Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances

Itoh, Nobutake

doi:10.1017/s0022112077000780

Cited by 37 publications

(25 citation statements)

References 14 publications

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“…All these authors used Landau theory. Itoh, how ever, suggested that the application of Landau theory to pipe flow might not be justified since Landau theo ry needs the vicinity of a linear instability [9], Later on Patera and Orszag [10] confirmed Itoh's suggestion by numerical simulations, and so do we: There is no non-linear instability of the kind imagined by Davey and Nguyen. Another interesting approach was the application of non-linear boundary-layer theory by Smith and Bodonyi [11], In contrast to Davey and Nguyen [7], these authors found that the onset of turbulence should be caused by a non-axisymmetric disturbance, and so do we (Sect.…”

Section: Even the Most Ordinary Things Are Not Understoodmentioning

confidence: 77%

“…This is a stringent test since also an analytical formula (A.12) is available for these inte grals. Numerical and analytical results coincide only if the checked functions satisfy at the same time differen tial equation (9) and boundary conditions (5-6).…”

Section: Numerics and Checksmentioning

confidence: 93%

“…With this normalization we obtain from (9) and from the rules of field theory another orthogonality j w* n>v dr = y.2 ößV, (12) (pipe) namely for the 'Stokes vorticities' wv(r): = Fx sv ,(r). The simultaneous validity of both relations, (11) and (12), eased all further work considerably.…”

Section: Derivation Of the Galerkin Equationsmentioning

confidence: 99%

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Onset of Turbulence in a Pipe

Boberg

Brösa

1988

Zeitschrift Für Naturforschung A

202

167

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Turbulence in a pipe is derived directly from the Navier-Stokes equation. Analysis of numerical simulations revealed that small disturbances called 'mothers' induce other much stronger distur bances called 'daughters'. Daughters determine the look of turbulence, while mothers control the transfer of energy from the basic flow to the turbulent motion. From a practical point of view, ruling mothers means ruling turbulence. For theory, the mother-daughter process represents a mechanism permitting chaotic motion in a linearly stable system. The mechanism relies on a property of the linearized problem according to which the eigenfunctions become more and more collinear as the Reynolds number increases. The mathematical methods are described, comparisons with experi ments are made, mothers and daughters are analyzed, also graphically, with full particulars, and the systematic construction of small systems of differential equations to mimic the non-linear process by means as simple as possible is explained. We suggest that more then 20 but less than 180 essential degrees of freedom take part in the onset of turbulence.PACS number: 47.25. Ae O n s e t o f T u rb u le n c e in a P ip e

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Section: Even the Most Ordinary Things Are Not Understoodmentioning

confidence: 77%

Section: Numerics and Checksmentioning

confidence: 93%

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Onset of Turbulence in a Pipe

Boberg

Brösa

1988

Zeitschrift Für Naturforschung A

202

167

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“…This means that in PCF two-dimensional waves with k ≈ 0.13( Re ) 1/2 are the most unstable. Later Davey (1978) found that the results of his paper with Nguyen relating to disturbances in PCF are very close to those which follow from the application to the same problem of another method of the same type proposed by Itoh (1977b). More detailed calculations of the values of A e (k, Re) and E e (k, Re) for numerous values of the arguments (k, Re), also based on a version of the ReynoldsPotter method, were made by Coffee (1977), whose results agree satisfactorily with earlier estimates by Ellingsen et al and Davey and Nguyen.…”

Section: Plane Couette and Circular Poiseuille Flowsmentioning

confidence: 60%

Stability to Finite Disturbances: Energy Method and Landau’s Equation

Yaglom

Frisch

2012

Fluid Mechanics and Its Applications

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The main part of Chap. 2 and the whole of Chap. 3 were devoted to topics of linear stability theory dealing with the evolution of very small flow disturbances satisfying the linearized fluid dynamics equations. In Chap. 2 it was shown that the classical normal-mode method of the linear theory of hydrodynamic stability often leads to results which strongly disagree with experimental data. It was also indicated there that these disagreements are apparently due to nonlinear effects, which make linearization of the equations of motion physically unjustified. In Chap. 3 it was explained that the necessity for consideration of the full nonlinear dynamic equations often follows from the fact that many solutions of the initial-value problems for linearized fluid dynamics equations grow considerably at small and moderate values of the time t even in the cases when the normal-mode analysis shows that these solutions decay asymptotically (i.e. at t → ∞).The nonlinear theory of hydrodynamic stability has achieved a high level of development. Although the theory is still far from being completed, it has elucidated many formerly mysterious properties of fluid flows which are interesting for physicists and important for engineers. There is now an enormous literature on this subject and only a small part of it, dealing with relatively simple flows of incompressible fluids, will be considered in this book. In the present Chapter two topics from the nonlinear stability theory will be discussed: the energy method of stability analysis (short introductory consideration of this method was included in Sect. 3.4 above) and Landau's approach to the weakly nonlinear stability theory which described the initial period of the nonlinear development of flow disturbances.

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“…The amplitude equation (2.44) is then in fact, when A u, exactly that which can be derived using multiple scale methods (Davey and Nguyen (1971)), although the correctness of such equations is in doubt (Itoh (1977), Davey (1978)). However, it does seem likely that some such unstable solution branch bifurcates from infinity, as is suggested by the work of Rosenblat and Davis (1979) and Smith and Bodonyi (1982).…”

Section: Closure Assumptionsmentioning

confidence: 90%

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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Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances

Cited by 37 publications

References 14 publications

Onset of Turbulence in a Pipe

Onset of Turbulence in a Pipe

Stability to Finite Disturbances: Energy Method and Landau’s Equation

Intermittency in the Transition to Turbulence

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