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Stability of plane Couette flow and pipe Poiseuille flow
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns the stability of plane Couette flow and pipe Poiseuille flow in three space dimensions. The mathematical model for both flows is the incompressible Navier--Stokes equations. Both analytical and numerical techniques are used.

We present new results for the resolvent corresponding to both flows. For plane Couette flow, analytical bounds on the resolvent have previously been derived in parts of the unstable half-plane. In the remaining part, only bounds based on numerical computations in an infinite parameter domain are available. Due to the need for truncation of this infinite parameter domain, these results are mathematically insufficient.

We obtain a new analytical bound on the resolvent at s=0 in all but a compact subset of the parameter domain. This is done by deriving approximate solutions of the Orr--Sommerfeld equation and bounding the errors made by the approximations. In the remaining compact set, we use standard numerical techniques to obtain a bound. Hence, this part of the proof is not rigorous in the mathematical sense.

In the thesis, we present a way of making also the numerical part of the proof rigorous. By using analytical techniques, we reduce the remaining compact subset of the parameter domain to a finite set of parameter values. In this set, we need to compute bounds on the solution of a boundary value problem. By using a validated numerical method, such bounds can be obtained. In the thesis, we investigate a validated numerical method for enclosing the solutions of boundary value problems.

For pipe Poiseuille flow, only numerical bounds on the resolvent have previously been derived. We present analytical bounds in parts of the unstable half-plane. Also, we derive a bound on the resolvent for certain perturbations. Especially, the bound is valid for the perturbation which numerical computations indicate to be the perturbation which exhibits largest transient growth. The bound is valid in the entire unstable half-plane.

We also investigate the stability of pipe Poiseuille flow by direct numerical simulations. Especially, we consider a disturbance which experiments have shown is efficient in triggering turbulence. The disturbance is in the form of blowing and suction in two small holes. Our results show the formation of hairpin vortices shortly after the disturbance. Initially, the hairpins form a localized packet of hairpins as they are advected downstream. After approximately $10$ pipe diameters from the disturbance origin, the flow becomes severely disordered. Our results show good agreement with the experimental results.

In order to perform direct numerical simulations of disturbances which are highly localized in space, parallel computers must be used. Also, direct numerical simulations require the use of numerical methods of high order of accuracy. Many such methods have a global data dependency, making parallelization difficult. In this thesis, we also present the process of parallelizing a code for direct numerical simulations of pipe Poiseuille flow for a distributed memory computer.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , ix, 31 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2007:07
Keyword [en]
Hydrodynamic stability, plane Couette flow, pipe Poiseuille flow, direct numerical simulations, resolvent
National Category
Fluid Mechanics and Acoustics
Identifiers
URN: urn:nbn:se:kth:diva-4368ISBN: 978-91-7178-651-7 (print)OAI: oai:DiVA.org:kth-4368DiVA: diva2:11981
Public defence
2007-05-25, D3, Huvudbyggnaden, Lindstedtsv 3, KTH, 10:15
Opponent
Supervisors
Note
QC 20100825Available from: 2007-05-11 Created: 2007-05-11 Last updated: 2010-08-25Bibliographically approved
List of papers
1. On a Rigorous Resolvent Estimate for Plane Couette Flow
Open this publication in new window or tab >>On a Rigorous Resolvent Estimate for Plane Couette Flow
2007 (English)In: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 9, no 2, 153-180 p.Article in journal (Refereed) Published
Abstract [en]

We derive a rigorous bound of the solution of the resolvent equation for plane Couette flow in three space dimensions. We combine analytical techniques with numerical computations. Compared to earlier results, our analytical techniques cover a larger part of the parameter domain consisting of wave numbers in two space directions and the Reynolds number. Numerical computations are needed only in a compact subset of the parameter domain.

Keyword
Hydrodynamical stability; Couette flow; resolvent estimate;
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-7073 (URN)10.1007/s00021-005-0194-2 (DOI)000247204500002 ()2-s2.0-34250202869 (Scopus ID)
Note
Tidigare titel: A rigorous resolvent estimate for plane Couette flow. Titel ändrad 20100825.Available from: 2007-05-11 Created: 2007-05-11 Last updated: 2017-12-14Bibliographically approved
2. On a computer-assisted method for proving existence of solutions of boundary value problems
Open this publication in new window or tab >>On a computer-assisted method for proving existence of solutions of boundary value problems
(English)Manuscript (Other academic)
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-7074 (URN)
Note
QC 20100825Available from: 2007-05-11 Created: 2007-05-11 Last updated: 2010-08-25Bibliographically approved
3. Resolvent bounds for pipe Poiseuille flow
Open this publication in new window or tab >>Resolvent bounds for pipe Poiseuille flow
2006 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 568, 451-471 p.Article in journal (Refereed) Published
Abstract [en]

We derive an analytical bound on the resolvent of pipe Poiseuille flow in large parts of the unstable half-plane. We also consider the linearized equations, Fourier transformed in axial and azimuthal directions. For certain combinations of the wavenumbers and the Reynolds number, we derive an analytical bound on the resolvent of the Fourier transformed problem. In particular, this bound is valid for the perturbation which numerical computations indicate to be the perturbation that gives the largest transient growth. Our bound has the same dependence on the Reynolds number as given by the computations.

Keyword
CHANNEL FLOWS; STABILITY; TRANSITION; GROWTH
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-7075 (URN)10.1017/S0022112006002618 (DOI)000242566600020 ()2-s2.0-33750981817 (Scopus ID)
Note
QC 20100825Available from: 2007-05-11 Created: 2007-05-11 Last updated: 2017-12-14Bibliographically approved
4. A parallel code for direct numerical simulations of pipe Poiseuille flow
Open this publication in new window or tab >>A parallel code for direct numerical simulations of pipe Poiseuille flow
(English)Manuscript (Other academic)
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-7076 (URN)
Note
QC 20100825Available from: 2007-05-11 Created: 2007-05-11 Last updated: 2010-08-25Bibliographically approved
5. Direct numerical simulations of localized disturbances in pipe Poiseuille flow
Open this publication in new window or tab >>Direct numerical simulations of localized disturbances in pipe Poiseuille flow
2010 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 39, no 6, 926-935 p.Article in journal (Refereed) Published
Abstract [en]

We consider pipe Poiseuille flow subjected to a disturbance which is highly localized in space. Experiments by Peixinho and Mullin have shown this disturbance to be efficient in triggering turbulence, yielding a threshold dependence on the required amplitude as R-1.5 on the Reynolds number, R. The experiments also indicate an initial formation of hairpin vortices, with each hairpin having a length of approximately one pipe radius, independent of the Reynolds number in the range of R = 2000-3000. We perform direct numerical simulations for R = 5000. The results show a packet of hairpin vortices traveling downstream, each having a length of approximately one pipe radius. The perturbation remains highly localized in space while being advected downstream for approximately 10 pipe diameters. Beyond that distance from the disturbance origin, the flow becomes severely disordered.

Keyword
Direct numerical simulations, Pipe Poiseuille flow, Hydrodynamical stability, Incompressible Navier-Stokes equations, CIRCULAR PIPE, SHEAR FLOWS, TRANSITION, STABILITY, INSTABILITY, THRESHOLDS, GROWTH, BOUNDS
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-7077 (URN)10.1016/j.compfluid.2009.09.016 (DOI)000277222100002 ()2-s2.0-77950865814 (Scopus ID)
Note
QC 20100825. Uppdaterad från Submitted till Published 20100825.Available from: 2007-05-11 Created: 2007-05-11 Last updated: 2017-12-14Bibliographically approved

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