Computational and Applied Mathematics 2019 38:121 https://doi.org/10.1007/s40314-019-0895-4 Structural bifurcation analysis of vortex shedding from shear flow past circular cylinder Atendra Kumar 1 · Rajendra K. Ray 1 Received: 31 May 2018 / Revised: 9 May 2019 / Accepted: 17 May 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019 Abstract In this paper, the unsteady flow separation of two-dimensional (2-D) incompressible shear flow past a circular cylinder is studied using theoretical structural bifurcation analysis based on topological equivalence. The stream function vorticity form of Navier–Stokes (N–S) equa- tions in cylindrical polar coordinates are considered as the governing equations. Numerical simulations are performed, using higher-order compact finite difference scheme (Kalita and Ray J Comput Phys 228:5207–5236 2009), for the Reynolds number ( Re) 100, 200 and shear parameter ( K )0.0, 0.05, 0.1 and 0.2. Through this structural bifurcation analysis, the exact location and time of occurrence of bifurcation points (flow separation points) associated with primary and secondary vortices are studied. In this process, the instantaneous vorticity contours and streakline patterns, center-line velocity fluctuation, lift and drag coefficients, phase diagram are also studied to confirm the theoretical results. Unlike the shear flow past a square cylinder, in this case, the structural bifurcation occurs from the downstream sur- face of the circular cylinder for the same Reynolds number. All the computed results very efficiently and very accurately reproduce the complex flow phenomena. Through this study, many noticeable and interesting results for this problem are reported for the first time. Keywords Shear flow · Circular cylinder · Vortex shedding · Structural bifurcation · HOC scheme · Navier–Stokes equations Mathematics Subject Classification 65N06 · 65Z05 · 65Y99 1 Introduction The unsteady flow separation leading to vortex shedding behind a circular cylinder has been one of the most important subjects in fluid mechanics due to its practical and theoretical Communicated by Corina Giurgea. B Rajendra K. Ray rajendra@iitmandi.ac.in Atendra Kumar atendra.iitd@gmail.com 1 School of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175005, India 123