On the Poisson-Source Basis of Logarithmic Wall-Pressure-Variance Growth

In high-Reynolds-number wall-bounded flows, the inner-scaled wall-pressure variance \ra{is often represented as a} logarithmic increase with frictional Reynolds number. We consider the two sources of the incompressible pressure--Poisson equation: a linear (rapid) term linked to mean shear and a nonlinear (slow) term composed of quadratic velocity fluctuations. This paper establishes a link between the sources and the coefficients in \rtwo{a logarithmic} inner-scaled variance \rtwo{representation}. \rone{To leading order} we \rone{posit that} the \ra{linear source provides a Reynolds-number-independent offset, while the nonlinear source contributes the logarithmic coefficient}. The illustrative dataset is direct numerical simulation (DNS) at \rtwo{frictional Reynolds number} $\delta^+\approx 550$, although the principal contribution is the establishment of a mechanistic link to well-known high-$\delta^+$ scalings of wall-bounded turbulence. Through consideration of the sources and the integral solution method of the Poisson equation, we find that the linear source contribution sits predominantly in the buffer layer and maps to the near-wall cycle. \rone{To leading order}, this contribution becomes $\delta^+$ invariant under inner scaling, thus contributing an offset in the \rtwo{logarithmic representation}. The interfacial regions between uniform momentum zones characteristic of the inertial layer (vortical fissures) spatially localise strain and vorticity contributions and contain an increasingly large proportion of the strain and vorticity. We show that fissures act as a compact carrier for the source terms, with the nonlinear term especially prominent in these regions. By considering the inertial layer statistics, we link the changing nonlinear contribution to \rtwo{the} $\ln \delta^+$ \rtwo{growth}, in agreement with previous empirical observations.