An Inner-Scaled Linear Contribution to Wall-Pressure Variance at High Reynolds Number

In canonical turbulent wall-bounded flows, the inner-scaled wall-pressure variance is empirically well described by a constant offset plus a slope logarithmic in the friction Reynolds number ($\text{dep}$). Because the fluctuating pressure is predominantly a Poisson response to only two source terms – a linear contribution from the mean shear coupled to a fluctuating velocity gradient, and a nonlinear contribution from the fluctuating velocity field – the origin of this growth can be pinned down by elimination: if the linear source saturates at a Reynolds-number-independent value, the nonlinear source must carry the logarithmic growth. Here we supply the complementary evidence for inner-scaled invariance of the linear source at $\text{dep}$ up to $O({10}^4)$, using the simultaneous velocity and velocity-gradient hot-wire measurements of Zimmermann $\mathit{\text{et}}\mathit{\text{al.}}$ (2019 $\mathit{\text{JFM}}$, vol. 869, pp. 182–213) acquired with a single eight-sensor probe in both a zero-pressure-gradient turbulent boundary layer and a high-Reynolds-number pipe flow. The inner-scaled factors entering the linear source collapse across Reynolds number, and the inertial-layer variance of the relevant fluctuating velocity gradient decays inversely with wall distance. Together with the established inner scaling of the mean shear, this is consistent with a linear wall-pressure contribution that, under inner normalisation, remains $O(1)$ as $\text{dep}\to \mathrm{\infty }$. Both source terms then trace to one structural mechanism: the near-wall depletion of mean spanwise vorticity that caps the linear source also feeds, through vortex stretching, the inertial-layer fissures that carry the growing nonlinear contribution.