# URANS Analysis of a Launch Vehicle Aero-Acoustic Environment
## 2 Methods
### Governing Equations  
The basic equations describing fluid dynamics are the three-dimensional Navier-Stokes equations as follows:  
\[\rho\frac{\partial u_{j}}{\partial t}+\rho\frac{\partial u_{j}u_{i}}{\partial x_{ j}}=-\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}(2\mu s_{ji}), \tag{1}\]  
where the vectors \(u_{i}\) and \(x_{i}\) are the velocity and position, \(\rho\) is the density, and \(s_{ij}\) is the strain-rate tensor. Turbulence is represented by the convective term in the Navier-Stokes equations. At present, there are three approaches to model turbulence that can be chosen depending on the requirements of the problem. Direct numerical simulation (DNS) resolves all scales of turbulence with extremely high computational cost out of reach for this problem. Large eddy simulation (LES) only computes larger scales of turbulence while modeling small scales. The computational cost is reduced, but it is still too high to capture large-scale eddies accurately. The lowest level corresponds to the Reynolds-averaged Navier-Stokes (RANS) model, in which all turbulent fluctuations are modeled. The characterization of turbulent scales in the impingement of the plume against the launchpad can be extremely hard to model due to the necessity of extremely fine meshes. Moreover, LES is extremely sensitive to the initial conditions of the flow and its peculiarities. Therefore, the URANS approach was chosen. The Reynolds decomposition, introduced by [PERSON] in 1895 [14], expresses all variables as the sum of their mean value, \(U_{i}\), and fluctuating part, \(u_{i}^{\prime}\). The time-averaged mean velocity is the same value \(\overline{U_{i}}=U_{i}\), while the time averaging of the fluctuating part is zero. The time averaging Equation (1) yields the RANS equations. \[\rho\frac{\partial U_{j}}{\partial t}+\rho U_{j}\frac{\partial U_{i}}{\partial x _{j}}=-\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}(2\mu s _{ij}-\rho\overline{u_{j}^{\prime}u_{i}^{\prime}}) \tag{2}\]  
In this work, we used the URANS equations since we retained the transient term, \(\partial U_{j}/\partial t\), during the computation so that the averaged components were still a function of time. The URANS equations are the usual RANS equations as Equation (2). The acoustic field that the launch vehicle must withstand is due primarily to the fluctuating component of the turbulence of the exhaust gas mixture layer. In the acoustic field, sound pressure (\(P_{s}\)) generated by a sound wave is understood as the pressure deviation from the ambient atmospheric pressure (\(P_{0}\)). Thus, the total pressure (\(P\)) is the sum of these two components. The nature of turbulent flows is chaotic, and therefore, they were analyzed in statistical terms. The Reynolds decomposition states that any flow magnitude is decomposed into a mean (\(P_{m}\)) and a fluctuating part (\(P_{f}\)), where \(P_{m}\) can be seen as an ensemble average of the pressure measured in an infinite number of experiments. The fluctuating part \(P_{f}\) can only be computed through expensive algorithms such as DNS. Here, the turbulence was modeled by a URANS model. This model is based on the characterization of the transport and dissipation of energy through an effective viscosity called eddy viscosity introduced by the Boussinesq approximation [14]. When using the URANS turbulence model, the pressure is simply:  
\[P=P_{0}+P_{m}+P_{f}\quad\quad\rightarrow\quad P_{URANS}=P_{0}+P_{m} \tag{3}\]  
but the numerical method, through the two-equation eddy viscosity model \(k-\omega\) SST [15], takes into account all fluctuations at non-resolved scales. This model was chosen because it is a two-layer model based on the Wilcox \(k-\omega\) model in the inner parts of the boundary layer and switches to the \(k-\epsilon\) model in the free-stream region. However, it is not possible to estimate \(P_{f}\) nor its standard deviation. Therefore, \(P_{m}\) is a smooth function where the high-frequency noise due to turbulent fluctuations is lost.