On the weakly compressible/incompressible limit applied to falling jets

Abstract

We consider the flow of a fluid falling from a vertical pipe, hitting a horizontal plate and flowing sideways. First the steady two-dimensional and irrotational flow of an inviscid and incompressible fluid is considered. A parametric study shows that the flow can either leave the pipe tangentially or detach from the edge of the pipe. Two dimensionless numbers come into play: the Froude number and the aspect ratio between the falling altitude and the pipe width. When the flow leaves tangentially, it can either be diverted immediately by the plate or experience squeezing before being diverted. The profile of the pressure exerted on the plate is computed and discussed. Then the same problem is revisited with the inclusion of compressibility effects, both for the falling liquid and for the gas surrounding it. An additional dimensionless number comes into play, namely the Mach number. Finally, an asymptotic analysis in the limit of the Mach number going to 0 is provided.