Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire's theorem

Physics of Fluids

Volumn 18, Issue 4, 2006, Pages

  Wu, Xuesong ^a,b   Luo, Jisheng ^a  

^a TIANJIN UNIVERSITY   (China)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; REYNOLDS NUMBER; SURFACE ROUGHNESS; WALL FLOW;

POISEUILLE FLOW; PRESSURE GRADIENT; SQUIRE'S THEOREM;

FLUID DYNAMICS;

FLUID DYNAMICS; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; REYNOLDS NUMBER; SURFACE ROUGHNESS; WALL FLOW;

EID: 33745617730     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2182002     Document Type: Article

References (44)

1. W. Heisenberg, "Uber stabilitate und turbulenz von Flussig-keitsstromen," Ann. Phys.ANPYA20003-3804 74, 577 (1924).
2. H. B. Squire, "On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls," Proc. R. Soc. London, Ser. APRLAAZ1364-5021 142, 621 (1933).
3. S. A. Orszag, "Accurate solution of the Orr-Sommerfeld stability equation," J. Fluid Mech.JFLSA70022-1120 50, 689 (1971).
4. M. Nishioka, S. Iida, and Y. Ichikawa, "An experimental investigation of the stability of plane Poiseuille flow," J. Fluid Mech.JFLSA70022-1120 72, 731 (1975).
5. D. R. Carlson, S. E. Widnall, and M. F. Peeters, "A flow visualization study of transition in plane Poiseuille flow," J. Fluid Mech.JFLSA70022-1120 121, 487 (1982).
6. M. Nishioka and M. Asai, "Some observations of the subcritical transition in plane Poiseuille flow," J. Fluid Mech.JFLSA70022-1120 150, 441 (1985).
7. F. Alavyoon, D. S. Henningson, and P. H. Alfredsson, "Turbulent spots in plane Poiseuille flow - flow visualization," Phys. FluidsPFLDAS0031-917110.1063/1.865884 29, 1328 (1986).
8. D. Meksyn and J. T. Stuart, "Stability of viscous motion between parallel plates for finite disturbances," Proc. R. Soc. London, Ser. APRLAAZ1364-5021 208, 517 (1951).
9. J. T. Stuart, "On the nonlinear mechanics of wave disturbances in stable and unstable shear flows. Part I. The basic behavior in plane Poiseuille flow," J. Fluid Mech.JFLSA70022-1120 9, 353 (1960).
10. W. C. Reynolds and M. C. Potters, "Finite-amplitude instability of parallel shear flows," J. Fluid Mech.JFLSA70022-1120 27, 465 (1967).
11. C. L. Pekeris and B. Shkoller, "Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve," J. Fluid Mech.JFLSA70022-1120 29, 31 (1967).
12. T. S. Chen and D. D. Joseph, "Subcritical bifurcation of plane Poiseuille flow," J. Fluid Mech.JFLSA70022-112010.1017/ S0022112073002624 58, 337 (1973).
13. J.-P. Zahn, J. Moore, E. A. Spiegel, and D. O. Gough, "Nonlinear cellular motions in Poiseuille channel flow," J. Fluid Mech.JFLSA70022-1120 64, 319 (1974).
14. I. Soibelman and D. Meiron, "Finite-amplitude bifurcation in plane Poiseuille flow: Two-dimensional Hopf bifurcation," J. Fluid Mech.JFLSA70022-1120 229, 389 (1991).
15. T. H. Herbert, "Periodic secondary motions in a plane channel," in Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics Vol. 59 (Springer, Berlin, 1976), p. 235.
16. T. H. Herbert, "Finite-amplitude stability of plane parallel flows," AGARD Conf. Proc.AGCPAV0549-7191 224, 3 (1977).
17. S. A. Orszag and A. T. Patera, "Secondary instability of wall-bounded shear flows," J. Fluid Mech.JFLSA70022-1120 128, 347 (1983).
18. S. A. Orszag and L. C. Kells, "Transitional to turbulence in plane Poiseuille and plane Couette flow," J. Fluid Mech.JFLSA70022-1120 96, 159 (1980).
19. K. M. Butler and B. F. Farrell, "Three-dimensional optimal perturbations in viscous shear flows," Phys. Fluids APFADEB0899-821310.1063/1.858386 4, 1637 (1993).
20. S. C. Reddy and D. S. Henningson, "Energy growth in viscous channel flows," J. Fluid Mech.JFLSA70022-1120 252, 209 (1993).
21. S. C. Reddy, P. J. Schmid, and D. S. Henningson, "Pseudospectra of the Orr-Sommerfeld operator," SIAM J. Appl. Math.SMJMAP0036-139910.1137/0153002 53, 15 (1994).
22. L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, "Hydrodynamic instability without eigenvalues," ScienceSCIEAS0036-8075 261, 578 (1993).
23. D. C. Henningson and S. C. Reddy, "On the role of linear mechanisms in transition to turbulence," Phys. FluidsPHFLE61070-663110.1063/ 1.868251 6, 1396 (1994).
24. F. Waleffe, "Hydrodynamics stability and turbulence: Beyond transient to a self-sustaining process," Stud. Appl. Math.SAPMB60022-2526 95, 319 (1995).
25. F. Waleffe, "Transition in shear flows: Nonlinear normality versus non-normal linearity," Phys. FluidsPHFLE61070-663110.1063/1.868682 7, 3060 (1995).
26. F. Waleffe, "On self-sustaining process in shear flows," Phys. FluidsPHFLE61070-663110.1063/1.869185 9, 883 (1997).
27. S. Reddy, P. J. Schmid, J. S. Baggett, and D. S. Henningson, "On stability of streamwise streaks and transition thresholds in plane channel flows," J. Fluid Mech.JFLSA70022-112010.1017/ S0022112098001323 365, 269 (1998).
28. F. Waleffe, "Exact coherent structures in channel flow," J. Fluid Mech.JFLSA70022-112010.1017/S0022112001004189 435, 93 (2001).
29. S. Toh and T. Itano, "A periodic-like solution in channel flow," J. Fluid Mech.JFLSA70022-112010.1017/S0022112003003768 481, 67 (2003).
30. H. Wedin and R. R. Kerswell, "Exact coherent structures in pipe flow: Travelling wave solutions," J. Fluid Mech.JFLSA70022-112010.1017/S0022112004009346 508, 333 (2004).
31. V. G. Priymak and T. Miyazaki, "Direct numerical simulation of equilibrium spatially localized structures in pipe flow," Phys. FluidsPHFLE61070-663110.1063/1.1804549 16, 4221 (2004).
32. J. Luo and X. Wu, "Influence of small imperfections on the stability of plane Poiseuille flow: A theoretical model and direct numerical simulation," Phys. FluidsPHFLE61070-663110.1063/1.1760100 16, 2852 (2004).
33. P. J. Schmid and D. Henningson, "A new mechanism for rapid transition involving a pair of oblique waves," Phys. Fluids APFADEB0899-821310.1063/1.858367 4, 1986 (1992).
34. P. A. Elofsson and P. H. Alfredsson, "An experimental study of oblique transition in plane Poiseuille flow," J. Fluid Mech.JFLSA70022-112010.1017/S0022112097008288 358, 177 (1998).
35. D. J. Benney and K. Chow, "A mean flow first harmonic theory for hydrodynamic instabilities," Stud. Appl. Math.SAPMB60022-2526 80, 37-73 (1989).
36. M. E. Goldstein and S.-W. Choi, "Nonlinear evolution of interacting oblique waves on two-dimensional shear layers," J. Fluid Mech.JFLSA70022-1120 207, 97 (1989); M. E. Goldstein and S.-W. Choi,corrigendum, J. Fluid Mech.JFLSA70022-1120 216, 659 (1990).
37. X. Wu, S. S. Lee, and S. J. Cowley, "On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: The Stokes layer as a paradigm," J. Fluid Mech.JFLSA70022-1120 253, 681 (1993).
38. P. Hall and F. T. Smith, "On strongly nonlinear vortex/wave interaction in boundary-layer transition," J. Fluid Mech.JFLSA70022-1120 227, 641 (1991).
39. F. T. Smith and P. Blennerhassett, "Nonlinear interaction of oblique three-dimensional Tollmien-Schlichting waves and longitudinal vortices, in channel flows and boundary layers," Proc. R. Soc. London, Ser. APRLAAZ1364-5021 436, 585 (1992).
40. J. Floryan, J. Szumbarski, and X. Wu, "Stability of a channel flow with vibrating walls," Phys. FluidsPHFLE61070-663110.1063/1.1511545 14, 3927 (2002).
41. H. Zhou, "A necessary modification for the amplitude equation of the weakly nonlinear theory of hydrodynamic instability," Chin. Sci. Bull.CSBUEF1001-6538 36, 373 (1991).
42. H. Zhou and X. You, "On problems in the weakly nonlinear theory of hydrodynamic stability and its improvement," Acta Mech. Sin.LHHPAE0459-1879 9, 1 (1993).
43. J. T. Stuart, "The production of intense shear layer by vortex stretching and convection," ARGARD Report No. 514 (1965).
44. M. M. Landahl, "Wave breakdown and turbulence," SIAM J. Appl. Math.SMJMAP0036-1399 26, 735 (1975).