Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-27T16:51:30.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Roll-cell instabilities in rotating laminar and trubulent channel flows

Published online by Cambridge University Press:  11 April 2006

Dietrich K. Lezius  and
James P. Johnston
Dietrich K. Lezius
Affiliation:
Department of Mechanical Engineering, Stanford University Present address: Lockheed Palo Alto Research Laboratory, California.
James P. Johnston
Affiliation:
Department of Mechanical Engineering, Stanford University
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of laminar and turbulent channel flow is examined for cases where Coriolis forces are introduced by steady rotation about an axis perpendicular to the plane of mean flow. Linearized equations of motion are derived for small disturbances of the Taylor type. Conditions for marginal stability in laminar Couette and Poiseuille flow correspond, in part, to the analogous solutions of buoyancy-driven convection instabilities in heated fluid layers, and to those of Taylor instabilities in the flow between rotating cylinders. In plane Poiseuille flow with rotation, the critical disturbance mode occurs at a Reynolds number of Rec = 88.53 and rotation number Ro = 0.5. At higher Reynolds numbers, unstable conditions canexist over the range of rotation numbers given by 0 < Ro < 3, provided the undisturbed flow remains laminar. A two-layer model is devised to investigate the onset of longitudinal instabilities in turbulent flow. The linear disturbance equations are solved essentially in their laminar form, whereby the velocity gradient of laminar flow is replaced by a numerically computed profile for the gradient of the turbulent mean velocity. The turbulent stress levels in the stable and unstable flow regions are represented by integrated averages of the eddy viscosity. Onset of instability for Reynolds numbers between 6000 and 35 000 is predicted to occur at Ro = 0.022, a value in remarkable agreement with the experimentally observed appearance of roll instabilities in rotating turbulent channel flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 77 , Issue 1 , 9 September 1976 , pp. 153 - 174
Copyright
© 1976 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

Bradshaw, P. 1969 The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech. 36, 177.Google Scholar
Cess, R. D. 1958 A survey of the literature on heat transfer in turbulent tube flow. Thesis, University of Pittsburgh, Pennsylvania.
Chandrasekhar, S. 1961a Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Chandrasekhar, S. 1961b Adioint differential systems in the theory of hydrodynamic stability. J. Math. Mech. 10, 683.Google Scholar
Debler, W. R. 1966 On the analogy between thermal and rotational hydrodynamic stability. J. Plwid Mech. 24, 165.Google Scholar
Halleen, R. M. & Johnston, J. P. 1967 The influence of rotation on flow in a long rectangular channel: an experimental study. Department of Mechanical Engineering, Stanford University, Rep. MD-18.Google Scholar
Hart, J. E. 1971 Instability and secondary motion in a rotating channel flow. J. Flwid Mech. 45, 341.Google Scholar
Hung, W. L., Joseph, D. D. & Munson, B. R. 1972 Global stability of spiral flow. Part 2. J. Fluid Mech. 51, 593.Google Scholar
Johnston, J. P., Halleen, R. M. & Lezius, D. K. 1972 Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech. 56, 533.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
Lezius, D. K. 1975 Finite difference solutions of Taylor instabilities in viscous plane flow. Computers & Fluids, 3, 103.Google Scholar
Lezius, D. K. & Joenston, J. P. 1971 The structure and stability of turbulent wall layers in rotating channel flow. Department of Mechanical Engineering, Stanford University, Rep. MD-29.Google Scholar
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97.Google Scholar
Pellew, A. & Southwell, R. V. 1940 On maintaining a convective motion in a fluid heated from below. Proc. Roy. Soc. A176, 312.Google Scholar