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

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 $δ^+$ 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 $δ^+$-invariant and an outer-scaled term whose amplitude broadens smoothly with $δ^+$. Calibrated against large-eddy simulations, direct numerical simulations, and recent high-$δ^+$ pipe data, it reproduces the {inner-scaled peak} and the emergence of an outer-scaled peak at large $δ^+$. 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 $δ^+$-dependence in smooth asymptotic functions yields a formulation that varies continuously with $δ^+$ {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.