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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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##### References:
 [1] Batchelor, G. K., An introduction to fluid dynamics, (1967), Cambridge University Press · Zbl 0152.44402 [2] Dennis, S. C.R.; Young, P. J.S., Steady flow past an elliptic cylinder inclined to the stream, J Eng Math, 47, 101-120, (2003) · Zbl 1038.76513 [3] Dutta, S.; Muralidhar, K.; Panigrahi, P. K., Influence of the orientation of a square cylinder on the wake properties, Exp Fluids, 34, 16-23, (2003) [4] Dutta, S.; Panigrahi, P. K.; Muralidhar, K., Experimental investigation of flow past a square cylinder at an angle of incidence, J Eng Mech, 134, 9, 788-803, (2008) [5] Huang, R. F.; Lin, B. H., Effects of flow patterns on aerodynamic forces of a square cylinder at incidence, J Mech, 27, 3, 347-355, (2011) [6] Hunt, J. C.R.; Abell, C. J.; Peterka, J. A.; Woo, H., Kinematical studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization, J Fluid Mech, 86, 1, 179-200, (1978) [7] Igarashi, T., Characteristics of the flow around a square prism, Bull JSME, 27, 231, 1858-1865, (1984) [8] Jahromi, J. A.; Nezhad, A. H.; Behzadmehr, A., Effects of inclination angle on the steady flow and heat transfer of power-law fluids around a heated inclined square cylinder in a plane channel, J NonNewton Fluid Mech, 166, 1406-1414, (2011) · Zbl 1282.76019 [9] Kumar, A.; Dhiman, A. K.; Bharti, R. P., Power-law flow and heat transfer over an inclined square bluff body: effect of blockage ratio, Heat Transf. Asian Res, 43, 2, 167-196, (2014) [10] Lighthill, M. J.; Rosenhead, L., Laminar boundary layers, 46-113, (1963), Dover Publications, Inc. [11] Moussaoui, M. A.; Jami, M.; Mezrhab, A.; Naji, H., MRT-lattice Boltzmann simulation of forced convection in a plane channel with an inclined square cylinder, Int J Therm Sci, 49, 131-142, (2010) [12] Park, J. K.; Park, S. O.; Hyun, J. M., Flow regimes of unsteady laminar flow past a slender elliptic cylinder at incidence, Int J Heat Fluid Flow, 10, 4, 311-317, (1989) [13] Perry, A. E.; Chong, M. S., A description of eddying motions and flow patterns using critical-point concepts, Annu Rev Fluid Mech, 19, 125-155, (1987) [14] Saad, Y.; Schultz, M. H., A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J Sci. Stat Comput, 7, 856-869, (1986) · Zbl 0599.65018 [15] Sen, S.; Mittal, S.; Biswas, G., Steady separated flow past a circular cylinder at low Reynolds numbers, J Fluid Mech, 611, 89-110, (2009) · Zbl 1156.76381 [16] Sen, S.; Mittal, S.; Biswas, G., Flow past a square cylinder at low Reynolds numbers, Int J Numer Methods Fluids, 67, 9, 1160-1174, (2011) · Zbl 1426.76303 [17] Sen, S.; Mittal, S.; Biswas, G., Steady separated flow past elliptic cylinders using a stabilized finite-element method, Comput Model in Eng Sci, 86, 1, 1-27, (2012) · Zbl 1356.76065 [18] Sheard, G. J.; Fitzgerald, M. J.; Ryan, K., Cylinders with square cross-section: wake instabilities with incidence angle variation, J Fluid Mech, 630, 43-69, (2009) · Zbl 1181.76069 [19] Sohankar, A.; Norberg, C.; Davidson, L., Low-Reynolds number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition, Int J Numer Methods Fluids, 26, 39-56, (1998) · Zbl 0910.76067 [20] Sumer, B. M.; Fredsoe, J., Hydrodynamics around cylindrical structures, (2006), World Scientific Publishing Pte. Ltd. · Zbl 1153.76003 [21] Tezduyar, T. E.; Mittal, S.; Ray, S. E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput Methods Appl Mech Eng, 95, 221-242, (1992) · Zbl 0756.76048 [22] Tong, X. H.; Luo, S. C.; Khoo, B. C., Transition phenomena in the wake of an inclined square cylinder, J Fluids Struct, 24, 7, 994-1005, (2008) [23] Tritton, D. J., Physical fluid dynamics, (1988), Oxford University Press · Zbl 0383.76001 [24] Williamson, C. H.K., Vortex dynamics in the cylinder wake, Annu Rev Fluid Mech, 28, 477-539, (1996) [25] Yoon, D. H.; Yang, K. S.; Choi, C. B., Flow past a square cylinder with an angle of incidence, Phys Fluids, 22, 4, 043603, (2010) · Zbl 1190.76135 [26] Zaki, T. G.; Gad-El-Hak, M., Numerical and experimental investigation of flow past a freely rotatable square cylinder, J Fluids Struct, 8, 7, 555-582, (1994) [27] Zdravkovich, M. M., Flow around circular cylinders, 1, (1997), Oxford University Press · Zbl 0882.76004
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