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
${\mathcal {M}}_\infty$
. JREs were carried out with terms polynomial in the inverse radius
$r^{-1}$
to high orders in two dimensions, but were limited to order
${\mathcal {M}}_\infty ^{4}$
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
${\mathcal {M}}_\infty ^{4}$
. Such terms are apparently absent in the 2-D disk, as we verify up to order
${\mathcal {M}}_\infty ^{100}$
, although they do appear in other dimensions (e.g. at order
${\mathcal {M}}_\infty ^{2}$
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.
Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences
