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Wave packets, resonant interactions and soliton formation in inlet pipe flow

Published online by Cambridge University Press:  26 April 2006

Igor V. Savenkov
Igor V. Savenkov
Affiliation:
Computer Center of the Academy of Sciences, 117 967 Vavilova, 40, Moscow, Russia, CIS
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Abstract

The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds number R → ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 1 - 30
Copyright
© 1993 Cambridge University Press

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References

Bogdanova, E. V. 1982 On the free oscillations of viscous incompressible fluid in a semi-infinite circular pipe. Dokl. Akad. Nauk.USSR 263 829–833 (in Russian).Google Scholar
Bogdanova, E. V. & Ryzhov, O. S. 1983 On free oscillations of viscous incompressible fluid in a semi-infinite channel. Prikl. Matem. Mekh. 47, 64–72.Google Scholar
Borodulin, V. I. & Kachanov, Yu. S. 1989 The series of harmonic and parametric resonances in the K-regime of breakdown of laminar boundary layer. Modelir. v. Mekh., Siberian Div. Akad. Sci. USSR, ITPM 3(20), No. 2 (in Russian).
Borodulin, V. I. & Kachanov, Yu. S. 1990 Experimental study of soliton-like coherent structures in boundary layer. In Proc. 19th Session Scientific and Methodological Seminar on Ship Hydrodynamics, 1–6 Oct. Varna, Bulgaria, vol. 2, pp. 99-1–99-10. Bulgarian Ship Hydrodynamics Centre.
Burggraf, O. R. & Duck, P. W. 1982 Spectral computation of triple-deck flow. In Numerical and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci), pp. 145–158. Academic.
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.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, pp. 297–312. Springer.
Duck, P. W. 1990 Triple-deck flow over unsteady surface disturbances: the three-dimensional development of Tollmien-Schlichting waves. Computers Fluids. 18, 1–34.Google Scholar
Goldstein, M. E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120.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. 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96.Google Scholar
Huang, L. M. & Chen, T. S. 1974 Stability of developing pipe flow subjected to non-axisymmetric disturbances. J. Fluid Mech. 63, 183–193.Google Scholar
Kachanov, Yu. S. 1987 On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 184, 43–74.Google Scholar
Kachanov, Yu. S. 1991a Resonant-soliton nature of boundary layer transition. Russian J. Theor. Appl. Mech. 1 (2)1 79–94.Google Scholar
Kachanov, Yu. S. 1991b The mechanism of formation and breakdown of soliton-like structures in boundary layer. In Advances in Turbulence (ed. A. V. Johansson & P. H. Alfredsson), pp. 42–51. Springer.
Kachanov, Yu. S. & Levchenko, V. Ya. 1982 The resonant interaction of disturbances at laminar - turbulent transition in a boundary layer. Preprint 10–82, Novosibirsk, ITPM SO AN SSSR (in Russian).
Kachanov, Yu. S. & Levchenko, V. Ya. 1984 The resonant interaction of disturbances at laminar turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.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
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
Messiter, A. F. 1970 Boundary layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18, 241–257.Google Scholar
Neiland, V. Ya. 1969 Towards the theory of separation of the laminar boundary layer in a supersonic flow. Izv. Akad. Nauk USSR, Mekh. Zhidk. i Gaza 4, 53–57 (in Russian).Google Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1992a 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. 1992b Vortex-induced boundary-layer separation. Part 2. Unsteady interacting boundary-layer theory. J. Fluid Mech. 232, 133–165.Google Scholar
Popov, S. P. 1992 On soliton perturbations excited by an oscillator in a boundary layer. Comput. Maths Mech. Phys. 32 (1)1 61–69.Google Scholar
Reynolds, O. 1983 Experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels. Phil. Trans. R. Soc. 174, 935–982.Google Scholar
Rothmayer, A. P. & Smith, F. T. 1987 Strongly nonlinear wave-packets in boundary layers. Trans. ASME I J. Fluids Engng June1 67.Google Scholar
Ryzhov, O. S. 1990 On the formation of organized vortex structures from unstable oscillations in a boundary layer. Zh. Vich. Matem. Mat. Fiz. 30 1804–1814 (in Russian).Google Scholar
Ryzhov, O. S. & Saveenkov, I. V. 1989 Asymptotic approach to hydrodynamic stability theory. Mathematical Simulation 1 (4), 61–86 (in Russian).Google Scholar
Ryzhov, O. S. & Saveenkov, I. V. 1992 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. Mir Publishers & CRC Press.
Ryzhov, O. S. & Terent'ev, E. D. 1986 On the transition regime, which characterizes the start-up process of vibrator oscillations in a subsonic boundary layer on a flat plate. Prikl, Matem. Mekh. 50 974–986 (in Russian).Google Scholar
Sarpkaya, T. 1975 A note on the stability of developing laminar pipe flow subjected to axisymmetric and non-axisymmetric disturbances. J. Fluid Mech. 68, 345–351.Google Scholar
Saveenkov, I. V. 1992 The resonant amplification of two-dimensional disturbances in a semi-infinite channel. Zh. Vich. Matem. Mat. Fiz. 32 1332–1339 (in Russian).Google Scholar
Smith, F. T. 1976 On entry-flow effects in bifurcating, blocked or constricted tubes. J. Fluid Mech. 78, 709–736.Google Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T. 1986 Two-dimensional disturbance travel, growth and spreading in boundary layers. J. Fluid Mech. 169, 353–377.Google Scholar
Smith, F. T. 1988 Finite-time break-up can occur in any unsteady interacting boundary layer. Mathematika 35, 256–273.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1980 On the stability of the developing flow in a channel or circular pipe. Q. J. Mech. Appl. Maths 33, 293–320.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, 463–489.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., Doorly, D. J. & Rothmayer, A. P. 1990 On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers, and slugs of vorticity in channel and pipe flows. Proc. R. Soc. Lond. A 428, 255–281.Google Scholar
Smith, F. T. & Stewart, P. A. 1987 The resonant-triad nonlinear interaction in boundary-layer transition. J. Fluid Mech. 179, 227–252.Google Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate-II. Mathematika 16, 106–121.Google Scholar
Tatsumi, T. 1952 Stability of the laminar inlet-flow prior to the formation of Poiseuille regime. J. Phys. Soc. Japan 7, 489.Google Scholar
Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59, 281–335.Google Scholar
Zhuk, V. I. 1993 On the nonlinear disturbances in a boundary layer with self-induced pressure on a flat plate in transonic flow. Prikl. Matem. Mekh. (to appear) (in Russian).Google Scholar
Zhuk, V. I. & Popov, S. P. 1989a On the solutions of inhomogeneous Benjamin-Ono equation. Zh. Vich. Matem. Mat. Fiz. 29 1852–1862 (in Russian).Google Scholar
Zhuk, V. I. & Popov, S. P. 1989b On the nonlinear development of longwave inviscid disturbances in a boundary layer. Zh. Prikl. Makh. Tekhn. Fiz. 3 101–108 (in Russian).Google Scholar
Zhuk, V. I. & Ryzhov, O. S. 1980 Free interaction and stability of an incompressible boundary layer. Dokl. Akad. Nauk. USSR 253 1326–1329 (in Russian).Google Scholar
Zhuk, V. I. & Ryzhov, O. S. 1982 On the locally inviscid disturbances in a boundary layer with selfinduced pressure. Dokl. Akad. Nauk. USSR 263 56–59 (in Russian).Google Scholar