Eddy population based model for the wall-pressure spectrum at high Reynolds number

TL;DR

This work introduces a two-eddy-population framework to model wall-pressure spectra in turbulent wall-bounded flows at high Reynolds numbers, decomposing the spectrum into inner and outer contributions that correspond to distinct eddy populations. It presents two complementary models: Model A, a parsimonious Log-Normal representation of the inner and outer components, and Model B, a physically grounded Modified Lorentzian form with δ^+-dependent outer-shape parameters, both calibrated against boundary-layer, pipe, and channel data spanning δ^+ from roughly 10^3 to 5×10^4 and extended to higher Reynolds numbers. Both models reproduce the pre-multiplied spectrum and the Reynolds-number–dependent variance, capturing the growth of the low-frequency (sub-convective) energy and the emergence of the outer-scaled peak, thereby improving upon Goody’s model for engineering predictions of wall-pressure fluctuations. The findings provide a unified, continuous framework for predicting wall-pressure fluctuations across canonical flows and offer pathways to extend the approach to more complex physics such as pressure gradients, roughness, and compressibility, with implications for noise mitigation and structural fatigue assessments.

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. Consistent with the approach outlined modelling the streamwise Reynolds stress in the recent work of Gustenyov et al. (\textit{J. Fluid Mech.} 1016, 2025, A23), the models are based on consideration of two eddy populations that broadly 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 populations: an inner-scaled term that is $δ^+$-invariant and an outer-scaled term whose amplitude broadens smoothly with $δ^+$. The model reproduces the 1-D convective signature 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 populations. 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, laying the groundwork for more accurate engineering predictions of wall-pressure fluctuations.

Figures (9)

• Figure 1: Pre-multiplied spectra of wall-pressure fluctuations. Left column: inner scaling. Right column: outer scaling. Top row: Boundary layers. Highly-resolved LES data from eitel-amor_simulation_2014 for $\delta^+=500$ to 2000, experimental data from fritsch_pressure_2020fritsch_fluctuating_2022 for $\delta^+=4021$ to 11,064. Middle row: pipes. Experiments by dacome_scaling_2025 for $\delta^+=4794$ to 47,015. Bottom row: channels. DNS data from lee_direct_2015 for $\delta^+=180$ to 5200.
• Figure 2: Spectra of wall-pressure fluctuations in boundary layers. Smooth wall data at $\delta^+=4\,021$ to 11 064 fritsch_pressure_2020fritsch_fluctuating_2022. Left: log-log form. Right: pre-multiplied form.
• Figure 3: Pre-multiplied spectra of wall-pressure fluctuations in boundary layers compared with Goody's (2004) model. Left: model as given. Right: model normalised so that the peak value is fixed at 3.36. Highly-resolved LES data from eitel-amor_simulation_2014 for $\delta^+=500$ to 2000, experimental data from fritsch_pressure_2020fritsch_fluctuating_2022 for $\delta^+=4021$ to 11,064. Model predictions are shown by the dashed lines colour-coded to the data.
• Figure 4: Comparison of Model A with data shown in figure \ref{['fig:data']}. Model constants listed in table \ref{['tab:A params']}. Top row: boundary layers. Left: all $\delta^+$. Right: $\delta^+= 1\,000$, 11 064, showing $g_1$ and $g_2$ ($g_1$ is identical for these $\delta^+$). Middle row: pipes. Left: all $\delta^+$. Right: $\delta^+= 4\,794$, 47 015, showing $g_1$ and $g_2$. Bottom row: channels. Left: all $\delta^+$. Right: $\delta^+= 550$, 5 200 showing $g_1$ and $g_2$.
• Figure 5: The modelled vs measured wall-pressure spectra at the range of Reynolds numbers measured in dacome_scaling_2025 (a-g). The solid lines are the measured spectra, the dashed lines are the modelled spectra. The purple dashed line is $g_1$ for model B, and the green dashed line is $g_2$. In panel (h), we show a prediction of the spectrum at $\delta^+=5\times 10^5$. Panels (i,j) correspond to the data and the adapted model B predictions at the labelled $\delta^+$ values.