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Interaction of a potential vortex with a local roughness on a smooth surface

Published online by Cambridge University Press:  26 April 2006

Oleg S. Ryzhov  and
Sergey V. Timofeev
Oleg S. Ryzhov
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180–3590, USA
Sergey V. Timofeev
Affiliation:
Computing Center, Russian Academy of Sciences, 40 Vavilov Street; 117333 Moscow, Russian Federation
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Abstract

Disturbances generated by a potential vortex moving past a small hump or dent on the otherwise smooth flat plate are considered. Features peculiar to this problem derive from the fact that the vortex is stuck with a fixed fluid particle; hence the nonlinear dependence of the pressure on the induced velocity field ensues even if the vortex intensity tends to zero. Formulation of the problem on a flow in the viscous wall sublayer given in canonical variables involves four similarity parameters for any particular shape of a roughness. The parallels between the process at hand and sound scattering from a boundary layer with a small obstacle at the bottom are indicated. Results from numerical integration of the boundary-value problem posed allow us to trace the evolution of the wave-packet structure depending on the potential vortex intensity. Overlapping of the peak wings and formation of an almost continuous spectrum in the Fourier decomposition of the signal serve as a guide for explaining the explosive development of the wave packet as distinct from the Tollmien–Schlichting wavetrain that has been registered experimentally.

The theory developed is applied to discussing the so-called bypass mode of transition provoked by external turbulence. Special emphasis is laid on flows in gas turbine engines where bypass transition plays a dominant role owing to extremely high free-stream turbulence levels.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 287 , 25 March 1995 , pp. 21 - 58
Copyright
© 1995 Cambridge University Press

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References

Bodonyi, R. J., Welch, W. J. C., Duck, P. W. & Tadjfar, M. 1989 A numerical study of the interaction between unsteady free-stream disturbances and localized variations in surface geometry. J. Fluid Mech. 209, 285–308.Google Scholar
Borodulin, V. I. & Kachanov, Y. S. 1988 Role of the mechanism of local secondary instability in K-breakdown of boundary layer. Proc. Siberian Div. USSR Acad. Sci., Ser. Tech. Sci. No. 18, 65–77 (in Russian; English translation: Sov. J. App. Phys. 3 (2), 70–81, 1989).Google Scholar
Breuer, K. S. & Haritonidis, J. H. 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 1. Weak disturbances. J. Fluid Mech. 220, 569–594.Google Scholar
Breuer, K. S. & Landahl, M. T. 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances. J. Fluid Mech. 220, 595–621.Google Scholar
Brotherton-Ratcliffe, R. V. & Smith, F. T. 1987 Complete breakdown of an unsteady interacting boundary layer (over a surface distortion or in a liquid layer). Mathematika 34, 86–100.Google Scholar
Burggraf, O. R. & Duck, P. W. 1982 Spectral computation of triple-deck flows. In Proc. Symp. on Numerical and Physical Aspects of Aerodynamic Flow (ed. T. Celeci), pp. 145–158. Springer.
Burov, A. A. & Ryzhov, O. S. 1992 The onset of stochastic pulsations at the early nonlinear stage of disturbance development in an incompressible near-wall jet. Prikl. Math. Mech. 56, 1016–1022 (in Russian; English translation: PMM USSR, 56, 921–927, 1992).Google Scholar
Caradonna, F. X., Strawn, R. C. & Bridgeman, J. O. 1988 An experimental and computational study of rotor–vortex interactions. Vertica, 12, 315–327.Google Scholar
Cebeci, T. 1989 The laminar boundary layer on a circular cylinder started impulsively from rest. J. Comput. Phys. 31, 153–172.Google Scholar
Chuang, F. S. & Conlisk, A. T. 1989 Effect of interaction on the boundary layer induced by a convected rectilinear vortex. J. Fluid Mech. 200, 337–365.Google Scholar
Cohen, J., Breuer, K. S. & Haritonidis, J. H. 1991 On the evolution of a wave packet in a laminar boundary layer. J. Fluid Mech. 225, 575–606.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1973 Flow past an impulsively started circular cylinder. Fluid Mech. 60, 105–127.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
Doligalski, T. L. & Walker, J. D. A. 1984 Boundary layer induced by a convected two-dimensional vortex. J. Fluid Mech. 139, 1–28.Google Scholar
Duck, P. W. 1985 Laminar flow over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465–498.Google Scholar
Duck, P. W. 1987 Unsteady triple-deck flows leading to instabilities. In Proc. IUTAM Symp. on Boundary Layer Separation (ed. F. T. Smith & S. N. Brown), pp. 297–312. Springer.
Duck, P. W. 1988 The effect of small surface perturbations on the pulsatile boundary layer on a semi-infinite flat plate. J. Fluid Mech. 197, 259–293.Google Scholar
Gaster, M. 1975 A theoretical model for the development of a wave packet in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 271–289.Google Scholar
Gaster, M. 1980 On wave packets in laminar boundary layers. In Proc. IUTAM Symp. on Laminar–Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 14–16. Springer.
Gaster, M. 1981 On transition to turbulence in boundary layers. In Proc. Symp. on Transition and Turbulence, University of Wisconsin-Madison, pp. 95–112. Academic.
Gaster, M. & Grant, I. 1975 An experimental investigation of the formation and development of a wave packet in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 253–269.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, 509–529.Google Scholar
Goldstein, M. E. & Hultgren, L. S. 1987 A note on the generation of Tollmien–Schlichting waves by sudden surface-curvature change. J. Fluid Mech. 181, 519–525.Google Scholar
Goldstein, M. E. & Hultgren, L. S. 1989 Boundary-layer receptivity to long-wave free-stream disturbances. Ann. Rev. Fluid Mech. 21, 137–166.Google Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1992 Distortion of a flat-plate boundary layer by free-stream vorticity normal to the plate. J. Fluid Mech. 237, 231–260.Google Scholar
Guckhenheimer, J. & Holmes, P. J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Heinrich, R. A. E. & Kerschen, E. J. 1989 Leading-edge boundary layer receptivity to various free-stream disturbance structure. Z. angew. Math. Mech. 69, 526–598.Google Scholar
Harvey, J. K. & Perry, F. J. 1971 Flowfield produced by trailing vortices in the vicinity of the ground. AIAA J. 9, 1659–1660.Google Scholar
Kachanov, Y. S., Ryzhov, O. S. & Smith, F. T. 1993 Formation of solitons in transitional boundary layers: theory and experiment. J. Fluid Mech. 251, 273–297.Google Scholar
Kerschen, E. 1989 Boundary layer receptivity. AIAA Paper 89-1109.
Kozlov, V. V. & Ryzhov, O. S. 1990 Receptivity of boundary layers: asymptotic theory and experiment. Proc. R. Soc. Lond. A 429, 341–373.Google Scholar
McCroskey, W. J. 1982 Unsteady airfoils. Ann. Rev. Fluid Mech. 14, 285–311.Google Scholar
Mayle, R. E. 1991 The role of laminar–turbulent transition in gas turbine engines. Trans. ASME J. Turbomachinery 113, 509–537.Google Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18, 241–257.Google Scholar
Milne-Thomson, L. M. 1962 Theoretical Hydrodynamics. Macmillan.
Morkovin, M. V. 1969 On the many faces of transition. In Viscous Drag Reduction (ed. C. S. Wells), pp. 1–31. Plenum.
Neiland, V. YA. 1969 Contribution to the theory of separation of a laminar boundary layer in a supersonic stream. Ihv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza No. 4, 53–57 (in Russian; English translation: Fluid Dyn. No. 4, 33–35, 1972).Google Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991a Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re → ∞ J. Fluid Mech. 232, 99–131.Google Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991b Vortex-induced boundary-layer separation. Part 2. Unsteady interacting boundary-layer theory. J. Fluid Mech. 232, 133–165.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8, 311–349.Google Scholar
Ruban, A. I. 1984 On the generation of Tollmien–Schlichting waves by sound. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza No. 5, 44–52 (in Russian; English translation: Fluid Dyn. 19, 709–716, 1985).Google Scholar
Ryzhov, O. S. 1989 Boundary value problems of the asymptotic theory of hydrodynamic stability. Proc. Steklov Inst. Maths 186, 124–131 (in Russian; English translation: Proc. Steklov Inst. Maths Issue 1, 143–151, 1991).Google Scholar
Ryzhov, O. S. 1990 The formation of well ordered structures from unstable oscillations in the boundary layer. Zh. vychisl. Mat. mat. Fiz. 30, 1804–1814 (in Russian; English translation: USSR Comput. Math Maths Phys, 30, No. 6, 146–154, 1990).Google Scholar
Ryzhov, O. S. 1991 Boundary-layer stability and paths to transition: theoretical concepts and experimental evidence. Invited Lecture, 1st European Conference on Fluid Mechanics, Cambridge, UK.
Ryzhov, O. S. 1993 An asymptotic approach to separation and stability problems of a transonic boundary layer. In Transonic Aerodynamics: Problems in Asymptotic Theory (ed. L. P. Cook), pp. 29–53. SIAM.
Ryzhov, O. S. & Savenkov, I. V. 1987 Asymptotic theory of a wave packet in a boundary layer on a plate. Prikl. Math. Mech. 51, 820–828 (in Russian; English translation: PMM USSR 51, 644–651, 1987).Google Scholar
Ryzhov, O. S. & Savenkov, I. V. 1989 Asymptotic approach in the hydrodynamic stability theory. Math. Modelling, 1 (4), 61–86 (in Russian).Google Scholar
Ryzhov, O. S. & Savenkov, I. V. 1991 On the nonlinear stage of perturbation growth in boundary layer. In Modern Problems in Computational Aerodynamics (ed. A. A. Dorodnicyn & P. I. Chushkin), pp. 81–92. CRC Press.
Ryzhov, O. S. & Terent'ev, E. D. 1984 Some properties of vortex spots in a boundary layer on a plate. Dokl. Akad. Nauk SSSR 276, 571–575 (in Russian; English translation: Sov. Phys. Dokl. 29 (5), 349–351, 1984).Google Scholar
Ryzhov, O. S. & Terent'ev, E. D. 1986 On the transition mode characterizing the triggering of a vibrator in the subsonic boundary layer on a plate. Prikl. Math. Mech. 50, 974–986 (in Russian; English translation: PMM USSR 50, 753–762, 1986).Google Scholar
Smith, C. R., Walker, J. D. A., Haidari, A. H. & Sobrun, U. 1991 On the dynamics of near-wall turbulence. Phil. Trans. R. Soc. Lond. A 336, 131–175.Google Scholar
Smith, F. T. 1988 Finite-time breakup can occur in any unsteady interacting boundary layer. Mathematika 35, 256–273.Google Scholar
Smith, F. T. 1991 Steady and unsteady 3-D interactive boundary layers. Comput. Fluids 20, 243–268.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. R. Aero. Soc. 36, 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
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate II. Mathematika 16, 106–121.Google Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181–206.Google Scholar
Straus, J. & Mayle, R. E. 1992 Boundary-layer measurements during a parallel blade–vortex interaction. AIAA Paper 92-262.
Tutty, O. R. & Cowley, S. J. 1986 On the stability and numerical solution of the unsteady interactive boundary-layer equation. J. Fluid Mech. 168, 431–456.Google Scholar
Van dommelen, L. L. & Shen, S. F. 1980 The spontaneous generation of the singularity in a separating boundary layer. J. Comput. Phys. 38, 125–140.Google Scholar
Van dommelen, L. L. & Shen, S. F. 1982 The genesis of separation. In Proc. Symp. on Numerical and Physical Aspects of Aerodynamic Flow (ed. T. Cebeci), pp. 283–311. Springer.
Walker, G. J. & Gostelow, J. P. 1990 Effects of adverse pressure gradients on the nature and length of boundary layer transition. Trans. ASME J. Turbomachinery 112, 196–205.Google Scholar
Walker, J. D. A. 1978 The boundary layer due to rectilinear vortex. Proc. R. Soc. Lond. A 359, 167–188.Google Scholar
Williams, J. C. 1977 Incompressible boundary-layer separation. Ann. Rev. Fluid Mech. 9, 113–144.Google Scholar
Zhuk, V. I. & Ryzhov, O. S. 1982 Locally nonviscous perturbations in a boundary layer with self-induced pressure. Dokl. Akad. Nauk SSSR 263, 56–59 (in Russian; English translation: Sov. Phys. Dokl. 27, (3), 177–179, 1982).Google Scholar
Zhuk, V. I. & Ryzhov, O. S. 1983 Asymptotic behaviour of solutios of the Orr–Sommerfeld equation that yield unstable oscillations at large Reynolds numbers. Dokl. Akad. Nauk SSSR 268, 1328–1332 (in Russian; English translation: Sov. Phys. Dokl. 28, (2), 87–89, 1983).Google Scholar