## 2 Fluidic Oscillator Main Characteristics  
The central part of the (FO) considered in the present study is introduced in Figure 1, the four internal geometries modified, the mixing chamber (MC) inlet and outlet widths and angles, are clearly shown. Figure 1 also introduces the positive and negative directions taken for each geometry modification. Regardless of the configuration evaluated, an orthogonal 3D mesh with 2,242,000 cells was used to evaluate the flow at Reynolds numbers up to 16,034, for the baseline case configuration the respective maximum x+, y+ and z+ were of 1.8, 4.7 and 1.2, very similar values were obtained for the rest of the configurations. A mesh with 5,933,900 cells was used to perform all simulations at Reynolds number 32,068. The maximum values of x+, y+ and z+ obtained with this mesh and for the baseline case at Reynolds number 32,068, were of 0.9, 1.2 and 0.7, respectively. The rest of the configurations evaluated at Reynolds number 32068 generated very similar x+, y+ and z+ values. Figure 2 presents the mesh used for the simulations, showing as well the entire computational domain. The mesh independence test performed to validate the meshes used in the present paper, was presented in the authors previous paper [12], where the fluidic oscillator baseline case was analyzed for a set of different Reynolds numbers, including the largest Reynolds numbers analyzed in the present manuscript. The boundary conditions employed in all simulations were, Dirichlet conditions for velocity and Neumann for pressure at the inlet. A relative pressure of \(10^{4}\) Pa and Neumann conditions for velocity were considered at the two outlets. Dirichlet boundary conditions for velocity and Neumann for pressure were set to all walls. Figure 1: Fluidic oscillator mixing chamber internal dimensions modifications. In the present study the flow was considered as turbulent, incompressible and isothermal, all simulations were three dimensional. The fluid used was water. Fluid dynamic viscosity was chosen to be 1.003 \(\times\) 10\({}^{-3}\) Kg/(m s), the fluid density was 998.2 Kg/m\({}^{3}\). The turbulence model used was the DDES, which is a hybrid RANS-LES model, since according to the research undertaken by [4], this model generates a very close approach to the experimental results. The Spalart-Allmaras turbulent model was used for the RANS approach, while the Subgrid Scale (SGS) model was employed as the LES one. The parameter defining which turbulent model RANS-LES needs to be used is the distance between a given cell and the nearest wall, \(d\). In the DES model, When \(d<(C_{DES}\Delta)\), being \(C_{DES}=0.65\) a constant of the model and \(\Delta\) the generic cell length, the RANS approach was used, and whenever \(d>(C_{DES}\Delta)\) then the SGS model was employed. The DDES model is a modified version of the DES one, in which a new formulation with a filter function \(f_{d}\) was introduced to avoid the so called grid-induced separation (GIS). In the DDES model, the switching mechanism between RANS and LES is not only dependent on the wall distance and grid spacing but also on the flow itself. The filter function \(f_{d}\) is designed to take a value of 0 in the region where the boundary layer is attached, under this conditions the RANS model is used, and a value of 1 in the region where the flow is separated, under these conditions the switching criteria to employ whether the RANS or the LES model, is the one just defined for the DES model. The different constants involved in the turbulent models are, \(c_{b1}=0.1355\); \(c_{b2}=0.622\); \(\sigma=\frac{2}{3}\); \(\kappa=0.41\); \(c_{w1}=\frac{c_{b1}}{\kappa^{2}}+\frac{1+c_{b2}}{\sigma}\); \(c_{w2}=0.3\); \(c_{w3}=2\); \(c_{v1}=7.1\). Further information of the DDES model mathematical background employed along with the different constants involved is to be found in [13; 14; 15; 16; 17]. The software OpenFOAM was considered for all 3D simulations, finite volume method is the approach OpenFOAM uses to discretize Navier Stokes equations.