Non-dimensional parameters are routinely used to classify different flow regimes. We propose a non-dimensional parameter, called Aneurysm number (An), which depends on both geometric and flow characteristics, to classify the flow inside aneurysm-like geometries (sidewalls and bifurcations). The flow inside aneurysm-like geometries can be widely classified into (i) the vortex mode in which a vortex ring is formed and (ii) the cavity mode in which a stationary shear layer acts similar to a moving lid of a lid-driven cavity. In these modes, two competing time scales exist: (a) a transport time scale, T t , which is the time scale to develop a shear layer by transporting a fluid particle across the expansion region, and (b) the vortex formation time scale, . Consequently, a relevant non-dimensional parameter is the ratio of these two time scales, which is called Aneurysm number: An = T t / . It is hypothesized, based on this definition, that the flow is in the vortex mode if the time required for vortex ring formation is less than the transport time T t (An ≳ 1). Otherwise, the flow is in the cavity mode (An ≲ 1). This hypothesis is systematically tested through numerical simulations on simplified geometries and shown to be true through flow visualizations and identification of the main vortex and shear layer. The main vortex is shown to evolve when An ≳ 1 but stationary when An ≲ 1. In fact, it is shown that the flows with An ≲ 1 (cavity mode) are characterized by much smaller fluctuations of wall shear stress and oscillatory shear index relative to flows with An ≳ 1 (vortex mode) because of their quasi-stationary flow pattern (cavity mode) compared to the evolution and breakdown of the formed vortex ring (vortex mode).