The triple decomposition of a fluctuating velocity field in a multiscale flow

A new method for the triple decomposition of a multiscale flow, which is based on the novel optimal mode decomposition (OMD) technique, is presented. OMD provides low order linear dynamics, which fits a given data set in an optimal way and is used to distinguish between a coherent (periodic) part of a flow and a stochastic fluctuation. The method needs no external phase indication since this information, separate for coherent structures associated with each length scale introduced into the flow, appears as the output. The proposed technique is compared against two traditional methods of the triple decomposition, i.e., bin averaging and proper orthogonal decomposition. This is done with particle image velocimetry data documenting the near wake of a multiscale bar array. It is shown that both traditional methods are unable to provide a reliable estimation for the coherent fluctuation while the proposed technique performs very well. The crucial result is that the coherence peaks are not observed within the spectral properties of the stochastic fluctuation derived with the proposed method; however, these properties remain unaltered at the residual frequencies. This proves the method’s capability of making a distinction between both types of fluctuations. The sensitivity to some prescribed parameters is checked revealing the technique’s robustness. Additionally, an example of the method application for analysis of a multiscale flow is given, i.e., the phase conditioned transverse integral length is investigated in the near wake region of the multiscale object array

An adaptive turbulence filter for decomposition of organized turbulent flows

A new decomposition has been developed in which turbulent processes in shear flows may be represented as a combination of organized and more random turbulent motions. Each component is modeled as a summation of its characteristic eddies, of strength that varies in time and space as a function of the entire process. The contribution of all turbulent eddies of the more random component are estimated with an adaptive turbulence filter, which recognizes this component as the orthogonal partner to organized motion, with a power density spectrum of appropriate shape. The decomposition recovers organized motion from time and space series of data in a physically meaningful way, and can be used to characterize interaction between coherent and more random motions. It also provides an estimate for the turbulence in shear flows that are too complex for a meaningful average motion to be identified.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69639/2/PHFLE6-6-5-1775-1.pd

Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows

In this paper, we propose a non-parametric method for state estimation of high-dimensional nonlinear stochastic dynamical systems, which evolve according to gradient flows with isotropic diffusion. We combine diffusion maps, a manifold learning technique, with a linear Kalman filter and with concepts from Koopman operator theory. More concretely, using diffusion maps, we construct data-driven virtual state coordinates, which linearize the system model. Based on these coordinates, we devise a data-driven framework for state estimation using the Kalman filter. We demonstrate the strengths of our method with respect to both parametric and non-parametric algorithms in three tracking problems. In particular, applying the approach to actual recordings of hippocampal neural activity in rodents directly yields a representation of the position of the animals. We show that the proposed method outperforms competing non-parametric algorithms in the examined stochastic problem formulations. Additionally, we obtain results comparable to classical parametric algorithms, which, in contrast to our method, are equipped with model knowledge.Comment: 15 pages, 12 figures, submitted to IEEE TS

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet

Galerkin spectral estimation of vortex-dominated wake flows

We propose a technique for performing spectral (in time) analysis of spatially-resolved flowfield data, without needing any temporal resolution or information. This is achieved by combining projection-based reduced-order modeling with spectral proper orthogonal decomposition. In this method, space-only proper orthogonal decomposition is first performed on velocity data to identify a subspace onto which the known equations of motion are projected, following standard Galerkin projection techniques. The resulting reduced-order model is then utilized to generate time-resolved trajectories of data. Spectral proper orthogonal decomposition (SPOD) is then applied to this model-generated data to obtain a prediction of the spectral content of the system, while predicted SPOD modes can be obtained by lifting back to the original velocity field domain. This method is first demonstrated on a forced, randomly generated linear system, before being applied to study and reconstruct the spectral content of two-dimensional flow over two collinear flat plates perpendicular to an oncoming flow. At the range of Reynolds numbers considered, this configuration features an unsteady wake characterized by the formation and interaction of vortical structures in the wake. Depending on the Reynolds number, the wake can be periodic or feature broadband behavior, making it an insightful test case to assess the performance of the proposed method. In particular, we show that this method can accurately recover the spectral content of periodic, quasi-periodic, and broadband flows without utilizing any temporal information in the original data. To emphasize that temporal resolution is not required, we show that the predictive accuracy of the proposed method is robust to using temporally-subsampled data.Comment: 35 pages, 12 figure