Pressure fluctuation caused by moderate acceleration

Pressure fluctuation caused by acceleration of a liquid column is observed in various important technologies, e.g. water-hammer in a pipeline. The magnitude of fluctuation can be estimated by two different approaches: When the duration time of acceleration is much shorter than the propagation time for a pressure wave to travel the length of the liquid column, e.g. sudden valve closure for a long pipe, Joukowsky equation is applied. In contrast, if the acceleration duration is much longer, the liquid is modeled as a rigid column, ignoring compressibility of the fluid. However, many of practical cases exist between these two extremes. In this study we propose a model describing pressure fluctuation when the duration of acceleration is in the same order of the propagation time for a pressure wave, i.e. under moderate acceleration. The novel model considers both temporal and spatial evolutions of pressure propagation as well as gradual pressure rise during the acceleration. We conduct experiments in which we impose acceleration to a liquid with varying the length of the liquid column, acceleration duration, and properties of liquids. The ratio between the acceleration duration and the propagation time is in the range of 0.02 - 2. The model agrees well with measurement results.