operating at design conditions. The inlet velocity boundary condition represents this mass flow rate. The wall roughness is 10−<sup>3</sup> m in the numerical model. An overview of solution parameters and boundary conditions is presented in Table 1.


**Table 1.** Selected parameters and boundary conditions for unsteady simulations.

Multiphase flow was implemented by enabling the particle transport solid model in CFX, also known as the discrete phase model (DPM). DPM uses the Eulerian–Lagrangian multiphase model to track the paths of individual particles as they travel through the domain. It is well-suited for situations such as in the present work, where the volume fraction of the solid phase is low. The one-way coupled fluid pair model was chosen to solve the fluid–particle interactions. This model is computationally cheaper than the two-way coupled model. As the sediment phase's effect on the fluid phase is negligible in this case, the one-way coupled model was found to produce satisfactory results. In the transient multiphase simulations, sediments were injected with uniform distribution over the inlet during the first 100 s. The mass flow of sediments was set to 1000 kg/s, leading to a total mass of sediments injected into the sand trap of 10<sup>5</sup> kg. The sediments were tracked as they travelled through the model, and the sand trap efficiency was given by the time-integrated mass flow report at the outlet by the end of the simulation. The sediment diameters have a normal distribution with a mean of 0.75 mm and a standard deviation of 0.25 mm.

By also ensuring that the solution is mesh independent, the accuracy of the results are further improved. Mesh independence is achieved when a defining value of the simulation no longer changes significantly. The purpose of a mesh-independence study is to minimise the discretisation error resulted from approximating the geometry during the meshing stage [18]. The model with both rakes and ribs was used for the meshindependence study. Three different mesh qualities were used. Following the procedure described by Celik I. B. et al. [19], an estimation of the discretisation error was obtained. In the calculations, *N* is the number of cells; *h* is the representative cell size; *r* is the refinement factor; *φ* is the pressure drop from inlet to outlet; *p* is the apparent order of the method; *φext* are the extrapolated values of *φ*; *ea* and *eext* are the approximate and extrapolated relative errors, respectively; and GCIfine is the fine-grid convergence index. The mesh discretisation error was found to be 1.4%. The results of the calculations are presented in Table 2. The fine mesh with 26.9 million cells is considered for further analysis and will be used to conduct the final numerical studies.


**Table 2.** Parameters of the mesh-independence study.

The simulation results are verified by ensuring that residuals reach a satisfactory convergence criteria and that there is a stable mass flow through the domain. This signifies that the solution is computationally correct. The hydraulic representation of the numerical model will be validated by comparison with PIV measurements on a physical scale model of the sand trap and to ADCP measurements from the prototype sand trap [4,20]. As the flow state is highly stochastic, the velocity distributions will never be identical. However, it is useful to make sure that the jet in the diffuser observed in the PIV measurements also exists in the simulated flow. It is also important to ensure that the simulated velocities are of reasonable magnitudes.
