# Analysis of Cavity Corner Geometry Effect on Recirculation Zone Structure
## 2 Materials and Methods
### Computational Approach Using Ansys Fluent  
Numerical simulations were performed using commercially available Ansys Fluent 2019 R2 (Ansys, Canonsburg, PA, USA) software with a 3D double-precision solver. The two-equation _k_-\(\varepsilon\) model was employed since it offers competitive efficiency and wide usage  
Figure 1: Investigated channels with given dimensions.  
compared to other Reynolds-averaged Navier-Stokes (RANS) models [8]. Further, the standard wall function was used due to its significant reduction in computational expense. The pressure-based solver used the semi-implicit method for pressure-linked equations (SIMPLE) segregated algorithm, which applies corrections to the pressure field using the continuity equation. Additionally, a second-order upwind scheme was applied to discretize the momentum equation. The simulations were performed with a convergence criterion of \(10^{-6}\) for continuity.  
The standard \(k\)-\(\epsilon\) turbulence model can be expressed as two equations as follows:  
For turbulent kinetic energy \(k\):  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku)}{\partial x}=\frac {\partial}{\partial x}\bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}}\bigg{)} \frac{\partial k}{\partial x}\bigg{]}+G_{k}+G_{b}-\rho\varepsilon-Y_{m}+S_{k}; \tag{1}\]  
For dissipation \(\varepsilon\):  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho\varepsilon u )}{\partial x}=\frac{\partial}{\partial x}\bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{ \sigma_{\varepsilon}}\bigg{)}\frac{\partial\varepsilon}{\partial x}\bigg{]} +C_{\varepsilon t}\frac{\varepsilon}{K}(G_{k}+C_{\varepsilon t}G_{b})-C_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{k}; \tag{2}\]  
where \(G_{k}\) and \(G_{b}\)--the generation of turbulence kinetic energy due to the mean velocity gradient and buoyancy, respectively; \(Y_{m}\)--contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate; \(S_{k}\) and \(S_{\varepsilon}\)--user-defined source terms; \(\sigma_{k}\) and \(\sigma_{\varepsilon}\)--Prandtl numbers for \(k\) and \(\varepsilon\), respectively; \(C_{1\varepsilon}\), \(C_{2\varepsilon}\) and \(C_{3\varepsilon}\) are adjustable constants. Constant values of adjustable constants were used: \(C_{1\varepsilon}\) = 1.44; \(C_{2\varepsilon}\) = 1.92; \(C_{3\varepsilon}\) = \(-\)0.33; \(\sigma_{k}\) = 1; \(\sigma_{\varepsilon}\) = 1.3.  
Several grids of different cell densities were tested to ensure grid independence, with a step-height-to-cell-size ratio range of \(7<h_{1}/\Delta<43\). A mesh independence test was performed at Re = 1000, and pressure difference was chosen as the reference parameter (Figure 2). As cell size decreased, the average relative change of pressure difference between inlet and outlet increased, becoming less dependent on cell size. Therefore, to perform computationally economic calculations, the grid was chosen at the point where average total pressure was no longer dependent on cell size. As a result, relative cell size was chosen \(h_{1}/\Delta_{c}\approx 21.43\).