# Computational Characterization of Turbulent Flow in a Microfluidic Actuator
## 2 Geometrical Configuration and Simulation Set-Up  
The geometry of the fluidic oscillator considered in this work is that investigated in [14; 15], which is presented in Figure 1, showing its constitutive parts and also the employed boundary conditions. A uniform velocity profile is imposed at the inlet, while a zero-pressure condition is set at the outlets; finally, at the walls, the no-slip condition is enforced. According to the experimental conditions in [14], the working fluid is water with density \(\rho=998.2\) kg/m\({}^{3}\) and kinematic viscosity \(\
u=1.004\times 10^{-6}\) m\({}^{2}\)/s; therefore, its dynamics will be described by the transient incompressible Navier-Stokes equations. The principle of operation of the fluidic oscillator is summarized as follows. The flow is injected at the inlet with constant velocity; it accelerates in the nozzle (also called power nozzle) due to the reduction in cross section and enters the mixing chamber. In principle, the generated jet goes straight to the end of the mixing chamber, crosses the exit nozzle and encounters the splitter, which divides the flow through the branches towards both outlets. This flow configuration is unstable, and the jet starts to perform oscillations in the cross flow direction. Such oscillations soon increase their amplitude, and the jet collides either with the upper or lower wall at the end of the mixing chamber with the effect that the flow divides, and part of it is directed through one of the feedback channels. The flow through that feedback channel reaches the beginning of the mixing chamber, pushing the main jet towards the opposite wall, generating the self-sustained jet oscillation dynamics. The flow Reynolds number based on the hydraulic diameter \(D_{h}\) is \(Re\approx 16,000\)[15]. This means that the flow inside the device is turbulent, which in this study is described using turbulence models under the Unsteady Reynolds Averaged Navier-Stokes (URANS) approximation. In the first instance and for validation purposes, the Shear Stress Transport (SST) k-\(\omega\) model [18] was used, as in [PERSON] et al. [15]. This model combines both k-\(\epsilon\) and k-\(\omega\) models through a blending function. The k-\(\epsilon\) model is used far away from the wall, whilst the k-\(\omega\) model is used near the wall up to the edge of the boundary layer. The SST k-\(\omega\) model performs better than other two-equation classical turbulence models in the case of flows with strong adverse pressure gradients and flow separation and reattachment, which are situations commonly found in fluidic oscillators. The governing equations of such a model can be found in the original paper of [PERSON] [18] and will not be repeated here. Since the fluidic oscillator has a planar geometry, two-dimensional simulations were performed with the ANSYS Fluent v. 19 commercial software. Preliminary results of a 3D simulation will also be presented in Section 6. The 2D geometrical domain was discretizedby means of a structured grid based on quads. To perform the mesh independence study, three grids have been considered in this work: a coarse one with around 10\({}^{5}\) elements, intermediate 3.43 \(\times\) 10\({}^{5}\) cells and refined with 11.75 \(\times\) 10\({}^{5}\) elements. Figure 2a shows a panoramic view of the intermediate mesh, whilst Figure 2b is a zoom-in of the red rectangle enclosing the mixing chamber: there, it is seen that the grid is pretty uniform. In this graph it can also be seen that the grid is substantially refined close to the walls in order to resolve the boundary layer, giving maximum values of \(y^{+}\sim 1\) (Table 1), which is a requirement of the turbulence model used. The employed spatial discretization schemes for all the differential equations were the second-order centered scheme for diffusive terms and the second-order upwind for the convective terms. Discretization in time used an implicit second-order scheme with a time step determined after performing a sensitivity study, which is presented below.