Hydraulic analogy and visualisation of two-dimensional compressible fluid flows: part 1: theoretical aspects

Abstract

The principles of the well-known, hydraulic analogy are explained (see Courant and Friedricks, 1948; Loh, 1969). The governing equations of two-dimensional compressible fluid two-dimensional flows in non-dimensional form, based on conservation laws are first discussed. Then, the isentropic flow condition is introduced to produce the isentropic Euler equations. On the other hand, the equations governing surface waves on thin water layers over a flat surface are derived in non-dimensional form, using the assumption of hydrostatic pressure across the water layer, hence the analogy between the two problems is established. The normalised density of the compressible flow corresponds to the normalised height of the thin water layer and the speed of sound corresponds to the speed of surface waves in water, hence, the Mach number corresponds to Froude number. Finally, it is shown that the analogy can be used to visualise supersonic and transonic two-dimensional flow patterns, including shock waves and expansion fans around airfoils and in convergent/divergent nozzles. Also, nonlinear water waves of finite amplitude, in dispersive media, are discussed. In part 2 of this study, water table experiments are presented together with qualitative and quantitative measurement techniques.