Compressible potential flows around round bodies: Janzen-Rayleigh expansion inferences

The subsonic, compressible, potential flow around a hypersphere can be derived using the Jansen-Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number M-infinity. JREs were carried out with terms polynomial in the inverse radius r(-1) to high orders in two dimensions, but were limited to order M-infinity(4) in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of ln(r) can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order M-infinity(4). Such terms are apparently absent in the 2-D disk, as we verify up to order M-infinity(100). although they do appear in other dimensions (e.g. at order M-infinity(2) in 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.

Automated summary

The study derives the potential flow around a hypersphere using Jansen-Rayleigh expansion, finding that logarithmic terms are essential in three-dimensional objects but absent in two-dimensional objects. It also reveals a connection between the appearance of logarithmic terms and the topological properties of different geometric shapes.