Amirkabir Journal of Mechanical Engineering

Amirkabir Journal of Mechanical Engineering

Stabilization of Reduced Order Model for Convection-Diffusion Problems Based on Dynamic Mode Decomposition at High Reynolds Numbers Using Eddy Viscosity Approach

Document Type : Research Article

Authors
Mohammad Kazem Moayyedi 1
Fatemeh Bigdeloo 2
Farshad Sabaghzadeghan 1
1 CFD, Turbulence and Combustion Research Lab., Department of Mechanical Engineering, University of Qom, Iran
2 Space Science and Earth Atmosphere Research Lab., Department of Mechanical Engineering, University of Qom, Iran
10.22060/mej.2022.20801.7327
Abstract
Since the analytical methods have a low accuracy and numerical algorithms are time-consuming with hardware limitations, therefore researchers are interested to develop models with high speed and efficiency. The reduced order model is the method that could be an alternative approach for simulating dynamical systems. These models are mainly developed based on the calculation of the dynamical systems' effective structures. The dynamic mode decomposition method is one of the methods for calculating these basic structures. In this study, using this model and based on the principles of dynamical systems, a reduced order model has been developed for the Burgers equation. The results show that if the Reynolds number increases then the effects of the viscous term in the governing equation are decreased, accordingly the required dissipation of the system to stabilize the numerical solution is reduced. Also, due to the incompleteness of the modes which are selected in the order reduction procedure, the dissipation level of the surrogate model is reduced more. Therefore, by creating an artificial dissipation called the eddy viscosity approach, the stability of the model is enhanced. Finally, by comparing the results obtained from the reduced order model and direct numerical simulation, the accuracy of this model is proven.
Keywords
Dynamic mode decomposition
Reduced order model
Eddy viscosity approach
Burgers equation
Dynamical system
Subjects
  1. M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, in: Selected Papers of JM Burgers, Springer, 1995, pp. 281-334.
  2. Bec, K. Khanin, Burgers turbulence, Physics reports, 447(1-2) (2007) 1-66.
  3. -P. Bouchaud, M. Mézard, Velocity fluctuations in forced Burgers turbulence, Physical Review E, 54(5) (1996) 5116.
  4. Bayona, J. Baiges, R. Codina, Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling, International Journal for Numerical Methods in Fluids, 86(5) (2018) 313-328.
  5. Lumley, The structure of inhomogeneous turbulence. in atmospheric turbulence and wave propagation. Ed. AM Yaglom, VI Tatarski, 1967, in, Moscow: Nauka.
  6. Wu, D. Laurence, S. Utyuzhnikov, I. Afgan, Proper orthogonal decomposition and dynamic mode decomposition of jet in channel crossflow, Nuclear Engineering and Design, 344 (2019) 54-68.
  7. I. Abreu, A.V. Cavalieri, P. Schlatter, R. Vinuesa, D.S. Henningson, Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows, Journal of Fluid Mechanics, 900 (2020).
  8. K. Moayyedi, F. Sabaghzadeghan, Development of parametric and time dependent reduced order model for diffusion and convection-diffusion problems based on proper orthogonal decomposition method, Amirkabir Journal of Mechanical Engineering, 53(7) (2021) 8-8. (In Persian)
  9. J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of fluid mechanics, 656 (2010) 5-28.
  10. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, D.S. Henningson, Spectral analysis of nonlinear flows, Journal of fluid mechanics, 641 (2009) 115-127.
  11. Grilli, P.J. Schmid, S. Hickel, N.A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction, Journal of Fluid Mechanics, 700 (2012) 16-28.
  12. Hong, G. Huang, Introducing DMD method to study dynamic structures of flow separation with and without control, Acta Aeronaut et Astronaut Sin, 38(8) (2017) 10-17.
  13. Duke, J. Soria, D. Honnery, An error analysis of the dynamic mode decomposition, Experiments in fluids, 52(2) (2012) 529-542.
  14. Seena, H.J. Sung, Spatiotemporal representation of the dynamic modes in turbulent cavity flows, International journal of heat and fluid flow, 44 (2013) 1-13.
  15. Kong, C. Li, C. Wang, Y. Zhang, J. Zhang, Short-term electrical load forecasting based on error correction using dynamic mode decomposition, Applied Energy, 261 (2020) 114368.
  16. Hu, C. Yang, W. Yi, K. Hadzic, L. Xie, R. Zou, M. Zhou, Numerical investigation of centrifugal compressor stall with compressed dynamic mode decomposition, Aerospace Science and Technology, 106 (2020) 106153.
  17. Sabaghzadeghan, M. Moayyedi, Reduced Order Model of Conduction Heat Transfer in a Solid Plate Based on Dynamic Mode Decomposition, Sharif Journal of Mechanical Engineering, 37(2) (2021) 3-12. (In Persian)
  18. Sabaghzadeghan, F. (2019). “Development the Reduce Order Model for Convection-Diffusion and Diffusion Problems Based on Proper Orthogonal Decomposition and Dynamic Mode Decomposition Methods.” Thesis for Degree of Master of Science (MSc) In Mechanical Engineering-Energy Conversion, Faculty of Engineering, University of Qom. (In Persian)
Amirkabir Journal of Mechanical Engineering
Volume 54, Issue 11
February 2023
Pages 2479-2498