We consider steady compressible Euler flow corresponding to the
compressible analogue
of the well-known incompressible Hill's spherical vortex (HSV). We
first derive
appropriate compressible Euler equations for steady homentropic flow and
show how
these may be used to define a continuation of the HSV to finite Mach number
M∞=U∞/C∞,
where U∞, C∞ are
the fluid velocity and speed of sound at infinity
respectively. This is referred to as the compressible Hill's spherical
vortex (CHSV).
It corresponds to axisymmetric compressible Euler flow in which, within
a vortical
bubble, the azimuthal vorticity divided by the product of the density and
the distance
to the axis remains constant along streamlines, with irrotational flow
outside the bubble.
The equations are first solved numerically using a fourth-order finite-difference
method, and then using a Rayleigh–Janzen expansion in powers of
M2∞ to order M4∞.
When M∞>0, the vortical bubble is
no longer spherical and its detailed shape must
be determined by matching conditions consisting of continuity of the fluid
velocity
at the bubble boundary. For subsonic compressible flow the bubble boundary
takes
an approximately prolate spheroidal shape with major axis aligned along
the flow
direction. There is good agreement between the perturbation solution and
Richardson
extrapolation of the finite difference solutions for the bubble boundary
shape up to
M∞ equal to 0.5. The numerical solutions
indicate that the flow first becomes locally
sonic near or at the bubble centre when M∞≈0.598
and a singularity appears to
form at the sonic point. We were unable to find shock-free steady CHSVs
containing
regions of locally supersonic flow and their existence for the present
continuation of
the HSV remains an open question.
Formation of Unstable Shocks for 2D Isentropic Compressible Euler
