Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T12:13:01.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Global stability of spiral flow

Published online by Cambridge University Press:  29 March 2006

D. D. Joseph  and
B. R. Munson
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis 55455
B. R. Munson
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis 55455 Present address: Department of Mechanical Engineering, Duke University, Durham, North Carolina.
Get access Rights & Permissions [Opens in a new window]

Abstract

Energy and linear limits are calculated for the Poiseuille–Couette spiral motion between concentric cylinders which rotate rigidly and rotate and slide relative to one another. The addition of solid rotation can bring the linear limit down to the energy limit with coincidence achieved in the limit of infinitely fast rotation. If the differential rotation is also added, the solid rotation rate need be only finite to achieve near coincidence. Sufficient conditions for non-existence of sub-linear instability are derived. The basic spiral character of the instability is analysed and the results compared with the experiments of Ludwieg (1964).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 43 , Issue 3 , 16 September 1970 , pp. 545 - 575
Copyright
© 1970 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carmi, S. 1970 Linear stability of apial flow in an annular pipe. Phys. Fluids, 13, 829.Google Scholar
Chandrasekhar, S. 1960 The hydrodynamic stability of inviscid flow between coaxial cylinders. Proc. Nat. Acad. Sci. U.S.A. 46, 137.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Datta, S. K. 1965 Stability of spiral flow between concentric cylinders at low axial Reynolds numbers. J. Fluid Mech. 21, 635.Google Scholar
Elder, J. W. 1960 An experimental investigation of turbulent spots and breakdown to turbulence. J. Fluid Mech. 9, 235.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463.Google Scholar
Hughes, T. H. & Reid, W. H. 1968 The stability of spiral flow between rotating cylinders. Phil. Trans. Roy. Soc. A 263, 57.Google Scholar
Joseph, D. D. 1965 On the stability of the Boussinesq equations by the method of energy. Arch. Rat. Mech. Anal. 20, 59.Google Scholar
Joseph, D. D. 1966 Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Rat. Mech. Anal. 22, 163.Google Scholar
Joseph, D. D. 1969 Eigenvalue bounds for the Orr — Sommerfeld equations. Part 2. J. Fluid Mech. 36, 721.Google Scholar
Joseph, D. D. & Carmi, S. 1969 Stability of Poiseuille flow in pipes, annuli, and channels Quart. Appl. Math. 26, 57.Google Scholar
Joseph, D. D. & Shir, C. C. 1967 Subcritical convective instability. Part 1. Fluid layers heated from below and internally. J. Fluid Mech. 26, 753.Google Scholar
Kiessling, I. 1963 Über das Taylorsche Stabilitatsproblem bei zusätzlicher axialer Durchströmung der Zylindern. Deutsche Versuchsantalt für Luft und Raumfahrt, Bericht 290.Google Scholar
Kirchgässner, K. & Sorger, P. 1968 Stability analysis of branching solutions of the Navier — Stokes equations. 12th Congress of the Int. Union of Theo. and Appl. Mech.Google Scholar
Krueger, E. R. & Di Prima, R. C. 1964 The stability of viscous fluid between rotating cylinders with an axial flow. J. Fluid Mech. 19, 528.Google Scholar
Ludwieg, H. 1961 Ergänzung zu der Arbeit: ‘Stabilität der Strömung in einem zylindrischen Ringraum’. Z. Flugwiss, 9, 359.Google Scholar
Ludweig, H. 1964 Experimentelle Nachprüfung der Stabilitätstheorien für reibungsfreie Strömungen mit Schraubenlinienformigen Stromlinien. 11th Int. Congress of Appl. Mech. 1045.Google Scholar
Mott, J. E. & Joseph, D. D. 1968 Stability of parallel flow between concentric cylinders. Phys. Fluids, 11, 2065.Google Scholar
Munson, B. R. 1970 Ph.D. Thesis, Dept. of Aerospace Engineering and Mechanics. University of Minnesota.
Nagib, H. M., Wolf, L., Lavan, Z. & Fejer, A. 1969 On the stability of flow in rotating pipes. Aero Res. Lab. Rep. 69–0176, Ill. Inst. Tech.Google Scholar
Nickerson, E. C. 1969 Upper bounds on the torque in cylindrical Couette flow. J. Fluid Mech. 38, 80.Google Scholar
Pedley, T. J. 1968 On the instability of rapidly rotating shear flows to non-axisymmetric disturbances. J. Fluid Mech. 31, 603.Google Scholar
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97.Google Scholar
Sattinger, D. H. 1969 On the linearization of the equations of hydrodynamics. J. Math. Mech. 19, 797.Google Scholar
Serrin, J. 1959 On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1.Google Scholar
Serrin, J. 1968 Edinburgh Lectures in Fluid Mechanics.
Sparrow, E. M., Munro, W. D. & Jonsson, V. K. 1964 Instability of the flow between rotating cylinders. J. Fluid Mech. 20, 35.Google Scholar
Yudovich, V. I. 1965 Stability of steady flows of viscous incompressible fluids. Soviet Physics, Doklady, 10, 293.Google Scholar