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Long-time Reynolds averaging of reduced order models for fluid flows: Preliminary results

1 Dipartimento di Matematica, Università di Pisa, Pisa, I-56127, Italy
2 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
3 Univ Rennes & INRIA, CNRS–IRMAR UMR 6625 & Fluminance team, Rennes, F-35042, France

This contribution is part of the Special Issue: Nonlinear models in applied mathematics
Guest Editor: Giuseppe Maria Coclite
Link: https://www.aimspress.com/newsinfo/1213.html

Special Issues: Nonlinear models in applied mathematics

We perform a preliminary theoretical and numerical investigation of the time-average of energy exchange among modes of Reduced Order Models (ROMs) of fluid flows. We are interested in the statistical equilibrium problem, and especially in the possible forward and backward average transfer of energy among ROM basis functions (modes). We consider two types of ROM modes: Eigenfunctions of the Stokes operator and Proper Orthogonal Decomposition (POD) modes. We prove analytical results for both types of ROM modes and we highlight the differences between them. We also investigate numerically whether the time-average energy exchange between POD modes is positive. To this end, we utilize the one-dimensional Burgers equation as a simplified mathematical model, which is commonly used in ROM tests. The main conclusion of our numerical study is that, for long enough time intervals, the time-average energy exchange from low index POD modes to high index POD modes is positive, as predicted by our theoretical results.
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Keywords reduced order model; long-time behavior; Reynolds equations; statistical equilibrium

Citation: Luigi C. Berselli, Traian Iliescu, Birgul Koc, Roger Lewandowski. Long-time Reynolds averaging of reduced order models for fluid flows: Preliminary results. Mathematics in Engineering, 2020, 2(1): 1-25. doi: 10.3934/mine.2020001


  • 1. Batchelor GK (1953) The Theory of Homogeneous Turbulence, Cambridge University Press.
  • 2. Berselli LC, Iliescu T, Layton WJ (2006) Mathematics of Large Eddy Simulation of Turbulent Flows, Berlin: Springer-Verlag.
  • 3. Berselli LC, Lewandowski R (2019) On the Reynolds time-averaged equations and the long-time behavior of Leray-Hopf weak solutions, with applications to ensemble averages. Nonlinearity 32: 4579-4608.    
  • 4. Berselli LC, Fagioli S, Spirito S (2019) Suitable weak solutions of the Navier-Stokes equations constructed by a space-time numerical discretization. J Math Pures Appl 125: 189-208.
  • 5. Rebollo TC, Lewandowski R (2014) Mathematical and Numerical Foundations of Turbulence Models and Applications, New York: Springer.
  • 6. Constantin P, Foias C (1988) Navier-Stokes Equations, Chicago: University of Chicago Press.
  • 7. Couplet M, Sagaut P, Basdevant C (2003) Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow. J Fluid Mech 491: 275-284.
  • 8. DeCaria V, Layton WJ, McLaughlin M (2017) A conservative, second order, unconditionally stable artificial compression method. Comput Methods Appl Mech Engrg 325: 733-747.
  • 9. DeCaria V, Iliescu T, Layton W, et al. (2019) An artificial compression reduced order model. arXiv preprint arXiv:1902.09061.
  • 10. Girault V, Raviart PA (1986) Finite Element Methods for Navier-Stokes Equations, Berlin: Springer-Verlag.
  • 11. Foiaş C (1972/73) Statistical study of Navier-Stokes equations. I, Ⅱ. Rend Sem Mat Univ Padova 48: 219-348; 49: 9-123.
  • 12. Foias C, Manley O, Rosa R, et al. (2001) Navier-Stokes Equations and Turbulence, Cambridge: Cambridge University Press.
  • 13. Guermond JL, Minev P, Shen J (2006) An overview of projection methods for incompressible flows. Comput Methods Appl Mech Engrg 195: 6011-6045.
  • 14. Guermond JL, Oden JT, Prudhomme S (2004) Mathematical perspectives on large eddy simulation models for turbulent flows. J Math Fluid Mech 6: 194-248.
  • 15. Hesthaven JS, Rozza G, Stamm B (2016) Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Berlin: Springer.
  • 16. Iliescu T, Wang Z (2014) Are the snapshot difference quotients needed in the proper orthogonal decomposition? SIAM J Sci Comput 36: A1221-A1250.
  • 17. Iliescu T, Liu H, Xie X (2018) Regularized reduced order models for a stochastic Burgers equation Int J Numer Anal Mod 15: 594-607.
  • 18. Jiang N, Layton WJ (2016) Algorithms and models for turbulence not at statistical equilibrium. Comput Math Appl 71: 2352-2372.
  • 19. Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluids for very large Reynolds number. Dokl Akad Nauk SSR 30: 9-13.
  • 20. Kunisch K, Volkwein S (1999) Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J Optim Theory Appl 102: 345-371.
  • 21. Kunisch K, Volkwein S (2001) Galerkin proper orthogonal decomposition methods for parabolic problems. Numer Math 90: 117-148.
  • 22. Kunisch K, Volkwein S, Xie L (2004) HJB-POD-based feedback design for the optimal control of evolution problems. SIAM J Appl Dyn Syst 3: 701-722.
  • 23. Kunisch K, Xie L (2005) POD-based feedback control of the Burgers equation by solving the evolutionary HJB equation. Comput Math Appl 49: 1113-1126.
  • 24. Kunisch K, Volkwein S (2008) Proper orthogonal decomposition for optimality systems. ESAIM: Math Model Numer Anal 42: 1-23.
  • 25. Lassila T, Manzoni A, Quarteroni A, et al. (2014) Model order reduction in fluid dynamics: Challenges and perspectives. In: Reduced order methods for modeling and computational reduction, Springer, 9: 235-273.
  • 26. Layton WJ (2014) The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow. Technical Report TR-MATH 14-07, Pittsburgh Univ.
  • 27. Layton WJ, Rebholz L (2012) Approximate Deconvolution Models of Turbulence Approximate Deconvolution Models of Turbulence, Heidelberg: Springer.
  • 28. Lewandowski R (2015) Long-time turbulence model deduced from the Navier-Stokes equations. Chin Ann Math Ser B 36: 883-894.
  • 29. Lions JL, (1969) Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Paris: Dunod.
  • 30. Málek J, Nečas J, Rokyta M, et al. (1996) Weak and Measure-valued Solutions to Evolutionary PDEs, London: Chapman & Hall.
  • 31. Park HM, Jang YD (2000) Control of Burgers equation by means of mode reduction. Int J of Eng Sci 38: 785-805.
  • 32. Prandtl L (1925) Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z Angew Math Mech 5: 136-139.    
  • 33. Prodi G (1960) Teoremi ergodici per le equazioni della idrodinamica, In: Sistemi Dinamici e Teoremi Ergodici, Berlin: Springer, 159-177.
  • 34. Prodi G (1961) On probability measures related to the Navier-Stokes equations in the 3-dimensional case. Technical Report AF61(052)-414, Trieste Univ.
  • 35. Quarteroni A, Manzoni A, Negri F (2016) Reduced Basis Methods for Partial Differential Equations, Berlin: Springer.
  • 36. Quarteroni A, Rozza G, Manzoni A (2011) Certified reduced basis approximation for parametrized partial differential equations and applications. J Math Ind 1: 3.
  • 37. Reynolds O (1895) On the dynamic theory of the incompressible viscous fluids and the determination of the criterion. Philos Trans Roy Soc London Ser A 186: 123-164.
  • 38. Rozza G (2014) Fundamentals of reduced basis method for problems governed by parametrized PDEs and applications, In: Separated Representations and PGD-based Model Reduction, Vienna: Springer, 153-227.
  • 39. Sagaut P (2001) Large Eddy Simulation for Incompressible Flows. Berlin: Springer-Verlag.
  • 40. San O, Maulik R (2018) Neural network closures for nonlinear model order reduction. Adv Comput Math 44: 1717-1750.
  • 41. Wells D, Wang Z, Xie X, et al. (2017) An evolve-then-filter regularized reduced order model for convection-dominated flows. Internat J. Numer Methods Fluids 84: 598-615.
  • 42. Xie X, Wells D, Wang Z, et al. (2017) Approximate deconvolution reduced order modeling. Comput Methods Appl Mech Engrg 313: 512-534.
  • 43. Xie X, Mohebujjaman M, Rebholz LG, et al. (2018) Data-driven filtered reduced order modeling of fluid flows. SIAM J Sci Comput 40: B834-B857.
  • 44. Xie X, Mohebujjaman M, Rebholz LG, et al. (2018) Lagrangian data-driven reduced order modeling of finite time Lyapunov exponents. arXiv preprint arXiv:1808.05635.


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