# Near-Field Aeroacoustics of Spanwise Forcing on a Transonic Wing: A DNS Study
## 2 Numerical Method and Description of the Dataset  
### Solver, Setup, and Procedures  
The analysis presented in this work derives stems from the database of 29 DNSs of the transonic flow around a wing slab performed in [PERSON] et al. [13]. The database has been marginally extended to consider longer integration times for a few selected cases. The main features of the dataset are reported here; the reader is referred to the original studies for further details.  
The numerical code adopted for the computations was presented and validated extensively in [PERSON] et al. [20]. The code solves the compressible Navier-Stokes equations for an ideal gas, with the Newtonian fluid hypothesis and Fourier's law adopted for describing the viscous stress tensor and the heat flux, respectively. The code employs a second-order, energy-consistent finite-volumes method [21], which switches to a third-order weighted essentially non-oscillatory scheme close to shock waves. The switch is governed by a modified version [21] of the Ducros sensor [22]. A low-storage, third order Runge-Kutta scheme is employed for temporal advancement.  
A C-grid with approximately \(536\times 10^{6}\) points, uniformly extruded in the spanwise direction, and with a hyperbolic tangent clustering in the wall-normal direction is adopted. The dimensions of the smallest, near-wall cells ensure that all the turbulent scales are well resolved, as \(\Delta x^{+}<10\), \(\Delta y^{+}<0.5\) and \(\Delta z^{+}<5\) (where \(\Delta\) denotes the grid spacing, and the '+' superscript indicates scaling with the inner viscous quantities, i.e., the friction velocity \(u_{\tau}\) and the viscous length scale \(\delta_{v}\)). The mesh features a radius of 25 chords (\(c\)) and a spanwise extension of \(0.1c\), while the outlet is placed \(25c\) downstream of the trailing edge of the wing. The temporal discretization uses a fixed time step \(\Delta t=1.5\times 10^{-4}c/U_{\infty}\), which ensures that the CFL number is maintained below unity, therefore granting numerical stability.  
The supercritical V2C airfoil, designed within the European project TFAST and purposely developed for transonic flow conditions, is studied. The free stream airflow is steady, with \(M_{\infty}\equiv U_{\infty}/a_{\infty}=0.7\) (where \(M\) is the Mach number, \(U\) and \(a\) are the velocity and speed of sound, respectively, and the '\(\infty\)' subscript denotes the undisturbed flow) and \(Re_{\infty}\equiv U_{\infty}c/\
u_{\infty}=3\times 10^{5}\) (where \(\
u_{\infty}\) is the kinematic viscosity). The angle of attack is fixed to \(\alpha=4^{\circ}\), at which the profile is known to have maximum aerodynamic efficiency; other incidences are considered when flow control alters the wing efficiency. The inflow is laminar. Since the value of \(Re_{\infty}\) adopted in the computations is not sufficiently high to grant a naturally well-developed turbulent boundary layer, turbulence is tripped numerically at \(x=0.1c\) downstream of the leading edge on both sides of the airfoil through a Gaussian blob of random wall-normal forces, following a well established practice [23].  
After the flow reaches stationary conditions, statistics for each simulation are accumulated; the initial transient length is determined with a case-by-case analysis. Convergence and uncertainty of the computed mean values are checked with the algorithm presented by [PERSON] and [PERSON] [24]; the standard deviation of the main integral aerodynamic coefficients remains below 1% of the coefficient's mean value across all simulations. Complete details on the numerical procedures are contained in Ref. [13].