# Single and Multivariate Statistics of Jet-Induced Pressure Fluctuations over an Infinite Plate
## 3 Results
### Two-Point Statistics  
The two-point statistics of the wall pressure fluctuations were characterized in the time domain in terms of the cross-correlation computed between two contiguous pressure transducers in both the stream-wise and span-wise directions. The cross-correlation function is defined as follows:  
\[R_{\rm pp}=<p(x,t),p(x+\xi,y+\eta,t+\tau)> \tag{2}\]  
where \(\xi\) and \(\eta\) are the stream-wise and span-wise separations, respectively (in the present study, \(\xi=\eta=1D\)), \(\tau\) is the time lag and the symbol \(<>\) denotes the ensemble average. The cross-correlation is normalized by dividing by the product of the standard deviations of the two pressure signals, providing the non-dimensional cross-correlation coefficient.  
The axial evolution of the cross-correlation coefficient at plate radial distances of H/D = 0.75 and H/D = 2 is reported in Figure 5. The oscillatory pseudo-periodic trend observed at the axial  
Figure 4: Span-wise evolution of the pressure autospectra for the jet–plate distance H/D = 0.75 and the jet Mach number M\({}_{\rm j}\) = 0.5: (a) x/D = 5; (b) x/D = 25.  
Figure 3: Mach number effect on the pressure autospectra for the jet–plate radial distance H/D = 0.75: (a) x/D = 5; (b) x/D = 25.  
distances x/D = 5 and x/D = 6, is probably related to the jet hydrodynamic field (i.e., Kelvin-Helmholtz instability). At any rate, further analyses are required to better clarify this issue.  
This effect is more evident at H/D = 0.75, because of the vicinity of the pressure transducers to the jet plume. For increasing axial distances, the negative bump observed for a small x/D progressively disappears and the time scale of the cross-correlations increases due to the development of large-scale structures in the jet plume. Comparing Figure 5a against Figure 5b, we note that a reduction of the flat plate radial position (H) induces a narrowing of the cross correlations. This is due to the breakup of the large-scale turbulent structures as the plate approaches the jet. In both configurations, the time delay of the cross-correlation peak increases for larger x/D, thus inducing a reduction of the phase speed of the associated flow structures. This effect is more perceptible at H/D = 0.75.  
The effect of the jet Mach number on the cross-correlation coefficient is analyzed at x/D = 5 and x/D = 25 and reported in Figure 6a,b, respectively. Figure 6a shows that the oscillatory trend detected at x/D = 5 seems to be slightly reduced with increasing Mach numbers. We observe a reduction in the time delay associated with the first peak for increasing jet Mach numbers, thus revealing a dependence of phase speed upon the jet velocity--an effect that will be discussed in the next section. As reported in [PERSON] et al. [15], a similar behavior was observed at H/D = 2.  
Cross-correlations were evaluated also in the span-wise direction at two different axial locations x/D = 5 and x/D = 25, reported in Figure 7a,b, respectively. We observe an oscillatory trend that is slightly influenced by the different span-wise location in Figure 7a. Figure 7b shows an increase of the cross-correlation peak amplitude for increasing y/D--a trend that can be ascribed to the reduction of the turbulence level of the jet and hence the randomness of the associated time signal.  
Figure 5: Axial evolution of the wall pressure cross-correlation coefficient at M\({}_{\text{j}}\) = 0.5: (**a**) H/D = 0.75; (**b**) H/D = 2.  
Figure 6: Mach effect on the wall pressure cross-correlation coefficient for the jet–plate distance H/D = 0.75: (**a**) x/D = 5; (**b**) x/D = 25.  
Furthermore, the time delay of the cross-correlation peak is close to zero and remains about constant in both Figure 7a,b, confirming that the convection effect is not relevant in the span-wise direction.