# Combined CFD-Stochastic Analysis of an Active Fluid Injection System for Jet Noise Reduction
## 2 Model Description and Validation  
### Model Description  
The SNGR approach assumes that the turbulent velocity field can be reconstructed as a summation of Fourier components, according to [PERSON] [8]. Therefore, the reconstructed turbulent velocity \(\mathbf{u}^{\prime}\) at point \(\mathbf{x}\) and time \(t\) reads:  
\[\mathbf{u}^{\prime}(\mathbf{x},t)=2\sum_{n=1}^{N_{p}}\hat{u}_{n}\cos\{\mathbf{k}_{n}\cdot( \mathbf{x}-\varrho\mathbf{U}t)+\psi_{n}\}\sigma_{n} \tag{1}\]  
where \(\hat{u}_{n}\), \(\psi_{n}\) and \(\sigma_{n}\) are the magnitude, phase and direction of the \(n\)th Fourier component, respectively. As proposed by [PERSON] and [PERSON] [11], each Fourier mode is supposed to be convected at the local mean-flow velocity \(\mathbf{U}\) corrected by the vortex convection velocity ratio \(\varrho\). This factor may account for the wall induction effect that reduces the vortex convection velocity with respect to the mean-flow velocity at the location of the vortex core. The factor may also account for the vertical induction in a jet shear layer, but this effect is negligible for low-speed subsonic jets. For jet noise prediction, the value \(\varrho=1\) has been used. Notice that the scalar product \(\varrho k_{n}\cdot\mathbf{U}\) accounts for the local time variation of the velocity field. Assuming incompressibility, the zero-divergence condition, applied to the Equation (1), results in the relationship \(k_{n}\cdot\mathbf{\sigma}_{n}=0\), stating that the wave vector is perpendicular to the velocity vector. By supposing that the turbulent flow field is isotropic, the magnitude of the \(n\)th Fourier mode is related to the mono-dimensional energy spectrum \(E(k)\) by the expression, \(\hat{u}_{n}=\sqrt{E(k_{n})Ak_{n}}\), where \(k_{n}\) and \(\Delta k_{n}\) are the wave number and the corresponding band of the n-th mode. The Von Karman--Pao isotropic turbulence spectrum is assumed [10]; that is:  
\[E(k)=A(2/3)(K/k_{e})(k/k_{e})^{4}exp\big{[}-2\big{(}k/k_{\eta}\big{)}\big{]} \big{[}1+(k/k_{e})^{2}\big{]}^{-17/6} \tag{2}\]  
where \(K\) is the turbulent kinetic energy, \(A\) is a numerical constant, \(k_{e}\) is the wave number of maximum energy, \(k_{\eta}=\varepsilon^{1/4}\upsilon^{-3/4}\) is the Kolmogorov wave number, \(\upsilon\) is the kinematical viscosity of the fluid and \(\varepsilon\) is the turbulent dissipation rate. The constants \(A\) and \(k_{e}\) can be determined by equating the integral energy and the integral length scale derived from turbulence spectrum to the RANS quantities \(K\) and \(L_{T}=c_{1}u^{\prime 3}/\varepsilon\), respectively, \(u^{\prime}=\sqrt{2K/3}\) being the isotropic turbulent velocity and \(c_{1}\) the first tuning parameter of the method. The aforementioned calculation provides [11]\(A=1.453\) and \(k_{e}=\frac{9\pi}{55}\frac{A}{L_{T}}=0.747/L_{T}\). The parameter \(c_{1}\) allows tuning of the RANS turbulent integral length scale of the large-scale eddies. Its value is, by definition, close to unity, but its optimal value depends on the turbulent flow structure and conditions and on the RANS turbulence model. The optimal value of the parameter \(c_{1}=2\) was found by [PERSON] and [PERSON] [13], and used in this work. The stochastic isotropic and homogeneous velocity perturbation field can be generated by choosing probability density functions for all the random variables involved in the Fourier decomposition [10]. These random variables are the angles \(\varrho_{n}\), \(\alpha_{n}\), \(\theta_{n}\) which define the direction of the wave vector \(k_{n}\), as sketched in Figure 1, the angle \(\alpha_{n}\), which defines the direction of the unit vector \(\mathbf{\sigma}_{n}\) in a plane orthogonal to \(k_{n}\), and finally the phase angle \(\psi_{n}\). Figure 1: Representation of the wave vector \(k_{n}\) and velocity direction vector \(\sigma_{n}\) and definition of the stochastic angles (Adapted from [13]).