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Steady separation of flow from an inclined square cylinder with sharp and rounded base. (English) Zbl 1410.76183
Summary: Initial separation of laminar boundary layer for steady flow around square cylinders at \(45^\circ\) incidence is investigated numerically using a blockage of 0.05. The cylinder shapes differ solely at the base region where corner rounding of various degrees is provided such that the base point approaches the center of the cylinder as the corner radius continues to increase. The normalized corner radius is varied between 0 and 0.25, in steps of 0.05. A very narrow regime of Reynolds number (Re) bounded by 6 and 8.2 is found to surprisingly accommodate a wide description of flow physics unforeseen in common geometries, i.e., circle, square and ellipse at \(0^\circ\) or \(90^\circ\) incidence, etc. These include secondary (no wake) followed by primary (wake) separation, simultaneous primary and secondary separation, vortex merger, degeneration of half-saddles, dual nature of a singular point, etc. A very interesting vortex structure forms when separation bubbles meet at the sharp base point, yet do not form a wake immediately. This unique structure however, disappears once the base is rounded. Two fundamental and novel flow topologies are proposed and it is demonstrated that the classical wake topology is a degenerated structure of the proposed topologies. Each of the proposed topologies satisfies the kinematic requirement of J. C. R. Hunt et al. [“Kinematical studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization”, J. Fluid Mech. 86, No. 1, 179–200 (1978; doi:10.1017/s0022112078001068)] implying that the intermediate vortical structures are stable. Overall, three distinct regimes of separation are identified – regime I for secondary separation, regime II for simultaneous primary and secondary separation and regime III for primary separation alone. A ‘flow separation map’ that completely specifies all the regimes of separation is presented for the first time for steady flow past a symmetric obstacle. The flow bifurcation is a function of corner radius. The maximum number of bifurcations equals three and this is associated with small values of radius of curvature. For secondary separation, the critical Re marking its onset is virtually constant at 7.3. The occurrence of secondary separation ceases to exist beyond a normalized corner radius of 0.15. Among the cylinder shapes considered, it is only for this cylinder that the number of singular points on the surface or number of no-slip critical points reaches a maximum value of 8.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D25 Wakes and jets
76D05 Navier-Stokes equations for incompressible viscous fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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