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 ($\delta^+$).

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 $\delta^+$ up to $O(10^4)$, using the simultaneous velocity and velocity-gradient hot-wire measurements of Zimmermann \textit{et al.} (2019 \textit{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 $\delta^+\to\infty$.

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.