Pressure Fluctuations in Turbulent Wall Layers

Pressure fluctuation profile data from the channel flow DNS of Lee and Moser [\textit{J. Fluid Mech.}, vol 774, 2015] extend to $Re_\tau \approx 5200$. In the outer region, with $Y=y/h$, the overlap layer pressure correlates very well by a log law; $\lim_{Y \rightarrow 0}\langle p^2 \rangle^+ \sim (1/\eta) \ln Y + D_o$. The constant $\eta = - 0.380$ is remarkable like the von K\'{a}rm\'{a}n value. In the inner region, the defect variable $\mathcal{P} (y^+) \equiv \langle p^2 \rangle^+ - \langle p^2 \rangle^+ \vert_{y=0}$ absorbs the $Re_\tau$ dependence. The inner overlap equation is; $\lim_{y^+ \rightarrow \infty}\mathcal{P} \sim (1/\eta) \ln y^+ + D_i$. Together, the overlap laws imply that the wall pressure relation is $\langle p^2 \rangle^+ \vert_{y=0} \sim (-1/\eta) \ln Re_\tau + D_i - D_o$. A completely equivalent expression, which is finite as $Re_\tau \rightarrow \infty$, is obtained by rescaling the pressure variable; $\langle p^2 \rangle^\# \vert_{y=0} \equiv (u_\tau / U_o)\langle p^2 \rangle^+ \vert_{y=0} \ = C_1 + C_2 (u_\tau/U_o)$. Here, the constants are related to $\eta, D_0$, and $D_i$ . Additionally, it was found that the wavenumber spectrum $E_{pp} \{k_x/h\}$ does not have a $k^{-1}$ region. However, the trends do not rule out this at higher Re.