Abstract
In 1953 James Lighthill, conducting an investigation into the potential mechanisms for upstream influence within boundary layers in supersonic flow, published a theoretical approach which explicitly took into account the influence of viscosity on a disturbance to an incident boundary-layer profile. In doing so he was able to predict a length-scale for upstream influence which scales with the global Reynolds number R, assumed large, as R −3/8. The physical process he identified is now referred to as a pressure–displacement (or viscous–inviscid) interaction. This article discusses Lighthill’s original paper and then proceeds to show how an appreciation of this interaction mechanism can help in the solution of many other problems in fluid mechanics and especially those of flow separation and late-stage laminar–turbulent transition. Finally, the article gives a brief description of the similarities between these two processes as seen from the unifying viewpoint of Lighthill’s pressure–displacement interaction.
Similar content being viewed by others
References
Prandtl L (1905) Über Flüsskigkeitsbewegung bei sehr kleiner Reibung. In: Vehr. III. Intern Math Kongr Heidelberg, pp 484–491
Goldstein S (1930) Concerning some solutions of the boundary-layer equations in hydrodynamics. Proc Camb Phil Soc 26:1–30
Goldstein S (1948) On laminar boundary-layer flow near a position of separation. Quart J Mech Appl Math 1:43–69
Stewartson K (1969) On the flow near the trailing edge of a flat plate. Mathematika 16:106–121
Messiter AF (1970) Boundary-layer flow near the trailing edge of a flat plate. SIAM J Appl Math 18:241–257
Neiland VYa (1969) Towards a theory of separation of the laminar boundary layer in a supersonic stream. Izv Akad Nauk SSSR, Mekh Zhidk i Gaza 4:53–57
Stewartson K (1970) Is the singularity at separation removable? J Fluid Mech 44:347–364
Stewartson K (1981) D’Alembert’s paradox. SIAM Rev 23:308–343
Lighthill MJ (1953) On boundary layers and upstream influence. II. Supersonic flows without separation. Proc Roy Soc London (A217):478–507
Lighthill MJ (1950) Reflection at a laminar boundary layer of a weak steady disturbance to a supersonic stream, neglecting viscosity and heat conduction. Quart J Mech Appl Math 3:303–325
Stewartson K (1951) On the interaction between shock waves and boundary layers. Proc Camb Phil Soc 47:545–553
Lighthill MJ (1958) Introduction to Fourier analysis and generalised functions. Cambridge University Press
Van Dyke M (1964) Perturbation methods in fluid mechanics. Academic Press, New York
Kaplun S (1957) Low Reynolds number flow past a circular cylinder. J Math Mech 6:595–603
Lagerstrom PA, Cole PA (1955) Examples illustrating expansion proceedures for the Navier-Stokes equations. J Rat Mech Anal 4:817–882
Lighthill MJ (1949) A technique for rendering approximate solutions to physical problems uniformly valid. Phil Mag 40:1179–1201
Smith FT (1976) Flow through constricted or dilated pipes and channels: Part 1. Quart J Mech Appl Math 29:343–364
Smith FT (1976) Flow through constricted or dilated pipes and channels: Part 2. Quart J Mech Appl Math 29:365–376
Gajjar J, Smith FT (1983) On hypersonic free interactions, hydraulic jumps and boundary layers with algebraic growth. Mathematika 30:77–93
Bowles RI, Smith FT (1992) The standing hydraulic jump: theory, computations and comparisons with experiments. J Fluid Mech 242:145–168
Stewartson K, Williams PG (1969) Self-induced separation. Proc Roy Soc London 312:181–206
Smith FT (1977) The laminar separation of an incompressible fluid streaming past a smooth surface. Proc Roy Soc London A365:433–463
Smith FT (1986) Steady and unsteady boundary-layer separation. Ann Rev Fluid Mech 18:197–220
Tollmien W (1929) Über die entstehung der Turbulenz. Nach Ges Wiss Göttingen, pp 79–114
Lin CC (1946) On the stability of two-dimensional parallel flows. Quart Appl Math 3:117–142, 213–234, 277–301
Smith FT (1979) On the non-parallel flow stability of the Blasius boundary layer. Proc Roy Soc London A366:91–109
Smith FT (1979) Nonlinear stability of boundary layers for disturbances of various sizes. Proc Roy Soc London 368: 573–589
Rosenhead L (1963) Laminar boundary layers. Dover Publications, New York
Hoyle JM, Smith FT (1994) On finite-time break-up in three-dimensional unsteady interacting boundary layers. Proc Roy Soc London A447:467–492
Stewart PA, Smith FT (1987) The resonant-triad nonlinear interaction in boundary-layer transition. J Fluid Mech 179: 227–252
Smith FT, Walton AG (1989) Nonlinear interaction of near-planar TS waves and longitudinal vortices in boundary-layer transition. Mathematika 36:262–289
Stewart PA, Smith FT (1992)Three-dimensional nonlinear blow-up from a nearly planar initial disturbance in boundary-layer transition; theory and experimental comparisons. J Fluid Mech 244:649–676
Sandham ND, Kleiser L (1992) The late stages of transition to turbulence in channel flow. J Fluid Mech 245:319–348
Kachanov YS (1994) Physical mechanisms of laminar-boundary-layer transition. Ann rev Fluid Mech 26:411–482
Bake S, Fernholz HH, Kachanov YS (2000) Resemblance of K- and N- regimes of boundary-layer transition at late stages. Eur J Mech B Fluids 19(1):1–22
Han G, Tumin A, Wygnanski I (1999) Laminar-turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall. Part 2. Late stage of transition. J Fluid Mech 419:1–27
Smith FT (1988) Finite-time break-up can occur in any unsteady interacting boundary layer. Mathematika 35:256–373
Smith FT and Bowles RI (1992) Transition theory and experimental comparisons on (a) amplification into streets and (b) a strongly nonlinear break-up criterion. Proc Roy Soc London A439:163–175
Li L, Walker JDA, Bowles RI, Smith FT (1998) Short-scale break-up in unsteady interactive layers: local development of normal pressure gradients and vortex wind-up. J Fluid Mech 374:335–378
Bowles RI (2000) Transition to turbulent flow in aerodynamics. Phil Trans Roy Soc London 358:245–260
Smith FT, Bowles RI, Walker JDA (2000) Wind-up of a spanwise vortex in deepening transition and stall. Theoret Comput Fluid Dynamics 14:135–165
Bowles RI (2000) On vortex interaction in the latter stages of boundary-layer transition. In: Fasel H, Saric WS (eds) Laminar-turbulent transition, IUTAM Symposium, Sedona, Arizona, USA, 1999. Springer-Verlag, Berlin, pp 275–280
Whitham GB (1974) Linear and nonlinear waves. Wiley-Interscience, New York
Lighthill MJ (2004) Waves in fluids. Cambridge University Press
Marshall TJ (2004) A study of three-dimensional effects in end-stage boundary layer transition. PhD thesis, University College London
Bowles RI, Davies C, Marshall JT, Smith FT (2005) Stall, transition and turbulence: a tribute to JDAW. AIAA paper 2005-4934. Presented at 4th AIAA Theoretical Fluid Mechanics Meeting, Toronto, Canada, 6–9 June 2005
Ryzhov OS (2006) Transition length in turbine/compressor blade flows. Proc Roy Soc London A462:2281–2298
Bowles RI, Davies C, Smith FT (2003) On the spiking stages in deep transition and unsteady separation. J Eng Math 45:227–245
van Dommelen LL, Shen SF (1980) The spontaneous generation of the singularity in a separating laminar boundary layer. J Comput Phys 38:125–140
van Dommelen LL (1981) Unsteady boundary-layer separation. PhD thesis, Cornell University, Ithaca, NY
Cowley SJ (1983) Computer extension and analaytic continuation of Blasius’ expansion for impulsive flow past a cylinder. J Fluid Mech 135:389–405
Elliott JW, Cowley SJ, Smith FT (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
Cassel KW, Smith FT, Walker JDA (1996) The onset of instability in unsteady boundary-layer separation. J Fluid Mech 315:223–256
Obabko AV, Cassel KW (2000) Large-scale and small-scale interaction in unsteady separation. In: Fluids 2000, Denver Colorado, June 19–22, number AIAA Paper 2000–2469
Cassel KW (2000) A comparison of Navier–Stokes solutions with the theoretical description of unsteady separation. Phil Trans Roy Soc London A 358:3207–3227
Brinkman KW, Walker JDA (2001) Instability in a viscous flow driven by streamwise vortices. J Fluid Mech 432:127–166
Obabko AV, Cassel KW (2002) Navier–Stokes solutions of unsteady separation induced by a vortex. J Fluid Mech 465:99–130
Obabko AV, Cassel KW (2005) On the ejection-induced instability in Navier–Stokes solutions of unsteady separation. Phil Trans Roy Soc London A 363:1189–1198
Obabko AV, Cassel KW A Rayleigh instability in a vortex induced unsteady boundary layer. J Fluid Mech (To be published)
Bodonyi RJ, Smith FT (1985) On the short-scale inviscid instabilities in flow past surface-mounted obstacles and other parallel motions. Aero J (June/July):205–212
Cowley SJ (2000) Laminar boundary-layer theory: a 20th century paradox?. In: Aref H, Phillips JW (eds) Proc 20th Int Congr of Theoret and Appl Mech, Chicago, IL, pp 389–411
Lighthill MJ (2000) Upstream influence in boundary layers 45 years ago. Phil Trans Roy Soc London 358:3047–3061
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bowles, R. Lighthill and the triple-deck, separation and transition. J Eng Math 56, 445–460 (2006). https://doi.org/10.1007/s10665-006-9093-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-006-9093-7