Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T12:57:16.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous starting jets

Published online by Cambridge University Press:  21 April 2006

Brian J. Cantwell
Brian J. Cantwell
Affiliation:
Stanford University, Stanford, CA 94305, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the transient motion produced when a viscous incompressible fluid is forced from an initial state of rest. The applied force is time dependent in the form of an impulse, step and ramp function applied at a point and along a line. These cases have been chosen because they form a logical progression for investigating the connection between the flow Reynolds number and the sequence of events leading to the creation of a starting vortex. Much of the structure of the starting process can be revealed through a study of boundary conditions, integrals of the motion and the invariance properties of the governing equations prior to the consideration of a particular solution. The method used to bring out the flow structure is applicable to flows that can be treated as self-similar over some interval in time. The equations for unsteady particle paths are written in terms of similarity variables and then analysed as a quasi-autonomous system with the, usually time-dependent, Reynolds number treated as a parameter. The structure of the flow is examined by finding and classifying critical points in the phase portrait of this system. Bifurcations in the phase portrait are found to occur at specific values of the Reynolds number of the flow in question. When exact solutions of the Stokes equations for the low-Reynolds-number limit are examined they are found to contain two critical Reynolds numbers and three distinct states of motion which culminate in the onset of a vortex roll-up. An interesting feature of the Stokes solutions for planar unsteady jets is that they are uniformly valid over 0 < r < ∞.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 159 - 189
Copyright
© 1986 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

Allen Jr., G. A. 1984 Transition and mixing in axisymmetric jets and vortex rings. Ph.D. thesis, Stanford University, Department of Aeronautics and Astronautics, and SUDAAR No. 541.
Cantwell, B. J. Coles, D. E. & Dimotakis, P. E. 1978 Structure and entrainment on the plane of symmetry of a turbulent spot. J. Fluid Mech. 87, 641–672.Google Scholar
Cantwell, B. J. 1981a Transition in the axisymmetric jet. J. Fluid Mech. 104, 369–386.Google Scholar
Cantwell, B. J. 1981b Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457–515.Google Scholar
Cantwell, B. J. & Allen Jr., G. A. 1984 Transition and mixing in impulsively started jets and vortex rings. In Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena in Fluids, Kyoto, Japan, pp. 123–132.
Glezer, A. 1981 An experimental study of a turbulent vortex ring. Ph.D. thesis, Graduate Aeronautical Laboratories, California Institute of Technology
Griffiths, R. W. 1986 Particle motions induced by spherical convective elements in Stokes flow. J. Fluid Mech. 166, 139–159.Google Scholar
Jackson, J. D. 1977 Classical Electrodynamics, pp. 140–181. Wiley.
Lamb, H. 1932 Hydrodynamics, pp. 214–216. Dover.
Landau, L. 1944 A new exact solution of the Navier—Stokes equations. C. R. Acad. Sci. Dokl. 43, 286–288.Google Scholar
Lighthill, M. J. 1963 In Laminar Boundary Layer (ed. L. Rosenhead), pp. 48–88. Oxford University Press.
Oseen, C. W. 1910 Über die Stokessche Formel und über die Verwandte Aufgabe in der Hydrodynamik. Arkiv for Mathematik, Astronomi och Fysik 6, No. 29.
Oswatitsch, K. 1958 Die Ablösungsbedingung von Grenzschichten. In Grenzschichtforschung (ed. H. Goertler), pp. 357–367. Springer. Also in K. Oswatitsch: Contributions to the Development of Gasdynamics—Selected Papers translated (to English) on the occasion of K. Oswatitsch's 70th Birthday (ed. W. Schneider & M. Platzer), pp. 6–18. Braunschweig, Wiesbaden: Vieweg 1980.
Perry, A. E. & Fairlie, B. D. 1974 Critical points in flow patterns. Adv. Geophys. 18, 299–315.Google Scholar
Perry, A. E., Lim, T. T. & Chong, M. S. 1980 Instantaneous vector fields in coflowing jets and wakes. J. Fluid Mech. 101, 243–256.Google Scholar
Perry, A. E. & Chong, M. S. 1987 Eddying motions and flow patterns. Ann. Rev. Fluid Mech. 19 (to appear).Google Scholar
Sozou, C. 1979 Development of the flow field of a point force in an infinite fluid. J. Fluid Mech. 91, 541–546.Google Scholar
Squire, H. B. 1951 The round laminar jet. Q. J. Mech. Appl. Maths 4, 321–329.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Proc. Camb. Phil. Soc. 9, 52–57.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Tobak, M. & Peake, D. 1982 Topology of three-dimensional separated flows. Ann. Rev. Fluid Mech. 14, 61–85.Google Scholar
Turner, J. S. 1964 The flow into an expanding spherical vortex. J. Fluid Mech. 18, 195–208.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics, pp. 151–156. Parabolic.