Subcritical Transition to Turbulence in Plane Channel Flows

Orszag, Steven A.; Patera, Anthony T.

doi:10.1103/physrevlett.45.989

Cited by 129 publications

(67 citation statements)

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“…Based, in part, on pioneering attempts at nonlinear theories e.g., Benney and Lin, 1960;Craik, 1971, Orszag and Patera 1980, 1983 by n umerical experiments and Herbert 1984 by theory discovered a secondary instability mechanism. Herbert's 1984 theory is based on Floquet theory and accounts for an experimentally observed three-dimensional 3-D instability.…”

Section: Introductionmentioning

confidence: 99%

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

Joslin

Streett

Chang

1993

Theoret. Comput. Fluid Dynamics

168

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Abstract. A study of instabilities in incompressible boundary-layer ow on a at plate is conducted by spatial direct numerical simulation DNS of the Navier-Stokes equations. Here, the DNS results are used to critically evaluate the results obtained using parabolized stability equations PSE theory and to study mechanisms associated with breakdown from laminar to turbulent o w. Three test cases are considered: two-dimensional TollmienSchlichting wave propagation, subharmonic instability breakdown, and oblique-wave breakdown. The instability modes predicted by PSE theory are in good quantitative agreement with the DNS results, except a small discrepancy is evident in the mean-ow distortion component of the 2-D test problem. This discrepancy is attributed to far-eld boundarycondition di erences. Both DNS and PSE theory results show several modal discrepancies when compared with the experiments of subharmonic breakdown. Computations that allow for a small adverse pressure gradient in the basic ow and a variation of the disturbance frequency result in better agreement with the experiments.

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Section: Introductionmentioning

confidence: 99%

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

Joslin

Streett

Chang

1993

Theoret. Comput. Fluid Dynamics

168

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“…A natural question is the significance of the critical Reynolds number Re * = 5,772 derived from linear stability analysis (21). This is the point at which the laminar flow loses linear stability, so that arbitrary perturbation will drive it to turbulence.…”

Section: Resultsmentioning

confidence: 99%

“…It is known that in 2D channel flow, there exist secondary flows in the form of traveling waves (20,21). The 2D secondary flows were discovered in the range Re ≥ 2,750 for the length L ≥ 6π (22).…”

Section: Two-dimensional Channel Flowmentioning

confidence: 99%

“…The flow is driven by a constant average pressure gradient f = ðf x , f y Þ = ð2=Re, 0Þ. The flow possesses a laminar state characterized by a parabolic velocity profile u P = 1 − y 2 (20)(21)(22)(23)(24). Although linear stability theory gives a critical Reynolds number Re * = 5,772 above which the laminar solution becomes linearly unstable, there are also stable traveling solutions at subcritical Reynolds numbers Re < Re * (21).…”

Section: Two-dimensional Channel Flowmentioning

confidence: 99%

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Study of the instability of the Poiseuille flow using a thermodynamic formalism

Wang

Weinan

2015

Proc. Natl. Acad. Sci. U.S.A.

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The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic NavierStokes equation with Gaussian random initial data. A unique critical Reynolds number, Re c ≈ 2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.Poiseuille flow | subcritical transition | phase transition | statistical mechanics | free energy T he instability of shear flows, of which the Poiseuille flow is a canonical example, is among the most classical and most challenging problems in fluid mechanics, and a huge amount of effort has been devoted to it (1-13). The most definitive advance has been the recent experimental work by Avila et al. (9): By measuring the puff decaying and splitting times, they obtained an estimate for the critical Reynolds number, at around 2,040, for the 3D pipe flow. On the theoretical side, although the normal mode analysis in the linear stability theory of Poiseuille flow is regarded as being among the most important chapters of theoretical fluid dynamics and is associated with the names of OrrSommerfeld, Heisenberg, C. C. Lin, etc., it is generally accepted that normal mode analysis is insufficient in describing the instability of Poiseuille flow. If anything, Poiseuille flow should be regarded as an example of subcritical instability, perhaps the most well-known one.Currently, we still lack theoretical tools for tackling such instabilities. One interesting proposal has been to draw an analogy with phase transitions (13-18): The bifurcation diagram for subcritical instabilities resembles the phase diagram for firstorder phase transitions. However, to develop some quantitative tools out of this analogy, one also faces some serious issues. The first is that shear flow is well known to be a nonconservative and nonequilibrium system. More importantly, what we are interested in is the relative stability, i.e., the basins of attraction for laminar and turbulent states. This is opposite to the ergodic hypothesis that is the cornerstone of statistical mechanics. In fact, at least for finite-size systems undergoing subcritical instabilities, the complication arises precisely because the system is not ergodic. This casts serious doubt on the analogy with phase transitions in statistical mechanics.While philosophical issues exist, we p...

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“…The constant k should represent the idea that the laminar-turbulent transition is associated with a rapid inviscid three-dimensional instability of slowly decaying twodimensional disturbances (Orszag and Patera (1980)). This suggests that t ∼ d/u ∼ 1/kA 2 u, and it is thus suitable to choose (for example)…”

Section: Closure Assumptionsmentioning

confidence: 99%

Intermittency in the Transition to Turbulence

Fowler¹,

Howell²

2003

SIAM J. Appl. Math.

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Abstract.It is commonly known that the intermittent transition from laminar to turbulent flow in pipes occurs because, at intermediate values of a prescribed pressure drop, a purely laminar flow offers too little resistance, but a fully turbulent one offers too much. We propose a phenomenological model of the flow, which is able to explain this in a quantitative way through a hysteretic transition between laminar and turbulent "states," characterized by a disturbance amplitude variable that satisfies a natural type of evolution equation. The form of this equation is motivated by physical observations and derived by an averaging procedure, and we show that it naturally predicts disturbances having the characteristics of slugs and puffs. The model predicts oscillations similar to those which occur in intermittency in pipe flow, but it also predicts that stationary "biphasic" states can occur in sufficiently short pipes.

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Subcritical Transition to Turbulence in Plane Channel Flows

Cited by 129 publications

References 10 publications

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

Study of the instability of the Poiseuille flow using a thermodynamic formalism

Intermittency in the Transition to Turbulence

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