# Computational Characterization of Turbulent Flow in a Microfluidic Actuator
## 6 Preliminary Three-Dimensional Results  
In this section, some preliminary results of the three-dimensional simulation are shown at \(Re\approx 16,000\). Figure 13 shows the structured grid, which consists of 2.85 million cells, a slightly refined mesh than in previous studies [15,16]. The simulation methodology and the numerical schemes used were the same as in the 2D computation. The chosen turbulence model was SST k-\(\omega\),  
Figure 11: Normalized mass flow versus time for the three analyzed geometrical configurations: original, C1 and C2. Figure 12: Illustration of the flow in the three configurations at approximately equivalent instants of the oscillation cycle: (**a**) original; (**b**) C1; (**c**) C2. Figure 13: Illustration of the computational grid in the three-dimensional simulation. found that it provided the closest results to the experimental measurements. Additionally, the time step was set to \(\Delta t=10^{-6}\) s, which is one order of magnitude smaller than in the two-dimensional case. As a result, the total computational time in 3D is nearly two orders of magnitude higher than in 2D. With the previous settings, the average value of y\({}^{+}\) was around 2, still an acceptable value that allows resolving the boundary layer, and the average Courant number was lower than 0.1 (maximum 0.9), which guarantees both the stability and accuracy of the numerical method. The normalized mass flow through outlet 1 as a function of time is shown in Figure 14. As a result, the computed operating frequency in the 3D case was somewhat higher than in 2D, similar to the findings of [PERSON] et al. [15]. It can be seen that the peaks of mass flow are smaller in the three-dimensional simulation than in the two-dimensional one, a fact probably related to the strong secondary flows observed in the outlet branches of the former, which are out of reach for the latter. Moreover, the position of the minimum of the curve is noticeably displaced in time in both simulations; however, the location of the maximum is quite similar in both curves. Another fact to be noticed is that the 3D simulation curve does not show the inflection points previously seen in the 2D computation. The three-dimensional simulation provides many more details about the flow structure inside the fluidic oscillator than 2D computations. In particular, the flow at the entrance of the feedback channels shows the typical counter rotating vortices, known as Dean vortices, that appear in pipes and channels when the flow moves along a 90\({}^{\circ}\) elbow [26; 27]. In such a configuration, the progress of the primary flow is strongly influenced by the generation of a secondary flow in the cross section of the duct, which results from a balancing among the existing pressure gradient between the inner and outer walls and the centrifugal acceleration experienced by the fluid. This is a typical 3D effect that is naturally absent in 2D and that intensifies the pressure loss of the flow within the recirculation loops. Just as an illustration, Figure 15 shows the vorticity in the stream-wise flow direction in the mid plane of the fluidic oscillator (Figure 15a) as well as some of the vortices present in the flow, represented by the grey structures (Figure 15b). Figure 15a shows the vorticity in the flow direction (\(x\)-axis) in the mid plane of the device. Moreover, some vertical planes cutting the feedback loops and the mixing chamber are included, which are also colored by the x-component of the vorticity, \(\omega_{x}\). In them, some streamlines of the secondary flow are drawn, illustrating the complex three-dimensional flow structure in both the mixing chamber and the recirculation channels. The highest absolute values of \(\omega_{x}\) appear in the mixing layer formed when the jet enters the mixing chamber, the strong recirculation zones in the mixing chamber, the entrance in the feedback channels, and upstream and downstream of the exit nozzle, including the corresponding outlet branch.