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Modal instability of the flow in a toroidal pipe
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0003-3211-4347
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0001-9627-5903
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0002-1663-3553
2016 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 792, 894-909 p.Article in journal (Refereed) Published
Resource type
Text
Abstract [en]

The modal instability encountered by the incompressible flow inside a toroidal pipe is studied, for the first time, by means of linear stability analysis and direct numerical simulation (DNS). In addition to the unquestionable aesthetic appeal, the torus represents the smallest departure from the canonical straight pipe flow, at least for low curvatures. The flow is governed by only two parameters: the Reynolds number (Formula presented.) and the curvature of the torus (Formula presented.), i.e. the ratio between pipe radius and torus radius. The absence of additional features, such as torsion in the case of a helical pipe, allows us to isolate the effect that the curvature has on the onset of the instability. Results show that the flow is linearly unstable for all curvatures investigated between 0.002 and unity, and undergoes a Hopf bifurcation at (Formula presented.) of about 4000. The bifurcation is followed by the onset of a periodic regime, characterised by travelling waves with wavelength (Formula presented.) pipe diameters. The neutral curve associated with the instability is traced in parameter space by means of a novel continuation algorithm. Tracking the bifurcation provides a complete description of the modal onset of instability as a function of the two governing parameters, and allows a precise calculation of the critical values of (Formula presented.) and (Formula presented.). Several different modes are found, with differing properties and eigenfunction shapes. Some eigenmodes are observed to belong to groups with a set of common characteristics, deemed ‘families’, while others appear as ‘isolated’. Comparison with nonlinear DNS shows excellent agreement, confirming every aspect of the linear analysis, its accuracy, and proving its significance for the nonlinear flow. Experimental data from the literature are also shown to be in considerable agreement with the present results.

Place, publisher, year, edition, pages
Cambridge University Press, 2016. Vol. 792, 894-909 p.
Keyword [en]
bifurcation, instability, nonlinear dynamical systems
National Category
Fluid Mechanics and Acoustics
Identifiers
URN: urn:nbn:se:kth:diva-187267DOI: 10.1017/jfm.2016.104ISI: 000379218400003Scopus ID: 2-s2.0-84960155237OAI: oai:DiVA.org:kth-187267DiVA: diva2:929738
Note

QC 20160519

Available from: 2016-05-19 Created: 2016-05-19 Last updated: 2016-10-03Bibliographically approved
In thesis
1. Numerical studies on flows with secondary motion
Open this publication in new window or tab >>Numerical studies on flows with secondary motion
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This work is concerned with the study of flow stability and turbulence control - two old but still open problems of fluid mechanics. The topics are distinct and are (currently) approached from different directions and with different strategies. This thesis reflects this diversity in subject with a difference in geometry and, consequently, flow structure: the first problem is approached in the study of the flow in a toroidal pipe, the second one in an attempt to reduce the drag in a turbulent channel flow.

The flow in a toroidal pipe is chosen as it represents the common asymptotic limit between spatially developing and helical pipes. Furthermore, the torus represents the smallest departure from the canonical straight pipe flow, at least for small curvatures. The interest in this geometry is twofold: it allows us to isolate the effect of the curvature on the flow and to approach straight as well as helical pipes. The analysis features a characterisation of the steady solution as a function of curvature and the Reynolds number. The problem of forcing fluid in the pipe is addressed, and the so-called Dean number is shown to be of little use, except for infinitesimally low curvatures. It is found that the flow is modally unstable and undergoes a Hopf bifurcation that leads to a limit cycle. The bifurcation and the corresponding eigenmodes are studied in detail, providing a complete picture of the instability.

The second part of the thesis approaches fluid mechanics from a different perspective: the Reynolds number is too high for a deterministic description and the flow is analysed with statistical tools. The objective is to reduce the friction exerted by a turbulent flow on the walls of a channel, and the idea is to employ a control strategy independent of the small, and Reynolds number-dependent, turbulent scales. The method of choice was proposed by Schoppa & Hussain [Phys. Fluids 10:1049-1051 (1998)] and consists in the imposition of streamwise invariant, large-scale vortices. The vortices are re-implemented as a volume force, validated and analysed. Results show that the original method only gave rise to transient drag reduction while the forcing version is capable of sustained drag reduction of up to 18%. An analysis of the method, though, reveals that its effectiveness decreases rapidly as the Reynolds number is increased.

Place, publisher, year, edition, pages
Stockholm: Kungliga Tekniska högskolan, 2016. 26 p.
Series
TRITA-MEK, ISSN 0348-467X ; 2016:16
Keyword
nonlinear dynamical systems, instability, bifurcation, flow control, skin-friction reduction
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-193537 (URN)978-91-7729-149-7 (ISBN)
Presentation
2016-10-28, D3, Lindstedtsvägen 5, Stockholm, 08:15 (English)
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Note

QC 20161004

Available from: 2016-10-04 Created: 2016-10-03 Last updated: 2016-10-04Bibliographically approved

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