Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-01T00:58:24.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Discontinuous solutions of the unsteady boundary-layer equations for a rotating disk of finite radius

Published online by Cambridge University Press:  18 March 2021

A. Djehizian Open the ORCID record for A. Djehizian [Opens in a new window]  and
A.I. Ruban Open the ORCID record for A.I. Ruban [Opens in a new window]
A. Djehizian*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, LondonSW7 2AZ, UK
A.I. Ruban
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the motion at large Reynolds number of an incompressible fluid around a thin circular disk of finite radius rotating in its plane. The disk is placed in a large tank filled with an initially stagnant fluid. Then it is brought into rotation about its centre with constant angular velocity. Due to viscosity, a layer of fluid adjacent to the disk gets involved in the circumferential motion. This activates centrifugal forces; the fluid particles start to deviate in the radial direction. When they cross the edge of the disk, a thin jet is formed. Firstly, we solve the classical boundary-layer equations for the flow in the boundary layer in the direct neighbourhood of the disk surface and in the jet. The solution is found to develop a discontinuity at the ‘head’ of the jet where the radial and circumferential velocity components experience a jump. This type of discontinuity, called a pseudo-shock, was previously observed by Ruban & Vonatsos (J. Fluid Mech., vol. 614, 2008, pp. 407–424). Then, we investigate the internal structure of the pseudo-shock. We find that the fluid motion is described by the Euler equations in the leading-order approximation. Their solution shows that, as the jet penetrates the stagnant fluid, it ejects the fluid from the boundary layer into the surrounding area. Analysis of the inviscid region outside the boundary layer reveals that the ejected fluid returns back to the boundary layer through the ‘entrainment process’. Finally, we conclude this paper with the study of the wake in the vicinity of the disk rim.

JFM classification

Boundary Layers: Boundary layer separation Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A75
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Allen, T. & Riley, N. 1994 The three-dimensional boundary layer on a yawed body of revolution. J. Engng Maths 28, 345–364.CrossRefGoogle Scholar
Burggraf, O.D. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113–151.CrossRefGoogle Scholar
Calabretto, S.A.W., Denier, J.P. & Levy, B. 2019 An experimental and computational study of the post-collisional flow induced by an impulsively rotated sphere. J. Fluid Mech. 881, 772–793.CrossRefGoogle Scholar
Douglas, J. & Gunn, J. 1964 A general formulation of alternating direction methods. Numer. Math. 6, 428–453.CrossRefGoogle Scholar
Glauert, M.B. 1956 The wall jet. J. Fluid Mech. 1, 625–643.CrossRefGoogle Scholar
Goldstein, S. 1930 Concerning some solutions of the boundary layer equations in hydrodynamics. Math. Proc. Camb. Phil. Soc. 26, 1–30.CrossRefGoogle Scholar
Hewitt, R.E. & Duck, P.W. 2011 Pulsatile jets. J. Fluid Mech. 670, 240–259.CrossRefGoogle Scholar
von Kármán, T. 1921 Über laminare und turbulente reibung. Z. Angew. Math. Mech. 1, 233–252.CrossRefGoogle Scholar
Kluwick, A. & Wohlfahrt, H. 1986 Hot-wire-anemometer study of the entry flow in a curved duct. J. Fluid Mech. 165, 335–353.CrossRefGoogle Scholar
Mallinson, G.D. & de Vahl Davis, G. 1973 The method of the false transient for the solution of coupled elliptic equations. J. Comput. Phys. 12, 435–461.CrossRefGoogle Scholar
Peaceman, D.W. & Rachford, H.H. J.. 1955 The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Maths 3, 28–41.CrossRefGoogle Scholar
Prandtl, L. 1904 Über flüssigkeitsbewegung bei sehr kleiner Reibung. In Verh. III. Intern. Math. Kongr., Heidelberg, pp. 484–491. Teubner, Leipzig, 1905.Google Scholar
Ruban, A.I. & Vonatsos, K.N. 2008 Discontinuous solutions of the boundary-layer equations. J. Fluid Mech. 614, 407–424.CrossRefGoogle Scholar
Schlichting, H. 1933 Laminare strahlausbreitung. Z. Angew. Math. Mech. 13, 260–263.CrossRefGoogle Scholar
Stewartson, K., Cebeci, T. & Chang, K.C. 1980 A boundary-layer collision in a curved duct. Q. J. Mech. Appl. Maths 33, 59–75.CrossRefGoogle Scholar
Turner, J.S. 1962 The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13, 356–368.CrossRefGoogle Scholar