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Studies on instability and optimal forcing of incompressible flows
KTH, School of Engineering Sciences (SCI), Mechanics, Stability, Transition and Control. KTH, Centres, SeRC - Swedish e-Science Research Centre. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0001-9446-7477
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers the hydrodynamic instability and optimal forcing of a number of incompressible flow cases. In the first part, the instabilities of three problems that are of great interest in energy and aerospace applications are studied, namely a Blasius boundary layer subject to localized wall-suction, a Falkner–Skan–Cooke boundary layer with a localized surface roughness, and a pair of helical vortices. The two boundary layer flows are studied through spectral element simulations and eigenvalue computations, which enable their long-term behavior as well as the mechanisms causing transition to be determined. The emergence of transition in these cases is found to originate from a linear flow instability, but whereas the onset of this instability in the Blasius flow can be associated with a localized region in the vicinity of the suction orifice, the instability in the Falkner–Skan–Cooke flow involves the entire flow field. Due to this difference, the results of the eigenvalue analysis in the former case are found to be robust with respect to numerical parameters and domain size, whereas the results in the latter case exhibit an extreme sensitivity that prevents domain independent critical parameters from being determined. The instability of the two helices is primarily addressed through experiments and analytic theory. It is shown that the well known pairing instability of neighboring vortex filaments is responsible for transition, and careful measurements enable growth rates of the instabilities to be obtained that are in close agreement with theoretical predictions. Using the experimental baseflow data, a successful attempt is subsequently also made to reproduce this experiment numerically.

In the second part of the thesis, a novel method for computing the optimal forcing of a dynamical system is developed. The method is based on an application of the inverse power method preconditioned by the Laplace preconditioner to the direct and adjoint resolvent operators. The method is analyzed for the Ginzburg–Landau equation and afterwards the Navier–Stokes equations, where it is implemented in the spectral element method and validated on the two-dimensional lid-driven cavity flow and the flow around a cylinder.

Place, publisher, year, edition, pages
Stockholm, Sweden: KTH Royal Institute of Technology, 2017. , p. 47
Series
TRITA-MEK, ISSN 0348-467X ; 2017:19
Keyword [en]
hydrodynamic stability, optimal forcing, resolvent operator, Laplace preconditioner, spectral element method, eigenvalue problems, inverse power method, direct numerical simulations, Falkner–Skan–Cooke boundary layer, localized roughness, crossflow vortices, Blasius boundary layer, localized suction, helical vortices, lid-driven cavity, cylinder flow
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-218172ISBN: 978-91-7729-622-5 (print)OAI: oai:DiVA.org:kth-218172DiVA, id: diva2:1159852
Public defence
2017-12-14, D3, Lindstedtsvägen 5, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20171124

Available from: 2017-11-24 Created: 2017-11-23 Last updated: 2017-11-27Bibliographically approved
List of papers
1. On the stability of a Blasius boundary layer subject to localized suction
Open this publication in new window or tab >>On the stability of a Blasius boundary layer subject to localized suction
2017 (English)Report (Other academic)
Abstract [en]

In this work the problem of premature transition in boundary layers due to localized suction is revisited. A thorough study involving nonlinear direct numerical simulations, a three-dimensional linear stability analysis, a sensitivity study and a Koopman analysis is presented. The ensemble of these different techniques enables the origins of oversuction to be studied in great detail and provides new insight into the transition process of the flow. The configuration considered consists of an infinite row of widely separated suction pipes that are mounted to the plate at right angles. For the parameter range investigated, the flow inside the pipe is seen to bifurcate at a lower suction ratio than the boundary layer and thus act as an oscillator that forces the external flow over the plate. At low levels of suction, this forcing is not enough to cause transition in the boundary layer, but as the suction level is increased beyond criticality, modes originating from the pipe and extending into the boundary layer are seen to destabilize as well. These modes enable the perturbations forced in the pipe to also amplify in the boundary layer, which leads to a rapid breakdown to turbulence in the wake of the suction hole.

Publisher
p. 25
Keyword
absolute/convective instability, boundary layer stability, transition to turbulence
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-218167 (URN)
Note

QC 20171124

Available from: 2017-11-23 Created: 2017-11-23 Last updated: 2017-11-24Bibliographically approved
2. Stability and sensitivity of a cross-flow-dominated Falkner-Skan-Cooke boundary layer with discrete surface roughness
Open this publication in new window or tab >>Stability and sensitivity of a cross-flow-dominated Falkner-Skan-Cooke boundary layer with discrete surface roughness
Show others...
2017 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 826, p. 830-850Article in journal (Refereed) Published
Abstract [en]

With the motivation of determining the critical roughness size, a global stability and sensitivity analysis of a three-dimensional Falkner-Skan-Cooke (FSC) boundary layer with a cylindrical surface roughness is performed. The roughness size is chosen such that breakdown to turbulence is initiated by a global version of traditional secondary instabilities of the cross-flow (CF) vortices instead of an immediate flow tripping at the roughness. The resulting global eigenvalue spectra of the systems are found to be very sensitive to numerical parameters and domain size. This sensitivity to numerical parameters is quantified using the epsilon-pseudospectrum, and the dependency on the domain is analysed through an impulse response, structural sensitivity analysis and an energy budget. It is shown that while the frequencies remain relatively unchanged, the growth rates increase with domain size, which originates from the inclusion of stronger CF vortices in the baseflow. This is reflected in a change in the rate of advective energy transport by the baseflow. It is concluded that the onset of global instability in a FSC boundary layer as the roughness height is increased does not correspond to an immediate flow tripping behind the roughness, but occurs for lower roughness heights if sufficiently long domains are considered. However, the great sensitivity results in an inability to accurately pinpoint the exact parameter values for the bifurcation, and the large spatial growth of the disturbances in the long domains eventually becomes larger than can be resolved using finite-precision arithmetic.

Place, publisher, year, edition, pages
Cambridge University Press, 2017
Keyword
absolute/convective instability, boundary layer stability, transition to turbulence
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:kth:diva-214322 (URN)10.1017/jfm.2017.466 (DOI)000407571200038 ()2-s2.0-85029412275 (Scopus ID)
Funder
Swedish e‐Science Research Center
Note

QC 20170914

Available from: 2017-09-14 Created: 2017-09-14 Last updated: 2017-11-23Bibliographically approved
3. Long-wave instabilities of two interlaced helical vortices
Open this publication in new window or tab >>Long-wave instabilities of two interlaced helical vortices
2017 (English)Manuscript (preprint) (Other academic)
Abstract [en]

We investigate theoretically and experimentally the stability of two interlaced helical vortices with respect to displacement perturbations having wavelengths that are large compared to the size of the vortex cores. First, existing theoretical results are recalled and applied to the present configuration. Various modes of unstable perturbations, involving different phase relationships between the two vortices, are identified and their growth rates are calculated. They lead to a local pairing of neighbouring helix loops, or to a uniform pairing with one helix expanding and the other one contracting. A relation is established between this instability and the three-dimensional pairing of arrays of straight parallel vortices, and a striking quantitative agreement concerning the growth rates is found, showing that the local pairing of vortices is the driving mechanism behind the instability of the helix system. Second, an experimental study designed to observe these instabilities in a real flow is presented. Two helical vortices are generated by a two-bladed rotor in a water channel and characterised through dye visualisations and PIV measurements. Unstable displacement modes are triggered individually, either by varying the rotation frequency of the rotor, or by imposing a small rotor eccentricity. The observed unstable mode structure, and the corresponding growth rates obtained from advanced processing of visualisation sequences, are in good agreement with theoretical predictions. The non-linear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent break-up of the vortices, uniform pairing results in a leapfrogging phenomenon, which intermittently restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.

Publisher
p. 34
Keyword
vortex flows, vortex instability, vortex interactions
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-218170 (URN)
Note

QC 20171124

Available from: 2017-11-23 Created: 2017-11-23 Last updated: 2017-11-24Bibliographically approved
4. A note on the numerical realization of helical vortices: application to vortex instability
Open this publication in new window or tab >>A note on the numerical realization of helical vortices: application to vortex instability
2017 (English)Report (Other academic)
Abstract [en]

The need to numerically represent a free vortex system arises frequently in fundamental and applied research. Many possible techniques for realizing this vortex system exist but most tend to prioritize accuracy either inside or outside of the vortex core, which therefore makes them unsuitable to for a stability analysis considering the entire flow field. In this article, a simple method is presented that is shown to yield an accurate representation of the flow inside and outside of the vortex core. The method is readily implemented in any incompressible Navier–Stokes solver using primitive variables and Cartesian coordinates. It can potentially be used to model a wide range of vortices but is here applied to reproduce a recent experiment by Quaranta et al. (2017) considering two helices. A three-dimensional stability analysis is performed and yields an eigenvalue spectrum that features both long- and short-wave instabilities.

Publisher
p. 19
Keyword
vortex dynamics, vortex instability
National Category
Fluid Mechanics and Acoustics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-218169 (URN)
Funder
Swedish Research Council
Note

QC 20171124

Available from: 2017-11-23 Created: 2017-11-23 Last updated: 2017-11-24Bibliographically approved
5. Computing Optimal Forcing Using Laplace Preconditioning
Open this publication in new window or tab >>Computing Optimal Forcing Using Laplace Preconditioning
2017 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 22, no 5, p. 1508-1532Article in journal (Refereed) Published
Abstract [en]

For problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on timestepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing timestepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential timestepping.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-214476 (URN)10.4208/cicp.OA-2016-0070 (DOI)000408436300012 ()
Note

QC 20171011

Available from: 2017-10-11 Created: 2017-10-11 Last updated: 2017-11-23Bibliographically approved

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