TY - JOUR
T1 - Compressible Potential Flows Around Round Bodies
T2 - Janzen-Rayleigh Expansion Inferences
AU - Wallerstein, Idan S.
AU - Keshet, Uri
N1 - Funding Information:
This research has received funding from the GIF (Grant No. I-1362-303.7 / 2016), and was supported by the Ministry of Science, Technology & Space, Israel, by the IAEC-UPBC joint research foundation (Grants No. 257/14 and 300/18) and by the Israel Science Foundation (Grant No. 1769/15).
Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.
PY - 2022/2/10
Y1 - 2022/2/10
N2 - 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.
AB - 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.
KW - general fluid mechanics
UR - http://www.scopus.com/inward/record.url?scp=85120922459&partnerID=8YFLogxK
U2 - 10.1017/jfm.2021.965
DO - 10.1017/jfm.2021.965
M3 - Article
SN - 0022-1120
VL - 932
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
M1 - A6
ER -