Linear Analysis of the Currents in a Pipe

Brösa, U.

doi:10.1515/zna-1986-0909

Cited by 10 publications

(5 citation statements)

References 9 publications

(11 reference statements)

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“…We use here methods of the general theory of ordinary differential equations with case discussions to obtain our results. Let us mention that our results agree with the results of [6] ( [ 7 ] ) , where the Stokes functions were obtained by the use of a representation formula.…”

Section: Prefacesupporting

confidence: 91%

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The Eigenfunctions of the Stokes Operator in Special Domains. I

Rummler

1997

Z Angew Math Mech

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We consider the eigenvalue problem of the Stokes operator in a bounded cylindrical domain of ℝ3 with homogeneous Dirichlet boundary conditions on the curved part of the boundary and periodical conditions in the main stream direction. We deduce by separation the corresponding systems of ordinary differential equations and solve them explicitly looking for bounded, solenoidal vector fields fulfilling the boundary conditions. The investigation of possible cases yields either the explicit eigenfunctions and eigenvalues, or equations for the determination of the eigenvalues, and a general representation of the eigenfunctions.

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Section: Prefacesupporting

confidence: 91%

“…Under these assumptions from (5)- (7) we get the following system of ordinary differential equations, where the differentiations with respect to r are denoted by ' and ", respectively, and the factor exp(ikz1 + inq) is omitted:…”

Section: Introductionmentioning

confidence: 99%

The Eigenfunctions of the Stokes Operator in Special Domains. I

Rummler

1997

Z Angew Math Mech

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“…Phase space becomes more densely filled with the stable and unstable manifolds of the saddles so that trajectories trapped in this tangle need more and more time before they can escape and relax to the laminar profile. This is in line with previous observations in large systems [19] and pipe flow [29] that long lived transients can imitate a permanent turbulent state. Clearly a measure of the thickness of the repeller or an estimate of the life time of transients as a function of Reynolds number would be highly desirable but is presently beyond our numerical or analytical abilities.…”

supporting

confidence: 93%

Fractal Stability Border in Plane Couette Flow

1997

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We study the dynamics of localised perturbations in plane Couette flow with periodic lateral boundary conditions. For small Reynolds number and small amplitude of the initial state the perturbation decays on a viscous time scale $t \propto Re$. For Reynolds number larger than about 200, chaotic transients appear with life times longer than the viscous one. Depending on the type of the perturbation isolated initial conditions with infinite life time appear for Reynolds numbers larger than about 270--320. In this third regime, the life time as a function of Reynolds number and amplitude is fractal. These results suggest that in the transition region the turbulent dynamics is characterised by a chaotic repeller rather than an attractor.Comment: 4 pages, Latex, 4 eps-figures, submitted to Phys. Rev. Le

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“…The unusual properties of the transition to turbulence in pipe flow are causally connected to the fact that the parabolic profile is linearly stable against infinitesimal perturbations (Salwen et al 1980, Brosa 1986, Meseguer and Trefethen 2003. Experimentally, the laminar flow has been maintained for Reynolds numbers as high as 100 000 (Pfenniger 1961).…”

Section: Observations and Elementary Propertiesmentioning

confidence: 99%

Turbulence transition in pipe flow: some open questions

Eckhardt¹

2007

Nonlinearity

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The transition to turbulence in pipe flow is a long standing problem in fluid dynamics. In contrast to many other transitions it is not connected with linear instabilities of the laminar profile and hence follows a different route. Experimental and numerical studies within the last few years have revealed many unexpected connections to the nonlinear dynamics of strange saddles and have considerably improved our understanding of this transition. The text summarizes some of these insights and points to a few outstanding problems in areas where nonlinear dynamics can be expected to provide useful insights.

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Linear Analysis of the Currents in a Pipe

Cited by 10 publications

References 9 publications

The Eigenfunctions of the Stokes Operator in Special Domains. I

The Eigenfunctions of the Stokes Operator in Special Domains. I

Fractal Stability Border in Plane Couette Flow

Turbulence transition in pipe flow: some open questions

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