Abstract
The triple-deck equations for the steady subsonic flow past a convex corner are solved numerically using a novel technique based on Chebychev collocation in the direction normal to the body combined with finite differences in the direction along the flow. The resulting set of nonlinear algebraic equations are solved with Newton linearization and using the GMRES method for the solution of the linear system of equations. The stability of the computed steady flows is then examined using global stability analysis. It is found that for small corner angles, the Tollmien–Schlichting modes are globally unstable and these persist to larger corner angles. Multiple steady state solutions also exist beyond a critical corner angle but these are globally unstable because of the presence of the Tollmien–Schlichting modes.
Similar content being viewed by others
References
Smith F.T.: On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A366, 91–109 (1979)
Smith F.T.: Two-dimensional disturbance travel, growth and spreading in boundary layers. J. Fluid Mech. 169, 353–377 (1986)
Sychev V.V., Ruban A.I., Sychev V.V., Korolev G.L.: Asymptotic Theory of Separated Flows. Cambridge University Press, Cambridge (1998)
Stewartson K.: On laminar boundary layers near corners. Q. J. Mech. Appl. Math. 23, 137–152 (1970)
Korolev G.L., Gajjar J.S.B., Ruban A.I.: Once again on the supersonic flow separation near a corner. J. Fluid Mech. 463, 173–199 (2002)
Boppana V.B.L., Gajjar J.S.B.: Global instability in a lid-driven cavity. Int. J. Numer. Methods Fluids 62(18), 827–853 (2009)
Kravtsova M.A., Zametaev V.B., Ruban A.I.: An effective numerical method for solving viscous-inviscid interaction problems. Phil. Trans. R. Soc. A 363, 1157–1167 (2005)
Saad Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, New York (1996)
Lehoucq R.B., Sorensen D.C., Yang C.: ARPACK User’s Guide. S.I.A.M., Philadelphia (1998)
Logue, R.P.: Stability and bifurcations governed by the triple-deck and related equations. PhD Thesis, University of Manchester (2008)
Ruban A.I.: On the theory of laminar flow separation of a fluid from a corner point on a solid surface. Uch. zap. TsaGI. 8(1), 6–11 (1976)
Smith F.T., Merkin J.H.: Triple-deck solutions for subsonic flow past humps, steps, concave or convex corners and wedged trailing edges. Comput Fluids 10, 7–25 (1982)
Turkyilmazoglu M.: Linear absolute and convective instabilities of Some two and three dimensional flows. PhD Thesis, University of Manchester (1998)
Korolev G.L.: Nonuniqueness of separated flow past nearly flat corners. Fluid Dyn. 27, 442–444 (1992)
Fasel H.: Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations. J. Fluid Mech. 78, 355–383 (1976)
Terentev E.D.: The linear problem of a vibrator in a subsonic boundary layer. Prikl. Matem. Mekhan. 45(6), 1049–1055 (1981)
Ryzhov O.S., Terentev E.D.: On the transition mode characterizing the triggering of a vibrator in the subsonic boundary layer on a plate. Prikl. Matem. Mekham. 50, 974–986 (1986)
Cowley, S.J.: Laminar boundary-layer theory: a 20th century paradox? In: Aref H., Phillips J.W. (eds.) Proceedings of ICTAM 2000, pp. 389–411. Kluwer (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Colonius
Rights and permissions
About this article
Cite this article
Logue, R.P., Gajjar, J.S.B. & Ruban, A.I. Global stability of separated flows: subsonic flow past corners. Theor. Comput. Fluid Dyn. 25, 119–128 (2011). https://doi.org/10.1007/s00162-010-0198-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-010-0198-2