Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-07T20:02:41.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Laminar flow of an incompressible fluid past a bluff body: the separation, reattachment, eddy properties and drag

Published online by Cambridge University Press:  19 April 2006

F. T. Smith
F. T. Smith
Affiliation:
Department of Mathematics, Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic theory for the laminar, incompressible, separating and reattaching flow past the bluff body is based on an extension of Kirchhoff's (1869) free-streamline solution. The flow field (only the upper half of which is discussed since we consider a symmetric body and flow) consists of two basic parts. The first is the flow on the body scale l*, which is described to leading order by the Kirchhoff solution with smooth inviscid separation, but with an $O(Re^{-\frac{1}{16}})$ modification to explain fully the viscous separation (here Re ([Gt ] 1) is the Reynolds number). The influence of this $O(Re^{-\frac{1}{16}})$ modification is determined for the circular cylinder. The second part is the large-scale flow, comprising mainly the eddy and the ultimate wake. The eddy has length scale O(Rel*), width O(Re½l*) and is of elliptical shape to keep the eddy pressure almost uniform. The ultimate wake is determined numerically and fixes the eddy length. The (asymptotically small) back pressure from the eddy acts (on the body scale) both in the free stream and in the eddy, and it has a marked effect at moderate Reynolds numbers; combined with the Kirchhoff solution, it predicts the pressure drag on a circular cylinder accurately, to within 10% when Re = 5 and to within 4% when Re = 50. Other predictions, for the eddy length and width, the front pressure and the eddy pressure, also show encouraging agreement with experiments and Navier-Stokes solutions at moderate Reynolds numbers (of about 30), both for the circular cylinder and the normal flat plate. Finally, an analysis in the appendix indicates that, in wind-tunnel experiments, the tunnel walls (even if widely spaced) can exert considerable influence on the eddy properties, eventually forcing an upper bound on the eddy width as Re increases instead of the O(l* Re½) growth appropriate to the unbounded flow situation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 92 , Issue 1 , 15 May 1979 , pp. 171 - 205
Copyright
© 1979 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

Acrivos, A., Leal, L. G., Snowden, D. D. & Pan, F. 1968 Further experiments on steady separated flow past bluff objects. J. Fluid Mech. 34, 25.Google Scholar
Acrivos, A., Snowden, D. D., Grove, A. S. & Petersen, E. E. 1965 The steady separated flow past a circular cylinder at large Reynolds numbers. J. Fluid Mech. 21, 737.Google Scholar
Batchelor, G. K. 1956 A proposal concerning laminar wakes behind bluff bodies at large Reynolds numbers. J. Fluid Mech. 1, 388.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, wakes and cavities. Appl. Math. Mech. 2.Google Scholar
Brodetsky, S. 1923 Discontinuous fluid motion past circular and elliptic cylinders. Proc. Roy. Soc. A 102, 542.Google Scholar
Burggraf, O. R. 1970 U.S. Air Force Aerospace Res. Lab. Rep. ARL 70-0275.
Burggraf, O. R. 1975 Asymptotic theory of separation and reattachment of a laminar boundary layer on a compression ramp. AGARD Paper no. 168 (on flow separation).Google Scholar
Dennis, S. C. R. & Chang, G.-Z. 1970 Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech. 42, 471.Google Scholar
Deshpande, M. D., Giddens, D. P. & Mabon, R. F. 1976 Steady laminar flow through modelled vascular stenoses. J. Biomech. 9, 165.Google Scholar
Dimopoulos, H. G. & Hanratty, T. J. 1968 Velocity gradients at the wall for flow around a cylinder for Reynolds numbers between 60 and 360. J. Fluid Mech. 33, 303.Google Scholar
Föppl, L. 1913 Wirbelbewegung hinter einem Kreiszylinder. Munich Akad. Wiss., Math.-Phys. Classe no. 1.Google Scholar
Gilbarg, D. & Serrin, J. 1950 Free boundaries and jets in the theory of cavitation. J. Math. Phys. 29, 1.Google Scholar
Goldstein, S. 1930 Concerning some solutions of the boundary layer equations in hydro-dynamics. Proc. Camb. Phil. Soc. 26, 1.Google Scholar
Goldstein, S. 1948 On laminar boundary layer flow near a point of separation. Quart. J. Mech. Appl. Math. 1, 43.Google Scholar
Grove, A. S., Shair, F. H., Petersen, E. E. & Acrivos, A. 1964 An experimental investigation of the steady separated flow past a circular cylinder. J. Fluid Mech. 19, 60.Google Scholar
Homann, F. 1936 Einfluss grösser Zähigkeit bei Strömung um Zylinder. Forsch. Ing. Wes. 7, 1.Google Scholar
Imai, I. 1953 Discontinuous potential flow as the limiting form of the viscous flow for vanishing viscosity. J. Phys. Soc. Japan 8, 399.Google Scholar
Jenson, R., Burggraf, O. R. & Rizzetta, D. P. 1974 Asymptotic solution for supersonic viscous flow past a compression corner. Proc. 4th. Int. Conf. Numer. Meth. in Fluid Mech. p. 218. Springer.
Kawaguti, M. 1953 Discontinuous flow past a circular cylinder. J. Phys. Soc. Japan. 8, 403.Google Scholar
Kawaguti, M. & Jain, P. C. 1966 Numerical study of a viscous flow past a circular cylinder. J. Phys. Soc. Japan 21, 2055.Google Scholar
Kirchhoff, G. 1869 Zur Theorie freir Flüssigkeitsstrahlen. J. reine angew. Math. 70, 289.Google Scholar
Kiya, M. & Arie, M. 1977 An inviscid bluff-body wake model which includes the far-wake displacement effect. J. Fluid Mech. 81, 593.Google Scholar
Lock, R. C. 1951 The velocity distribution in the laminar boundary layer between parallel streams. Quart. J. Mech. 4, 42.Google Scholar
Lighthill, M. J. 1949 A note on cusped cavities. Aero. Res. Counc. R. & M. no. 2328.Google Scholar
Messiter, A. F. 1975 Laminar separation - a local asymptotic flow description for constant pressure downstream. AGARD Paper no. 168 (on flow separation).Google Scholar
Messiter, A. F. 1978 Boundary-layer separation. U.S. Appl. Mech. Cong., University of California, Los Angeles.Google Scholar
Messiter, A. F. & Enlow, R. L. 1973 A model for laminar boundary layer flow near a separation point. SIAM J. 25, 655.Google Scholar
Messiter, A. F., Hough, G. R. & Feo, A. 1973 Base pressure in laminar supersonic flow. J. Fluid Mech. 60, 605.Google Scholar
Parkinson, G. V. & Jandali, T. 1970 A wake source model for bluff body potential flow. J. Fluid Mech. 40, 577.Google Scholar
Patel, V. A. 1976 Time-dependent solutions of the viscous incompressible flow past a circular cylinder by the method of series truncation. Computers & Fluids 4, 13.Google Scholar
Riabouchinsky, D. 1919 On the steady flow motions with free surfaces. Proc. Lond. Math. Soc. 19, 206.Google Scholar
Roshko, A. 1954 A new hodograph for free streamline theory. N.A.C.A. Tech. Note no. 3168.Google Scholar
Roshko, A. 1955 On the wake and drag of bluff bodies. J. Aero. Sci. 22, 124.Google Scholar
Roshko, A. 1967 A review of concepts in separated flow. Proc. Can. Cong. Appl. Mech., Quebec, vol. 3, p. 81.Google Scholar
Shair, F. H. 1963 Ph.D. thesis, University of California, Berkeley.
Smith, F. T. 1974 Boundary layer flow near a discontinuity in wall conditions. J. Inst. Math. Appl. 13, 127.Google Scholar
Smith, F. T. 1977 The laminar separation of an incompressible fluid streaming past a smooth surface. Proc. Roy. Soc. A 356, 443.Google Scholar
Smith, F. T. 1978 Three-dimensional viscous and inviscid separation of a vortex sheet from a smooth non-slender body surface. Three-dimensional viscous and inviscid separation of a vortex sheet from a smooth non-slender body surface.Rep. TR 78095.Google Scholar
Smith, F. T. 1979 The separating flow through a severely constricted symmetric tube. J. Fluid Mech. 90, 725.Google Scholar
Smith, F. T. & Duck, P. W. 1977 Separation of jets or thermal boundary layers from a wall. Quart. J. Mech. Appl. Math. 30, 143.Google Scholar
Smith, J. H. B. 1977 Behaviour of a vortex sheet separating from a smooth surface. Behaviour of a vortex sheet separating from a smooth surface.Rep. TR 77058.Google Scholar
Son, J. S. & Hanratty, T. J. 1969 Numerical solution for the flow around a cylinder at Reynolds numbers of 40, 200 and 500. J. Fluid Mech. 35, 369.Google Scholar
Southwell, R. V. & Vaisey, G. 1946 Fluid motions characterized by free streamlines. Phil. Trans. Roy. Soc. A 240, 117.Google Scholar
Squire, H. B. 1934 On the laminar flow of a viscous fluid with vanishing viscosity. Phil. Mag. 17, 1150.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145.Google Scholar
Sychev, V. Ya. 1967 On laminar fluid flow behind a blunt body at high Reynolds numbers. Rep. to 8th Symp. Recent Problems in Mech. Liquids & Gases, Tarda, Poland.Google Scholar
Sychev, V. Ya. 1972 Concerning laminar separation. Izv. Akad. Nauk SSSR, Mekh. Zh. Gaza 3, 47.Google Scholar
Takami, H. & Keller, H. B. 1969 Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder. Phys. Fluids Suppl. 12, II 51.Google Scholar
Tritton, D. J. 1959 Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6, 547.Google Scholar
Woods, L. C. 1955 Two dimensional flow of a compressible fluid past given curved obstacles with infinite wakes. Proc. Roy. Soc. A 227, 367.Google Scholar
Wu, T. Y. 1956 A free streamline theory for two-dimensional fully cavitated hydrofoils. J. Math. Phys. 35, 236.Google Scholar
Wu, T. Y. 1962 A wake model for free streamline flow theory. J. Fluid Mech. 13, 161.Google Scholar
Wu, T. Y. 1968 Inviscid cavity and wake flows. Basic Develop. in Fluid Dyn. 2, 1.Google Scholar
Wu, T. Y. 1972 Cavity and wake flows. Ann. Rev. Fluid Mech. 4, 243.Google Scholar