# Numerical and Experimental Characterization of a Coanda-Type Industrial Air Amplifier
## 4 Numerical Methodology
### Governing Equations and Boundary Conditions  
Let us consider the steady three-dimensional turbulent compressible fluid equations, closing the system with the \(SST-k\omega\) turbulence model [27]. This eddy viscosity turbulence model has provided very good results when comparing the numerical data with the experimental data of air amplifiers [13; 22], although other authors [14] have successfully implemented the Realizable \(k-\varepsilon\) model [28]. Additionally, assuming the air as an ideal gas (i.e., \(p=\rho R_{g}T\)), the continuity, momentum, and energy equations for a steady, compressible, and turbulent flow are as follows:  
\[\frac{\partial\rho u_{i}}{\partial x_{i}} = 0, \tag{8}\] \[\frac{\partial\rho u_{i}u_{j}}{\partial x_{j}} = -\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j} }\left[\frac{1}{Re}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\right)-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}} \right],\] (9) \[\frac{\partial\rho u_{i}T}{\partial x_{i}} = -\frac{\partial}{\partial x_{i}}\left[-\frac{1}{RePr}\frac{ \partial T}{\partial x_{i}}+\rho\overline{u_{i}^{\prime}T}\right]+\frac{ \partial}{\partial x_{j}}\left[u_{i}^{\prime}\left(\frac{1}{Re}\left(\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right)\right], \tag{10}\]  
where \(\rho\) is the fluid density, \(u_{i}\) is the fluid velocity in the \(i\) direction, \(p\) is the fluid pressure, \(u_{i}^{\prime}\) is the turbulent deviation of the fluid velocity, and \(T\) is the fluid temperature. Compressibility effects cannot be neglected, and thus, the energy equation cannot be decoupled from the mass and momentum equations.  
The computational domain extends further from the device in all three dimensions to avoid external boundary conditions' interactions with the numerical results. A schematic description of the volumetric domain for the isolated air amplifier and the far field boundary conditions of the problem is given in Figure 13. The stagnation pressure inlet conditions are fixed upstream from the device, and static pressure conditions are fixed at the outlet, setting an atmospheric pressure, \(p=p_{atm}\), at both sections. The difference between them is that the inflow is treated as a stagnation section, where the fluid is at rest, while a flow velocity field is allowed in the outflow section. The upper, lower, lateral, and air amplifier walls are solid walls where no slip boundary conditions are imposed. In the suction and blowing configurations, the Venturi geometry is added to the computational domain, where the walls are also imposed as a no slip boundary condition. The pressurized air inlet of the air amplifier is modeled as a stagnation pressure inlet boundary condition, with \(p=p_{c}\). Finally, we assume that all boundaries are thermally isolated \(\frac{\partial T}{\partial n}=0\), with the only exception of the reference temperature of \(T_{\infty}=300\) K, which is imposed at the inlet section.  
Figure 13: Sketch of the computational domain and boundary conditions considered for the isolated case.  
The STAR-CCM+ software is used to solve the stationary fluid problem using finite-volume discretization. STAR-CCM+ is a complete engineering software package capable of solving the compressible Navier-Stokes equations for turbulent flows. A semi-implicit method, known as the SIMPLE algorithm, is used to handle the velocity-pressure coupling of the Navier-Stokes equations and to calculate primitive variables such as density, pressure, temperature, and velocity at the steady state. For these steady simulations, an iterative process is used to ensure that the steady-state condition is satisfied with low residuals.