Efficient stability analysis of flows using complex mapping techniques

Global linear stability analysis of open flows leads to difficulties associated to boundary conditions, leading to either spurious wave reflections (in compressible cases) or to non-local feedback due to the elliptic nature of the pressure equation (in incompressible cases). A novel approach is introduced to address both these problems. The approach consists of solving the problem using a complex mapping of the spatial coordinates, in a way that can be directly applicable in an existing code without any additional auxiliary variable. The efficiency of the method is first demonstrated for a simple 1D equation modeling incompressible Navier–Stokes, and for a linear acoustics problem. The application to full linearized Navier–Stokes equation is then discussed. A criterion on how to select the parameters of the mapping function is derived by analyzing the effect of the mapping on plane wave solutions. Finally, the method is demonstrated for three application cases, including an incompressible jet, a compressible hole-tone configuration and the flow past an airfoil. The examples allow to show that the method allows to suppress the artificial modes which otherwise dominate the spectrum and can possibly hide the physical modes. Finally, it is shown that the method is still efficient for small truncated domains, even in cases where the computational domain is comparable to the dominant wavelength

Bifurcation scenario in the two-dimensional laminar flow past a rotating cylinder

The aim of this paper is to provide a complete description of the bifurcation scenario of a uniform flow past a rotating circular cylinder up to Re=200 . Linear stability theory is used to depict the neutral curves and analyse the arising unstable global modes. Three codimension-two bifurcation points are identified, namely a Takens–Bogdanov, a cusp and generalised Hopf, which are closely related to qualitative changes in orbit dynamics. The occurrence of the cusp and Takens–Bogdanov bifurcations for very close parameters (corresponding to an imperfect codimension-three bifurcation) is shown to be responsible for the existence of multiple steady states, as already observed in previous studies. Two bistability regions are identified, the first with two stable fixed points and the second with a fixed point and a cycle. The presence of homoclinic and heteroclinic orbits, which are classical in the presence of Takens–Bogdanov bifurcations, is confirmed by direct numerical simulations. Finally, a weakly nonlinear analysis is performed in the neighbourhood of the generalised Hopf, showing that above this point the Hopf bifurcation is subcritical, leading to a third range of bistability characterised by both a stable fixed point and a stable cycle

On the properties of high-order least-squares finite-volume schemes

High-order finite-volume schemes based on polynomial least-squares methods are an active research topic for the discretization of hyperbolic equations as they allow to obtain high-order spatial discretization schemes in arbitrary meshes. However, few studies have analyzed their performance in good-quality/near-to-uniform meshes, which are commonly used as a meshing strategy in zones where turbulent effects are important. In this paper, the theoretical numerical properties of commonly used least-squares (LSQ) k-exact high-order finite volume schemes are studied in one-dimensional and in several two- dimensional meshes (with some remarks regarding their properties in three-dimensional meshes). These results are compared to those obtained using fully-constrained polynomial reconstructions only compatible with structured meshes. The numerical properties of the schemes are investigated through the von Neumann analysis methodology applied to the one-dimensional and two-dimensional finite volume formulation, including temporal discretization errors. This analysis is also extended to non-uniform and unstructured two- dimensional meshes. At last, the schemes are tested with several numerical experiments using the linear advection, the Euler equations and the Navier-Stokes equations. The analysis of both studies yields similar conclusions regarding the numerical errors and stability of the different studied schemes showing that the high-order least-squares finite volume schemes yield stable and robust results across different uniform and non-uniform unstructured meshes. However, their performance is heavily degraded in simulations with low mesh resolution compared to schemes specially catered to structured meshes. On the other hand, the latter schemes lack stability and robustness in general structured meshes and its formulation cannot be straightforwardly extended to unstructured meshes. Moreover, this work shows that the use of weighted-LSQ can drastically improve the results of LSQ schemes in under-resolved simulations

A practical review on linear and nonlinear global approaches to flow instabilities

This paper aims at reviewing linear and nonlinear approaches to study the stability offluid flows. We provide a concise but self-contained exposition of the main concepts andspecific numerical methods designed for global stability studies, including the classicallinear stability analysis, the adjoint-based sensitivity, and the most recent nonlineardevelopments. Regarding numerical implementation, a number of ideas making resolu-tion particularly efficient are discussed, including mesh adaptation, simple shift-invertstrategy instead of the classical Arnoldi algorithm, and a simplification of the recent non-linear self-consistent (SC) approach proposed by Manticˇ-Lugo et al. (2014, “Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding inthe Cylinder Wake,” Phys. Rev. Lett., 113(8), p. 084501). An open-source software imple-menting all the concepts discussed in this paper is provided. The software is demon-strated for the reference case of the two-dimensional (2D) flow around a circularcylinder, in both incompressible and compressible cases, but is easily customizable to avariety of other flow configurations or flow equations

Mode interactions in external flows

In this thesis, we study the formation of coherent structures in the early stages of transition from a laminar state towards a chaotic attractor. We aim to unveil the rich array of dynamical characteristics appearing in the laminar-chaos transition of external flows, and to determine some of the universal ingredients in the transition. In particular, we analyse the formation of large coherent structures under the competition among multiple global instabilities. These techniques are applied, to a different extent, to multiple physical problems: acoustics, wake and jet flows and problems with moving interfaces. In acoustic problems, the instability of the jet flow past a hole is analysed in terms of its transfer function, the impedance, which allows us to consider the incompressible problem, as long as the flow is acoustically compact. The continuation of the emerging limit cycle from the self-sustained instability is carried out following a fixed-point method for limit cycles, which also provides information about the sensitivity of the cycle characteristics to localised variations in the flow-field. The flow problems with moving interfaces require the development of novel numerical techniques to analyse the issuing instabilities. Herein, we explore the techniques of a linearised diffusive Immerse Boundary Method in a case with Vortex-Induced-Vibrations (VIV), and the linearised Arbitrary Lagrangian Eulerian technique for the dynamics of a bubble in a straining flow. In the case of multiple interacting flow instabilities, we analyse the organising centres: the steady-steady bifurcation with 1:2 resonance condition in the configuration of two concentric jets, a codimension-three Takens-Bogdanov bifurcation in the case of the wake flow behind a spinning cylinder, the steady-Hopf bifurcation in a case of an axisymmetric wake flow, and the triple Hopf in two cases, the wake flow dynamics behind a rotating particle (non-resonant) and the sound emissions of a rounded impinging jet (resonant). In addition to the qualitative description of the dynamics and the formation of spatio-temporal patterns in the flow, the analysis of these organising centres offers information about the connections between the underlying baseflow/perturbations and the nonlinear transitions. For instance, the existence of a (near-)resonance condition in the case of the rounded impinging jet suggests for the existence of a non-local feedback mechanism, which is herein analysed using a decomposed notion of the structural sensitivity tensor.Dans cette thèse, nous étudions la formation de structures cohérentes dans les premiers stades de transition de l'état laminaire vers un attracteur chaotique. Nous visons à dévoiler la riche gamme de caractéristiques dynamiques apparaissant dans la transition laminaire-chaotique pour les écoulements externes, et ainsi déterminer certains ingrédients universels de la transition. En particulier, nous analysons la formation des structures cohérentes sous la compétition entre plusieurs instabilités globales. Ces techniques s'appliquent, à des degrés divers, à de multiples problèmes physiques : acoustique, écoulements de sillage et jet et à des problèmes avec des interfaces mobiles. Dans les problèmes acoustiques, on analyse l'instabilité de l'écoulement à travers d'un trou à partir de sa fonction de transfert, l'impédance, ce qui permet de considérer une modélisation incompressible, tant que l'écoulement est acoustiquement compact. La continuation du cycle limite émergeant à partir de l'instabilité auto-entretenue est effectuée selon une méthode de point-fixe pour les cycles limites, une méthode qui fournit également des informations sur les grandeurs de sensibilité. Les problèmes d'écoulement à interfaces mobiles nécessitent le développement de nouvelles techniques numériques pour analyser les instabilités. Ici, nous explorons une technique Immersed Boundary Method (IMB) linéarisée dans un cas avec des vibrations induites par vortex (VIV), et la technique Arbitrary Lagrangian/Eulerian (ALE) linéarisée pour la dynamique d'une bulle dans un flux de déformation. Dans les cas des écoulements avec des multiples instabilités qui intéragissent, nous analysons les bifurcations à codimension élevée suivantes: interaction de modes stationnaire-stationnaire avec une résonance 1:2 dans le cas de jets concentriques ; bifurcation de Takens-Bogdanov de codimension trois dans le cas d'un écoulement de sillage derrière un cylindre en rotation ; interaction mode stationnaire/mode instationnaire dans le cas d'un écoulement de sillage axisymétrique, et pour finir la bifurcation triple Hopf dans deux cas, la dynamique de l'écoulement de sillage derrière une sphère en rotation (non-résonnante) et les émissions sonores d'un jet incident (résonant). En plus de la description qualitative de la dynamique et de la formation de structures spatio-temporelles dans l'écoulement, l'analyse de ces bifurcations à codimension élevée fournit des informations concernant les mécanismes physiques de l'instabilité. Par exemple, l'existence d'une condition de (quasi-)résonance dans le cas d'un jet axisymétrique impactant une paroi suggère l'existence d'un mécanisme de rétroaction non local, qui est ici analysé en utilisant une décomposition du tenseur de la sensibilité structurelle

Mode interactions in external flows

Dans cette thèse, nous étudions la formation de structures cohérentes dans les premiers stades de transition de l'état laminaire vers un attracteur chaotique. Nous visons à dévoiler la riche gamme de caractéristiques dynamiques apparaissant dans la transition laminaire-chaotique pour les écoulements externes, et ainsi déterminer certains ingrédients universels de la transition. En particulier, nous analysons la formation des structures cohérentes sous la compétition entre plusieurs instabilités globales. Ces techniques s'appliquent, à des degrés divers, à de multiples problèmes physiques : acoustique, écoulements de sillage et jet et à des problèmes avec des interfaces mobiles. Dans les problèmes acoustiques, on analyse l'instabilité de l'écoulement à travers d'un trou à partir de sa fonction de transfert, l'impédance, ce qui permet de considérer une modélisation incompressible, tant que l'écoulement est acoustiquement compact. La continuation du cycle limite émergeant à partir de l'instabilité auto-entretenue est effectuée selon une méthode de point-fixe pour les cycles limites, une méthode qui fournit également des informations sur les grandeurs de sensibilité. Les problèmes d'écoulement à interfaces mobiles nécessitent le développement de nouvelles techniques numériques pour analyser les instabilités. Ici, nous explorons une technique Immersed Boundary Method (IMB) linéarisée dans un cas avec des vibrations induites par vortex (VIV), et la technique Arbitrary Lagrangian/Eulerian (ALE) linéarisée pour la dynamique d'une bulle dans un flux de déformation. Dans les cas des écoulements avec des multiples instabilités qui intéragissent, nous analysons les bifurcations à codimension élevée suivantes: interaction de modes stationnaire-stationnaire avec une résonance 1:2 dans le cas de jets concentriques ; bifurcation de Takens-Bogdanov de codimension trois dans le cas d'un écoulement de sillage derrière un cylindre en rotation ; interaction mode stationnaire/mode instationnaire dans le cas d'un écoulement de sillage axisymétrique, et pour finir la bifurcation triple Hopf dans deux cas, la dynamique de l'écoulement de sillage derrière une sphère en rotation (non-résonnante) et les émissions sonores d'un jet incident (résonant). En plus de la description qualitative de la dynamique et de la formation de structures spatio-temporelles dans l'écoulement, l'analyse de ces bifurcations à codimension élevée fournit des informations concernant les mécanismes physiques de l'instabilité. Par exemple, l'existence d'une condition de (quasi-)résonance dans le cas d'un jet axisymétrique impactant une paroi suggère l'existence d'un mécanisme de rétroaction non local, qui est ici analysé en utilisant une décomposition du tenseur de la sensibilité structurelle.In this thesis, we study the formation of coherent structures in the early stages of transition from a laminar state towards a chaotic attractor. We aim to unveil the rich array of dynamical characteristics appearing in the laminar-chaos transition of external flows, and to determine some of the universal ingredients in the transition. In particular, we analyse the formation of large coherent structures under the competition among multiple global instabilities. These techniques are applied, to a different extent, to multiple physical problems: acoustics, wake and jet flows and problems with moving interfaces. In acoustic problems, the instability of the jet flow past a hole is analysed in terms of its transfer function, the impedance, which allows us to consider the incompressible problem, as long as the flow is acoustically compact. The continuation of the emerging limit cycle from the self-sustained instability is carried out following a fixed-point method for limit cycles, which also provides information about the sensitivity of the cycle characteristics to localised variations in the flow-field. The flow problems with moving interfaces require the development of novel numerical techniques to analyse the issuing instabilities. Herein, we explore the techniques of a linearised diffusive Immerse Boundary Method in a case with Vortex-Induced-Vibrations (VIV), and the linearised Arbitrary Lagrangian Eulerian technique for the dynamics of a bubble in a straining flow. In the case of multiple interacting flow instabilities, we analyse the organising centres: the steady-steady bifurcation with 1:2 resonance condition in the configuration of two concentric jets, a codimension-three Takens-Bogdanov bifurcation in the case of the wake flow behind a spinning cylinder, the steady-Hopf bifurcation in a case of an axisymmetric wake flow, and the triple Hopf in two cases, the wake flow dynamics behind a rotating particle (non-resonant) and the sound emissions of a rounded impinging jet (resonant). In addition to the qualitative description of the dynamics and the formation of spatio-temporal patterns in the flow, the analysis of these organising centres offers information about the connections between the underlying baseflow/perturbations and the nonlinear transitions. For instance, the existence of a (near-)resonance condition in the case of the rounded impinging jet suggests for the existence of a non-local feedback mechanism, which is herein analysed using a decomposed notion of the structural sensitivity tensor