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 hows 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. 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 similar to 0.1), the how becomes unstable. In the case with vanishing thickness, the accretion shock formed to stop the incoming how 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 inward toward the central mass and regenerated as it moves outward. We suggest a possible explanation for the instability. The phenomenon may be related to the quasi-periodic oscillations observations in accreting galactic sources
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