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On a Rigorous Resolvent Estimate for Plane Couette Flow
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
2007 (Engelska)Ingår i: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 9, nr 2, s. 153-180Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2007. Vol. 9, nr 2, s. 153-180
Nyckelord [en]
Hydrodynamical stability; Couette flow; resolvent estimate;
Nationell ämneskategori
Strömningsmekanik och akustik
Identifikatorer
URN: urn:nbn:se:kth:diva-7073DOI: 10.1007/s00021-005-0194-2ISI: 000247204500002Scopus ID: 2-s2.0-34250202869OAI: oai:DiVA.org:kth-7073DiVA, id: diva2:11976
Anmärkning
Tidigare titel: A rigorous resolvent estimate for plane Couette flow. Titel ändrad 20100825.Tillgänglig från: 2007-05-11 Skapad: 2007-05-11 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
Ingår i avhandling
1. Stability of plane Couette flow and pipe Poiseuille flow
Öppna denna publikation i ny flik eller fönster >>Stability of plane Couette flow and pipe Poiseuille flow
2007 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2007. s. ix, 31
Serie
Trita-CSC-A, ISSN 1653-5723 ; 2007:07
Nyckelord
Hydrodynamic stability, plane Couette flow, pipe Poiseuille flow, direct numerical simulations, resolvent
Nationell ämneskategori
Strömningsmekanik och akustik
Identifikatorer
urn:nbn:se:kth:diva-4368 (URN)978-91-7178-651-7 (ISBN)
Disputation
2007-05-25, D3, Huvudbyggnaden, Lindstedtsv 3, KTH, 10:15
Opponent
Handledare
Anmärkning
QC 20100825Tillgänglig från: 2007-05-11 Skapad: 2007-05-11 Senast uppdaterad: 2010-08-25Bibliografiskt granskad

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Åsén, Per-OlovKreiss, Gunilla
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