# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 2 Experimental and Numerical Approaches
### Numerical Approach  
#### 2.2.1 High-Fidelity ILES Model  
The employed ILES code ([PERSON] and [PERSON] [11]) solves a set of the compressible Navier-Stokes equations represented in a strong, conservative, time-dependent form in the generalized curvilinear computational coordinates (\(\xi\), \(\eta\), \(\zeta\), \(\tau\)) transformed from the physical coordinates (\(x\), \(y\), \(z\), \(t\)), where \(i\) and \(v\) denote inviscid and viscous terms, respectively. \[\frac{\partial}{\partial\tau}(\frac{\overset{\rightarrow}{J}}{J})+\frac{ \partial\overset{\rightarrow}{F}_{i}}{\partial\xi}+\frac{\partial\overset{ \rightarrow}{G}_{i}}{\partial\eta}+\frac{\partial\overset{\rightarrow}{H}_{i }}{\partial\xi}+\frac{1}{\text{Re}}[\frac{\partial\overset{\rightarrow}{F}_ {v}}{\partial\xi}+\frac{\partial\overset{\rightarrow}{G}_{v}}{\partial\eta}+ \frac{\partial\overset{\rightarrow}{H}_{v}}{\partial\xi}]=\overset{ \rightarrow}{S} \tag{1}\]  
Figure 1: Locations of the Remote Microphone Probes (RMPs) on the NACA-0012 mock-up (red symbols) and the reference points for the analysis of wall-pressure fluctuations in the experiment ([PERSON] et al. [8]). The solution vector \(\overset{\rightarrow}{Q}=(\rho,\rho u,\rho v,\rho w,\rho e)\) is defined in terms of the flow density \(\rho\), Cartesian flow velocity components (\(u\), \(v\), \(w\)), and flow specific energy,  
\[e=\frac{T}{\gamma(\gamma-1)M_{\infty}^{2}}+\frac{1}{2}(u^{2}+v^{2}+w^{2}) \tag{2}\]  
with the assumed perfect gas relationship \(p=\rho T/\gamma M_{\infty}^{2}\) connecting the flow pressure \(p\), temperature \(T\), and the freestream Mach number \(M_{\infty}\) (\(\gamma\) is the specific heat ratio). The other variables in Equation (1) include the inviscid flux vectors defined by  
\[\overset{\rightarrow}{F}_{i}=\left[ \begin{array}{c}\rho\hat{u}\\ \rho u\hat{u}+\hat{\xi}_{x}p\\ \rho v\hat{u}+\hat{\xi}_{y}p\\ \rho w\hat{u}+\hat{\xi}_{z}p\\ (\rho e+p)\hat{u}-\hat{\xi}_{t}p\end{array} \right],\overset{\rightarrow}{G}_ {i}=\left[ \begin{array}{c}\rho\hat{v}\\ \rho u\hat{v}+\hat{\eta}_{x}p\\ \rho v\hat{v}+\hat{\eta}_{y}p\\ \rho w\hat{v}+\hat{\xi}_{z}p\\ (\rho e+p)\hat{v}-\hat{\xi}_{t}p\end{array} \right],\overset{\rightarrow}{H}_ {i}=\left[ \begin{array}{c}\rho\hat{w}\\ \rho v\hat{w}+\hat{\xi}_{x}p\\ \rho v\hat{w}+\hat{\xi}_{y}p\\ \rho w\hat{w}+\hat{\xi}_{z}p\\ (\rho e+p)\hat{w}-\hat{\xi}_{t}p\end{array} \right] \tag{3}\]  
where the transformation Jacobian is \(J=\partial(\xi,\eta,\zeta,\tau)/\partial(x,y,z,t)\), the metric quantities defined, e.g., as \(\hat{\xi}_{x}=(J^{-1})\partial\xi/\partial x\), etc., and the transformed flow velocity components are  
\[ \begin{array}{l}\hat{u}=\hat{\xi}_{t}+\hat{\xi}_{x}u+\hat{\xi}_{y}v+\hat{\xi }_{z}w\\ \hat{v}=\hat{\eta}_{t}+\hat{\eta}_{x}u+\hat{\eta}_{y}v+\hat{\eta}_{z}w\\ \hat{w}=\hat{\xi}_{t}+\hat{\xi}_{x}u+\hat{\xi}_{y}v+\hat{\xi}_{z}w\end{array}  \tag{4}\]  
The viscous flux vectors, \(\overset{\rightarrow}{F}_{v}\), \(\overset{\rightarrow}{G}_{v}\) and \(\overset{\rightarrow}{H}_{v}\), are defined, e.g., in [PERSON] et al. (2012), while \(\overset{\rightarrow}{S}\) represents the source term that allows incompressible unsteady vortical perturbation to be introduced into the flow field. All flow variables are normalized by their respective reference freestream values except for pressure, which is nondimensionalized by \(\rho_{\infty}u_{\infty}^{2}\). Note that the governing equations are represented in the original unfiltered form, used unchanged in the laminar, transitional or fully turbulent regions of the flow, with [PERSON] et al. (2013) providing further details on the code's employed ILES procedure in which a high-order, low-pass filter operator is applied to the dependent variables during the solution process, in contrast to the standard LES addition of sub-grid stress (SGS) and heat flux terms. The resulting filter selectively damps the evolving poorly resolved high-frequency content of the solution.