Hostname: page-component-5db6c4db9b-bhjbq Total loading time: 0 Render date: 2023-03-25T18:33:13.184Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Discontinuous solutions of the boundary-layer equations

Published online by Cambridge University Press:  16 October 2008

Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK


Since 1904, when Prandtl formulated the boundary-layer equations, it has been presumed that due to the viscous nature of the boundary layers the solution of the Prandtl equations should be sought in the class of continuous functions. However, there are clear mathematical reasons for discontinuous solutions to exist. Moreover, under certain conditions they represent the only possible solutions of the boundary-layer equations.

In this paper we consider, as an example, an unsteady analogue of the laminar jet problem first studied by Schlichting in 1933. In Schlichting's formulation the jet emerges from a narrow slit in a flat barrier and penetrates into a semi-infinite region filled with fluid which would remain at rest if the slit were closed. Assuming the flow steady, Schlichting was able to demonstrate that the corresponding solution to the Prandtl equations may be written in an explicit analytic form. Here our concern will be with unsteady flow that is initiated when the slit is opened and the jet starts penetrating into the stagnant fluid. To study this process we begin with the numerical solution of the unsteady boundary-layer equations. Since discontinuities were expected, the equations were written in conservative form before finite differencing. The solution shows that the jet has a well-established front representing a discontinuity in the velocity field, similar to the shock waves that form in supersonic gas flows.

Then, in order to reveal the ‘internal structure’ of the shock we turn to the analysis of the flow in a small region surrounding the discontinuity. With Re denoting the Reynolds number, the size of the inner region is estimated as an order Re−1/2 quantity in both longitudinal and lateral directions. We found that the fluid motion in this region is predominantly inviscid and may be treated as quasi-steady if considered in the coordinate frame moving with the jet front. These simplifications allow a simple formula for the front speed to be deduced, which proved to be in close agreement with experimental observation of Turner (J. Fluid Mech. vol. 13 (1962), p. 356).

Copyright © Cambridge University Press 2008

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.)



Allen, T. & Riley, N. 1994 The three-dimensional boundary layer on a yawed body of revolution. J. Engng Maths 28 (5), 345364.CrossRefGoogle Scholar
Degani, A. T., Li, Q. & Walker, J. D. A. 1996 Unsteady separation from the leading edge of an airfoil. Phys. Fluids 8 (3), 181206.CrossRefGoogle Scholar
Goldstein, S. 1948 On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1 (1), 4369.CrossRefGoogle Scholar
Kirchhoff, G. 1869 Zur theorie freier flüssigkeitsstrahlen. J. Reine Angew. Math. 70 (4), 289298.CrossRefGoogle Scholar
Kluwick, A. & Wohlfahrt, H. 1986 Hot-wire-anemometer study of the entry flow in a curved duct. J. Fluid Mech. 165, 335353.CrossRefGoogle Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18 (1), 241257.CrossRefGoogle Scholar
Neiland, V. Ya. 1969 Theory of laminar boundary layer separation in supersonic flow. Izv. Akad. Nauk SSSR, Mech. Zhidk. Gaza (4), 5357.Google Scholar
Nikolskii, A. A. 1957 a The force produced by the ‘second form’ of fluid motion on two-dimensional bodies (dynamics of two-dimensional separated flows). Dokl. Akad. Nauk SSSR 116 (3), 365368.Google Scholar
Nikolskii, A. A. 1957 b On the ‘second form’ of the motion of an ideal fluid around a body (a study of separated vortex flows). Dokl. Akad. Nauk SSSR 116 (2), 193196.Google Scholar
Pera, L. & Gebhart, B. 1975 Laminar plume interactions. J. Fluid Mech. 68, 259271.CrossRefGoogle Scholar
Prandtl, L. 1904 Über flüssigkeitsbewegung bei sehr kleiner Reibung. In Verh. III. Intern. Math. Kongr., Heidelberg, pp. 484491. Teubner, Leipzig, 1905.Google Scholar
Schlichting, H. 1933 Laminare strahlausbreitung. Z. Angew. Math. Mech. 13, 260.CrossRefGoogle Scholar
Schneider, W. 1985 Decay of momentum flux in submerged jets. J. Fluid Mech. 154, 91110.CrossRefGoogle Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate. Mathematika 16 (1), 106121.CrossRefGoogle Scholar
Stewartson, K., Cebeci, T. & Chang, K. C. 1980 A boundary layer collision in a curved duct. Q. J. Mech. Appl. Maths 33, 5975.CrossRefGoogle Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181206.CrossRefGoogle Scholar
Sudakov, G. G. 1974 Calculation of the separated flow past a small aspect ratio delta wing. Uch. Zap. TsAGI 5 (2), 1018.Google Scholar
Sychev, V. V. 1972 Laminar separation. Izv. Akad. Nauk SSSR, Mech. Zhidk. Gaza (3), 4759.Google Scholar
Sychev, V. V., Ruban, A. I., Sychev, Vic. V. & Korolev, G. L. 1998 Asymptotic Theory of Separated Flows. Cambridge University Press.CrossRefGoogle Scholar
Tulin, M. P. 1964 Supercavitating flows – Small perturbation theory. J. Ship Res. 7 (3), 1637.Google Scholar
Turner, J. S. 1962 The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13, 356368.CrossRefGoogle Scholar
Zakharov, S. B. 1976 Calculation of inviscid separated flow past a thin circular cone at large angle of attack. Uch. Zap. TsAGI 7 (6), 111116.Google Scholar