Compressible Potential Flows Around Round Bodies: Janzen-Rayleigh Expansion Inferences

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen-Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number M_∞. JREs were carried out with terms polynomial in the inverse radius r^-1 to high orders in two dimensions, but were limited to order in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of In(r) can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order M⁴_∞. Such terms are apparently absent in the 2-D disk, as we verify up to order M¹⁰⁰_∞, although they do appear in other dimensions (e.g. at order M_∞²in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.