Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-04T18:35:38.343Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability and the numerical solution of the unsteady interactive boundary-layer equation

Published online by Cambridge University Press:  21 April 2006

O. R. Tutty  and
S. J. Cowley
O. R. Tutty
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW Present address: Department of Aeronautics and Astronautics, The University, Highfield, Southampton SO9 5NH.
S. J. Cowley
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT Present address: Department of Mathematics, Imperial College of Science and Technology, Huxley Building, 180 Queen's Gate, London SW7 2BZ.
Get access Rights & Permissions [Opens in a new window]

Abstract

By means of a high-frequency analysis it is shown that a Rayleigh instability is possible within the interactive boundary-layer formulation. This instability reflects the tendency of large-Reynolds-number flows to be unstable. For most, but not all, pressure-displacement relations both Rayleigh's and Fjørtoft's theorems hold, although Fjørtoft's criterion is not a sufficient condition for instability. However, for two pressure-displacement relations neither theorem could be proved, and for one of these, unstable flows exist which are free of inflexion points. Analytically, the existence of this instability may result in a finite-time singularity, while numerically the presence of Rayleigh modes often leads to accuracy problems which cannot be overcome by simple grid refinement. A test integration resulted in the generation of small grid-dependent eddies. It is suggested that the instability may be a possible cause of the eddy splitting observed in experiments on unsteady flows through distorted channels. This Rayleigh instability is also possible within the ‘inverse’ boundary-layer formulation, but is absent from classical boundary-layer problems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 168 , July 1986 , pp. 431 - 456
Copyright
© 1986 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

Ahmed, S. A. & Giddens D. P.1983 Velocity measurements in steady flow through axisymmetric stenoses at moderate Reynolds numbers. J. Biomech. 16, 505–516.Google Scholar
Belotserkovskii O. M., Gushchin, V. A. & Shchennikov V. V.1975 Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid. Zh. vychisl. Mat. mat. Fiz. 15, 197–207.Google Scholar
Benjamin T. B.1961 The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane. J. Fluid Mech. 10, 401–419.Google Scholar
Betchov R.1960 On the mechanisms of turbulent transition. Phys. Fluids 3, 1026–1027.Google Scholar
Bodonyi, R. J. & Smith F. T.1981 The upper branch stability of the Blasius boundary layer, including non-parallel flow effects Proc. R. Soc. Lond. A 375, 65–92.Google Scholar
Bodonyi R. J., Smith, F. T. & Gajjar J.1983 Amplitude-dependent stability of a boundary layer flow with a strongly non-linear critical layer. IMA J. Appl. Maths 30, 1–19.Google Scholar
Bogdanova, E. V. & Ryzhov O. S.1983 The free and induced oscillations in the Poiseuille flow. Q. J. Mech. Appl. Maths 36, 271–287.Google Scholar
Bouard, R. & Coutanceau M.1980 The early stage of development of the wake behind an impulsively started cylinder for 40 < Re < 104 J. Fluid Mech. 101, 583–607.Google Scholar
Bradshaw P., Cebeci, T. & Whitelaw J. H.1981 Engineering Calculation Methods for Turbulent Flow. Academic.
Burggraf, O. R. & Duck P. W.1982 Spectral computation of triple-deck flows. In Numerical and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci). Springer.
Cassanova, R. A. & Giddens D. P.1978 Disorder distal to modelled stenoses in steady and pulsatile flow. J. Biomech. 11, 441–453.Google Scholar
Cebeci T.1979 The laminar boundary layer on a circular cylinder started impulsively from rest. J. Comp. Phys. 31, 153–172.Google Scholar
Cebeci T.1983 A time-dependent approach for calculating steady inverse boundary layer flows with separation Proc. R. Soc. Lond. A 389, 171–178.Google Scholar
Corkill, R. W. & Stewart J. M.1983 Numerical relativity. II. Numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space-times with two Killing vectors Proc. R. Soc. Lond. A 386, 373–391.Google Scholar
Cowley S. J.1985 High frequency Rayleigh instability of Stokes layers. Proc. ICASE Workshop on the Stability of Time Dependent and Spatially Varying Flows. To be published.
Cowley S. J., Hocking, L. M. & Tutty O. R.1985 On the stability of solutions of the unsteady classical boundary-layer equation. Phys. Fluids 28, 441–443.Google Scholar
Dennis, S. C. R. & Smith F. T.1980 Steady flow through a channel with a symmetrical constriction in the form of a step Proc. R. Soc. Lond. A 372, 393–414.Google Scholar
Dijkstra D.1979 Separating, incompressible, laminar boundary layer flow over a smooth step of small height. In Proc. 6th Intl Conf. Num. Meth. Fluid Dyn. Springer.
Drazin, P. G. & Reid W. H.1981 Hydrodynamic Stability. Cambridge University Press.
Duck P. W.1979 Viscous flow through unsteady symmetric channels. J. Fluid Mech. 95, 635–653.Google Scholar
Duck P. W.1985a Laminar flow over unsteady humps: the formation of growing waves. J. Fluid Mech. 160, 465–498.Google Scholar
Duck P. W.1985b Pulsatile flow through constricted or dilated channels: Part II. Q. J. Mech. Appl. Maths 38, 621–653.Google Scholar
Gajjar, J. & Smith F. T.1983 On hypersonic self induced separation, hydraulic jumps and boundary layers with algebraic growth. Mathematika 30, 77–93.Google Scholar
Gill, A. E. & Davey A.1969 Instabilities of a buoyancy driven system. J. Fluid Mech. 35, 775–798.Google Scholar
Greenspan, H. P. & Benny D. J.1963 On shear-layer instability, breakdown and transition. J. Fluid Mech. 15, 133–153.Google Scholar
Hall, P. & Bennett J.1986 Taylor—Görtler instabilities of Tollmien—Schlichting waves and other flows governed by the interactive boundary-layer equations. J. Fluid Mech. (to appear).Google Scholar
Jobe, C. E. & Burggraf O. R.1974 The numerical solution of the asymptotic equations of trailing edge flow Proc. R. Soc. Lond. A 340, 91–111.Google Scholar
Killworth, P. D. & McIntyre M. E.1985 Do Rossby-wave critical-layers absorb, reflect or over-reflect? J. Fluid Mech. 161, 449–491.Google Scholar
Klebanoff P. S., Tidstrom, K. D. & Sargent L. M.1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.Google Scholar
Krasny R.1986 A study of singularity formation in a vortex sheet by the point vortex approximation. J. Fluid Mech. (to appear).Google Scholar
Longuet-Higgins, M. S. & Cokelet E. D.1976 The deformation of steep surface waves on water Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Moore D. W.1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet Proc. R. Soc. Lond. A 365, 105–119.Google Scholar
Moore D. W.1981 On the point vortex method. SIAM J. Sci. Stat. Comput. 2, 65–84.Google Scholar
Nishioka M., Asai, M. & Iida S.1980 An experimental investigation of the secondary instability. In Laminar—Turbulent Transition (ed. R. Eppler & H. Fasel). Springer.
Pedley, T. J. & Stephanoff K. D.1985 Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves. J. Fluid Mech. 160, 337–367.Google Scholar
Rizzetta D. P., Burggraf, O. R. & Jenson R.1978 Triple-deck solutions for viscous supersonic and hypersonic flow past corners. J. Fluid Mech. 89, 535–552.Google Scholar
Ruban A. I.1978 Numerical solution of the local asymptotic problem of the unsteady separation of a laminar boundary layer in a supersonic flow. Zh. vychisl. Mat. mat. Fiz. 18, 1253–1265.Google Scholar
Ryzhov, O. S. & Zhuk V. I.1982 Stability and separation of freely interacting boundary layers. In Proc. 7th Intl Conf. Num. Meth. Fluid Dyn. Springer.
Secomb T. W.1979 Flows in tubes and channels with indented and moving walls. Ph.D. thesis, University of Cambridge.
Smith F. T.1974 Boundary layer flow near a discontinuity in wall conditions. J. Inst. Maths Applics 13, 127–145.Google Scholar
Smith F. T.1976a Flow through constricted or dilated pipes and channels, Parts 1 & 2. Q. J. Mech. Appl. Maths 29, 343–364 & 365–376.Google Scholar
Smith F. T.1976b Pipeflows distorted by non-symmetric indentation or branching. Mathematika 23, 62–83.Google Scholar
Smith F. T.1979a Instability of flow through pipes of general cross section: Parts I & II. Mathematika 26, 187–210 & 211–223.Google Scholar
Smith F. T.1979b On the non-parallel flow stability of the Blasius boundary layer Proc. R. Soc. Lond. A 336, 91–109.Google Scholar
Smith F. T.1979c Nonlinear stability for boundary layers for disturbances of various sizes Proc. R. Soc. Lond. A 368, 573–589 (and corrections, 1980, A 371, 439–440).Google Scholar
Smith F. T.1982a On the high Reynolds number theory of laminar flows. IMA J. Appl. Maths 28, 207–281.Google Scholar
Smith F. T.1982b Concerning dynamic stall. Aero. Quart. Nov./Dec. 331–352.Google Scholar
Smith F. T.1984 Concerning upstream influence in separating boundary layers and downstream influence in channel flow. Q. J. Mech. Appl. Maths 37, 389–399.Google Scholar
Smith, F. T. & Bodonyi R. J.1982 Amplitude-dependent neutral modes in the Hagen—Poiseuille flow through a circular pipe Proc. R. Soc. Lond. A 384, 436–489.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. J. 89, 205–212.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, 25–55.Google Scholar
Smith, F. T. & Daniels P. G.1981 Removal of Goldstein's singularity at separation in flow past obstacles in wall layers. J. Fluid Mech. 110, 1–38.Google Scholar
Smith, F. T. & Elliott J. W.1985 On the abrupt turbulent reattachment downstream of leading-edge laminar separation Proc. R. Soc. Lond. A 401, 1–27.Google Scholar
Smith F. T., Papageorgiou, D. & Elliott J. W.1984 An alternative approach to linear and nonlinear stability calculations at finite Reynolds numbers. J. Fluid Mech. 146, 313–330.Google Scholar
Sobey I. J.1980 On flow through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96, 1–26.Google Scholar
Stephanoff K. D., Pedley T. J., Lawrence, C. J. & Secomb T. W.1983 Fluid flow along a channel with an asymmetric oscillating constriction. Nature 305, 692–695.Google Scholar
Stewartson K.1981 D'Alembert's Paradox. SIAM Rev. 23, 308–343.Google Scholar
Tollmien W.1936 General instability criterion of laminar velocity distributions. Tech. Mem. Nat. Adv. Comm. Aero. Wash. No. 792.Google Scholar
van Dommelen, L. L. & Shen, S. F. 1980 The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comp. Phys. 38, 125–140.Google Scholar
Veldman, A. E. P. & Dijkstra D.1980 A fast method to solve incompressible boundary layer interaction problems. in Proc. 7th Intl Conf. Num. Meth. Fluid Dyn. Springer.Google Scholar