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No Access Submitted: 02 September 2016 Accepted: 26 May 2017 Published Online: 20 June 2017

  • Low-order dynamical system model of a fully developed turbulent channel flow

Physics of Fluids 29, 065107 (2017); https://doi.org/10.1063/1.4985203
Nicholas Hamilton1, Murat Tutkun2,3, and Raúl Bayoán Cal1
more...View AffiliationsView Contributors
  • Nicholas Hamilton
  • Murat Tutkun
  • Raúl Bayoán Cal
ABSTRACT
A reduced order model of a turbulent channel flow is composed from a direct numerical simulation database hosted at the Johns Hopkins University. Snapshot proper orthogonal decomposition (POD) is used to identify the Hilbert space from which the reduced order model is obtained, as the POD basis is defined to capture the optimal energy content by mode. The reduced order model is defined by coupling the evolution of the dynamic POD mode coefficients through their respective time derivative with a least-squares polynomial fit of terms up to third order. Parameters coupling the dynamics of the POD basis are defined in analog to those produced in the classical Galerkin projection. The resulting low-order dynamical system is tested for a range of basis modes demonstrating that the non-linear mode interactions do not lead to a monotonic decrease in error propagation. A basis of five POD modes accounts for 50% of the integrated turbulence kinetic energy but captures only the largest features of the turbulence in the channel flow and is not able to reflect the anticipated flow dynamics. Using five modes, the low-order model is unable to accurately reproduce Reynolds stresses, and the root-mean-square error of the predicted stresses is as great as 30%. Increasing the basis to 28 modes accounts for 90% of the kinetic energy and adds intermediate scales to the dynamical system. The difference between the time derivatives of the random coefficients associated with individual modes and their least-squares fit is amplified in the numerical integration leading to unstable long-time solutions. Periodic recalibration of the dynamical system is undertaken by limiting the integration time to the range of the sampled data and offering the dynamical system new initial conditions. Renewed initial conditions are found by pushing the mode coefficients in the end of the integration time toward a known point along the original trajectories identified through a least-squares projection. Under the recalibration scheme, the integration time of the dynamical system can be extended to arbitrarily large values provided that modified initial conditions are offered to the system. The low-order dynamical system composed with 28 modes employing periodic recalibration reconstructs the spatially averaged Reynolds stresses with similar accuracy as the POD-based turbulence description. Data-driven reduced order models like the one undertaken here are widely implemented for control applications, derive all necessary parameters directly from the input, and compute predictions of system dynamics efficiently. The speed, flexibility, and portability of the reduced order model come at the cost of strict data requirements; the model identification requires simultaneous realizations of mode coefficients and their time derivatives, which may be difficult to achieve in some investigations.

ACKNOWLEDGMENTS

The authors would like to acknowledge the U.S. National Science Foundation for supporting the above work (Grant Nos. NSF-ECCS-1032647 and NSF-CBET-1034581). In addition, Hamilton is grateful for support from National Science Foundation IGERT Grant No. 0966376. Work by M. Tutkun has been partially financed by the research project DOMT, funded by the Research Council of Norway with Project No. 231491 under the FRINATEK program. This work also benefited greatly from the constructive and detailed comments of the reviewers.

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