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

TL;DR

The paper addresses the logarithmic growth of the inner-scaled wall-pressure variance in high-$δ^+$ wall-bounded turbulence by decomposing the fluctuating pressure via the Poisson equation into a linear rapid term $\mathcal{L}$ and a nonlinear slow term $\mathcal{Q}$. Through spectral analysis, Green's-function elliptic attenuation, and a fissure-based inertial-layer model, it links the near-wall offset $B_{\mathcal{L}}$ to the linear source and the $A_{\mathcal{Q}}\ln δ^+$ growth to the nonlinear source, predicting $A_{\mathcal{Q}}$ of order unity (approximately 2) in agreement with observations. DNS data at $δ^+ \approx 550$ support that $\mathcal{L}$ is localized in the buffer layer while $\mathcal{Q}$ is concentrated in the inertial layer, with fissures acting as compact carriers for $\mathcal{Q}$ to the wall. The framework provides a mechanistic bridge between wall-pressure statistics and inertial-layer vorticity structures (UMZs and fissures), offering insights for control strategies that target either the linear offset or the nonlinear growth.

Abstract

In high-Reynolds-number wall-bounded flows, including zero-pressure-gradient turbulent boundary layers, channels, and pipes, the inner-scaled wall-pressure variance follows a logarithmic increase with frictional Reynolds number of the form $\langle p_w^{+2}\rangle = B_{\mathcal{L}} + A_{\mathcal{Q}} \ln δ^+$. We consider the two sources of the pressure Poisson equation: a linear (rapid) term linked to mean shear and a nonlinear (slow) term composed of 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 to well-known high-$δ^+$ 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 $δ^+$ invariant under inner scaling, thus contributing the offset $B_{\mathcal{L}}$. The interfacial regions between uniform momentum zones characteristic of the log layer (vortical fissures) spatially localise strain and vorticity contributions in the log layer 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. Then, by considering the inertial layer statistics, we associate the changing nonlinear contribution to $A_{\mathcal{Q}}\ln δ^+$.

Figures (6)

• Figure 1: (a) Schematic of the wall-pressure spectra from massey_eddy_2025, showing a universal---$\delta^+$-independent---inner function and a $\delta^+$-dependent outer function responsible for modelling the contribution from outer-scaled motions. (b) $\langle p_w^{+2}\rangle$ for various DNS and experiments, showing the $\ln\delta^+$ growth. Blue triangles represent boundary layer data from fritsch_pressure_2020fritsch_fluctuating_2022, green circles represent pipe data from dacome_scaling_2025, and orange squares represent channel data from lee_direct_2015. The red dotted line is the variance from the universal inner function, showing that it contributes an $O(1)$ offset, $B_{\mathcal{L}}$, when the inner function is fully developed.
• Figure 2: Instantaneous and mean r.m.s. profiles of the pressure across a half-channel width at $\delta^+\approx 550$ and a comparison to the r.m.s. reported by lee_direct_2015.
• Figure 3: Ring-averaged $y^+$--$k$ spectra of (a) linear $\mathcal{L}$ and (b) nonlinear $\mathcal{Q}$ sources, as well as the corresponding wall-pressure attribution maps to the bottom wall for (c) linear pressure and (d) nonlinear pressure. (c,d) The black dashed line indicates $k\,y=1$, and the two gray lines indicate $\xi=0.3$ and $\xi=3$. The kink in the lines is due to the change from log to linear scaling at $y^+=10$ toward the channel wall.
• Figure 4: Profiles of the r.m.s. of the Poisson source components versus $y^+$. (a) The individual components, $\mathcal{L}$, $\mathcal{T}$, $-\mathcal{S}$, and $\mathcal{Q}$, and the sign-split parts, $\mathcal{Q}^\oplus$ and $\mathcal{Q}^\ominus$ (Eqs. \ref{['eq:pp']},\ref{['eq:pp_vorticity']}, and \ref{['eq:strain enstrophy domination']}). (b) Log--log view of selected curves.
• Figure 5: (a) Wall-normal trends at $\delta^+\approx 550$: ${\rm r.m.s.}(\partial_x v)$ (blue), $|2\Omega_z|$ (green), and ${\rm r.m.s.} (\mathcal{L})$ (black). Shaded bands mark the buffer and logarithmic windows. (b) Premultiplied, inner-scaled wall-pressure spectrum carried by $\mathcal{L}$, shown as $k^+E^{+}_{p_{\mathcal{L}}}(k^+)$, when the integral in equation \ref{['eq:poisson_integral_walls']} is restricted to three slabs on the lower half-channel: $5\le y^+<30$ (blue), $30\le y^+<2.6\sqrt{\delta^+}$ (orange; $\sim$60 at $\delta^+\!\approx\!550$), and $2.6\sqrt{\delta^+}\le y^+<\delta^+$ (gray).