| 1. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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A method for computing optimal forcing of convectively unstable flows using Laplace preconditioning
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Other publication (other academic/artistic)abstract
- 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 problem of finding 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 optimal forcing function. Implementation of the method requires only minor modifications of an existing time-stepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss it in the context of the linear Ginzburg-Landau equation.
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| 2. |
- Brynjell-Rahkola, Mattias, et al.
(author)
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A note on the numerical realization of helical vortices: application to vortex instability
- 2017
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Reports (other academic/artistic)abstract
- 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.
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| 3. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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Computing Optimal Forcing Using Laplace Preconditioning
- 2017
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In: Communications in Computational Physics. - : Cambridge University Press. - 1815-2406 .- 1991-7120. ; 22:5, s. 1508-1532
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Journal article (peer-reviewed)abstract
- 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.
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| 4. |
- Brynjell-Rahkola, Mattias, et al.
(author)
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Global Stability Analysis of a Roughness Wake in a Falkner-Skan-Cooke Boundary Layer
- 2015
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In: Procedia IUTAM. - : Elsevier. - 2210-9838. ; , s. 192-200
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Conference paper (peer-reviewed)abstract
- A global stability analysis of a Falkner-Skan-Cooke boundary layer with distributed three-dimensional surface roughness is per- formed using high-order direct numerical simulations. Computations have been performed for different sizes of the roughness elements, and a time-stepping method has been used to find the instability modes. The study shows that a critical roughness height beyond which a global instability is excited does exist. Furthermore, the origins of this instability is examined by means of an energy analysis, which reveals the production and dissipation terms responsible for the instability, as well as the region in space where the instability originates.
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| 5. |
- Brynjell-Rahkola, Mattias, 1986-
(author)
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Global stability analysis of three-dimensional boundary layer flows
- 2015
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Licentiate thesis (other academic/artistic)abstract
- This thesis considers the stability and transition of incompressible boundary layers. In particular, the Falkner–Skan–Cooke boundary layer subject to a cylindrical surface roughness, and the Blasius boundary layer with applied localized suction are investigated. These flows are of great importance within the aviation industry, feature complex transition scenarios, and are strongly three-dimensional in nature. Consequently, no assumptions regarding homogeneity in any of the spatial directions are possible, and the stability of the flow is governed by an extensive three-dimensional eigenvalue problem.The stability of these flows is addressed by high-order direct numerical simulations using the spectral element method, in combination with a Krylov subspace projection method. Such techniques target the long-term behavior of the flow and can provide lower limits beyond which transition is unavoidable. The origin of the instabilities, as well as the mechanisms leading to transition in the aforementioned cases are studied and the findings are reported.Additionally, a novel method for computing the optimal forcing of a dynamical system is developed. This type of analysis provides valuable information about the frequencies and structures that cause the largest energy amplification in the system. The method is based on the inverse power method, and is discussed in the context of the one-dimensional Ginzburg–Landau equation and a two-dimensional flow case governed by the Navier–Stokes equations.
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| 6. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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Modal analysis of roughness-induced crossflow vortices in a Falkner-Skan-Cooke boundary layer
- 2013
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In: International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2013. - : TSFP-8. - 9780000000002
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Conference paper (peer-reviewed)abstract
- A three-dimensional global stability analysis using high-order direct numerical simulations is performed to investigate the effect of surface roughness with Reynolds number (based on roughness height) Rek above and below the critical value for transition, on the eigenmodes of a Falkner-Skan-Cooke boundary layer. The surface roughness is introduced with the immersed boundary method and the eigenvalues and eigenfunctions are solved using an iterative time-stepper method. The study reveals a global instability for the case with higher Reynolds number that causes the flow in the non-linear simulations to break down to turbulence shortly downstream of the roughness. Examination of the unstable linear global modes show that these are the same modes that are observed in experiments immediately before breakdown due to secondary instability, which emphasizes the importance of these modes in transition.
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| 7. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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Numerical realization of helical vortices : application to vortex instability
- 2019
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In: Theoretical and Computational Fluid Dynamics. - : Springer. - 0935-4964 .- 1432-2250.
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Journal article (peer-reviewed)abstract
- 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 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 the case of two helices, which is of renewed interest due to its relevance for wind turbines and helicopters. Three-dimensional stability analysis is performed in both a rotating and a translating frame of reference, which yield eigenvalue spectra that feature both mutual inductance and elliptic instabilities. Comparison of these spectra with available theoretical predictions is used to validate the proposed baseflow model, and new insights into the elliptic instability of curved Batchelor vortices are presented. Furthermore, it is shown that the instabilities in the rotating and the translating reference frames have the same structure and growth rate, but different frequency. A relation between these frequencies is provided.
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| 8. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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On the stability of a Blasius boundary layer subject to localised suction
- 2019
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In: Journal of Fluid Mechanics. - : CAMBRIDGE UNIV PRESS. - 0022-1120 .- 1469-7645. ; 871, s. 717-741
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Journal article (peer-reviewed)abstract
- In this study the origins of premature transition due to oversuction in boundary layers are studied. An infinite row of circular suction pipes that are mounted at right angles to a flat plate subject to a Blasius boundary layer is considered. The interaction between the flow originating from neighbouring holes is weak and for the parameters investigated, the pipe is always found to be unsteady regardless of the state of the flow in the boundary layer. A stability analysis reveals that the appearance of boundary layer transition can be associated with a linear instability in the form of two unstable eigenmodes inside the pipe that have weak tails, which extend into the boundary layer. Through an energy budget and a structural sensitivity analysis, the origin of this flow instability is traced to the structures developing inside the pipe near the pipe junction. Although the amplitudes of the modes in the boundary layer are orders of magnitude smaller than the corresponding amplitudes inside the pipe, a Koopman analysis of the data gathered from a nonlinear direct numerical simulation confirms that it is precisely these disturbances that are responsible for transition to turbulence in the boundary layer due to oversuction.
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| 9. |
- Brynjell-Rahkola, Mattias, et al.
(author)
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On the stability of a Blasius boundary layer subject to localized suction
- 2017
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Reports (other academic/artistic)abstract
- 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.
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| 10. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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On the stability of flat plate boundary layers subject to localized suction
- 2015
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Reports (other academic/artistic)abstract
- The stability of the Blasius boundary layer subject to localized suction is revisited. Using tools of global stability analysis, the leading direct and adjoint eigenpairs are determined, and novel insight into the sensitivity and receptivity of the flow is obtained. The problem is addressed through high-order spectral element simulations, which enables the inclusion of a suction pipe into the domain. Due to this, a detailed investigation of the connection between the pipe flow and the boundary layer flow is possible. For all cases investigated, the former always turns out to transition for a lower Reynolds number and suction rate than the latter, and the transition scenario is found to be due to a global instability originating inside a separation bubble at the pipe inlet. Identification of such regions, provides information that is valuable in further development of algorithms for laminar flow control.
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| 11. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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Onset of global instability behind distributed surface roughness in a Falkner–Skan–Cooke boundary layer
- 2015
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Reports (other academic/artistic)abstract
- A three-dimensional linear global stability analysis of a Falkner–Skan–Cooke boundary layer with distributed three-dimensional surface roughness is performed. The Falkner–Skan–Cooke boundary layer models the flow over swept airplane wings, and investigation of the critical roughness size for which a global instability emerges is thus of great importance within aeronautical applications. The study considers high-order direct numerical simulations and shows that such a critical roughness height exists for the Falkner–Skan–Cooke boundary layer. The roughness Reynolds number and roughness element aspect ratio for which this happens is comparable to the transition data reported in the literature for two-dimensional boundary layers. This demonstrates the importance of the local flow conditions in the vicinity of the roughness for triggering a global instability, although the resulting breakdown scenario is completely different from that of two-dimensional boundary layers. This breakdown scenario is studied in detail, and a global energy analysis is used to reveal the structures and mechanisms responsible for production and dissipation of perturbation energy.
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| 12. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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Stability and sensitivity of a cross-flow-dominated Falkner-Skan-Cooke boundary layer with discrete surface roughness
- 2017
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In: Journal of Fluid Mechanics. - : Cambridge University Press. - 0022-1120 .- 1469-7645. ; 826, s. 830-850
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Journal article (peer-reviewed)abstract
- 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.
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| 13. |
- Brynjell-Rahkola, Mattias, 1986-, et al.
(author)
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Stability and sensitivity of a crossflow-dominated Falkner–Skan–Cooke boundary layer with discrete surface roughness
- 2016
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In: Journal of Fluid Mechanics. - : Cambridge University Press. - 0022-1120 .- 1469-7645.
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Journal article (peer-reviewed)abstract
- 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 crossflow (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 "-pseudospectrum, and the dependency on the domain is analysed through an impulse response and an energy budget. It is shown that 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 su ciently 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 what can be resolved using finite precision arithmetics.
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| 14. |
- Brynjell-Rahkola, Mattias, 1986-
(author)
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Studies on instability and optimal forcing of incompressible flows
- 2017
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Doctoral thesis (other academic/artistic)abstract
- 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.
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| 15. |
- Lundström, T. Staffan, et al.
(author)
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Evaluation of Guiding Device for Downstream Fish Migration with In-field Particle Tracking Velocimetry and CFD
- 2015
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In: Journal of Applied Fluid Mechanics. - : Academic World Research. - 1735-3572 .- 1735-3645. ; 8:3, s. 579-589
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Journal article (peer-reviewed)abstract
- The performance of a fish guiding device located just upstream a hydropower plant is scrutinized. The device is designed to redirect surface orientated down-stream migrating fish (smolts) away from the turbines towards a spillway that act as a relatively safe fishway. Particles are added up-stream the device and the fraction particles going to the spillway is measured. A two-frame Particle Tracking Velocimetry algorithm is used to derive the velocity field of the water. The experimental results are compared to simulations with CFD. If the smolts move passively as the particles used in the study the guiding device works very well and some modifications may optimize its performance. In-field Particle Tracking Velocimetry is a suitable technique for the current case and the results compare well with numerical simulations.
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| 16. |
- Quaranta, Hugo Umberto, et al.
(author)
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Local and global pairing instabilities of two interlaced helical vortices
- 2019
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In: Journal of Fluid Mechanics. - : CAMBRIDGE UNIV PRESS. - 0022-1120 .- 1469-7645. ; 863, s. 927-955
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Journal article (peer-reviewed)abstract
- 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 global 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 and frequencies is found. This shows 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 particle image velocimetry 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 nonlinear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent breakup of the vortices, global pairing results in a leapfrogging phenomenon, which temporarily restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.
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| 17. |
- Quaranta, Hugo Umberto, et al.
(author)
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Long-wave instabilities of two interlaced helical vortices
- 2017
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Other publication (other academic/artistic)abstract
- 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.
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| 18. |
- Quaranta, H. U., et al.
(author)
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Long-wave instabilities of two interlaced helical vortices
- 2016
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In: Journal of Physics, Conference Series. - : Institute of Physics (IOP). - 1742-6588 .- 1742-6596. ; 753:3
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Journal article (peer-reviewed)abstract
- We present a comparison between experimental observations and theoretical predictions concerning long-wave displacement instabilities of the helical vortices in the wake of a two-bladed rotor. Experiments are performed with a small-scale rotor in a water channel, using a set-up that allows the individual triggering of various instability modes at different azimuthal wave numbers, leading to local or global pairing of successive vortex loops. The initial development of the instability and the measured growth rates are in good agreement with the predictions from linear stability theory, based on an approach where the helical vortex system is represented by filaments. At later times, local pairing develops into large-scale distortions of the vortices, whereas for global pairing the non-linear evolution returns the system almost to its initial geometry.
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