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

Abstract

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 (2D), but were limited to order \(\mathcal{M}_\infty^4\) in three dimensions (3D). We derive general JRE formulae to arbitrary order, adiabatic index, and dimension. We find that powers of \(\ln(r)\) can creep into the expansion, and are essential in 3D beyond order \(\mathcal{M}_\infty^4\). Such terms are apparently absent in the 2D disk, as we confirm up to order \(\mathcal{M}_\infty^{100}\), although they do show in other dimensions (e.g. at order \(\mathcal{M}_\infty^2\) in 4D) and in non-circular 2D bodies. This suggests that the disk, which was extensively used to study basic flow properties, has additional symmetry. Our results 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.