Vassilis Theofilis

Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two (and periodic in one) or all three spatial directions. After a brief exposition of the theory, some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton-Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers.

Finite Element Methods for Viscous Incompressible BiGlobal Instability Analysis on Unstructured Meshes

Viscous linear 3-D BiGlobal instability analyses of incompressible flows have been performed using finite element numerical methods, with a view to extend the scope of application of this analysis methodology to flows over complex geometries. The initial value problem (IVP), based on the linearized Navier-Stokes equations (LNSE), as well as the real and the complex partial-differential-equation-based generalized eigenvalue problems (EVP), have been solved. A mixed P 2 P 1 finite element spatial discretization on unstructured meshes for both the LNSE and the EVP approaches has been used. For the time-discretization of the LNSE a characteristics method has been used for the first time in the context of flow stability analysis; the complex BiGlobal EVP has also been solved for the first time in the context of a finite element numerical discretization. In either its real or its complex form, the EVP has been solved without the need to introduce pseudocompressibility into the incompressible equations, which has simplified the systems to be solved without sacrificing accuracy. An Arnoldi approach has been used to recover the most significant eigenvalues. In this context, the associated solutions to the resulting linear systems were obtained by iterative methods: generalized minimal residual with incomplete lower-upper preconditioning or conjugate gradient with I-Cholesky preconditioning, depending on whether the coefficient matrix was symmetric or not. The 3-D instability of the classic 2-D lid-driven cavity flow and that of the rectangular duct flow were used as validation cases for the real and complex EVP, respectively. New results have been obtained for the 3-D BiGlobal instability of two closed and one open flow, namely, the regularized lid-driven cavity of rectangular and triangular shape and flow in the wake of a model bluff body.

Spectral multi-domain methods for BiGlobal instability analysis of complex flows over open cavity configurations

The potential geometric complexity of a full-bay open cavity configuration calls for the development of novel numerical methods which permit addressing the appropriate linear instability problem in the open cavity. Here results are reported of the first-ever non-conforming spectral multidomain approach for the numerical solution for the twodimensional BiGlobal eigenvalue problem. Analytical model problems 1 and the linear BiGlobal instability of the lid-driven cavity 2 have been used to validate the developed algorithms. New results have been obtained in open cavities of variable aspect ratio, both empty and in the presence of a model store, and the strong dierences of the respective eigenspectra has been documented. The latter result should serve as guidance for theoretical/numerical eorts aiming at flow control of full-bay configurations, using potentially restictive instability analysis models which neglect the presence of stores in the cavity.

BiGlobal Instability Analysis of Turbulent Flow Over an Airfoil at an Angle of Attack

Stability analysis can provide insight to aerodynamic flow control studies by numerical means. It is advantageous to use spectral numerical methods for the stability analysis as they are, at the same level of numerical effort, more accurate than standard finite volume or finite element alternatives. The disadvantage with classic spectral collocation methods, however, is the difficulty in handling geometry. While spectrally accurate and geometrically flexible methods (TSB) exist, based on the spectral/hp−element concept, this paper will present a means of undertaking a fluid mechanical instability analysis using spectral collocation numerical methods on a rectangular grid and conformal mapping techniques in order to represent the geometry of the problem. The flow control configuration of interest in this study is the leading edge separation of a NACA 0015 airfoil, at an angle of attack α = 18. Water tunnel experiments of this configuration, were undertaken by for a Rec ≡ U∞c/ν = 3× 10, where c is the length of the airfoil chord, U∞ is the freestream velocity, and ν the kinematic viscosity. The flow was perturbed via a zero-net-mass-flux (ZNMF) jet, normal to the surface, and spanned the entire leading edge. Frequencies F ≡ fc/U∞ = 0.65 and from F = 1.1 → 1.4 were found to enhanced the lift by more than 45%. A Large Eddy Simulation (LES) of the uncontrolled case was undertaken, and the largest frequency component of the lift force history was F = 0.63, corresponding to one of the frequencies that were found to significantly enhance the lift in the experimental study. The work presented within will introduce the development work of the conformal mapping and numerical techniques required to enable the spectral analysis of this flow configuration. In order to separate the conformal mapping development procedure from that of unsteadiness in the flow, here a lower Rec = 200 is first adopted at which the flow is laminar and steady, then the analysis is repeated at a slightly higher Rec = 300 at which the flow is laminar and unsteady. Results at the target Reynolds number of the turbulent flow at Rec = 3× 10 will be presented elsewhere. The paper will be organised as follows. Firstly an overview of the experimental study will be presented, followed by a comparison of experimental results with those obtained by applying the finite-volume Stanford CDP LES solver to this problem. In addition, a second order finite-element solver (ADFC) has been used to obtain basic states. Next, the derivation of the stability linear operator will be outlined, including a discussion of numerical aspects on the general curvilinear coordinate system, in particular the means of transforming the geometry and velocities between coordinate systems. Following this, two specific geometries are chosen in order to highlight the proposed analysis methodology. An 8:1 ellipse placed at an angle of attack α = 18 to the oncoming flow is first analysed, as the analytical derivatives required for the conformal mapping

Global and Local Hydrodynamic Stability Analysis as a Tool for Combustor Dynamics Modeling

Coherent flow structures in shear flows are generated by instabilities intrinsic to the hydrodynamic field. In a combustion environment, these structures may interact with the flame and cause unsteady heat release rate fluctuations. Prediction and modeling of these structures is thereby highly wanted for thermo-acoustic prediction models. In this work we apply hydrodynamic linear stability analysis to the time-averaged flow field of swirl-stabilized combustors obtained from experiments. Recent fundamental investigations have shown that the linear eigenmodes of the mean flow accurately represent the growth and saturation of the coherent structures. In this work biglobal and local stability analysis is applied to the reacting flow in an industry-relevant combustion system. Both the local and the biglobal analysis accurately predicts the onset and structure of a self-excited global instability that is known in the combustion community as a precessing vortex core (PVC). However, only the global analysis accurately predicts a globally stable flow field for the case without the oscillation, while the local analysis wrongly predicts an unstable global growth rate. The predicted spatial distribution of the amplitude functions using both analysis agree very well to the experimentally identified global mode. The presented tools are considered as very promising for the understanding of the PVC and physics based flow control.Copyright

Stability analysis of the flow around a cylinder fitted with helical strakes

Helical strakes are the most employed devices to mitigate or suppress vortex shedding behind circular cylinders. Although several investigations have been performed in order to predict the performance of these devices, proving its efficiency in specific configurations, little is understood regarding the physical mechanisms leading to the efficiency of these devices. The present work addresses this question from a global linear instability analysis point of view. Direct Numerical Simulation, three-dimensional global (TriGlobal) and Floquet stability analysis of the flow around a cylinder fitted with helical strakes is performed at low and moderate Reynolds number in order to understand more deeply the flow instabilities and physical mechanisms that mitigate and suppress the vortex-shedding.

Structural analysis on a hemisphere-cylinder at moderate Reynolds number and high angle of attack

Three-dimensional DNS has been performed on a hemisphere-cylinder at Reynolds number Re= 1000 and angle of attack AoA= 20◦ in order to analyze flow structures and wake frequencies. PIV experiments have also been carried out over the same geometry and flow conditions to validate the numerical results. Critical point theory has been applied in order to determine the topology patterns over the surface of the body. Critical points and separation lines on the body surface show the presence of three different flow patterns: separation bubble, ”horn vortices” and ”leeward vortices”. Both, ”horn vortices” and ”leeward vortices” are found to be asymmetric and unsteady. The frequency related to ”leeward vortices” oscillations has been identified both experimentally and numerically. Two more dominant frequencies, related to two different wake shedding modes have been found. On the other hand, POD has been performed and the four most energetic POD modes is found to be composed of a mixture of these three frequencies. They are modes on the wake shedding. Finally, DMD modes associated with these three frequencies are found on the symmetry plane close to the nose area. These modes represent different shear layer instabilities. Flow separation was found to be intrinsically linked with the observed shear-layer instability.

Effect of the Trailing Edge Geometry on the Unsteadiness of the Flow Around a Stalled NACA 0015 Airfoil

The effect on the flow unsteadiness on a NACA 0015 airfoil at Reynolds number \(Re = 200\) and Angle of Attack \(AoA = 18^{\circ }\) is investigated numerically. Four different geometries based on the NACA 0015 airfoil and with trailing-edge modifications are compared. Long-time integration of the incompressible two-dimensional Navier-Stokes equations shows that the recovered flow field is steady independently of the trailing-edge geometry.

On Global Instability of Variable Aspect Ratio Channel Flows

The linear global stability of a flow of an incompressible fluid through a diverging channel is studied. Stability of such flows in a local context has been studied a long time ago, using the Jeffery-Hamel (JH) similarity profiles and a locally parallel flow approximation. The velocity profiles thus obtained include a parabolic profile of plane Poisuelle flow (PPF), when the divergence angle α = 0. At high α these profiles present a single region of flow reversal at each of the walls. The JH profiles are solutions to steady, laminar, two-dimensional flow of incompressible fluid within an infinite wedge driven by a line source/sink situated at the intersection of the rigid planes that form the wedge. However, these similarity solutions are not valid in finite channels. In this paper the base flow is obtained by solving for the two-dimenisonal Navier-Stokes equation (NSE) using a multi-grid Poisson equation solver (MPES) which employs the vorticity stream-function formulation. Here the stability is carried out using a global approach. The two-dimensional linearized Navier-Stokes equations (LNSE) are solved as a generalized eigenvalue problem using Arnoldi iteration and validated against previously obtained solutions employing the QZ algorithm. The Arnoldi iteration allows for faster computing of the eigenspectrum. This permits complementing our earlier global instability analyses in the small angle-of-divergence limit and extending the work to finite values of this parameter.

Molecular Dynamics Simulations of Couette Flow

In this work, the first steps towards developing a continuum-molecular coupled simulation technique are presented, for the purpose of computing macroscopic systems of confined fluids. The idea is to compute the interface wall-fluid by Molecular Dynamics (MD) simulations, where Lennard-Jones potential (and others) have been employed for the molecular interactions, so the usual no slip boundary condition is not specified. Instead, a shear rate can be imposed at the wall, which allows the calculation of wall material properties by means of an iterative method. The remaining fluid region will be computed by a spectral hp method. We present MD simulations of a Couette flow, and the results of the developed boundary conditions from the wall fluid interaction.