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Unsteady separation leading to secondary and tertiary vortex dynamics: the sub-$\alpha $- and sub-$\beta $-phenomena

Published online by Cambridge University Press:  30 July 2013

Jiten C. Kalita  and
Shuvam Sen
Jiten C. Kalita*
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India
Shuvam Sen
Affiliation:
Department of Mathematical Sciences, Tezpur University, Tezpur 784028, India
*
†Email address for correspondence: [email protected]
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Abstract

Studies on the $\alpha $- and $\beta $-phenomena, terms coined by Bouard & Coutanceau (J. Fluid Mech., vol. 101, 1980, pp. 583–607) for the flow past an impulsively started circular cylinder, have been confined only to the very early stages of the flow. In this paper, besides making a comprehensive in-depth analysis of these phenomena for a much longer period of time, we report the existence of some tertiary vortex phenomena for the first time, which we term the sub-$\alpha $- and sub-$\beta $-phenomena. The mechanism of unsteady flow separation at high Reynolds numbers for the flow past a circular cylinder developed in the last two decades has been used to understand these flow phenomena. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier–Stokes equations for the range of Reynolds number $500\leq \mathit{Re}\leq 10\hspace{0.167em} 000$. We specifically choose $\mathit{Re}= 5000$ to describe the interplay among the primary, secondary and tertiary vortices leading to these interesting vortex dynamics. We also report a $\beta $-like phenomenon which is very similar to the $\beta $-phenomenon, but slightly differs in details. We offer a new perception of the $\alpha $-phenomenon by defining its existence in a strong and weak sense along with a clearer characterization of the $\beta $-phenomenon. Apart from numerical computation, a detailed theoretical characterization using topological aspects of the boundary layer separation leading to the secondary and tertiary vortex phenomena has also been carried out. We compare our numerical results with established experimental and numerical results wherever available and an excellent match with the experimental results is obtained in all cases.

JFM classification

Aerodynamics: Aerodynamics Mathematical Foundations: Computational methods Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 730 , 10 September 2013 , pp. 19 - 51
Copyright
©2013 Cambridge University Press 

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References

Bakker, P. G. 1989 Bifurcation in flow patterns. PhD thesis, Technical University of Delft, Netherlands.Google Scholar
Ben-Artzi, M., Croisille, J. P. & Fishelov, D. 2006 Convergence of a compact scheme for the pure streamfunction formulation of the unsteady Navier–Stokes system. SIAM J. Numer. Anal. 44, 1997–2024.CrossRefGoogle Scholar
Ben-Artzi, M., Croisille, J. P., Fishelov, D. & Trachtenberg, S. 2005 A pure-compact scheme for the streamfunction formulation of Navier–Stokes equations. J. Comput. Phys. 205, 640–664.Google Scholar
Bouard, R. & Coutanceau, M. 1980 The early stage of development of the wake behind an impulsively started cylinder for $40\leq Re\leq 1{0}^{4} $ . J. Fluid Mech. 101, 583–607.Google Scholar
Brinckmanand, K. W. & Walker, J. D. A. 2001 Instability in a viscous flow driven by streamwise vortices. J. Fluid Mech. 432, 127–166.Google Scholar
Brinkerhoff, J. R. 2011 Interaction of viscous and inviscid instability modes in separation-bubble transition. Phys. Fluids 23, 124102.CrossRefGoogle Scholar
Cassel, K. W. 2000 A comparison of Navier–Stokes solutions with the theoretical description of unsteady separation. Phil. Trans. R. Soc. Lond. A 358, 3207–3227.CrossRefGoogle Scholar
Cassel, K. W., Smith, F. T. & Walker, J. D. 1996 The onset of instability in unsteady boundary-layer separation. J. Fluid Mech. 315, 223–256.Google Scholar
Chang, C.-C. & Chern, R.-L. 1991 A numerical study of flow around an impulsively started circular cylinder by a deterministic vortex method. J. Fluid Mech. 233, 243–263.CrossRefGoogle Scholar
Chou, M.-H. & Huang, W. 1996 Numerical study of high-Reynolds-number flow past a bluff object. Intl J. Numer. Meth. Fluids 23, 711–732.Google Scholar
Coutanceau, M. & Defaye, J.-R. 1991 Circular cylinder wake configurations: a flow visualization survey. Appl. Mech. Rev. 44, 255–305.Google Scholar
Dommelen, L. L. & Shen, S. F. 1980 The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comput. Phys. 38, 125–140.CrossRefGoogle Scholar
Evans, L. C. 2002 Partial Differential Equations. American Mathematical Society.Google Scholar
Ghil, M., Liu, J. G., Wang, C. & Wang, S. 2004 Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow. Physica D 197, 149–173.Google Scholar
Ghil, M., Ma, T. & Wang, S. 2005 Structural bifurcation of 2-d nondivergent flows with Dirichlet boundary conditions applications to boundary-layer separation. SIAM J. Appl. Maths 65, 1576–1596.CrossRefGoogle Scholar
Goodrich, J. W. & Sox, W. Y. 1989 Time-dependent viscous incompressible Navier–Stokes equations: the finite difference Galerkin formulation and streamfunction algorithms. J. Comput. Phys. 84, 207–241.CrossRefGoogle Scholar
Gupta, M. M. & Kalita, J. C. 2005 A new paradigm for solving Navier–Stokes equations: streamfunction-velocity formulation. J. Comput. Phys. 207, 52–68.Google Scholar
Gurcan, F., Deliceoglu, A. & Bakker, P. G. 2005 Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms. J. Fluid Mech. 539, 299–311.Google Scholar
Kalita, J. C. & Gupta, M. M. 2010 A streamfunction-velocity approach for the 2D transient incompressible viscous flows. Intl J. Numer. Meth. Fluids 62, 237–266.Google Scholar
Kalita, J. C. & Ray, R. K. 2009 A transformation-free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder. J. Comput. Phys. 228, 5207–5236.CrossRefGoogle Scholar
Kalita, J. C. & Sen, S. 2012a The biharmonic approach for unsteady flow past an impulsively started circular cylinder. Commun. Comput. Phys. 12, 1163–1182.Google Scholar
Kalita, J. C. & Sen, S. 2012b Triggering asymmetry for flow past circular cylinder at low Reynolds numbers. Comput. Fluids 59, 44–60.Google Scholar
Koumoutsakos, P. & Leonard, A. 1995 High-resolution simulations of the flow around an impulsively started cylinder using vortex methods. J. Fluid Mech. 296, 1–38.Google Scholar
Kupferman, R. 2001 A central difference scheme for a pure streamfunction formulation of incompressible viscous flow. SIAM J. Sci. Comput. 23, 1–18.Google Scholar
Loc, T. P. 1980 Numerical analysis of unsteady secondary vortices generated by an impulsively started circular cylinder. J. Fluid Mech. 100, 111–128.Google Scholar
Loc, T. P. & Bouard, R. 1985 Numerical solution of the early stage of the unsteady viscous flow around a circular cylinder: a comparison with experimental visualization and measurements. J. Fluid Mech. 160, 93–117.CrossRefGoogle Scholar
Ma, T. & Wang, S. 2002 Topology of 2D incompressible flows and applications to geophysical fluid dynamics. RACSAM, Rev. R. Acad. Cien. A Mat. 96, 447–459.Google Scholar
Obabko, A. V. & Cassel, K. W. 2002 Navier–Stokes solutions of unsteady separation induced by a vortex. J. Fluid Mech. 465, 99–130.Google Scholar
Obabko, A. V. & Cassel, K. W. 2005 On the ejection-induced instability in Navier–Stokes solutions of unsteady separation. Phil. Trans. R. Soc. Lond. A 363, 1189–1198.Google ScholarPubMed
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991 Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem $Re\rightarrow \infty $ . J. Fluid Mech. 232, 99–131.CrossRefGoogle Scholar
Sanyasiraju, Y. V. S. S. & Manjula, V. 2005 Flow past an impulsively started circular cylinder using a higher-order semi compact scheme. Phys. Rev. E 72, 1–10.CrossRefGoogle Scholar
Sarpkaya, T. & Schoafft, R. L. 1979 Inviscid model of two-dimensional vortex shedding by a circular cylinder. AIAA J. 17, 1193–1200.Google Scholar
Sengupta, T. K. & Sengupta, R. 1994 Flow past an impulsively started circular cylinder at high Reynolds number. Comput. Mech. 14, 298–310.CrossRefGoogle Scholar
Singh, S. P. & Mittal, S. 2004 Energy spectra of flow past a circular cylinder. Intl J. Comput. Fluid Dyn. 18, 671–679.Google Scholar
Stephenson, J. W. 1984 Single cell discretization of order two and four for biharmonic problems. J. Comput. Phys. 55, 65–80.CrossRefGoogle Scholar
Thoman, D. C. & Szewczyk, A. A. 1969 Time-dependent viscous flow over a circular cylinder. Phys. Fluids Suppl. II 12, 76–86.Google Scholar