The effect of a small initial distortion of the basic flow on the subcritical transition in plane Poiseuille flow

Author:
Pu Sun

Journal:
Quart. Appl. Math. **59** (2001), 667-699

MSC:
Primary 76E30; Secondary 35Q30, 76F06

DOI:
https://doi.org/10.1090/qam/1866553

MathSciNet review:
MR1866553

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Abstract: This paper presents an instability theory in which a mean flow and multiple wave interactions in the Poiseuille flow transition process are studied. It is shown that not only can this mean flow term come as the result of the Fourier decomposition of a general disturbance, it can also come as an exact solution to the unsteady Navier-Stokes equations. The presence of this term, though small, can produce totally different linear and nonlinear stability behavior for the flow at subcritical Reynolds numbers. In the linear stability case, with the presence of this mean flow perturbation term, the instabilities are obtained well below the critical value of 5772. When this mean flow term is introduced into the interactions with other harmonic perturbation waves, for the plane Poiseuille flow case with Reynolds numbers around 1200, the nonlinear interactions rapidly modify the total mean flow profile toward the mean flow profile observed in turbulence while the other two- and three-dimensional waves remain small. The initial energies needed to trigger the instabilities are much smaller than those reported by previous investigators. The intermittent character of the disturbance observed in transition experiments is also captured.

**[1]**M. Nishioka, S. Iida, and Y. Ichikawa,*An experimental investigation of the stability of plane Poiseuille flow*, J. Fluid Mech.**72**, 731-751 (1975)**[2]**L. Landau,*On the vibrations of the electronic plasma*, Acad. Sci. USSR. J. Phys.**10**(1946), 25–34. MR**0023765****[3]**D. Meksyn and J. T. Stuart,*Stability of viscous motion between parallel planes for finite disturbances*, Proc. Roy. Soc. London. Ser. A.**208**(1951), 517–526. MR**0046817**, https://doi.org/10.1098/rspa.1951.0177**[4]**J. T. Stuart,*On the non-linear mechanics of hydrodynamic stability*, J. Fluid Mech.**4**(1958), 1–21. MR**0093243**, https://doi.org/10.1017/S0022112058000276**[5]**J. T. Stuart,*On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. I. The basic behaviour in plane Poiseuille flow*, J. Fluid Mech.**9**(1960), 353–370. MR**0128228**, https://doi.org/10.1017/S002211206000116X**[6]**J. Watson,*On the non-linear mechanics of wave distrubances in stable and unstable parallel flows. II. The development of a solution for plane Poiseuille flow and for plane Couette flow*, J. Fluid Mech.**9**(1960), 371–389. MR**0128229**, https://doi.org/10.1017/S0022112060001171**[7]**W. C. Reynolds and M. C. Potter,*Finite-amplitude instability of parallel shear flows*, J. Fluid Mech.**27**, 465-492 (1967)**[8]**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.**29**, 31-38 (1967)**[9]**D. Grohne,*Die Stabilität der ebenen Kanalsträmung gegenuber dreidimensionalen Störungen von endlicher Amplitude*, AVA Göttingen Report, 69-A-30 (1969)**[10]**T. A. Zang and S. E. Krist,*Numerical experiments on stability and transition in plane channel flow*, Theoret. Comput. Fluid Dynamics**1**, 41-64 (1989)**[11]**T. Herbert,*Periodic Secondary Motions in a Plane Channel*, in Proc. Internat. Conf. Numer. Methods Fluid Dynamics, eds., A. I. van de Vooren and P. J. Zandbergen, Springer-Verlag, Berlin, 1976**[12]**T. Herbert,*Die neutrale Fläche der ebenen Poiseuille-Strömung*, Habiltationsschrift. Univ. Stuttgart, W. Germany, 1978**[13]**L. Wolf, Z. Lavan, and H. J. Nielsen,*Numerical computation of the stability of plane Poiseuille flow*, J. Appl. Mech.**45**, 13-18 (1978)**[14]**S. A. Orszag and A. T. Patera,*Subcritical transition to turbulence in plane channel flows*, Phys. Rev. Lett.**45**, 989-993 (1980)**[15]**L. Kleiser and U. Schumann,*Spectral simulations of the laminar-turbulent transition process in plane Poiseuille flow*, Spectral methods for partial differential equations (Hampton, Va., 1982) SIAM, Philadelphia, PA, 1984, pp. 141–164. MR**758265****[16]**S. A. Orszag and L. C. Kells,*Transition to turbulence in plane Poiseuille flow and plane Couette flow*, J. Fluid Mech.**96**, 159-205 (1980)**[17]**S. A. Orszag and A. T. Patera,*Secondary instability of wall-bounded shear flows*, J. Fluid Mech.**128**, 347-385 (1983)**[18]**Thorwald Herbert,*On perturbation methods in nonlinear stability theory*, J. Fluid Mech.**126**(1983), 167–186. MR**699550**, https://doi.org/10.1017/S0022112083000099**[19]**B. J. Bayly, S. A. Orszag, and T. Herbert,*Instability mechanisms in shear-flow transition*, Annual Reviews of Fluid Mech.**20**, 359-391 (1988)**[20]**David John Benney,*ON THE SECONDARY MOTION INDUCED BY OSCILLATIONS IN A SHEAR FLOW*, ProQuest LLC, Ann Arbor, MI, 1960. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR**2939165****[21]**P. S. Klebanoff, K. D. Tidstrom, and L. M. Sargent,*The three-dimensional nature of a boundarylayer instability*, J. Fluid Mech.**12**, 1-34 (1962)**[22]**D. J. Benney,*The evolution of disturbances in shear flows at high Reynolds numbers*, Stud. Appl. Math.**70**(1984), no. 1, 1–19. MR**728941**, https://doi.org/10.1002/sapm19847011**[23]**R. C. DiPrima and J. T. Stuart,*Hydrodynamic stability*, Trans. ASME Ser. E J. Appl. Mech.**50**(1983), no. 4, 983–991. MR**726555****[24]**M. Nishioka and M. Asai,*Some observations of the subcritical transition in plane Poiseuille flow*, J. Fluid Mech.**150**, 441-450 (1985)**[25]**O. Reynolds,*An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels*, Philos. Trans. Roy. Soc. London**174**, 935 (1883)

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DOI:
https://doi.org/10.1090/qam/1866553

Article copyright:
© Copyright 2001
American Mathematical Society