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On the Lagrangian description of unsteady boundary-layer separation. Part 1. General theory

Published online by Cambridge University Press:  26 April 2006

Leon L. Van Dommelen  and
Stephen J. Cowley
Leon L. Van Dommelen
Affiliation:
Department of Mechanical Engineering, FAMU/FSU College of Engineering, PO Box 2175, Tallahassee, FL 32316-2175, USA
Stephen J. Cowley
Affiliation:
Department of Mathematics, Imperial College of Science, Technology and Medicine, Huxley Building, 180 Queen's Gate, London SW7 2BZ, UK
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Abstract

Although unsteady, high-Reynolds-number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary-layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady three-dimensional separating flows follow and depend on the symmetry properties of the flow (e.g. line symmetry, axial symmetry). In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi two-dimensional with a displacement thickness in the form of a crescent-shaped ridge. Physically the singularities can be understood in terms of the behaviour of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 210 , January 1990 , pp. 593 - 626
Copyright
© 1990 Cambridge University Press

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References

Awang, M. A. O. & Riley, N. 1983 Unsteady free convection from a heated sphere at high Grashof number. J. Engng Maths 17, 355–365.Google Scholar
Banks, W. H. H. & Zaturska, M. B. 1979 The collision of unsteady laminar boundary layers. J. Engng Maths 13, 193–212.Google Scholar
Banks, W. H. H. & Zaturska, M. B. 1981 The unsteady boundary-layer development on a rotating disc in counter rotating flow. Acta Mech. 38, 143–155.Google Scholar
Blasius, H. 1908 Grenzschichten in Flussigkeiten mit kleiner Reibung. Z. Math. Phys. 56, 1–37.Google Scholar
Bodonyi, R. J. & Stewartson, K. 1977 The unsteady laminar boundary layer on a rotating disk in a counter-rotating fluid. J. Fluid Mech. 79, 669–688.Google Scholar
Bouard, R. & Coutanceau, M. 1980 The early stage of development of the wake behind an impulsively started cylinder for 40 Re. J. Fluid Mech. 101, 583–607.Google Scholar
Brown, S. N. 1965 Singularities associated with separating boundary layers.. Phil. Trans. R. Soc. Lond. A 257, 409–444.Google Scholar
Brown, S. N. & Simpson, C. J. 1982 Collision phenomena in free-convective flow over a sphere. J. Fluid Mech. 124, 123–127.Google Scholar
Cebeci, T. 1982 Unsteady separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci), pp. 265–277. Springer.
Cebeci, T. 1986 Unsteady boundary layers with an intelligent numerical scheme. J. Fluid Mech. 163, 129–140.Google Scholar
Cebeci, T., Khattab, A. K. & Schimke, S. M. 1983 Can the singularity be removed in time-dependent flows? In Proc. A.F. Workshop, Colorado Springs.
Cebeci, T., Khattab, A. K. & Stewartson, K. 1981 Three-dimensional boundary layers and the ok of accessibility. J. Fluid Mech. 107, 57–87.Google Scholar
Cebeci, T., Stewartson, K. & Schimke, S. M. 1984 Unsteady boundary layers close to the stagnation region of slender bodies. J. Fluid Mech. 147, 315–332.Google Scholar
Cowley, S. J. 1983 Computer extension and analytic continuation of Blasius’ expansion for impulsive flow past a circular cylinder. J. Fluid Mech. 135, 389–405.Google Scholar
Didden, N. & Ho, C.-M. 1985 Unsteady separation in a boundary layer produced by an impinging jet. J. Fluid Mech. 160, 235–256.Google Scholar
Ece, M. C., Walker, J. D. A. & Doligalski, T. L. 1984 The boundary layer on an impulsively started rotating and translating cylinder. Phys. Fluids 27, 1077–1089.Google Scholar
Elliott, J. W., Cowley, S. J. & Smith, F. T. 1983 Breakdown of boundary layers:, i. on moving surfaces, ii. in semi-similar unsteady flow, iii. in fully unsteady flow. Geophys. Astrophys. Fluid Dyn. 25, 77–138.Google Scholar
Goldstein, S. 1948 On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1, 43–69.Google Scholar
Henkes, R. A. W. M. & Veldman, A. E. P. 1987 On the breakdown of the steady and unsteady interacting boundary-layer description. J. Fluid Mech. 179, 513–530.Google Scholar
Hudson, J. A. 1980 The Excitation and Propagation of Elastic Waves. Cambridge University Press.
Ingham, D. B. 1984 Unsteady separation. J. Comput. Phys. 53, 90–99.Google Scholar
Lam, S. T. 1988 On high-Reynolds-number laminar flows through a curved pipe, and past a rotating cylinder. Ph.D. dissertation, University of London.
Lamb, H. 1945 Hydrodynamics. Dover.
Ludwig, G. R. 1964 An experimental investigation of laminar separation from a moving wall. AIAA Paper 64-6.Google Scholar
Matsushita, M., Murata, S. & Akamatsu, T. 1984 Studies on boundary-layer separation in unsteady flow using an integral method. J. Fluid Mech. 149, 477–501.Google Scholar
Mccroskey, W. J. & Pucci, S. L. 1982 Viscous-inviscid interaction on oscillating airfoils in subsonic flow. AIAA J. 20, 167–174.Google Scholar
Moore, F. K. 1958 On the separation of the unsteady laminar boundary layer. In Boundary-Layer Research (ed. H. G. Görtler), pp. 296–311. Springer.
Nagata, H., Minami, K. & Murata, Y. 1979 Initial flow past an impulsively started circular cylinder. Bull. JSME 22, 512–520.Google Scholar
Ockendon, H. 1972 An asymptotic solution of steady flow above an infinite rotating disk with suction. Q. J. Mech. Appl. Maths 25, 291–301.Google Scholar
Prandtl, L. 1904 Uber Flüssigkeitsbewegung bei sehr kleiner Reibung In Ludwig Prandtl gesammelte Abhandlüngen, pp. 575–584. Springer, 1961.
Ragab, S. A. 1986 The laminar boundary layer on a prolate spheroid started impulsively from rest at high incidence. AIAA Paper 86-1109.Google Scholar
Rosenhead, L. 1963 Laminar Boundary Layers. Oxford University Press.
Rott, N. 1956 Unsteady viscous flows in the vicinity of a separation point. Q. Appl. Maths 13, 444–451.Google Scholar
Russell, J. M. & Landahl, M. T. 1984 The evolution of a flat eddy near a wall in an inviscid shear flow. Phys. Fluids 27, 557–570.Google Scholar
Sears, W. R. 1956 Some recent developments in airfoil theory. J. Aeronaut. Sci. 23, 490–499.Google Scholar
Sears, W. R. & Telionis, D. P. 1975 Boundary-layer separation in unsteady flow. SIAM J. Appl. Maths 23, 215–234.Google Scholar
Shen, S. F. 1978a Unsteady separation according to the boundary-layer equation. Adv. Appl. Mech. 13, 177–220.Google Scholar
Shen, S. F. 1978b Unsteady separation of three-dimensional boundary layers from the Lagrangian viewpoint. In Nonsteady Fluid Dynamics (ed. D. E. Crow & J. A. Miller), pp. 47–51. ASME.
Shen, S. F. & Wu, T. 1988 Unsteady separation over maneuvering bodies. AIAA Paper 88-3552-CP.Google Scholar
Simpson, C. J. & Stewartson, K. 1982a A note on a boundary-layer collision on a rotating sphere. Z. Angew. Math. Phys. 33, 370–378.Google Scholar
Simpson, C. J. & Stewartson, K. 1982b A singularity in an unsteady free-convection boundary layer. Q. J. Mech. Appl. Maths 35, 291–304.Google Scholar
Smith, F. T. 1978 Three-dimensional viscous and inviscid separation of a vortex sheet from a smooth non-slender body. RAE Tech. Rep. 78095.Google Scholar
Smith, F. T. 1987 Break-up in unsteady separation In Forum on Unsteady Flow Separation, ASME FED-Vol. 52, pp. 55–64.
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized short-scaled disturbances in boundary layers.. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Stern, M. E. & Paldor, N. 1983 Large amplitude long waves in a shear flow. Phys. Fluids 26, 906–919.Google Scholar
Stewartson, K., Simpson, C. J. & Bodonyi, R. J. 1982 The unsteady boundary layer on a rotating disk in a counter rotating fluid. Part 2. J. Fluid Mech. 121, 507–515.Google Scholar
Stuart, J. T. 1988 Nonlinear Euler partial differential equations: singularities in their solution. In Proc. Symp. Honor of C. C. Lin (ed. D. J. Benney, Chi Yuan & F. H. Shu), pp. 81–95. World Scientific.
Sychev, V. V. 1979 Asymptotic theory of non-stationary separation. Izv. Akad. Nauk. SSSR, Mekh. Zhid. i Gaza No. 6, 21–32. Also Fluid Dyn. 14, 829–838.Google Scholar
Sychev, V. V. 1980 On certain singularities in solutions of equations of boundary layer on a moving surface. Prikl. Math. Mech. 44, 830–838.Google Scholar
Telionis, D. P. & Tsahalis, D. Th. 1974 Unsteady laminar separation over impulsively moved cylinders. Acta Astronaut. 1, 1487–1505.Google Scholar
Van Dommelen, L. L. 1981 Unsteady boundary-layer separation. Ph.D. thesis, Cornell University.
Van Dommelen, L. L. 1987 Computation of unsteady separation using Lagrangian procedures. In Proc. IUTAM Symp. on Boundary-Layer Separation (ed. F. T. Smith & S. N. Brown), pp. 73–87. Springer.
Van Dommelen, L. L. 1989 On the Lagrangian description of unsteady boundary-layer separation. Part 2. The spinning sphere. J. Fluid Mech. 210, 627–645.Google Scholar
Van Dommelen, L. L. & Shen, S. F. 1977 Presented at XIIIth Symp. Bienn. Symp. Adv. Meth. Prob. Fluid Mech., Olsztyn-Kortewo, Poland.
Van Dommelen, L. L. & Shen, S. F. 1980a The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comput. Phys. 38, 125–140.Google Scholar
Van Dommelen, L. L. & Shen, S. F. 1980b The birth of separation. Presented at the XVth Intl Congl Theo. Appl. Mech., Toronto, Canada. IUTAM.
Van Dommelen, L. L. & Shen, S. F. 1982 The genesis of separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci), pp. 293–311. Springer.
Van Dommelen, L. L. & Shen, S. F. 1983a Boundary-layer separation singularities for an upstream moving wall. Acta Mech. 49, 241–254.Google Scholar
Van Dommelen, L. L. & Shen, S. F. 1983b An unsteady interactive separation process. AIAA J. 21, 358–362.Google Scholar
Van, Dyke, 1975 Perturbation Methods in Fluid Mechanics, pp. 86. Stanford: Parabolic.
Vasantha, R. & Riley, N. 1988 On the initiation of jets in oscillatory viscous flows.. Proc. R. Soc. Lond. A 419, 363–378.Google Scholar
Walker, J. D. A. 1988 Mechanism of turbulence production near a wall. ICOMP seminar series, NASA-Lewis, July 26.
Wang, K. C. 1979 Unsteady boundary-layer separation. Tech. Rept. MML TR 79-16c. Baltimore: Martin Marietta Lab.
Wang, K. C. & Fan, Z. Q. 1982 Unsteady symmetry-plane boundary layers and three-dimensional unsteady separation. Part I. High incidence. San Diego State Univ. Tech. Rep. AE&EM TR-82-01.Google Scholar
Williams, J. C. 1977 Incompressible boundary-layer separation. Ann. Rev. Fluid Mech. 9, 113–144.Google Scholar
Williams, J. C. 1978 On the nature of unsteady three-dimensional laminar boundary-layer separation. J. Fluid Mech. 88, 241–258.Google Scholar
Williams, J. C. & Stewartson, K. 1983 Flow development in the vicinity of the trailing edge on bodies impulsively set into motion. Part 2. J. Fluid Mech. 131, 177.Google Scholar
Wu, T. 1989 Ph.D. dissertation, Cornell University.