Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-06T09:08:47.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Removal of Goldstein's singularity at separation, in flow past obstacles in wall layers

Published online by Cambridge University Press:  20 April 2006

F. T. Smith  and
P. G. Daniels
F. T. Smith
Affiliation:
Mathematics Department, Imperial College, London SW7 2BZ
P. G. Daniels
Affiliation:
Mathematics Department, City University, St John Street, London EC1V 4PB
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that, in the flow of a viscous wall layer past a relatively steep obstacle at the wall, the Goldstein (1948) singularity generated in the classical boundary-layer approach to separation is removable in a physically sensible fashion. The removal is effected by means of a sequence of local double structures, the last of which arises just beyond separation owing to the occurrence of a further singularity which is also removable and describes the necessary complete breakaway of the viscous layer from the wall. The novel forms of the local pressure–displacement relations are the key elements allowing the solution to retain physical reality throughout. Beyond the breakaway the reattachment process takes place only at a relatively large distance downstream, before the motion returns to its original uniform shear form. The present flow configuration, the first we know of where Goldstein's singularity proves to be removable, has important applications in both internal and external flows at high Reynolds numbers and these are also discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 110 , September 1981 , pp. 1 - 37
Copyright
© 1981 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

Brown, S. N. & Stewartson, K. 1969 Ann. Rev. Fluid Mech. 1, 45.
Burggraf, O. R. 1975 Agard Conf. Proc. 168.
Catherall, D. & Mangler, K. W. 1966 J. Fluid Mech. 26, 163.
Daniels, P. G. 1974 J. Fluid Mech. 63, 641.
Daniels, P. G. 1976 J. Inst. Maths Applics 17, 367.
Daniels, P. G. 1979 J. Fluid Mech. 90, 289.
Dijkstra, D. 1978 Proc. 6th Int. Conf. Num. Meth. Fluid Dyn., Tbilisi, U.S.S.R.
Dennis, S. C. R. & Smith, F. T. 1980 Proc. Roy. Soc. A 372, 393.
Goldstein, S. 1948 Quart. J. Mech. Appl. Math. 1, 43.
Jenson, R. 1975 Ph. D. thesis, Ohio State University.
Kirchhoff, G. 1869 J. reine angew. Math. 70, 289.
Messiter, A. F. 1970 SIAM J. Appl. Math. 18, 241.
Messiter, A. F. 1975 AGARD Conf. Proc. p. 168.
Messiter, A. F. 1979 Proc. 8th U.S. Nat. Cong. of Appl. Mech., 1978, Los Angeles, U.S.A.
Messiter, A. F., Hough, G. R. & Feo, A. 1973 J. Fluid Mech. 60, 605.
Proudman, I. 1960 J. Fluid Mech. 9, 593.
Reyhner, T. A. & Flügge-Lotz, I. 1968 Int. J. Nonlinear Mech. 3, 173.
Smith, F. T. 1972 D.Phil, thesis, Oxford University.
Smith, F. T. 1973 J. Fluid Mech. 57, 803.
Smith, F. T. 1974 J. Inst. Maths Applics 13, 127.
Smith, F. T. 1976a b Quart. J. Mech. Appl. Math. 29, 343; 29, 365.
Smith, F. T. 1976c Mathematika 23, 62.
Smith, F. T. 1977a Proc. Roy. Soc. A 356, 443.
Smith, F. T. 1977b J. Fluid Mech. 79, 631.
Smith, F. T. 1978 J. Inst. Maths Applics 21, 145.
Smith, F. T. 1979a J. Fluid Mech. 90, 725.
Smith, F. T. 1979b J. Fluid Mech. 92, 171.
Smith, F. T. 1979c Lecture course, Asymptotic Methods in Appl. Mech., C.I.S.M., Udine, Italy. (Also to appear in J. Inst. Math. Applic., 1981.)Google Scholar
Smith, F. T. 1980 J. Fluid Mech. 99, 185.
Smith, F. T., Brighton, P. W. M., Jackson, P. S. & Hunt, J. C. R. 1981 J. Fluid Mech. (to appear).
Smith, F. T. & Duck, P. W. 1977 Quart. J. Mech. Appl. Math. 30, 143.
Smith, F. T. & Duck, P. W. 1980 J. Fluid Mech. 90, 727.
Smith, F. T. & Stewartson, K. 1973 J. Fluid Mech. 58, 143.
Stewartson, K. 1970a J. Fluid Mech. 44, 347.
Stewartson, K. 1970b Mathematika 16, 106.
Stewartson, K. 1974a Adv. Appl. Mech. 14, 145.
Stewartson, K. 1974b J. Fluid Mech. 62, 289.
Stewartson, K. 1980 The von Neumann Lecture, Washington D.C., U.S.A.
Stewartson, K. & Williams, P. G. 1969 Proc. Roy. Soc. A 312, 181.
Stewartson, K. & Williams, P. G. 1973 Mathematika 20, 98.
Sychev, V. V. 1967 Rep. 8th. Symp. Recent Problems in Mech. Liquids & Gases, Tarda, Poland.
Sychev, V. V. 1972 Izv. Akad. Nauk S.S.S.R. Meth. Zhidk. i Gaza 3, 47.
Sykes, R. I. 1980 Proc. Roy. Soc. A 373, 311.
Williams, P. C. 1975 Proc. 4th Int. Conf. Num. Meths. in Fluid Dyn., Boulder 1974. Springer.