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Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases

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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.

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Notes

  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.

  2. Repeated indices denote summation throughout this paper; subscript indices reference modal sums and superscript indices reference spatial inner products.

  3. Important characteristics of time, ensemble averages and the provided signal are not elaborated upon here; they are explored in detail in the literature [3, 14].

  4. The authors recognize the important differences between 2D and 3D turbulence and therefore refer to the flow as chaotic, in which it is known to be in a formal sense. One commonality between 2D and 3D turbulence is the cross-scale energy transfer; this characteristic is what is captured better in the two-basis POD approach and a 2D chaotic flow is therefore an adequate test case.

  5. The authors again nod to the SPOD and its capability to organize modes by frequency; the time-average-based POD modes in this and much existing work each present several frequencies, though their individual coherence is dynamically meaningful in other ways.

  6. The inclusion of this \(\epsilon \) term was found to modestly but notably improve the accuracy of the ROMs, especially at higher Reynolds numbers.

  7. The velocity gradient norm is still a global norm and thus does not ideally focus on exclusively on small-scale dynamics, but it is an improvement at identifying small scales relative to the energy norm.

  8. As has been discussed, the velocity gradient and the enstrophy norms are qualitatively comparable, albeit not identical.

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Acknowledgements

The authors extend their appreciation to A. Bragg, H. Gavin, K. Hall, B. Mann, M. Balajewicz, and T. Witelski for their insight and counsel. The first author is supported by NSF Grant DGE-1644868.

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  1. Duke University, 144 Hudson Hall, Durham, NC, 27708, USA

    Michael W. Lee & Earl H. Dowell

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  1. Michael W. Lee
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Correspondence to Michael W. Lee.

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Lee, M.W., Dowell, E.H. Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases. Nonlinear Dyn 101, 1457–1471 (2020). https://doi.org/10.1007/s11071-020-05833-x

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