Formation of solitons in transitional boundary layers: theory and experiment

Kachanov, Y. S.; Ryzhov, O. S.; Smith, F. T.

doi:10.1017/s0022112093003416

Cited by 64 publications

(47 citation statements)

References 35 publications

(62 reference statements)

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“…As it is reported in Kachanov et al (1993) the theory describes well quantitatively the process of spike-soliton formation through a definite stage of the nonlinear disturbance development. It is remarkable that this stage extends in the experiment up to x· :;:,j 410 -420mm, where the oscillation swing reaches values of 25 -30%.…”

Section: Benjamin-dno Equationsupporting

confidence: 63%

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The Development of Nonlinear Oscillations in a Boundary Layer and the Onset of Random Disturbances

Ryzhov

1994

Nonlinear Instability of Nonparallel Flows

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A theoretical approach is discussed to elucidate the appearance of spikes in oscilloscope traces featuring an essentially nonlinear stage of TollmienSchlichting wave propagation. On the assumption that the Reynolds number is large enough an asymptotic analysis of the N avier-Stokes equations reveals a complicated four-deck oscillation pattern with a relatively simple mathematical description of each sublayer. The leading role is played by one of the two intermediate regions where disturbances are of inviscid nature giving rise to the Benjamin-Ono integral-differential equation and algebraic solitons. If vibrations of a plate itself are taken into consideration the governing equation becomes nonhomogeneous. It admits of a travelling wave-type solutions provided that the external forcing agency is of the same form.The forced Korteweg-de Vries model appears in a relevant asymptotic study of the boundary layer on a heated plate placed vertically into quiescent fluid in the gravity field. As a result, the simplest problem of the dynamical systems theory can be posed for travelling waves using the Hamiltonian formulation. The important point is that the separatrix in the phase plane is represented by a soliton of the original KdV equation. With the forcing term included, solitons lead to the stable and unstable manifolds with an ensuing homoclinic tangle in the Poincare map. The existence in the phase space of trajectories with unpredictable behaviour is a direct consequence of the study. '

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Section: Benjamin-dno Equationsupporting

confidence: 63%

“…The subject as a whole has been recently discussed from both theoretical and experimental viewpoints in Kachanov, Ryzhov & Smith (1993) under the same underlying assumption. Following the analysis presented in the latter paper we rewrite (3.2) as…”

Section: Benjamin-dno Equationmentioning

confidence: 99%

The Development of Nonlinear Oscillations in a Boundary Layer and the Onset of Random Disturbances

Ryzhov

1994

Nonlinear Instability of Nonparallel Flows

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“…A related point is that markedly different flow structures might be set up sufficiently far downstream of the initial disturbance as mentioned in Section 1, although there is little work in connecting these structures with an initial-value problem as here. Likewise, mention should be made of the near-planar case set up experimentally by Kachanov et al [51], the 2D inviscid theory of which is much easier to handle, but our emphasis is on the 3D nonlinear development which represents the more common case.…”

Section: Further Applications Comparisons and Commentsmentioning

confidence: 99%

On global and internal dynamics of spots: A theoretical approach

1994

Self Cite

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Abstract. Nonlinear effects on the free evolution of three-dimensional disturbances are discussed, these disturbances having a spot-like character sufficiently far downstream of the initial disturbance. The inviscid initial-value formulation taken involving the three-dimensional unsteady Euler equations offers hope of considerable analytical progress on the nonlinear side, as well as being suggested by some of the experimental evidence on turbulent spots and by engineering modelling and previous related theory. The large-time large-distance behaviour is associated with the two major length scales, proportional to (time) 1/2 and to (time), in the evolving spot; within the former scale the Euler flow exhibits a three-dimensional triple-deck-fike structure; within the latter scale, in contrast, there are additional time-independent scales in operation. As the typical disturbance amplitude increases, nonlinear effects first enter the reckoning in edge layers near the spot's wing-tips. The nonlinearity is mostly due to interplay between the fluctuations present and the three-dimensional mean-flow correction which varies relatively slowly. The resulting amplitude interaction points to a subsequent flooding of nonlinear effects into the middle of the spot. There it is suggested that the fluctuation/mean-flow interaction becomes strongly nonlinear, substantially altering the mean properties in particular. A new global viscous-inviscid interaction between the short and long scales present, involving Reynolds stresses, is also identified. The additional significance of viscous sublayer bursts is also noted, along with comments on links with experiments and direct numerical simulations, on channel flows and jets, and on further research.

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“…In general, this would involve a matching between the possibly diverging interior solutions and the critical layer dynamics (see the review by Maslowe 1986 for instance). We note, however, that for certain two-dimensional boundary layer flows, the leading-order dynamics is not affected by the detailed critical layer dynamics (see Kachanov, Ryzhov & Smith 1993 for instance). A similar conclusion was reached by Shrira (1989) and Voronovich, Shrira & Stepanyants (1998) for certain classes of free surface flows.…”

Section: Introductionmentioning

confidence: 86%

On vorticity waves propagating in a waveguide formed by two critical layers

Derzho

Grimshaw

2009

J. Fluid Mech.

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A theoretical model for long vorticity waves propagating on a background shear flow is developed. The basic flow is assumed to be confined between two critical layers, respectively, located near the lower and upper rigid boundaries. In these critical layers even small disturbances will break, and eventually a thin zone of mixed fluid will appear. We derive a nonlinear evolution equation for the amplitude of a wave-like disturbance in this configuration, based on the assumption that the critical layers are replaced by thin recirculation zones attached to the lower and upper rigid boundaries, where the flow is very weak. The dispersive and time-evolution terms in this equation are typical for Korteweg-de Vries theory, but the nonlinear term is more complicated. It comprises nonlinearity associated with the shear across the waveguide, and the nonlinearity due to the flow over the recirculation zones. The coefficient of the quadratic nonlinear term may change sign, depending on the presence or otherwise of recirculation zones at the upper or lower boundary of the waveguide. We then seek steady travelling wave solutions, and show that there are no such steady solutions if the waveguide contains no density stratification. However, steady solutions including solitary waves and bores can exist if the fluid between the critical layers is weakly density stratified.

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Formation of solitons in transitional boundary layers: theory and experiment

Cited by 64 publications

References 35 publications

The Development of Nonlinear Oscillations in a Boundary Layer and the Onset of Random Disturbances

The Development of Nonlinear Oscillations in a Boundary Layer and the Onset of Random Disturbances

On global and internal dynamics of spots: A theoretical approach

On vorticity waves propagating in a waveguide formed by two critical layers

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