Exact Solution of the Linear Hyperbolic Four-Equation System in Axial Liquid-Pipe Vibration

The so-called “FSI four-equation model” describes the axial vibration of liquid-filled pipes. Two equations for the liquid are coupled to two equations for the pipe, through terms proportional to the Poisson contraction ratio, and through mutual boundary conditions. Skalak (1955/1956ab) defined this basic model, which disregards friction and damping effects. The four equations can be solved with the method of characteristics (MOC). The standard approach is to cover the distance-time plane with equidistantly spaced grid-points and to time-march from a given initial state. This approach introduces error, because either numerical interpolations or wave speed adjustments are necessary. This paper presents a method of exact calculation in terms of a simple recursion. The method is valid for transient events only, because the calculation time grows exponentially with the duration of the event. The calculation time is proportional to the temporal and spatial resolution. The exact solutions are used to investigate the error due to numerical interpolations and wave speed adjustments, with emphasis on the latter.