**4. Numerical Analysis**

In the present work, the commercial CFD software ANSYS CFX 15.0® [31] was used for the flow analysis. In both the analyses of the internal flow of the fluidic oscillator and the external flow on the airfoil, three-dimensional URANS equations with the shear stress transport (SST) turbulence model and continuity equation were calculated numerically. Pandey and Kim [28] reported that URANS analysis with the SST model predicted the internal flow of a fluidic oscillator better than LES with the WALE model. The SST model is also known to predict the flow separation well under adverse pressure gradients [32,33].

For the computational domain inside the fluidic oscillator (Figure 1), the following boundary conditions were used. Uniform velocity was assigned at the oscillator inlet, and no-slip conditions were used at the walls. For the external flow domain shown in Figure 4, a uniform velocity of 25 m/s was assigned at the inlet of the domain, which corresponds to a Reynolds number of 5.0 × <sup>10</sup><sup>5</sup> based on the inlet velocity and chord length. Constant pressure is assigned at the outlet of the domain, and no-slip boundary conditions are used at the wall. In both the analyses, the working fluid is air at 25 ◦C, which is assumed to be an ideal gas.

**Figure 4.** Computational domain using periodic boundary conditions [29].

For the external flow, Melton et al. [18] measured the jet frequency of the fluidic oscillator in a range of mass flow rates, . *m* = 0 − 1.3 g/s, and evaluated the aerodynamic performance of the airfoil in a range of momentum coefficients (*Cμ*), 0–6.63%, for different flap angles. They found that as the mass flow rate increased, the frequency increased but converged to a value. The performance of flow control generally improved as *Cμ* increased, and a larger value of *Cμ* is required for a larger flap angle to achieve effective flow control. The momentum coefficient is defined as follows:

$$\mathbb{C}\_{\mu} = \frac{n \,\, \rho\_{j\text{et}} \, \text{LI}\_{j\text{et}} \, A\_{nozzle} \, \text{LI}\_{j\text{et}}}{\frac{1}{2} \, \rho\_{\infty} \, \text{LI}\_{\infty}^{2} \, \text{c} \, n \, \text{l}} = 2 \frac{A\_{nozzle}}{c \, \text{l}} \left(\frac{\text{LI}\_{j\text{et}}}{\text{LI}\_{\infty}}\right)^{2} \tag{6}$$

$$\dot{M}\_{j\varepsilon l} = \frac{\dot{m}}{\rho\_{j\varepsilon l} A\_{nozzle}} = \frac{\dot{m}}{\rho\_{\infty} A\_{nozzle}} \tag{7}$$

where *Ujet* is the velocity at the oscillator outlet, . *m* is the mass flow rate through the oscillator, and *Anozzle* is the area of the oscillator outlet. *l* and *n* are the space between adjacent oscillators and the number of oscillators. The fluid density in the fluidic oscillator (*ρjet*) was assumed to be the same as the density in the external flow (*ρ*∞).

In the present study, the bent oscillator was tested in a range of mass flow rates, . *m* =0.19–0.72 g/s, and the aerodynamic performance of the airfoil was evaluated in a *C<sup>μ</sup>* range of 0.41–5.94%.

Grid structures in the internal and external domains are shown in Figures 5 and 6, respectively. In both grids, prism meshes were constructed near a wall, and unstructured tetrahedral meshes were used in the other regions. To adopt the SST model developed for low Reynolds numbers, the first grid points near the wall were located at y+ < 2. In both the unsteady analyses of the internal and external flows, the time step was 5 × <sup>10</sup>−<sup>6</sup> s. As a convergence criterion, the root-mean-square of the relative residuals was kept less than 1.0 × <sup>10</sup>−4. The total time interval calculated was 0.03 s in each case. The results of the steady RANS analysis were used as the initial assumption for the URANS analysis. The simulation was carried out using a supercomputer employing an Intel Xeon Phi 7250/1.4 GHz processor with 68 CPU cores. The internal flow analysis of the fluidic oscillator took about 8 h, and the analysis of the external flow and three internal flows in the computational domain took about 48 h.

**Figure 5.** Grid system in the fluidic oscillator.

**Figure 6.** Grid system in internal and external domains (α = 8◦ and *δ<sup>f</sup>* = 40◦).
