Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T18:33:12.707Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of small surface perturbations on the pulsatile boundary layer on a semi-infinite flat plate

Published online by Cambridge University Press:  21 April 2006

P. W. Duck
P. W. Duck
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The laminar pulsatile flow over a semi-infinite flat plate, on which is located a small (steady) surface distortion is investigated; triple-deck theory provides the basis for the study. The problem is of direct relevance to the externally imposed acoustic excitation of boundary layers. The investigation is primarily numerical and involves the solution of the nonlinear, unsteady boundary-layer equations which arise from the lower deck. The numerical method involves the use of finite differencing in the transverse direction, Crank-Nicolson marching in time, and Fourier transforms in the streamwise direction, and as such is an extension of the spectral method of Burggraf & Duck (1982). Supersonic and incompressible flows are studied. A number of the computations presented suggest that the small surface distortion can excite a large-wavenumber, rapidly growing instability, leading to a breakdown of the solution, with the wall shear at a point seeming to increase without bound as a finite time is approached. Rayleigh modes for the basic (undisturbed) velocity profile are computed and there is some correlation between the existence and magnitude of the growth rate of these unstable modes, and the occurrence of the apparent singularity. Streamline plots indicate that this phenomenon is linked to the formation of closed (or ‘cats-eye’) eddies in the main body of the boundary layer, away from the wall. Tollmien-Schlichting instabilities are clearly seen in the case of incompressible flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 259 - 293
Copyright
© 1988 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

Ackerberg, R. C. & Phillips, J. H.1972 The unsteady boundary layer on a semi-infinite flat plate due to small fluctuations in the magnitude of a free-stream velocity. J. Fluid Mech. 51, 137.Google Scholar
Betchov, R.1960 On the mechanisms of turbulent transition, Phys. Fluids 3, 1026.Google Scholar
Bogdanova, E. V. & Ryzhov, O. S.1983 Free and induced oscillations in Poiseuille flow. Q. J. Mech. Appl. Maths 36, 271.Google Scholar
Brotherton-Ratcliffe, R. V. & Smith, F. T. 1987 Complete breakdown of an unsteady interactive boundary layer (over a surface distortion or in liquid layer). Mathematika 34, 86.Google Scholar
Brown, S. N. & Cheng, H. K.1981 Correlated and steady laminar trailing edge flows. J. Fluid Mech. 108, 171.Google Scholar
Brown, S. N. & Daniels, P. G.1975 On the viscous flow about the trailing edge of a rapidly oscillating plate. J. Fluid Mech. 67, 743.Google Scholar
Brown, S. N. & Stewartson, K.1973a On the propagation of disturbances in a laminar boundary layer I. Proc. Camb. Phil. Soc. 73, 493.Google Scholar
Brown, S. N. & Stewartson, K.1973b On the propagation of disturbances in a laminar boundary layer II. Proc. Camb. Phil. Soc. 73, 505.Google Scholar
Burggraf, O. R. & Duck, P. W.1981 Spectral computation of triple-deck flows. In Numerical and Physical Aspects of Aerodynamic flows (ed. T. Cebeci). Springer.
Clarion, C. & Pelissier, R.1975 A theoretical and experimental study of the velocity distribution and transition to turbulence in free oscillatory flow. J. Fluid Mech. 70, 59.Google Scholar
Cooley, J. W. & Tukey, J. W.1965 An algorithm for the machine computation of complex Fourier series. Math. Comp. 19, 297.Google Scholar
Cowley, S. J.1981 High Reynolds number flows through channels and flexible tubes. Ph.D. dissertation, University of Cambridge.
Cowley, S. J.1985 Pulsatile flow through distorted channels; low-Strouhal-number and translating-critical-layer effects. Q. J. Mech. Appl. Maths 38, 589.Google Scholar
Cowley, S. J.1987 High frequency Rayleigh instability of Stokes layers. In Proc. ICASE Workshop on the Stability of Time Dependent and Spatially Varying Flows. Springer.
Davis, S. H. & Rosenblat, S.1972 On bifurcating periodic solutions at low frequency. Stud. Appl. Maths 57, 59.Google Scholar
Duck, P. W.1978 Laminar flow over a small unsteady hump on a flat plate. Mathematika 25, 24.Google Scholar
Duck, P. W.1980 Pulsatile flow in constricted or dilated channels. Q. J. Mech. Appl. Maths 33, 78.Google Scholar
Duck, P. W.1981 Laminar flow over a small unsteady three-dimensional hump. Z. angew. Math. Phys. 32, 62.Google Scholar
Duck, P. W.1985a Laminar flow over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465.Google Scholar
Duck, P. W.1985b Pulsatile flow through constricted or dilated channels: Part II. Q. J. Mech. Appl. Maths 38, 621.Google Scholar
Duck, P. W.1986 Unsteady triple deck flows leading to instabilities. In Proc. IUTAM Symp. on Boundary Layer Separation (ed. S. N. Brown & F. T. Smith). Springer.
Gad-El-Hak, M., Davis, S. H., McMurray, J. T. & Orszag, S. A. 1984 On the stability of the decelerating laminar boundary layer. J. Fluid Mech. 138, 297.Google Scholar
Gibson, W. E.1957 Unsteady boundary layers. Ph.D. dissertation, M.I.T.
Goldstein, M. E.1983 The evolution of Tollmien-Schlichting waves near a leading edge. J. Fluid Mech. 127, 59.Google Scholar
Goldstein, M. E.1985 Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 309.Google Scholar
Goldstein, M. E., Pockol, P. M. & Sanz, J.1983 The evolution of Tollmien-Schlichting waves near a leading edge. Part 2. Numerical determination of amplitudes. J. Fluid Mech. 129, 443.Google Scholar
Greenspan, H. P. & Benney, D. J.1963 On shear-layer instability, breakdown and transition. J. Fluid Mech. 15, 133.Google Scholar
Hall, P.1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151.Google Scholar
Hall, P.1983 On the nonlinear stability of slowly varying time-dependent viscous flows. J. Fluid Mech. 126, 357.Google Scholar
Hino, M., Sawamoto, M. & Takasu, S.1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193.Google Scholar
Iguchi, M., Ohmi, M. & Megawa, K.1982 Analysis of free oscillating flow in a U shaped tube. Bull. JSME 25, 1398.Google Scholar
Illingworth, C. R.1958 The effects of a sound wave on the compressible boundary layer on a flat plate. J. Fluid Mech. 3, 471.Google Scholar
Lam, S. H. & Rott, N.1960 Theory of linearized time-dependent boundary layers. Cornell Univ. GSAE Rep. AF0512, TN-60–1100.Google Scholar
Li, H.1954 Stability of oscillatory laminar flow along a wall. Beach Erosion Board, OS. Army Corps Eng., Washington, D.C. Tech. Memo 47.
Lighthill, M. J.1954 The response of laminar skin friction and heat transfer to fluctuations in the stream velocity. Proc. R. Soc. Lond. A 224, 1.Google Scholar
Lin, C. C.1956 Motion in the boundary layer with a rapidly oscillating external flow. Proc. IXth Int. Cong. Appl. Mech. Brussels, vol. 4, p. 155.
Merkli, P. & Thomann, H.1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567.Google Scholar
Moore, F. K.1951 Unsteady laminar boundary layer flow. NACA Tech. Note 2471.Google Scholar
Moore, F. K.1957 Aerodynamic effects of boundary-layer unsteadiness. Proc. 6th Anglo. Amer. Conf. Roy. Aero. Soc., Folkestone, p. 439.
Murdoch, J. W.1980 The generation of a Tollmien-Schlichting wave by a sound wave. Proc. R. Soc. Lond. A 372, 517.Google Scholar
Nerem, R., Seed, W. A. & Wood, N. B.1972 An experimental study of the velocity distribution and transition to turbulence in the aorta. J. Fluid Mech. 52, 137.Google Scholar
Ohmi, M., Iguchi, M., Kakehashi, K. & Masuda, T. 1982 Transition to turbulence and velocity distribution in an oscillating pipe flow. Bull. JSME 25, 365.Google Scholar
Pedley, T. J.1972 Two-dimensional boundary layers in a free stream which oscillates without reversing. J. Fluid Mech. 55, 359.Google Scholar
Pelissier, R.1979 Stability of a pseudo-periodic flow. Z. angew. Math. Phys. 30, 577.Google Scholar
Rosenblat, S.1968 Centrifugal instability of time-dependent flows. Part 1. Inviscid, periodic flows. J. Fluid Mech. 33, 321.Google Scholar
Rosenblat, S. & Herbert, D. M.1970 Low frequency modulation of thermal instability. J. Fluid Mech. 43, 385.Google Scholar
Rosenhead, L. (ed.) 1963 Laminar Boundary Layers. Oxford University Press.
Ryzhov, O. S. & Zhuk, V. I.1980 Internal waves in a boundary layer with the self-induced pressure. J. Méc. 19, 561.Google Scholar
Sergeev, S. I.1966 Fluid oscillations in pipes at moderate Reynolds numbers. Mekh. Zhidk. Gaza 1, 168 (Transl. Fluid Dyn. 1, 121).Google Scholar
Sobey, I. J.1980 On the flow through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96, 1.Google Scholar
Sobey, I. J.1982 Oscillatory flows at intermediate Strouhal numbers in asymmetric channels. J. Fluid Mech. 125, 359.Google Scholar
Sobey, I. J.1985 Observation of waves during oscillatory channel flow. J. Fluid. Mech. 151, 395.Google Scholar
Smith, F. T.1973 Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803.Google Scholar
Smith, F. T.1979a Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573. (See also Proc. R. Soc. Lond. A 371 (1980), 439.)Google Scholar
Smith, F. T.1979b On the non-parallel flow stability of the Blasius boundary layers. Proc. R. Soc. Lond. A 366, 91.Google Scholar
Smith, F. T.1985 Non-linear effects and non-parallel flows: the collapse of separating motion. United Technologies Research Centre Rep.Google Scholar
Smith, F. T.1986 Two-dimensional disturbance travel, growth and spreading of boundary layers. J. Fluid Mech. 169, 353.Google Scholar
Smith, F. T. & Bodonyi, R. J.1985 On short scale inviscid instabilities in flow past surface-mounted obstacles and other non-parallel motions. Aero. Q. 36, 205.Google Scholar
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.Google Scholar
Smith, F. T. & Stewart, P. A.1987 The resonant-triad nonlinear interaction in boundary-layer transition. J. Fluid Mech. 179, 227.Google Scholar
Stewart, P. A. & Smith, F. T.1987 Three-dimensional instabilities in steady and unsteady non-parallel boundary layers including effects of Tollmien-Schlichting disturbances and cross flow. Proc. R. Soc. Lond. A 409, 299.Google Scholar
Stewartson, K.1969 On the flow near the trailing edge of a flat plate - II. Mathematika 16, 106Google Scholar
Stewartson, K. & Williams, P. G.1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181.Google Scholar
Terent'Ev, E. D.1978 On an unsteady boundary layer with self-induced pressure in the vicinity of a vibrating wall in a supersonic flow (in Russian). Dokl. Akad. Nauk. SSSR 240, 1046.Google Scholar
Terent'Ev, E. D.1984 Linear problem on vibrator on a flat plate in subsonic boundary layer. In Proc. IUTAM Symp. Laminar-Turbulent Transition Novosibrisk 1984 (ed. V. V. Kozlov). Springer.
Tutty, O. R. & Cowley, S. J.1986 On the stability and numerical solution of the unsteady interactive boundary-layer equation. J. Fluid Mech. 168, 431.Google Scholar
Von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753.Google Scholar
Yang, W. H. & Yih, C.-S. 1977 Stability of time-periodic flows in a circular pipe. J. Fluid Mech. 82, 497.Google Scholar
Zhuk, V. I. & Ryzhov, O. S.1978 On one property of the linearised boundary-layer equations with a self-induced pressure (in Russian). Dokl. Akad. Nauk SSSR 240, 1042.Google Scholar