Developments in Stability Theory
Part of the
International Centre for Mechanical Sciences
book series (CISM, volume 369)
In these notes we describe some of the recent developments in the linear theory of hydrodynamic stability. We start by distinguishing between temporal and spatial stability, in the context of the very familiar model problem of the Kelvin-Helmholtz theory for a vortex sheet. The general prescription for determining the stability of causal solutions for initial-value problems is then described, which introduces the distinction between convective and absolute instabilities. These ideas have found very wide application, and we describe a number of different situations in which absolute instability in particular is seen to play an important role; this includes the response of an elastic fluid-loaded plate, the boundary-layer flow on a rotating disk, and the stability of a range of wake flows. All these situations apply very much to parallel flows, however, and in order to handle non-parallel flows one must find ways of connecting the local, quasi-parallel properties of the flow to the global dynamics.
KeywordsVortex Sheet Absolute Instability Pinch Point Plane Poiseuille Flow Lower Deck
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Huerre, P. and Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech.
, 473–537.CrossRefMathSciNetGoogle Scholar
Goldstein, M.E. and Hultgren, L.S. 1989 Boundary-layer receptivity to long-wave free-stream disturbances. Ann. Rev. Fluid Mech.
, 137–66.CrossRefMathSciNetGoogle Scholar
Stewartson, K. and Stuart, J.T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech.
, 529–45.CrossRefMATHMathSciNetGoogle Scholar
Newell, A.C. and Whitehead, J.A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech.
, 279–303.CrossRefMATHGoogle Scholar
Hunt, R.E. 1995
Spatially varying flows with localised forcing. Ph.D. Thesis, University of Cambridge.Google Scholar
Briggs, R.J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.Google Scholar
Bers, A. Space-time evolution of plasma instabilities — absolute and convective. In Handbook of Plasma Physics
ed. M.N. Rosenbluth; 451–517.
Brazier-Smith, P.R. and Scott, J.F. 1984 Stability of fluid flow in the presence of a compliant surface. Wave Motion
, 547–560.CrossRefGoogle Scholar
Crighton, D.G. and Oswell, J.E. 1991 Fluid loading with mean flow. I. Response of an elastic plate to localised excitation. Phil. Trans. R. Soc. Lond., 335
, 557–592.CrossRefMATHMathSciNetGoogle Scholar
Lighthill, M.J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Gregory, N., Stuart, J.T. and Walker, W.S. 1955 On the stability of three-dimensional boundary layers with application to the flow down to a rotating disk. Phil. Trans. R. Soc. Lond., 248
, 155–199.CrossRefMATHMathSciNetGoogle Scholar
Rosenhead, L. 1963Laminar Boundary Layers. Dover.Google Scholar
Lingwood, R.J. Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech
., to appear.Google Scholar
Crighton, D.G. 1972 Radiation properties of the semi-infinite vortex sheet. Proc. Roy. Soc Lond.
, 185–98.CrossRefMATHGoogle Scholar
Orszag, S.A. and Crow, S.C. 1970 Instability of a vortex sheet leaving a semi-infinite plate. Stud. Appl.Math.
, 167–81.MATHGoogle Scholar
Noble, B. 1958Methods Based on the Wiener-Hopf Technique. Chelsea.Google Scholar
Crighton, D.G. and Leppington F.G. Radiation properties of the semi-infinite vortex sheet: the initial-value problem. J. Fluid Mech.
, 393–414.Google Scholar
Crighton, D.G. 1985 The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech., 17
, 411–45.Google Scholar
Monkewitz, P.A. 1988 The absolute and convective nature of instability in two dimensional wakes at low Reynolds numbers. Phys. Fluids
, 999–1006.CrossRefGoogle Scholar
Chomaz, J.M., Huerre, P. and Redekopp, L.G. 1987 Bifurcations to local and global modes in spatially developing flows. Phys. Review Letters
, 25–28.CrossRefGoogle Scholar
Drazin, P.G. and Reid, W.H. 1981 Hydrodynamic Stability.
Cambridge University Press.Google Scholar
Mack, L.M. Boundary-layer linear stability theory. IN AGARD Report 709.Google Scholar
Monkewitz, P.A., Huerre, P. and Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech.
, 1–20.CrossRefMATHMathSciNetGoogle Scholar
Huerre, P., Chomaz, J.M. and Redekopp, L.G. 1988 A frequency selection mechanism in spatially-developing flows. Bull. Am. Phys. Soc.
, 2283.Google Scholar
Pierrehumbert, R.T. 1984 Local and global baroclinic instability of zonally varying flow. J. Atmos. Sci.
, 2146–62.Google Scholar
Koch, W. 1985 Local instability characteristics and frequency determination of self-excited wake flows. J. Sound Vib.
, 55–83.Google Scholar
Woodley, B.M. and Peake, N. Estimation of vortex shedding frequencies for cascades. In preparation.Google Scholar
Kachanov, Y.S. 1994 Experiments on boundary-layer transition. Ann. Review Fluid Medi.
, 411–82.CrossRefMathSciNetGoogle Scholar
Kerschen, E.J. 1989 Boundary layer receptivity. AIAA Paper 89–1109.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer, J. Fluid Mech.
, 433–441.CrossRefMATHMathSciNetGoogle Scholar
Goldstein, M.E. 1983
The evolution of Tollmien—Schlichting waves near a leading edge. J. Fluid Mech.
, 127, 59–81.Google Scholar
Ackberg, R.C. and Phillips, J.H. 1972 The unsteady boundary layer on a semi-infinite plate due to small fluctuations in the magnitude of the free-stream velocity. J. Fluid Mech., 51, 137–157.CrossRefMathSciNetGoogle Scholar
Hammerton, P.W. and Kerschen, E.J. J. Fluid Mech., to appear.Google Scholar
Goldstein, M.E. 1985 Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech.
, 509–30.CrossRefMATHGoogle Scholar
Goldstein, M.E., Leib, S.J. and Cowley, S.J. 1987 Generation of Tollmien-Schlichting waves on interactive marginally separated flows. J. Fluid mech.
, 485–517.CrossRefMATHGoogle Scholar
Bodonyi, R.J. and Duck, P.W. 1992 Boundary-layer receptivity due to a wall suction and control of Tollmien-Schlichting waves. Phys. Fluids
, 1206–1214.CrossRefMATHMathSciNetGoogle Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate, II. Mathematika
, 106–21.CrossRefMATHGoogle Scholar
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