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Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0002-4346-4732
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0001-7864-3071
2010 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 650, 181-214 p.Article in journal (Refereed) Published
Abstract [en]

The global linear stability of the flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of an optimization technique. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimization problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearized Navier-Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. In addition, the use of matrix-free methods enables us to extend the present framework to any geometrical configuration. The optimal initial condition for spanwise wavelengths of the order of the boundary-layer thickness are finite-length streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths, it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. This mechanism is dominant for the long computational domain and thus for the relatively high Reynolds number considered here. Three-dimensional localized optimal initial conditions are also computed and the corresponding wave packets examined. For short optimization times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream direction without significant spreading in the lateral direction. For long optimization times, we find the optimal disturbances with the largest energy amplification. These are wave packets of Tollmien-Schlichting waves with low streamwise propagation speed and faster spreading in the spanwise direction. The pseudo-spectrum of the system for real frequencies is also computed with matrix-free methods. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the largest response to forcing is also associated with the Orr/oblique wave mechanism, however less so than in the case of the optimal initial condition. The lift-up mechanism is most efficient at zero frequency and degrades slowly for increasing frequencies. The response to localized upstream forcing is also discussed.

Place, publisher, year, edition, pages
2010. Vol. 650, 181-214 p.
Keyword [en]
Asymptotic response, Blasius, Computational domains, Disturbance growth, Eigen modes, Energy amplification, Flat plate, Flow perturbations, Geometrical configurations, High Reynolds number, Initial conditions, Lateral directions, Lift-up mechanism, Linear Stability, Linearized navier-stokes equations, matrix, Objective functions, Oblique wave, Optimal disturbances, Optimal time, Optimization problems, Optimization techniques, Periodic forcing, Propagation speed, Spatial structure, Streaky structure, Streamwise vortices, Tollmien-Schlichting waves, Unstable flows, Zero frequency, Amplification, Boundary layer flow, Boundary layers, Frequency response, Lagrange multipliers, Navier Stokes equations, Reynolds number, Three dimensional, Wave packets, Waves
National Category
Fluid Mechanics and Acoustics
Identifiers
URN: urn:nbn:se:kth:diva-10648DOI: 10.1017/S0022112009993703ISI: 000278212500007Scopus ID: 2-s2.0-77952398610OAI: oai:DiVA.org:kth-10648DiVA: diva2:222640
Note
QC 20100924. Uppdaterad från submitted till published (20100924).Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2017-12-13Bibliographically approved
In thesis
1. Optimisation and control of boundary layer flows
Open this publication in new window or tab >>Optimisation and control of boundary layer flows
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Both optimal disturbances and optimal control are studied by means of numerical simulations for the case of the flat-plate boundary-layer flow. The optimisation method is the Lagrange multiplier technique where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Control is applied to the bypass-transition scenario with high levels of free-stream turbulence. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to the non-modal growth. These so-called streaks are growing in amplitude until they reach high enough energy levels and breakdown into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, within the region that it is active, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced.

 

 

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. iii, 23 p.
Series
Trita-MEK, ISSN 0348-467X ; 2009:09
Keyword
boundary layer, control, estimation, optimal disturbances, Lagrange method
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:kth:diva-10652 (URN)978-91-7415-368-2 (ISBN)
Presentation
2009-06-15, Sal D42, KTH, Lindstedtsvägen 5, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2010-10-20Bibliographically approved
2. Optimisation and control of shear flows
Open this publication in new window or tab >>Optimisation and control of shear flows
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Transition to turbulence and flow control are studied by means of numerical simulations for different simple shear flows. Linear and non-linear optimisation methods using the Lagrange multiplier technique are employed.

In the linear framework as objective function the standard disturbance kinetic energy is chosen and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest disturbance energy growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response for the case of the flat plate boundary layer excluding the leading edge. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Also linear optimal disturbances are computed around a leading edge and the effect of the geometry is considered. It is found that two-dimentional disturbances originating upstream, relative to the leading edge of the plate are inefficient at generating a viable disturbance, while three dimentional disturbances are more amplified.

In the non-linear framework a new approach using ideas from non-equilibrium thermodynamics is developed. We determine the initial condition on the laminar/turbulent boundary closest to the laminar state. Starting from the general evolution criterion of non-equilibrium systems we propose a method to optimise the route to the statistically steady turbulent state, i.e. the state characterised by the largest entropy production. This is the first time information from the fully turbulent state is included in the optimisation procedure. The method is applied to plane Couette flow. We show that the optimal initial condition is localised in space for realistic flow domains, while the disturbance visits bent streaks before breakdown.

Feedback control is applied to the bypass-transition scenario with high levels of free-stream turbulence. The flow is the flat-plate boundary layer. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to non-modal growth. The so-called streaky structures are growing in amplitude until they reach high enough energy levels and break down into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced.

Open-loop control with wall travelling waves by means of blowing and suction is applied to a separating boundary layer. For downstream travelling waves we obtain a mitigation of the separation of the boundary layer while for upstream travelling waves a significant delay in the transition location accompanied by a modest reduction of the separated region.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. ix, 37 p.
Series
Trita-MEK, ISSN 0348-467X ; 2011:04
Keyword
shear flows, flow control, optimal disturbances, Lagrange method, transition to turbulence, non-linear dynamics
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:kth:diva-33771 (URN)978-91-7415-987-5 (ISBN)
Public defence
2011-05-27, D3, Lindstedtsvägen 5, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research CouncilSwedish e‐Science Research Center
Note
QC 20110518Available from: 2011-05-18 Created: 2011-05-17 Last updated: 2012-05-24Bibliographically approved

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Brandt, LucaHenningson, Dan S.

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