# Recent Advances in Acoustics of Transitional Airfoils with Feedback-Loop Interactions: A Review
## 3 Laminar Separation Bubble and Acoustic Feedback Loop Process
### Numerical Approaches
#### 3.2.1 Governing Equations for AFL Analysis  
To accurately resolve airfoil AFL interactions, a high-fidelity approach is required to solve a set of the compressible unsteady Navier-Stokes equations represented in strong, conservative, time-dependent form in the generalized curvilinear computational coordinates (\(\xi\), \(\eta\), \(\zeta\), \(\tau\)) transformed from the physical coordinates (\(x\), \(y\), \(z\), \(\hat{t}\)),  
\[\frac{\partial}{\partial\tau}(\frac{\overset{\rightarrow}{Q}}{J})+\frac{ \partial\overset{\rightarrow}{F}_{i}}{\partial\xi}+\frac{\partial\overset{ \rightarrow}{G}_{i}}{\partial\eta}+\frac{\partial\overset{\rightarrow}{H}_{ i}}{\partial\xi}+\frac{1}{\text{Re}}[\frac{\partial\overset{\rightarrow}{F}_{v}}{ \partial\xi}+\frac{\partial\overset{\rightarrow}{G}_{v}}{\partial\eta}+\frac{ \partial\overset{\rightarrow}{H}_{v}}{\partial\xi}]=\overset{\rightarrow}{S} \tag{1}\]  
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 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,  
\[\overset{\rightarrow}{F}_{i}=\begin{bmatrix}2&\rho\dot{u}&3&2&2&\rho\dot{w}& 3\\ &\rho u\dot{u}+\dot{\xi}_{x}p&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}&\overset{\rightarrow}{7}\\ &\rho v\dot{u}+\dot{\xi}_{x}p&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}\\ &\rho w\dot{u}+\dot{\xi}_{x}p&\overset{\rightarrow}{7}&\overset{\rightarrow }{7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}\\ &(\rho e+p)\dot{u}-\dot{\xi}_{t}p&\overset{\rightarrow}{7}&\overset{\rightarrow }{7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}&\overset{\rightarrow} {7}&\overset{\rightarrow}{7}&\overset{\rightarrow}{7}&\overset{\rightarrow }{7}\\ \end{bmatrix} \tag{3}\]  
where the transformation Jacobian, \(J=\partial(\xi,\eta,\zeta,\tau)/\partial(x,y,z,t)\), the metric quantities defined, e.g., as \(\dot{\xi}_{x}=(J^{-1})\partial\xi/\partial x\), etc., and the transformed flow velocity components,  
\[\dot{u} =\dot{\xi}_{t}+\dot{\xi}_{x}u+\dot{\xi}_{y}v+\dot{\xi}_{z}w \tag{4}\] \[\dot{v} =\dot{\eta}_{t}+\dot{\eta}_{x}u+\dot{\eta}_{y}v+\dot{\eta}_{z}w\] \[\dot{w} =\dot{\xi}_{t}+\dot{\xi}_{x}u+\dot{\xi}_{y}v+\dot{\xi}_{z}w\]  
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. [31], while \(\overset{\rightarrow}{S}\) represents the source term that, e.g., allows an upstream unsteady vortical perturbation field (such as synthetic turbulence) to be introduced into the flow field analysis. 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 laminar, transitional, or fully turbulent regions of the flow. The DNS approach (with no modeling of unresolved turbulence scales) was selected in numerical studies of [16; 17], while a wall-resolved large-eddy simulation (LES) was implemented in [18; 28; 30].