Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows

Abstract

Turbulence modeling has been always a challenge, given the degree of underlying spatial and temporal complexity. In this paper, we propose the use of a partitioned reduced order modeling (ROM) approach for efficient and effective approximation of turbulent flows. A piecewise linear subspace is tailored to capture the fine flow details in addition to the larger scales. We test the partitioned ROM for a decaying two-dimensional (2D) turbulent flow, known as 2D Kraichnan turbulence. The flow is initiated using an array of random vortices, corresponding to an arbitrary energy spectrum. We show that partitioning produces more accurate and stable results than standard ROM based on a global application of modal decomposition techniques. We also demonstrate the predictive capability of partitioned ROM through an energy spectrum analysis, where the recovered energy spectrum significantly converges to the full order model's statistics with increased partitioning. Although the proposed approach incurs increased memory requirements to store the local basis functions for each partition, we emphasize that it permits the construction of more compact ROMs (i.e., of smaller dimension) with comparable accuracy, which in turn significantly reduces the online computational burden. Therefore, we consider that partitioning acts as a converter which reduces the cost of online deployment at the expense of offline and memory costs. Finally, we investigate the application of closure modeling to account for the effects of modal truncation on ROM dynamics. We illustrate that closure techniques can help to stabilize the results in the inertial range, but over-stabilization might take place in the dissipative range. Fluids 2020, 5, 26 2 of 25Challenges in the development of ROM for turbulent flows are also related to the Kolmogorov n-width for these systems. The Kolmogorov n-width is a concept from approximation theory that determines the linear reducibility of a system [65,66]. Mathematically, it is defined as [66][67][68] where S n is a linear n-dimensional subspace, M is the solution manifold, and Π S n is the orthogonal projector onto S n . In other words, d n (M) quantifies the maximum possible error that might arise from the projection of solution manifold onto the best-possible n-dimensional linear subspace. When the decay of d n (M) with increasing n is fast, this means that the energy cascade and modal hierarchy allow the reduction of the system and the solution trajectory can be adequately approximated to evolve on a reduced linear subspace. However, for systems with strong nonlinearity, like turbulent flows, the intense interactions between different modes reduce the decay rate of the Kolmogorov n-width.Consequently, the linear reducibility is hindered. In ROM context, this is denoted as the "Kolmogorov barrier" because it necessitates involving more modes in ROM to guarantee sufficient approximation of the underlying dynamics. In short, turbulent flows are generally characterized by a slow decay of the Kolmogorov n-width, which ra...

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Reduced-order modeling for Ablowitz-Ladik equation

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Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning

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