Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction to Flow Instabilities Read chapter Read first chapter

2023 | OriginalPaper | Chapter

2. Global Stability Approach

Authors : Rameshkumar Bhoraniya, Gayathri Swaminathan, Vinod Narayanan

Published in: Global Stability Analysis of Shear Flows

Publisher: Springer Nature Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This chapter introduces the global stability approach and the need for such an approach to study the stability of non-parallel flows. We first introduce the two commonly used approximations which result in a problem solvable locally at one streamwise location. These are the parallel flow assumption and the weakly non-parallel flow assumption. Some applications of each are described. Then we introduce strongly non-parallel flows and the global stability analysis with a few examples. The problem formulation and the numerical discretization schemes used in the present work are discussed next. This is followed by the method of solving the problem numerically, the techniques used and the validation.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
previous chapter Introduction to Flow Instabilities
next chapter Diverging Channel
Literature
1.
Akervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.S.: Global two-dimensional stability measures of the flat plate boundary-layer flow. Euro. J. Mech. B/Fluids 27, 501–513 (2008)MathSciNetMATHCrossRef
2.
Akervik, E., Hoepffner, J., Ehrenstein, U., Henningson, D.S.: Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314 (2007)MathSciNetMATHCrossRef
3.
Alizard, F., Robinet, J.C.: Spatially convective global modes in a boundary layer. Phys. Fluids 19, 114105 (2007)MATHCrossRef
4.
Barbagallo, A., Sipp, D., Schmid, P.J.: Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. (2009)
5.
Barkley, D., Gomes, M.G.M., Henderson, R.D.: Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190 (2002)MathSciNetMATHCrossRef
6.
Bers, A.: Space-time evolution of plasma instabilities—absolute and convective, vol. 1. In: Rosenbluth, M.N., Sagdeev, R.Z. (eds.) Handbook of Plasma Physics, pp. 451–517 (1983)
7.
Bertolotti, F.P., Herbert, T.: Analysis of the linear stability of compressible boundary layer using the PSE. Theor. Comp. Fluid Dyn. 3, 117–24 (1991)MATHCrossRef
8.
Bertolotti, F.P., Herbert, T., Spalart, P.R.: Linear and nonlinear stability of Blasius boundary layer. J. Fluid Mech. 242, 441–474 (1992)MathSciNetMATHCrossRef
9.
Betchov, R., Criminale, W.O.: Spatial instability of the inviscid jet and wake. Phys. Fluids 9(2), 359–362 (1966)CrossRef
10.
Bhattacharya, P., Manoharan, M., Govindarajan, R., Narasimha, R.: The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory. J. Fluid Mech. 565, 105–114 (2006)MathSciNetMATHCrossRef
11.
12.
Cebeci, T., Shao, J.P., Kafyeke, F.: Computational Fluid Dynamics for Engineers: From Panel to Navier-Stokes Methods with Computer Programs. Springer (2005)
13.
Chedevergne, F., Casalis, G., Feraille, T.: Biglobal linear stability analysis of the flow induced by wall injection. Phys. Fluids 18, 014103 (2006)CrossRef
14.
Christodolou, K.N., Scriven, L.E.: Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3(4), 355–406 (1988)MATHCrossRef
15.
Deissler, R.J.: The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30(8), 2303–2305 (1987)MathSciNetCrossRef
16.
Dijkstra, H.A., Molemaker, M.J., Ploeg, A.V.D., Botta, E.F.F.: A efficient code to compute non-parallel steady flows and their linear stability. Comput. Fluids 24(4), 415–434 (1995)MATHCrossRef
18.
Drazin, P.G., Reid, W.H.: Hydrodynamic stability. Cambridge University Press (2004)
17.
19.
Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005)MathSciNetMATHCrossRef
20.
Gaster, M.: Growth of disturbances in both space and time. Phys. Fluids 11(4), 723–727 (1968)CrossRef
21.
Gaster, M.: On the effect of boundary layer growth on flow stability. J. Fluid Mech. 66, 465–480 (1974)MATHCrossRef
22.
Gaster, M.: A theoretical model of a wavepacket in the boundary layer on a flat plate. Proc. R. Soc. London A 347, 271–289 (1975)CrossRef
23.
Gaster, M.: On the growth of waves in boundary layers. J. Fluid Mech. 424, 367–377 (2000)MATHCrossRef
24.
Gonzalez, L.M., Theofilis, V., Gomez-Blanco, R.: Finite element methods for viscous incompressible biglobal instability analysis on unstructured meshes. AIAA J. 45(4), 840 (2007)CrossRef
25.
Govindarajan, R.: Effect of miscibility on linear instability of two fluid channel flow. Int. J. Multiphase flow 30, 1177–1192 (2004)MATHCrossRef
26.
Govindarajan, R., Narasimha, R.: Stability of spatially developing boundary layers in pressure gradients. J. Fluid Mech. 300, 117–147 (1995)MathSciNetMATHCrossRef
27.
Govindarajan, R., Narasimha, R.: A low order theory for stability of non-parallel boundary layer flows. Proc. Math. Phys. Eng. Sci. 453(1967), 2537–2549 (1997)
28.
Henningson, D.S.: Comment on “transition in shear flows: non-linear normality versus non-normal linearity” [phys. fluids 7,3060 (1995)]. Phys. Fluids 8, 2257–2258 (1996)
29.
Henningson, D.S., Akervik, E.: The use of global modes to understand transition and perform flow control. Phys. Fluids 20, 031302 (2008)MATHCrossRef
30.
31.
32.
Huerre, P., Monkewitz, P.A.: Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473–537 (1990)MathSciNetMATHCrossRef
33.
Jackson, C.P.: A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45 (1987)MATHCrossRef
34.
Lee, N.Y., Schultz, W.W., Boyd, J.P.: Stability of fluid in a rectangular enclosure by spectral method. J. Heat Mass Transf. 32(3), 513–520 (1989)CrossRef
35.
Lin, R.S., Malik, M.R.: On the stability of attachment-line boundary layers. part 2. The incompressible swept Hiemenz flow. J. Fluid Mech. 311, 239–255 (1996)
36.
Lin, R.S., Malik, M.R.: On the stability of attachment-line boundary layers. part 2. The effecting of leading-edge curvature. J. Fluid Mech. 333, 125–137 (1997)
37.
38.
Marquet, O., Lombardi, M., Chomaz, J.M., Sipp, D., Jacquin, L.: Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21 (2009)MathSciNetMATHCrossRef
39.
Martin, D., Carriere, P., Monkewitz, P.A.: Three dimensional global instability modes associated with a localized hot spot in Rayleigh-Benard-Poiseuille convection. J. Fluid Mech. 551, 275–301 (2006)MathSciNetMATHCrossRef
40.
41.
Mittal, S., Kumar, B.: A stabilized finite element method for global analysis of convective instabilities in nonparallel flows. Phys. Fluids 19, 008105 (2007)MATHCrossRef
42.
Mittal, S., Singh, S.: Vortex induced vibrations at subcritical re. J. Fluid Mech. 534, 185–194 (2005)MATHCrossRef
43.
44.
Morzynski, M., Afanasiev, K., Thiele, F.: Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Methods Appl. Mech. Eng. 169, 161–176 (1999)MATHCrossRef
45.
Nayar, N., Ortega, J.M.: Computation of selected eigenvalues of generalized eigenvalue problems. J. Comput. Phys. 108, 8–14 (1993)MathSciNetMATHCrossRef
46.
Noack, B.R., Eckelmann, H.: A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330 (1994)MATHCrossRef
47.
Noack, B.R., Konig, M., Eckelmann, H.: Three-dimensional stability analysis of the periodic flow around a circular cylinder. Phys. Fluids A 5(6), 1279–1281 (1993)CrossRef
48.
Orszag, S.A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, part 4, 689–703 (1971)
49.
Pier, B.: Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J. Eng. Math. 57, 237–251 (2007)MathSciNetMATHCrossRef
50.
51.
Pierrehumbert, R.T.: Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Letters 57(17), 2157–60 (1986)CrossRef
52.
Saad, Y.: Variations of Arnoldi’s method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl. 34, 269–295 (1980)MathSciNetMATHCrossRef
53.
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)MathSciNetMATHCrossRef
54.
Sahu, K.C.: Numerical computation of spatially developing flows by full-multigrid technique. Master’s thesis, Jawaharlal Nehru Centre for Advanced Scientific Research (2003)
55.
56.
57.
Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer (2001)
58.
59.
Srinivasan, S., Klika, M., Ludwig, M.H., Ram, V.V.: A Beginner’s Guide to the Use of the Spectral Collocation Method for Solving Some Eigenvalue Problems in Fluid Mechanics. Tech. Rep. National Aerospace Laboratories, Bangalore, India (1994)
60.
61.
Tatsumi, T., Yoshimura, T.: Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449 (1990)MATHCrossRef
62.
Theofilis, V.: On the verification and extension of the Gortler-Hammerlin assumption in three-dimensional incompressible swept attachment-line Boundary layer flow. Tech. report. IB 223-97, A 44, DLR (1997)
63.
Theofilis, V.: Advances in global linear instability analysis of non-parallel and three-dimensional flows. Progr. Aerospace Sci. 39, 249–315 (2003)CrossRef
64.
Theofilis, V., Duck, P.W., Owen, J.: Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286 (2004)MathSciNetMATHCrossRef
65.
Theofilis, V., Federov, A., Obrist, D., Dallmann, U.C.: The extended Gortler-Hammerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313 (2003)MathSciNetMATHCrossRef
66.
Theofilis, V., Hein, S., Dallmann, U.: On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phi. Tran. R. Soc. Lond. A 358, 3229–3246 (2000)MATHCrossRef
67.
Theofilis, V., Sherwin, S.J., Abdessemed, N.: On global instabilities of separated bubble flows and their control in external and internal aerodynamic applications. NATO RTO-AVT-111, 21–1 to 21–9 (2005)
68.
Tollmien, W.: Über die entstehung der turbulenz. 1. mitteilung, nachr. ges. wiss. göttingen. Math. Phys. Klasse, 21–44 (1929)
69.
Venkatesh, T.N., Sarasamma, V.R., Rajalakshmy, S., Sahu, K.C., Govindarajan, R.: Super-linear speedup of a parallel multigrid Navier-stokes solver on Flosolver. Curr. Sci. 88(4), 589–593 (2006)
70.
71.
Zebib, A.: Stability of viscous flow past a circular cylinder. J. Eng. Math. 21, 155–165 (1987)MATHCrossRef
Metadata
Title
Global Stability Approach
Authors
Rameshkumar Bhoraniya
Gayathri Swaminathan
Vinod Narayanan
Copyright Year
2023
Publisher
Springer Nature Singapore
Book
Global Stability Analysis of Shear Flows

Print ISBN: 978-981-19-9573-6

Electronic ISBN: 978-981-19-9574-3

Copyright Year: 2023

DOI
https://doi.org/10.1007/978-981-19-9574-3_2

Premium Partners

    Image Credits