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Exact analytical solutions for steady three-dimensional inviscid vortical flows

Published online by Cambridge University Press:  15 October 2007

S. BHATTACHARYA
S. BHATTACHARYA*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
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Abstract

Vortical flows with an axial (z-axis) swirl and a toroidal circulation (in the (rho,z)-plane) can be observed in a wide range of fluid mechanical phenomena such as flow around rotary machines or natural vortices like tornadoes and hurricanes. In this paper, we obtain exact analytical solutions for a general class of steady systems with such three-dimensional circulating structures. Assuming incompressible ideal fluid, a general single-variable equation, known as the Squire–Long equation, can be constructed which can uniquely describe the velocity fields with steady axial and toroidal circulations. In this paper, we consider the case where this type of flow can be analysed by solving a linear homogeneous partial differential equation. The derived equation resembles the governing equation of the hydrogen problem. As a result, we obtain a quantization relation which is similar to the expression for the quantized energy states in a hydrogen atom.

For circulating flows, this formalism provides a complete set of orthogonal basis functions which are regular and localized. Hence, each of the basis solutions can be used as a simplified model for a realistic phenomenon. Moreover, an arbitrary circulating field can be expanded in terms of these orthogonal functions. Such an expansion can be potentially useful in the study of more general vortices. As illustrations, we present a few examples where we solve the linear homogeneous equation to analyse fluid mechanical systems which can be models for circulating flow in confined geometry. First, we consider three-dimensional vortices confined between two parallel planar walls. Our examples include flows between two infinite planar walls, inside and outside a vertical cylinder bounded at the ends by horizontal plates, and in an axially confined annular region. Then we describe the special way in which the basis functions should be superposed so that a complicated steady velocity-field with three-dimensional vortical structures can be constructed. Two such cases are discussed to indicate that the derived solutions can be used for complicated fluid mechanical modelling.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 147 - 162
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Arnol'd, V. I. 1966a On an a priori estimate in the theory of hydrodynamical stability. Izv. Vyssh. Ucheb. Zaved. Matematika 79, 267–269.Google Scholar
Arnol'd, V. I. 1966b Sur un principe variationnel pour les ecoulement stationnaires des liquides parfaits et ses applications aux problemes de stabilité non lineaires. J. Méc. 5, 29–43.Google Scholar
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.CrossRefGoogle Scholar
Bazzant, M. Z. & Moffat, H. K. 2005 Exact solutions of the Navier–Stokes equations having steady vortex structure. J. Fluid Mech. 541, 55–64.Google Scholar
Bragg, S. L. & Hawthorne, W. R. 1950 Some exact solutions of the flow through annular cascade actuator disks. J. Aero. Sci. 17, 243–249.Google Scholar
Buntine, J. D. & Saffman, P. G. 1995 Inviscid swirling flow and vortex breakdown. Proc. R. Soc. Lond. 449, 139–153.Google Scholar
Burggraf, O. R. & Foster, M. R. 1977 Continuation or breakdown in tornado-like vortices. J. Fluid Mech. 80, 685–704.Google Scholar
Church, C. R., Snow, J. T., Baker, G. L. & Agee, E. M. 1979 Characteristics of tornado-like vortices as a function of swirl ratio: a laboratory investigation. J. Atmos. Sci. 36, 1755–1776.2.0.CO;2>CrossRefGoogle Scholar
Davidson, P. A. 1994 Global stability of two-dimensional and axisymmetric Euler flows. J. Fluid Mech. 276, 273–305.Google Scholar
Delbende, I., Chomaz, J. M. & Huerre, P. 1998 Absolute/convective instability in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.Google Scholar
Emanuel, K. A. 1991 The theory of hurricanes. Annu. Rev. Fluid Mech. 23, 177–196.Google Scholar
Escudier, M. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exps. Fluids 2, 189–196.Google Scholar
Escudier, M. 1987 Confined vortices in flow machinery. Annu. Rev. Fluid Mech. 19, 27–52.Google Scholar
Fernandez-Feria, R., de la Mora, J. F. & Barrero, A. 1995 Solution breakdown in a family of self-similar nearly inviscid axisymmetric vortices. J. Fluid Mech. 305, 77–91.Google Scholar
Fraenkel, L. E. 1956 On the flow of rotating field past bodies in a pipe. Proc. R. Soc. Lond. 233, 506–526.Google Scholar
Fukada, R., Nigim, H. & Koyama, H. 1996 Measurements and visualization in the flowfield behind a model propeller. J. Aircraft 33, 407–413.Google Scholar
Gelfgat, A. Y., Bar Yoseph, P. Z. & Solan, A. 1996 Stability of confined swirling flow with and without vortex breakdown. J. Fluid Mech. 311, 1–36.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1990 Collapse in conical viscous flows. J. Fluid Mech. 218, 483–508.Google Scholar
Gollub, J. P. & Swinney, H. L. 1975 Onset of turbulency in a rotating fluid. Phys. Rev. Lett. 35, 927–930.Google Scholar
Griffiths, D. J. 1994 Introduction to Quantum Mechanics. Prentice-Hall.Google Scholar
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. A 185, 213–245.Google Scholar
Keller, J. J. 1994 Entrainment effects in vortex flows. Phys. Fluids 6, 3028–3934.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.Google Scholar
Lepicovsky, J. & Bell, W. A. 1984 Aerodynamic measurements about a rotating propeller with a laser velocitymeter. J. Aircraft 21, 264–271.Google Scholar
Long, R. R. 1961 A vortex in an infinite viscous fluid. J. Fluid Mech. 11, 611–623.Google Scholar
Lopez, J. M. 1998 Characteristics of endwall and sidewall boundary layers in a rotating cylinder with a differentially rotating endwall. J. Fluid Mech. 359, 49–79.Google Scholar
Malik, M. & Chang, C. L. 1994 Crossflow disturbances in three-dimensional boundary layers: nonlinear development, wave interaction, and secondary instability. J. Fluid Mech. 268, 1–36.Google Scholar
Mestel, A. J. 1989 On the stability of high-Reynolds-number flows with closed streamlines. J. Fluid Mech. 200, 19–38.CrossRefGoogle Scholar
Mitsua, Y. & Monji, N. 1984 Development of a laboratory simulator for small-scale atmospheric vortices. Nat. Disaster Sci. 6, 43–54.Google Scholar
Moffatt, H. K. 1981 Some developments in the theory of turbulence. J. Fluid Mech. 106, 27–47.Google Scholar
Moffatt, H. K. 1990 Structure and stability of solutions of the Euler equations: a Lagrangian approach. Phil. Trans. R. Soc. Lond. A 333, 321–342.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shtern, V. N. & Hussain, F. 1999 Collapse, symmetry breaking and hysteresis in swirling flows. Annu. Rev. Fluid Mech. 31, 537–566.Google Scholar
Sozou, C., Wilkinson, L. L. & Shtern, V. N. 1994 On conical swirling flows in an infinite fluid. J. Fluid Mech. 276, 261–271.Google Scholar
Squire, H. B. 1956 Surveys in Mechanics: Rotating Fluid. Cambridge University Press.Google Scholar
Wang, C. Y. 1991 Exact solutions of the steady-state Navier–Stokes equation. Annu. Rev. Fluid Mech. 23, 159–177.Google Scholar
Ward, N. B. 1972 The explanation of certain features of tornado dynamics using a laboratory model. J. Atmos. Sci. 29, 1194.Google Scholar