# Combined CFD-Stochastic Analysis of an Active Fluid Injection System for Jet Noise Reduction
## 2 Model Description and Validation
### Model Description
The requirement that the wave vector is uniformly distributed in the 3D wave-number space provides the following probability densities:  
\[\mathcal{P}(\varphi_{n})=\left(2\pi\right)^{-1}\text{, with }-\pi\leq\varphi_{n}\leq\pi \tag{3}\]  
\[\mathcal{P}(\theta_{n})=cos(\theta_{n})/2\text{, with }-\pi/2\leq\theta_{n}\leq \pi/2\]  
Analogously, by supposing that the \(\sigma_{n}\) vector is uniformly distributed in the plane normal to \(k_{n}\) yields:  
\[\mathcal{P}(\alpha_{n})=\left(2\pi\right)^{-1}\text{, with }-\pi\leq\alpha_{n}\leq\pi \tag{4}\]  
Finally, the phase angle \(\psi_{n}\) is also supposed to be uniformly distributed in the \(2\pi\) range, that is:  
\[\mathcal{P}(\psi_{n})=\left(2\pi\right)^{-1}\text{, with }-\pi\leq\psi_{n}\leq\pi \tag{5}\]  
The extremes of the wave number interval are related, at each node of the CFD mesh, to the value of the wave number \(k_{e}\) through the relationships:  
\[k_{1}=c_{2}k_{e}\text{; }k_{N_{F}}=min\big{(}c_{3}k_{e},k_{\eta},2\pi/\left(N_ {\omega}\Delta_{i}\right)\big{)} \tag{6}\]  
where \(\Delta_{i}=max\big{(}\big{|}U\cdot\big{(}x_{j}\text{-}x_{i}\big{)}\big{|}/U_{ i}\big{)}\) is the maximum distance between node \(i\) and its neighboring stencil nodes \(j\), projected in the direction of the local mean-flow velocity \(U_{i}\); \(N_{\omega}\) is the number of grid points for the Fourier component, set to 6 in the present work, as suggested by [PERSON] and [PERSON] [11]. The optimal value of the Fourier modes number, \(N_{F}=50\), was found by [PERSON] and [PERSON] [13], and used in this work. The constants \(c_{2}\) and \(c_{3}\) are two additional tuning constants of the stochastic model: \(c_{2}=0.1\), \(c_{3}=10\). The use of an adaptive wave-number interval allows reducing the number of Fourier modes. Furthermore, in order to obtain a better discretization of the energy spectrum at the lower energy-containing Fourier modes, a logarithmic distribution of the \(k_{n}\) values is used, say:  
\[k_{n}=\exp[\ln(k_{1})+D(n-1)]\text{; }D=\big{[}\ln(k_{N_{F}})-\ln(k_{1})\big{]} /(N-1) \tag{7}\]  
The energy spectrum \(E(k)\) is integrated over each band \(n\) to obtain the magnitude of the velocity component \(\hat{u}_{n}\) by discretizing the interval \(D\) in a number of constant subintervals, from the value \(k_{n}^{-}=\exp[\ln(k_{1})+D(n-1.5)]\) up to the value \(k_{n}^{-}=\exp[\ln(k_{1})+D(n-0.5)]\). A crucial point of the SNGR approach is the definition of the two-point space correlation of the velocity fluctuations at each node of the CFD mesh by means of an artificial numerical approach able to reproduce the physical behaviour of the turbulent structures. The simplest approach, used for instance in Bechara [10], would consist of segmenting the bounding box of the active source region in square paths with edges equal to the average value of the correlation length over the whole source region. Then, a set of stochastic angles are sampled in each patch and these values are finally attributed to all the mesh nodes falling in the patch. The main drawback of this approach is that the correlation length is the same over all the source region, and therefore, does not account for the local Reynolds stresses and size of the turbulent structures. Conversely, two different SNGR approaches, based on the the correlation length of the CFD solution, have been proposed. The first one consists in dividing the domain in blobs whose dimension is proportional to the correlation length of the CFD cells falling inside the blob. In the first step, the RANS correlation lengths \({l_{k}}^{i}=\big{|}-0.18\big{(}K^{2}/\epsilon\big{)}\partial{U_{k}}/\partial{ x_{k}}+2K/3\big{|}^{3/2}/\epsilon\) along the three Cartesian directions, each one denoted by the subscript \(k\), are evaluated at each node, \(i\), of the CFD mesh. Thus, random angles are assigned constant for each blob [13] as depicted in Figure 2a.