# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 3 Governing Equations and Numerical Methods
### Numerical Methods  
First, a steady-state simulation was performed. The Reynolds-averaged Navier-Stokes equations (RANS) were used as the governing equations in the following form:  
\[ \begin{array}{l}\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i }}(\rho u_{i})=0\\ \frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\left( \rho u_{i}u_{j}\right)=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\left[\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{i}}{ \partial x_{1}}\right)\right]+\frac{\partial}{\partial x_{j}}\left(-\rho \overline{u_{i}^{\prime}u_{j}^{\prime}}\right)\end{array}  \tag{19}\]  
Equation (19) could be solved using the \(k\)-\(\omega\) SST turbulence model, where \(k\) represents the turbulent kinetic energy, and \(\omega\) denotes the specific turbulent dissipation rate. The \(k\)-\(\omega\) SST model uses the \(k\)-\(\omega\) model near the wall and a \(k\)-\(\varepsilon\) model in the region far from the boundary for accurate and stable calculations. This turbulence model is known to be the most suitable model for a supersonic internal flow16. The equations are written in the following form:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\left(\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\right)+\widetilde{G}_{k}-Y_{k}+S_{k} \tag{20}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{i}}(\rho \omega u_{i})=\frac{\partial}{\partial x_{j}}\left(\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\right)+G_{\omega}-Y_{\omega}+D_{\omega}+S_{\omega} \tag{21}\]  
where \(\widetilde{G}_{k}\) is the production term for the turbulent kinetic energy \(k\) due to the velocity gradient, and \(G_{\omega}\) is the generation term for the specific dissipation rate \(\omega\), \(\Gamma\) and \(Y\) are the effective diffusivity and dissipation terms, respectively. The \(k\)-\(\omega\) SST model requires a cross-diffusion term \(D_{\omega}\), which contributes to the conversion relationship between the \(k\)-\(\omega\) model and the \(k\)-\(\varepsilon\) model. \(S_{k}\) and \(S_{\omega}\) are source terms. The density-based implicit solver was used to solve the RANS equation numerically. The flow was discretized using a second-order upwind scheme, and the turbulent kinetic energy and dissipation rate were found using a first-order upwind scheme. Then, the large eddy simulation (LES) method with a Smagorinsky-Lilly model was used for the unsteady simulation to predict the fluctuating pressure waves inside a pipe with a high resolution. The turbulent viscosity of the Smagorinsky-Lilly model was calculated as follows. \[\mu_{t}=\rho L_{s}^{2}\bigg{|}\overline{S}\bigg{|}\textit{ where }\bigg{|} \overline{S}\bigg{|}=\sqrt{2\overline{S}_{ij}\overline{S}_{ij}} \tag{22}\]  
where \(\overline{S}_{ij}\) and \(L_{s}\) are the rate of the strain tensor and the mixing length, respectively. These are expressed in the following form:  
\[\overline{S}_{ij}=\frac{1}{2}\bigg{(}\frac{\partial\overline{u}_{i}}{ \partial x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\bigg{)} \tag{23}\]  
\[L_{s}=\min(\kappa\,d,C_{s}\Delta) \tag{24}\]  
where \(\kappa\) is the von Karman constant, \(d\) is the distance to the nearest wall, \(C_{s}\) is the Smagorinsky constant, and \(\Delta\) is the local grid size, which is defined as follows:  
\[\Delta=V^{1/3} \tag{25}\]  
Although Lilly proposed \(C_{s}\) = 0.23 for a homogeneous isotropic turbulence, because it is known that excessive damping occurs for large pressure fluctuation components, \(C_{s}\approx 0.1\) was used to obtain an accurate unsteady pressure in the current numerical simulation. The second-order upwind scheme was used as the spatial discretization scheme. The maximum Courant number was set to 1 and the time step was 0.05 ms, for which the Nyquist frequency was 10,000 Hz.