Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T05:31:04.307Z Has data issue: false hasContentIssue false
  • English
  • Français

On the wave excitation and the formation of recirculation eddies in an axisymmetric flow of uniformly rotating fluids

Published online by Cambridge University Press:  26 April 2006

Hideshi Hanazaki
Hideshi Hanazaki
Affiliation:
National Institute for Environmental Studies, Tsukuba, Ibaraki 305, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The inertial waves excited in a uniformly rotating fluid passing through a long circular tube are studied numerically. The waves are excited either by a local deformation of the tube wall or by an obstacle located on the tube axis. When the flow is subcritical, i.e. when the phase and group velocity of the fastest wave mode in their long-wave limit are larger than the incoming axial flow velocity, the excited waves propagate upstream of the excited position. The non-resonant waves have many linear aspects, including the upstream-advancing speed of the wave and the coexisting lee wavelength. When the flow is critical (resonant), i.e. when the long-wave velocity is nearly equal to the axial flow velocity, the large-amplitude waves are resonantly excited. The time development of these waves is described well by the equation derived by Grimshaw & Yi (1993). The integro-differential equation, which describes the strongly nonlinear waves until the axial flow reversal occurs, can predict the onset time and position of the recirculation eddies observed in the solutions of the Navier-Stokes equations. The numerical results and the theory both show that the flow reversal most probably occurs on the tube axis and also when the waves are excited by a contraction of the tube wall. The structure of the recirculation eddies obtained in the solutions of the Navier-Stokes equations at Re = 105 is similar to the axisymmetric or ‘bubble-type’ breakdown observed in the experiments of the vortex-breakdown which used a different non-uniform (Burgers-type) rotation. In uniformly rotating fluids the formation of the recirculation eddies has not been observed in the previous numerical studies of vortex breakdown where a straight tube was used and thus the inertial waves were not excited. This shows that the generation of the recirculation eddies in this study is genuinely explained by the topographically excited large-amplitude inertial ‘waves’ and not by other ‘instability’ mechanisms. Since the wave cannot be excited in a straight tube even in the non-uniformly rotating flows, the generation mechanism of the recirculation eddies in this study is different from the previous numerical studies for the vortex breakdown. The occurrence of the recirculation eddies depends not only on the Froude number and the strength of the excitation source but also on the Reynolds number since the wave amplitude generally decreases by the viscous effects. Some relations to the experiments of vortex breakdown, which have been exclusively done for non-uniformly rotating fluids but done in a ‘non-uniform tube’, are discussed. The flow states, which are classified as supercritical, subcritical or critical in hydraulic terminology, changes along the flow when the upstream flow is near resonant conditions and a non-uniform tube is used.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 322 , 10 September 1996 , pp. 165 - 200
Copyright
© 1996 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

Akylas, T. R. 1984 On the excitation of long nonlinear waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466.Google Scholar
Baines, P. G. 1979 Observations of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus 31, 351–371.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.Google Scholar
Benjamin, T. B. 1967 Some developments in the theory of the vortex breakdown. J. Fluid Mech. 28, 65–84.Google Scholar
Benjamin, T. B. 1970 Upstream influence. J. Fluid Mech. 40, 49–79.Google Scholar
Benny, D. J. 1979 Large amplitude Rossby waves. Stud. Appl. Maths 60, 1–10.Google Scholar
Beran, P. S. 1987 Numerical simulations of trailing vortex bursting. AIAA Paper 87-1313.
Beran, P. S. & Culick, F. E. 1992 The role of non-uniqueness in the development of vortex breakdown in tubes. J. Fluid Mech. 242, 491–527.Google Scholar
Castro, I. P. & Snyder, W. H. 1988 Upstream motions in stratified flow. J. Fluid Mech. 187, 487–506.Google Scholar
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 1385–1400.Google Scholar
Faler, J. H. & Leibovich, S. 1978 An experimental map of the internal structure of a vortex breakdown. J. Fluid Mech. 86, 313–335.Google Scholar
Grabowski, W. J. & Berger, S. A. 1976 Solutions of the Navier-Stokes equations for vortex breakdown. J. Fluid Mech. 75, 525–544.Google Scholar
Grimshaw, R. 1990 Resonant flow of rotating fluid past an obstacle: The general case. Stud. Appl. Maths 83, 249–269.Google Scholar
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Grimshaw, R. & Yi, Z. 1991 Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography. J. Fluid Mech. 229, 603–628.Google Scholar
Grimshaw, R. & Yi, Z. 1993 Resonant generation of finite-amplitude waves by the uniform flow of a uniformly rotating fluid past an obstacle. Mathematika, 40, 30–50.Google Scholar
Hafez, M., Ahmad, J., Kuruvila, G. & Salas, M. D. 1987 Vortex breakdown simulation. AIAA paper 87-1343.
Hafez, M., Kuruvila, G. & Salas, M. D. 1986 Numerical study of vortex breakdown. AIAA paper 86-0558.
Hall, M. G. 1972 Vortex breakdown. Ann. Rev. Fluid Mech. 4, 195–218.Google Scholar
Hanazaki, H. 1989 Upstream advancing columnar disturbances in two-dimensional stratified flow of finite depth.. Phys. Fluids A 1, 1976–1987.Google Scholar
Hanazaki, H. 1991 Upstream-advancing nonlinear waves in an axisymmetric resonant flow of rotating fluid past an obstacle.. Phys. Fluids A 3, 3117–3120.Google Scholar
Hanazaki, H. 1992 A numerical study of nonlinear waves in a transcritical flow of stratified fluid past an obstacle.. Phys. Fluids A 4, 2230–2243.Google Scholar
Hanazaki, H. 1993a Upstream-advancing nonlinear waves excited in an axisymmetric transcritical flow of rotating fluid.. Phys. Fluids A 5, 568–577.Google Scholar
Hanazaki, H. 1993b On the nonlinear internal waves excited in the flow of a linearly stratified Boussinesq fluid.. Phys. Fluids A 5, 1201–1205.Google Scholar
Harvey, J. K. 1962 Some observations of the vortex breakdown phenomenon. J. Fluid Mech. 14, 585–592.Google Scholar
Kawamura, T., Takami, H. & Kuwahara, K. 1986 Computation of high Reynolds number flow around a circular cylinder with surface roughness. Fluid Dyn. Res. 1, 145–162.Google Scholar
Kirkpatrick, D. L. I. 1965 Experimental investigation of the breakdown of a vortex in a tube. Aeronaut. Res. Counc. CP no. 821.Google Scholar
Kopecky, R. M. & Torrance, K. E. 1973 Initiation and structure of axisymmetric eddies in a rotating stream. Comput. Fluids 1, 289–300.Google Scholar
Lee, S. J. Yates, G. T. & Wu, T. Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569–593.Google Scholar
Leibovich, S. 1969 Evolution of nonlinear waves in rotating fluids. Phys. Fluids 12, 1124–1126.Google Scholar
Leibovich, S. 1970 Weakly nonlinear waves in rotating fluids. J. Fluid Mech. 42, 803–822.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 221–246.Google Scholar
Leibovich, S. 1983 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Leibovich, S. 1991 Vortex breakdown: a coherent transition trigger in concentrated vortices. In Turbulence and Coherent Structures (ed. O. Metais & M. Lesieur). Kluwer.
Leibovich, S. & Randall, J. D. 1973 Amplification and decay of long nonlinear waves. J. Fluid Mech. 53, 481–493.Google Scholar
Long, R. R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. J. Met. 10, 197–203.Google Scholar
McIntyre, M. E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating fluid. J. Fluid Mech. 52, 209–243.Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, K. G. 1985 Wave motions on vortex cores. J. Fluid Mech. 151, 141–165.Google Scholar
Melville, W. K. & Helfrich, K. R. 1987 Transcritical two-layer flow over topography. J. Fluid Mech. 178, 31–52.Google Scholar
Menne, S. 1988 Vortex breakdown in an axisymmetric flow. AIAA Paper 88-0506.
Pritchard, W. G. 1968 A study of wave motions in rotating fluids. PhD dissertation, University of Cambridge.
Pritchard, W. G. 1969 The motion generated by a body moving along the axis of a uniformly rotating fluid. J. Fluid Mech. 39, 443–464.Google Scholar
Randall, J. D. & Leibovich, S. 1973 The critical state: A trapped wave model of vortex breakdown. J. Fluid Mech. 53, 495–515.Google Scholar
Rottman, J. W., Broutman, D. & Grimshaw, R. 1996 Numerical simulations of the flow of a uniformly stratified, inviscid Boussinesq fluid over long topography in a channel of finite depth. J. Fluid Mech. 306, 1–30.Google Scholar
Salas, M. D. & Kuruvila, G. 1989 Vortex breakdown simulation: a circumspect study of the steady, laminar, axisymmetric model. Comput. Fluids 17, 247–262.CrossRefGoogle Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.Google Scholar
Smyth, N. F. 1988 Dissipative effects on the resonant flow of a stratified fluid over topography. J. Fluid Mech. 192, 287–312.Google Scholar
Taylor, G. I. 1922 The motion of a sphere in a rotating liquid.. Proc. R. Soc. Lond. A 102, 180–189.Google Scholar
Thames, F. C. Thompson, J. F., Mastin, C. W. & Walker, R. L. 1977 Numerical solutions for viscous and potential flow about arbitrary two-dimensional bodies using body-fitted coordinate systems. J. Comput. Phys. 24, 245–273.Google Scholar
Wu, T. Y. 1981 Long waves in ocean and coastal waters. J. Engng Mech. Div. ASCE 107, 501–522.Google Scholar
Yi, Z. & Warn, T. 1987 A numerical method for solving the evolution equation of solitary Rossby waves on a weak shear. Adv. Atmos. Sci. 4, 43–54.Google Scholar