Two-component inner--outer scaling model for the wall-pressure spectrum at high Reynolds number

TL;DR

This paper introduces two semi-empirical models for the wall-pressure spectrum in turbulent boundary layers, capturing high Reynolds number effects and improving upon existing models by considering inner and outer spectral components.

Contribution

The paper develops two new models that incorporate inner and outer scale contributions, calibrated with simulations and data, to better predict wall-pressure spectra at high Reynolds numbers.

Findings

Models accurately reproduce inner-scaled peak and outer-scaled peak emergence.

Both models capture the logarithmic growth of spectrum variance.

The second model generalizes beyond calibration data using theoretical spectral shapes.

Abstract

Wall-pressure fluctuations beneath turbulent boundary layers drive noise and structural fatigue through interactions between fluid and structural modes. Conventional predictive models for the spectrum--such as the widely accepted Goody model (\textit{AIAA Journal} 42 (9), 2004, 1788--1794)--fail to capture the energetic growth in the {low-frequency range} that occurs at high Reynolds number, while at the same time over-predicting the variance. To address these shortcomings, two semi-empirical models are proposed for the wall-pressure spectrum in canonical turbulent boundary layers, pipes and channels for friction Reynolds numbers $\delta^+$ ranging from 180 to 47 000. The models are based on consideration of two {spectral components} that represent the contributions to the wall pressure fluctuations from inner-scale motions and outer-scale motions. The first model expresses the pre-multiplied spectrum as the sum of two overlapping log-normal {components}: an inner-scaled term that is $\delta^+$-invariant and an outer-scaled term whose amplitude broadens smoothly with $\delta^+$. Calibrated against large-eddy simulations, direct numerical simulations, and recent high-$\delta^+$ pipe data, it reproduces the {inner-scaled peak} and the emergence of an outer-scaled peak at large $\delta^+$. The second model, developed around newly available pipe data, uses theoretical arguments to prescribe the spectral shapes of the inner and outer {components}. Embedding the $\delta^+$-dependence in smooth asymptotic functions yields a formulation that varies continuously with $\delta^+$ {and generalises beyond the calibration range}. Both models capture the full spectrum and {recover} the observed logarithmic growth of its variance, {providing a compact, physics-informed empirical representation} for more accurate engineering predictions of wall-pressure fluctuations.