Behind the logarithmic growth of inner-scaled wall-pressure variance

Abstract: In high-Reynolds-number zero-pressure-gradient turbulent channel flow, the inner-scaled wall-pressure variance follows a logarithmic increase with frictional Reynolds number of the form $$\langle p_w^{+2}\rangle = B_{\mathcal{L}} + A_{\mathcal{Q}} \ln δ<sup>+$$. We consider the two sources of the pressure Poisson equation: a linear (i.e., rapid) term linked to mean shear and a nonlinear (i.e., slow) term from quadratic velocity fluctuations. The goal of this paper is to provide a mechanistic link between the sources of the pressure Poisson equation and the coefficients in the inner-scaled variance form above. We tie the offset $B_{\mathcal{L}}$ to the linear source and connect the coefficient $A_{\mathcal{Q}}$ to the nonlinear source. The illustrative dataset is direct numerical simulation (DNS) at $δ^+\approx 550$, although the principal contribution is the establishment of a mechanistic link that allows us to connect well-known high-$δ<sup>+$ scalings of wall-bounded turbulence. Through consideration of the spectral content 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. This contribution becomes $δ<sup>+$ invariant under inner scaling, thus contributing the offset $B_{\mathcal{L}}$. The interfacial region between uniform momentum zones starting in the log layer (fissures) concentrates strain and vorticity contributions in the log layer and contains 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 this region. Then, by considering the inertial layer statistics, we tie the changing nonlinear contribution to $A_{\mathcal{Q}}$.