Relation between pressure and spanwise velocity fluctuations in turbulent channel flows

We study the logarithmic behavior of the mean-square pressure fluctuations ⟨p⁺²⟩ in fully developed turbulent channel flows. Data of direct numerical simulation of channel flow is used in the present research for Reynolds number flows up to Re_τ=4000 . The profile of ⟨p⁺²⟩ is found to follow a logarithmic behavior at some distances from the wall which is similar to that of the mean-square spanwise velocity fluctuations ⟨w⁺²⟩ at the same Reynolds number. The results are also emphasized for the higher-order moments of  p and w where the logarithmic behavior is confirmed. Spectral analysis in the streamwise and spanwise directions indicate that the ridge of the spectra of p and w  are overlapping each other within the regions where ⟨p⁺²⟩ and ⟨w²⟩ show the log-trends. In addition, within these regions, the shape of the two-dimensional spectra of p and w collapse with each other, and the contours of the spectra at a constant level are bounded by a linear relationship between the streamwise and spanwise length scales This linear relation indicates that p and w may be associated with geometrically shaped self-similar eddies. The shape of these self-similar eddies is investigated, and the scaling of their sizes with the distance from the wall are discussed.