## Appendix A
### Time Stepping Scheme  
To alleviate the computational resource requirements imposed by the CFL condition based on the sound speed, the dual time stepping method, which was proposed by [PERSON] [27], was employed to solve the Navier-Stokes formulation in Equation (1). Thus, the governing equation becomes  
\[\frac{\partial U_{p}}{\partial\tau}+\frac{\partial U}{\partial t}+\frac{ \partial F_{1}}{\partial x_{1}}+\frac{\partial F_{2}}{\partial x_{2}}+\frac{ \partial F_{3}}{\partial x_{3}}=0\] (A7)  
where \(U_{p}\) is the primitive form of \([P,u_{1},u_{2},u_{3},T]\); \(\tau\) and \(t\) are the artificial and physical times, respectively; and \(U\) is the conservative form of \([\rho,\rho u_{1},\rho u_{2},\rho u_{3},\rho e]\). \[ \begin{array}{l}\frac{U_{p}^{k+1}-U_{p}^{k}}{\Delta\tau}+\frac{3U^{k+1}-4U^ {n}+U^{n-1}}{2\Delta t}+\frac{1}{\Delta x_{1}}(F_{1(i+1/2,j,k)}^{k+1}-F_{1(i- 1/2,j,k)})\\ +\frac{1}{\Delta x_{2}}(F_{2(i,j+1/2,k)}^{k+1}-F_{2(i,j-1/2,k)})+\frac{1}{ \Delta x_{3}}(F_{3(i,j+k+1/2)}^{k+1}-F_{3(i,j,k-1/2)}^{k+1})=0\end{array} \] (A8)  
Equation (A6) can be arranged with implicit form as  
\[[\frac{I}{\Delta\tau}+M\frac{3}{2\Delta t}+(\delta_{x_{1}}A_{p}^{k}+\delta_{x _{3}}B_{p}^{k}+\delta_{x_{3}}C_{p}^{k})]\Delta U_{p}=R^{k},\] (A9)  
where \(\delta_{x_{i}}\) is the central difference operator and \(A_{p}=\partial F_{i}^{k}/\partial U_{p}\) is the flux Jacobian. With the dual time stepping, only \(\Delta\tau\) is restricted by the CFL condition based on the sound speed, and \(\Delta t\) can be free of the CFL condition. However, the inclusion of an artificial time term in the equation leads to Newton linearization error, resulting in reduced accuracy. Therefore, using Equation (A9) is not suitable for aeroacoustic simulations performed in this study as these require high levels of accuracy. The easiest way to solve this problem is to remove the artificial time term. Equation (A9) becomes  
\[[M\frac{3}{2\Delta t}+(\delta_{x_{1}}A_{p}^{k}+\delta_{x_{3}}B_{p}^{k}+\delta_ {x_{3}}C_{p}^{k})]\Delta U_{p}=R^{k}.\] (A10)  
However, a large \(\Delta t\) in Equation (A10) can easily cause divergence because of the lack of the artificial time term. To address this issue, [PERSON] et al. [21] proposed an adaptively switched time stepping (ASTS) scheme from LUSGS [28]; however, their work focused solely on two-dimensional simulations. In this study, we extend the ASTS scheme to 3D simulations. The concept behind the ASTS scheme is that when the iteration is not yet stable, Equation (A9) should be employed. However, once the iteration reaches a stable stage, Equation (A10) can be used. By automatically switching between the governing equations with and without the artificial time term, we can simultaneously maintain a large time step and high accuracy. To determine whether the iteration is stable, Equation (A11) suggested by [PERSON] et al. [29] is used as an indicator:  
\[\Delta Q_{est}=-\Delta\tau M^{-1}[-(3U^{k}-4U^{n}+U^{n-1})/(2\Delta t)-(\delta_{x _{1}}\bar{F}_{1}^{k}+\delta_{x_{2}}\bar{F}_{2}^{k}+\delta_{x_{3}}\bar{F}_{3}^{k })].\] (A11)  
Equation (A12) can be solved only when \(|\Delta Q_{est}|<\Delta Q_{p}^{ref}\), which implies that the iteration is sufficiently stable and that the artificial time term can be eliminated. Conversely, if \(|\Delta Q_{est}|>\alpha\Delta Q_{p}^{ref}\), Equation (A12) is first solved to obtain the allowable CFL number for each grid. \[CFL_{allowable}=CFL_{specified}\times\alpha\Delta Q_{p}^{ref}/|\Delta Q_{est}|.\] (A12)  
Then, Equation (A9) is solved to proceed to the next iteration. \(\Delta Q_{p}^{ref}\) for 3D simulations is expressed in Equation (A13):  
\[\Delta Q_{p}^{ref}=