Abstract
This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.
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Akervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.S.: Global two-dimensional stability measure of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids 27, 501–513 (2008)
Alizard, F., Robinet, J.C.: Speatially convective global modes in a boundary layer. Phys. Fluids 19, 114105 (2007)
Bert, P.: Universal short wave instability of two dimensional eddies in inviscid fluid. Phys. Rev. Lett. 57, 2157–2159 (1986)
Christodoulou, K.N., Scriven, L.E.: Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3, 355–406 (1988)
Costa, B., Don, W., Simas, A.: Spatial resolution properties of mapped spectral Chebyshev methods. In: Proceedings SCPDE: Recent Progress in Scientific Computing, pp. 179–188 (2007)
Crighton, D.G., Gaster, M.: Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413 (1976)
Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge Univerisity Press, Cambridge (2004)
Duck, P.W.: The effect of a surface discountinuty on a axisymmetric on an axisymmetric boundary layer. Q. J. Mech. Appl. Math. 37, 57–74 (1984)
Duck, P.W.: The inviscid axisymmetric stability of the supersonic flow along a circular cylinder. J. Fluid Mech. 214, 611–637 (1990)
Duck, P.W., Hall, P.: On the interaction of Tollmein-Shlichting waves in axisymmetric supersonic flows. Q. J. Mech. Appl. Math. 42, 115–130 (1989)
Duck, P.W., Shaw, S.J.: The inviscid stability of supersonic flow past a sharp cone. Theor. Comput. Fluid Dyn. 2, 139–163 (1990)
Ehrenstein, U., Gallaire, F.: On two-dimensioanl temporal modes in spatially evolving open flow: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005)
Fasel, H., Rist, U., Konzelmann, U.: Numericla investigation of the three-dimensional development in boundary layer transition. AIAA J. 28, 29–37 (1990)
Glauert, M.B., Lighthill, M.J.: The axisymmetric boundary layer on a thin cylinder. Proc. R. Soc. A 230, 1881 (1955)
Herrada, M.A., Del Pino, C., FernandezFeria, R.: Stability of the boundary layer flow on a long thin rotating cylinder. Phys. Fluids 20, 034105 (2008)
Jackson, C.P.: A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 2345 (1987)
Joseph, D.D.: Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Ration. Mech. 22, 163–184 (1966)
Kao, K., Chow, C.: Stability of the boundary layer on a spining semi-infinite circular cylinder. J. Spacecr. Rockets 28, 284–291 (1991)
Lin, R.S., Malik, M.R.: On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow. J. Fluid Mech. 311, 239–255 (1996)
Lin, R.S., Malik, M.R.: On the stability of attachment-line boundary layers. Part 2. The incompressible swept Hiemenz flow. J. Fluid Mech. 333, 125–137 (1997)
Mack, L.M.: Stability of axisymmetric boundary layers on sharp cones at hypersonic mach numbers. In: 19th AIAA, Fluid Dynamics, Plasma Dynamics, and Lasers Conference, p. 1413 (1987)
Malik, M.R.: Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86, 376–412 (1990)
Malik, M.R., Poll, D.I.A.: Effect of curvature on three dimensional boundary layer stability. AIAA J. 23, 1362–1369 (1985)
Malik, M.R., Spall, R.E.: On the stability of compressible flow past axisymmetric bodies. J. Fluid Mech. 228, 443–463 (1987)
Monokrousos, A., Akervik, E., Brandt, L., Heningson, H.: Global three-dimensional optimal disturbances in the Blasius boundary layer flow using time stepers. J. Fluid Mech. 650, 181–214 (2010)
Muralidhar, S.D., Pier, B., Scott, J.F., Govindarajan, R.: Flow around a rotating, semi-infinite cylinder in an axial stream. Proc. R. Soc. A 472, 20150850 (2016)
Petrov, G.V.: Boundary layer on rotating cylinder in axial flow. J. Appl. Mech. Tech. Phys. 17, 506–510 (1976)
Rao, G.N.V.: Mechanics of transition in an axisymmetric laminar boundary layer on a circular cylinder. J. Appl. Math. Phys. 25, 6375 (1974)
Rempfer, D.: On boundary conditions for incompressible Navier–Stokes problems. App. Mech. Rev. 59(3), 107–125 (2006)
Roache, P.J.: A method for uniform reporting of grid refinment study. J. Fluids Eng. 116, 405413 (1994)
Swaminathan, G., Shahu, K., Sameen, A., Govindarajan, R.: Global instabilities in diverging channel flows. Theor. Comput. Fluid Dyn. 25, 53–64 (2011)
Tatsumi, T., Yoshimura, T.: Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449 (1990)
Tezuka, A., Suzuki, K.: Three-dimensional global linear stability analysis of flow around a spheroid. AIAA J. 44, 1697–1708 (2006)
Theofilis, V.: Advances in global linear instability analysis of nonparallel and three dimensional flows. Prog. Aerosp. Sci. 39, 249315 (2003)
Theofilis, V.: Global liniear instability. Annu. Rev. Fluid Mech. 43, 319–352 (2011)
Theofilis, V.: The linearized pressure Poisson equation for global instability analysis of incompressible flows. Theor. Comput. Fluid Dyn. 31, 623–642 (2017)
Theofilis, V., Duck, P.W., Owen, J.: Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286 (2004)
Theofilis, V., Fedorov, A., Obrist, D., Dallman, U.C.: The extended Görtler–Hämmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313 (2003)
Theofilis, V., Stefan, H., Dallmann, U.: On the origins of unsteadiness and three dimensionality in a laminar separation bubble. Proc. R. Soc. A 358, 1777 (2000)
Tutty, O.R., Price, W.G.: Boundary layer flow on a long thin cylinder. Phys. Fluids 14, 628–637 (2002)
Vinod, N.: Stability and transition in boundary layers: effect of transverse curvature and pressure gradient. Ph.D. Thesis, Jawaharlal Nehru Center for Advanced Scientific Research (2005)
Vinod, N., Govindarajan, R.: Secondary instabilities in incompressible axisymmetric boundary layers: effect of transverse curvature. J. Fliud Eng. 134, 024503 (2012)
Zebib, A.: Stability of viscous flow past a circular cylinder. J. Eng. Math. 21, 155–165 (1987)
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Bhoraniya, R., Vinod, N. Global stability analysis of axisymmetric boundary layer over a circular cylinder. Theor. Comput. Fluid Dyn. 32, 425–449 (2018). https://doi.org/10.1007/s00162-018-0461-5
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DOI: https://doi.org/10.1007/s00162-018-0461-5