Bulletin of the American Physical Society
67th Annual Meeting of the APS Division of Fluid Dynamics
Volume 59, Number 20
Sunday–Tuesday, November 23–25, 2014; San Francisco, California
Session H17: Nonlinear Dynamics IV: Model Reduction |
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Chair: Clancy Rowley, Princeton University Room: 2002 |
Monday, November 24, 2014 10:30AM - 10:43AM |
H17.00001: On the relationship between Koopman Mode Decomposition and Dynamic Mode Decomposition Igor Mezic, Hassan Arbabi We discuss several issues in theory and applications of Koopman modes in fluid mechanics. We show an explicit relationship between a basic -- companion matrix - version of the Dynamic Mode Decomposition (DMD) and the Koopman Mode Decomposition (KMD) of dynamical systems, that allows for estimates of validity of approximation of Koopman modes by DMD modes. As a side result we link the recently introduced Generalized Laplace Analysis and the inverse of the Vandermonde matrix. Using these theoretical results, a new method for computation of Koopman modes is presented that avoids inversion of the Vandermonde matrix. Application of this method to analysis of dynamic stall are shown. [Preview Abstract] |
Monday, November 24, 2014 10:43AM - 10:56AM |
H17.00002: Applications of Koopman Operator Theory to Model Reduction in Fluid Mechanics Hassan Arbabi, Igor Mezi\'c We discuss some applications of the Koopman operator theory to the problems in fluid mechanics. These applications involve the Koopman mode decomposition (KMD), which describes the nonlinear evolution of the flow field observables, such as velocity or vorticity field, in terms of a linear expansion - analogous to the normal mode analysis in linear oscillations. By applying KMD to the incompressible flow in a 2D rectangular cavity, we identify the spectrum of the flow with the associated global modes, both in periodic and aperiodic regime. We also apply KMD to in-vivo measurements of the blood velocity field inside human's heart, and extract the Koopman modes and frequencies based on the assumption of evolution on an attractor. The dominant Koopman modes are then combined to create a low-dimensional model for both the cavity and heart flow. The mesochronic analysis shows that those reduced models capture the mixing topology with an accuracy comparable to that of the original data. Comparison in the $L^2$-norm also shows that the reduced models obtained by KMD could give a more accurate representation of the flow field compared to POD. [Preview Abstract] |
Monday, November 24, 2014 10:56AM - 11:09AM |
H17.00003: Extending Dynamic Mode Decomposition: A Data--Driven Approximation of the Koopman Operator Matthew Williams, Ioannis Kevrekidis, Clarence Rowley In recent years, Koopman spectral analysis has become a popular tool for the decomposition and study of fluid flows. One benefit of the Koopman approach is that it generates a set of spatial modes, called Koopman modes, whose evolution is determined by the corresponding set of Koopman eigenvalues. Furthermore, these modes are valid globally, and not only in some small neighborhood of a fixed point. A popular method for approximating the Koopman modes and eigenvalues is Dynamic Mode Decomposition (DMD). In this talk, we show that DMD approximates the Koopman {\em eigenfunctions}, but uses linear monomials to do so; this may be limiting in certain applications. We then introduce an extension of DMD, which we refer to as Extended DMD (EDMD), that uses a richer set of user determined basis functions to approximate the Koopman eigenfunctions. We demonstrate the impact this difference has on the eigenvalues and modes by applying DMD and EDMD to some simple example problems. Although the algorithms for DMD and EDMD appear to be similar, modifications like the ones we will present can be important if the resulting eigenvalues, eigenfunctions, and modes are to accurately approximate those of the Koopman operator. [Preview Abstract] |
Monday, November 24, 2014 11:09AM - 11:22AM |
H17.00004: Efficient simulation of detached flows at high Reynolds number Jose M. Vega, Victor Asensio, Raul Herrero, Fernando Varas A method is presented for the computationally efficient simulation of quasi-periodic detached flows in multi-parameter problems at very large Reynolds numbers, keeping in mind a variety of applications, including helicopter flight simulators, control and certification of unmanned aerial vehicles, control of wind turbines, conceptual design in aeronautics, and civil aerodynamics. In many of these applications, the large scale flows (ignoring the smaller turbulent scales) are at most quasi-periodic, namely the Fourier transform exhibits a finite set of concentrated peaks resulting from the nonlinear passive interaction of periodic wakes. The method consists in an offline preprocess and the online operation. In the preprocess, a standard CFD solver (such as URANS) is used in combination with several ingredients such as an iterative combination proper orthogonal decomposition and fast Fourier transform. The online operation is made with a combination of high order singular value decomposition and interpolation. The performance of the method is tested considering the ow over a fairly complex urban topography, for various free stream intensities and orientations, seeking real time online simulations. [Preview Abstract] |
Monday, November 24, 2014 11:22AM - 11:35AM |
H17.00005: Stability Analysis of Non-Newtonian Rotational Flow with Hydromagnetic Effect Nariman Ashrafi Stability of the magnetorheological rotational flow in~the~presence of~a~magnetic excitation in the tangential direction is examined. The conservation of mass and momentum equations for an isothermal Carreau fluid between coaxial cylinders are numerically solved while mixed boundary conditions are assumed. In~the~absence of magnetic excitation, the base flow loses its radial flow stability to the vortex structure at a critical Taylor number. The emergence of the vortices corresponds to the onset of a supercritical bifurcation. The Taylor vortices, in turn, lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. The tangential magnetic field turns out to be a controlling parameter as it alters the critical points throughout the bifurcation diagram. Also, the effect of the Hartmann number, the Deborah number and the fluid elasticity on the flow parameters were investigated. [Preview Abstract] |
Monday, November 24, 2014 11:35AM - 11:48AM |
H17.00006: Dynamics and Control of a Reduced Order System of the 2-d Navier-Stokes Equations Nejib Smaoui, Mohamed Zribi The dynamics and control problem of a reduced order system of the 2-d Navier-Stokes (N-S) equations is analyzed. First, a seventh order system of nonlinear ordinary differential equations (ODE) which approximates the dynamical behavior of the 2-d N-S equations is obtained by using the Fourier Galerkin method. We show that the dynamics of this ODE system transforms from periodic solutions to chaotic attractors through a sequence of bifurcations including a period doubling scenarios. Then three Lyapunov based controllers are designed to either control the system of ODEs to a desired fixed point or to synchronize two ODE systems obtained from the truncation of the 2-d N-S equations under different conditions. Numerical simulations are presented to show the effectiveness of the proposed controllers. [Preview Abstract] |
Monday, November 24, 2014 11:48AM - 12:01PM |
H17.00007: Network-theoretic approach to model vortex interactions Aditya Nair, Kunihiko Taira We present a network-theoretic approach to describe a system of point vortices in two-dimensional flow. By considering the point vortices as nodes, a complete graph is constructed with edges connecting each vortex to every other vortex. The interactions between the vortices are captured by the graph edge weights. We employ sparsification techniques on these graph representations based on spectral theory to construct sparsified models of the overall vortical interactions. The edge weights are redistributed through spectral sparsification of the graph such that the sum of the interactions associated with each vortex is maintained constant. In addition, sparse configurations maintain similar spectral properties as the original setup. Through the reduction in the number of interactions, key vortex interactions can be highlighted. Identification of vortex structures based on graph sparsification is demonstrated with an example of clusters of point vortices. We also evaluate the computational performance of sparsification for large collection of point vortices. [Preview Abstract] |
Monday, November 24, 2014 12:01PM - 12:14PM |
H17.00008: Sparsified-dynamics modeling of discrete point vortices with graph theory Kunihiko Taira, Aditya Nair We utilize graph theory to derive a sparsified interaction-based model that captures unsteady point vortex dynamics. The present model builds upon the Biot-Savart law and keeps the number of vortices (graph nodes) intact and reduces the number of inter-vortex interactions (graph edges). We achieve this reduction in vortex interactions by spectral sparsification of graphs. This approach drastically reduces the computational cost to predict the dynamical behavior, sharing characteristics of reduced-order models. Sparse vortex dynamics are illustrated through an example of point vortex clusters interacting amongst themselves. We track the centroids of the individual vortex clusters to evaluate the error in bulk motion of the point vortices in the sparsified setup. To further improve the accuracy in predicting the nonlinear behavior of the vortices, resparsification strategies are employed for the sparsified interaction-based models. The model retains the nonlinearity of the interaction and also conserves the invariants of discrete vortex dynamics; namely the Hamiltonian, linear impulse, and angular impulse as well as circulation. [Preview Abstract] |
Monday, November 24, 2014 12:14PM - 12:27PM |
H17.00009: Global Model Reduction for the Aerodynamics of Coupled Fluid-Structure Systems Haotian Gao, Mingjun Wei We have recently developed a global approach for model order reduction of dynamic problems involving coupled fluid-structure systems. The approach is based on but different from traditional POD-Galerkin projection method, which is usually applied on fluid flow with fixed solid boundaries (or infinite domain). To consider moving boundaries/structures, instead, we work on a modified Navier-Stokes equation for the combined fluid-solid domain where body forcing terms are added for the description of solid motion. Then, POD modes can be easily computed in the combined fluid-solid domain, and so is the Galerkin projection. However, our earlier model required time-consuming integration at every time steps to count for the contribution from solid motion. In the current work, we decompose the solid motion to base functions and reduce the integration time from the number of time steps to a much lower number of representative modes of solid motion. A separate dynamic equation is developed to describe the evolution of these modes of solid motion to further simplify the process and allow fully-coupled fluid-structure interaction to be considered. The accuracy and efficiency of the new approach are demonstrated in both canonical cases (e.g. oscillatory cylinder) and practical applications. [Preview Abstract] |
Monday, November 24, 2014 12:27PM - 12:40PM |
H17.00010: Solitary states in the Taylo-Couette system with a radial temperature gradient Cl\'ement Savaro, Arnaud Prigent, Innocent Mutabazi The vertical Taylor-Couette system with a radial temperature gradient exhibits a rich variety of states since the base flow state is a combination of the circular Couette flow and an axial baroclinic flow. Two main control parameters characterize the flow: the Taylor number ($Ta$) for the rotation and the Grashof number ($Gr$) for the temperature difference. For small values of $Gr$, the critical state is the Taylor vortices, and for large values of $Gr$, the critical states appear either in form of helicoidal vortices or modulated waves. For a fixed value of $Gr$, increasing $Ta$ leads to the appearance of higher instability modes where helicoidal vortices or traveling waves bifurcate into contrarotating vortices. A special attention will be focused on the states observed for $|Gr| > 1500$ and $Ta \simeq 12$ when the base state bifurcates to a state of modulated wave. A small increase of $Ta$ leads to the appearance of a solitary wave which is superimposed to the modulated wave state. Using visualization technique and particle image velocimetry (PIV) coupled with liquid crystal thermography (TLC), we have measured the amplitude of the solitary structure from velocity and temperature fields. The spatial and temporal localizations give the signature of the solitary wave. [Preview Abstract] |
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