  The properties of axisymmetric accretion flows of cold adiabatic gas with
zero total energy in the vicinity of a Newtonian point mass are characterized
by a single dimensionless parameter, the thickness of incoming flow. In the
limit of thin accretion flows with vanishing thickness, we show that the
governing equations become self-similar, involving no free parameters. We study
numerically thin accretion flows with finite thickness as well as those with
vanishing thickness. Mass elements of the incoming flow enter the computational
regime as thin rings. In the case with finite thickness, after a transient
period of initial adjustment, an almost steady-state accretion shock with a
small oscillation amplitude forms, confirming the previous work by Molteni,
Lanzafame, \& Chakrabarti (1994). The gas in the region of vorticity between
the funnel wall and the accretion shock follows closed streamlines, forming a
torus. This torus, in turn, behaves as an effective barrier to the incoming
flow and supports the accretion shock which reflects the incoming gas away from
the equatorial plane. The postshock flow, which is further accelerated by the
pressure gradient behind the shock, goes through a second shock which then
reflects the flow away from the symmetry axis to form a conical outgoing wind.
As the thickness of the inflowing layer decreases (or if the ratio of the half
thickness to the distance to the funnel wall along the equatorial plan is
smaller than $\sim0.1$), the flow becomes unstable. In the case with vanishing
thickness, the accretion shock formed to stop the incoming flow behind the
funnel wall oscillates quasi-periodically with an amplitude comparable to the
thickness. The structure between the funnel wall and the accretion shock is
destroyed as the shock moves inwards toward the central mass and re-generated
