Pressure fluctuations of liquids under short-time acceleration

Abstract

This study experimentally investigates the pressure fluctuations of liquids in a column under short-time acceleration and demonstrates that the Strouhal number $St$ [$=L/(c\Delta t)$, where $L$, $c$, and $\Delta t$ are the liquid column length, speed of sound, and acceleration duration, respectively] provides a measure of the pressure fluctuations both for limiting cases (i.e. $St\ll1$ or $St = \infty$) and for intermediate $St$ values. Incompressible fluid theory and water hammer theory respectively imply that the magnitude of the averaged pressure fluctuation $\overline{P}$ becomes negligible for $St\ll1$ (i.e., in the condition where the duration of acceleration $\Delta t$ is large enough compared to the acoustic timescale) and tends to $\rho cu_0$ (where $u_0$ is the change in the liquid velocity) for $St\geq O(1)$ (i.e., in the condition where $\Delta t$ is small enough). For intermediate $St$ values, there is no consensus on the value of $\overline{P}$. In our experiments, $L$, $c$, and $\Delta t$ are varied so that $0.02 \leq St \leq 2.2$. The results suggest that the incompressible fluid theory holds only up to $St\sim0.2$ and that $St$ governs the pressure fluctuations under different experimental conditions for higher $St$ values. The data relating to a hydrogel also tend to collapse to a unified trend. The inception of cavitation in the liquid starts at $St\sim 0.2$ for various $\Delta t$, indicating that the liquid pressure becomes negative. To understand this mechanism, we employ a one-dimensional wave propagation model with a pressure wavefront of finite thickness that scales with $\Delta t$. The model provides a reasonable description of the experimental results as a function of $St$. The slight discrepancy between the model and experimental results reveals additional contributing factors such as the container motion and the profile of the pressure wavefront.