Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases
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ORIGINAL PAPER
Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases Michael W. Lee
· Earl H. Dowell
Received: 9 April 2020 / Accepted: 18 July 2020 © Springer Nature B.V. 2020
Abstract A fundamental limitation of fluid flow reduced-order models (ROMs) which utilize the proper orthogonal decomposition is that there is little capability to determine one’s confidence in the fidelity of the ROM a priori. One reason why fluid ROMs are plagued by this issue is that nonlinear fluid flows are fundamentally multi-scale, often chaotic dynamical systems and a single linear spatial basis, however carefully selected, is incapable of ensuring that these characteristics are captured. In this paper, the velocity and the velocity gradient were decomposed using differently optimized linear bases. This enabled an optimization for several dynamically significant flow characteristics within the modal bases. This was accomplished while still ensuring the resulting model is accurate and without iterative methods for constructing the modal bases. Keywords Proper orthogonal decomposition · Reduced-order modeling · Fluid mechanics
1 Introduction Endeavors to simulate turbulent fluid flows can very generally be organized into two categories: predictive and reconstructive. Where predictive models aim to M. Lee (B) · E. Dowell Duke University, 144 Hudson Hall, Durham, NC 27708, USA e-mail: [email protected]
capture fluid dynamics with no a priori knowledge of a system’s behavior aside from its governing equations, reconstructive models aim to recapitulate a system’s dynamics (given some reference information) in a more insightful, efficient, or intuitive way such that additional insight can be gained about the observed physics. Methods including Reynolds-averaged Navier–Stokes simulations (RANS) [1] and Kraichnan’s direct interaction approximation [1,2] fall into the former category; reduced-order models like those based on the proper orthogonal [3,4] or dynamic mode [5] decomposition often fall into the latter category.1 Whether primarily predictive or reconstructive, a model’s stability, accuracy and computational cost must be understood in order for its utility to be established. Predictive models like RANS require turbulence models in order for their accuracy to be acceptable; these turbulence models must be carefully designed in order to yield stable behavior. Kraichnan’s direct interaction approximation is more intrinsically stable and accurate, but also requires a great deal of computational (and intellectual) effort to apply to even canonical flow problems [6]. The more recently developed variational multiscale methods [7,8], named aptly for their focus on the multiscale characteristics of fluid flows, present appealing stability and accuracy characteristics that do 1
Many models of course do not reside exclusively in one category or another; a researcher’s goals in using a model are often just as important as the model itself when identifying an approach in this context.
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