# Single and Multivariate Statistics of Jet-Induced Pressure Fluctuations over an Infinite Plate
## 1 Introduction  
One of the most important topics of current research into aircraft engines concerns the reduction of noise and pollutant emissions. The strategy used by aircraft manufacturers to mitigate CO\({}_{2}\) release is to reduce the mass-flow rate passing in the engine's combustion chamber, thus exhausting from the primary nozzle. In order to keep the same level of thrust, the reduction of jet velocity must be compensated by increasing the mass-flow rate passing outside of the engine primary body. Given that the noise emissions from the engine jets are essentially proportional to the eighth power of the velocity according to Lighthill [1], the mass-flow increase in the secondary flow must be achieved by increasing the nozzle size rather than the exhausting jet velocity. The ultra-high by-pass ratio (UHBPR) engine concept provides this solution, achieving a reduction of jet velocity and an increase of the fan/nacelle diameter. Design constraints in terms of ground clearance cause a more aggressive close-coupled architecture for the under-wing installation of the engine and thus a stronger jet-wing/flap interaction. The interaction between the exhausting jet flow and the wing surface induces an increase of the radiated noise, thus jeopardizing the noise reduction due to the jet velocity decrease. On the other hand, the increasing size of the fan/nacelle diameter will lead to stronger jet-fuselage interactions as well, thus generating more severe concerns in terms of panel stress and vibrations. These vibrations underpin the transmission of interior noise in the fuselage, causing passenger annoyance, as well as the re-emission of vibration noise in the exterior aeroacoustic field. Aircraft and engine manufacturers are investing in research to mitigate these installation effects in the design of future under-wing engine configurations. For the above-mentioned reasons, installation effects have been the subject of several works in the past and recent literature [2; 3; 4; 5]. In this context, an important link between the wall pressure fluctuations and the far-field noise was addressed by [PERSON] [6], who developed a model to predict the far-field noise from the wall pressure fluctuations. The effect of a surface on the emitted far-field noise was provided by[7], and the scattering effect of a tangential flat plate was also investigated by [PERSON] et al. [8] using a wavepacket noise-source model. The modification induced by a tangential flat plate on the jet aerodynamics was studied by [9; 10] who outlined how the jet bent towards the surface when the mutual distance between the jet and the flat plate was of the order of the nozzle diameter. The tonal dynamics that occur when an isothermal turbulent jet interacts with a flat-surface edge were investigated through hydrodynamic and acoustic pressure measurements [11]. Studies on small-scale jet wing configurations were performed numerically [12] and experimentally [13]. The investigation of the wall pressure field induced by a jet over a surface located nearby seems to be the missing piece of this puzzling problem. This configuration has been studied in incompressible conditions by [14], laying the foundations for wall pressure fluctuation modeling by deriving scaling laws for pressure autospectra and coherence functions. The effects of the jet-plate distance on the cross and conditioned statistics between velocity and wall pressure fields were explored in [9]. More recently, the effect on the wall pressure statistics of different jet Mach numbers spanning the compressible subsonic regime was investigated in [15], whereas the effect of the jet Reynolds number was addressed by [16] (see also [17] for a general overview).