Scaling of pressure fluctuation in turbulent internal flows

Previous studies on the scaling of pressure fluctuations in wall-bounded turbulent flows have typically employed the same frameworks as those used for mean flow, with inner scaling based on frictional velocity and viscous length scales, and outer scaling relying on boundary layer thickness or displacement thickness. These traditional scales primarily reflect the characteristics of the mean streamwise velocity profile and momentum balance. In this work, we propose novel scaling frameworks for pressure fluctuations in turbulent channel and pipe flows, derived from the Poisson equation for pressure fluctuations. Applying the scaling patch approach, we analyse the rapid and slow terms in the Poisson equation, and introduce new scaling for pressure fluctuation variance in both the inner and outer regions. These new scales are designed to better capture the influence of Reynolds stresses by incorporating their peak values. Additionally, we establish a strong correlation between the root mean square (r.m.s.) of pressure fluctuations and the Reynolds shear stress, resulting in an empirical equation that accurately predicts their ratio. This equation provides a practical method for estimating the r.m.s. of pressure fluctuations in the flow, which remains challenging to measure in experimental investigations.