Long-time Reynolds averaging of reduced order models for fluid flows: Preliminary results. (English) Zbl 1511.76016
Summary: 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.
Keywords:
reduced-order model; long-time behavior; statistical equilibrium problem; eigenfunction; Stokes operator; proper orthogonal decomposition; time-average energy exchange; Burgers equationSoftware:
redbKITReferences:
| [1] | Batchelor GK, The Theory of Homogeneous Turbulence, Cambridge University Press (1953) · Zbl 0053.14404 |
| [2] | Berselli LC; Iliescu T.; Layton WJ, Mathematics of Large Eddy Simulation of TurbulentFlows, Berlin: Springer-Verlag (2006) · Zbl 1089.76002 |
| [3] | Berselli LC; Lewandowski R., 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 (2019) · Zbl 1423.76089 · doi:10.1088/1361-6544/ab32bc |
| [4] | Berselli LC; Fagioli S.; Spirito S., Suitable weak solutions of the Navier-Stokes equations constructed by a space-time numerical discretization, J Math Pures Appl, 125, 189-208 (2019) · Zbl 1414.35143 |
| [5] | Rebollo TC; Lewandowski R., Mathematical and Numerical Foundations of TurbulenceModels and Applications, New York: Springer (2014) · Zbl 1328.76002 |
| [6] | Constantin P.; Foias C., Navier-Stokes Equations, Chicago: University of Chicago Press (1988) · Zbl 0687.35071 |
| [7] | Couplet M.; Sagaut P.; Basdevant C., Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow, J Fluid Mech, 491, 275-284 (2003) · Zbl 1063.76570 |
| [8] | DeCaria V.; Layton WJ; McLaughlin M., A conservative, second order, unconditionally stable artificial compression method, Comput Methods Appl Mech Engrg, 325, 733-747 (2017) · Zbl 1439.76059 |
| [9] | DeCaria V.; Iliescu T.; Layton W.; et al., An artificial compression reduced order model, arXiv preprint arXiv:1902.09061. (2019) |
| [10] | Girault V.; Raviart PA, Finite Element Methods for Navier-Stokes Equations, Berlin: Springer-Verlag (1986) · Zbl 0585.65077 |
| [11] | Foiaş C. (1972/73) Statistical study of Navier-Stokes equations, I, Ⅱ, Rend Sem Mat Univ Padova, 48, 219-348 (0) · Zbl 0283.76017 |
| [12] | Foias C.; Manley O.; Rosa R.; et al., Navier-Stokes Equations and Turbulence, Cambridge: Cambridge University Press (2001) · Zbl 0994.35002 |
| [13] | Guermond JL; Minev P.; Shen J., An overview of projection methods for incompressible flows, Comput Methods Appl Mech Engrg, 195, 6011-6045 (2006) · Zbl 1122.76072 |
| [14] | Guermond JL; Oden JT; Prudhomme S., Mathematical perspectives on large eddy simulation models for turbulent flows, J Math Fluid Mech, 6, 194-248 (2004) · Zbl 1094.76030 |
| [15] | Hesthaven JS; Rozza G.; Stamm B., Certified Reduced Basis Methods for ParametrizedPartial Differential Equations, Berlin: Springer (2016) · Zbl 1329.65203 |
| [16] | Iliescu T.; Wang Z., Are the snapshot difference quotients needed in the proper orthogonal decomposition, SIAM J Sci Comput, 36, A1221-A1250 (2014) · Zbl 1297.65092 |
| [17] | Iliescu T.; Liu H.; Xie X., Regularized reduced order models for a stochastic Burgers equation Int J Numer Anal Mod 15: 594-607 (2018) · Zbl 1397.35349 |
| [18] | Jiang N.; Layton WJ, Algorithms and models for turbulence not at statistical equilibrium, Comput Math Appl, 71, 2352-2372 (2016) · Zbl 1443.76138 |
| [19] | Kolmogorov AN, The local structure of turbulence in incompressible viscous fluids for very large Reynolds number, Dokl Akad Nauk SSR, 30, 9-13 (1941) |
| [20] | Kunisch K.; Volkwein S., Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J Optim Theory Appl, 102, 345-371 (1999) · Zbl 0949.93039 |
| [21] | Kunisch K.; Volkwein S., Galerkin proper orthogonal decomposition methods for parabolic problems, Numer Math, 90, 117-148 (2001) · Zbl 1005.65112 |
| [22] | Kunisch K.; Volkwein S.; Xie L., HJB-POD-based feedback design for the optimal control of evolution problems, SIAM J Appl Dyn Syst, 3, 701-722 (2004) · Zbl 1058.35061 |
| [23] | Kunisch K.; Xie L., POD-based feedback control of the Burgers equation by solving the evolutionary HJB equation, Comput Math Appl, 49, 1113-1126 (2005) · Zbl 1080.93012 |
| [24] | Kunisch K.; Volkwein S., Proper orthogonal decomposition for optimality systems, ESAIM: Math Model Numer Anal, 42, 1-23 (2008) · Zbl 1141.65050 |
| [25] | Lassila T.; Manzoni A.; Quarteroni A.; et al., Model order reduction in fluid dynamics: Challenges and perspectives, In: Reduced order methods for modeling and computationalreduction, Springer, 9, 235-273 (2014) · Zbl 1395.76058 |
| [26] | Layton WJ, The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow, Technical Report TR-MATH, 14-07 (2014) |
| [27] | Layton WJ; Rebholz L., Approximate Deconvolution Models of Turbulence ApproximateDeconvolution Models of Turbulence, Heidelberg: Springer (2012) · Zbl 1241.76002 |
| [28] | Lewandowski R., Long-time turbulence model deduced from the Navier-Stokes equations, Chin Ann Math Ser B, 36, 883-894 (2015) · Zbl 1327.35301 |
| [29] | Lions JL; , Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Paris: Dunod (1969) · Zbl 0189.40603 |
| [30] | Málek J.; Nečas J.; Rokyta M.; et al., Weak and Measure-valued Solutions to EvolutionaryPDEs, London: Chapman & Hall (1996) · Zbl 0851.35002 |
| [31] | Park HM; Jang YD, Control of Burgers equation by means of mode reduction, Int J of Eng Sci, 38, 785-805 (2000) · Zbl 1210.49032 |
| [32] | Prandtl L., Bericht über Untersuchungen zur ausgebildeten Turbulenz, Z Angew Math Mech, 5, 136-139 (1925) · JFM 51.0666.08 · doi:10.1002/zamm.19250050212 |
| [33] | Prodi G., Teoremi ergodici per le equazioni della idrodinamica, In: Sistemi Dinamici eTeoremi Ergodici, Berlin: Springer, 159-177 (1960) · Zbl 0117.10504 |
| [34] | Prodi G., On probability measures related to the Navier-Stokes equations in the 3-dimensional case, Technical Report AF61(052)-414, Trieste Univ. (1961) |
| [35] | Quarteroni A.; Manzoni A.; Negri F., Reduced Basis Methods for Partial DifferentialEquations, Berlin: Springer (2016) · Zbl 1337.65113 |
| [36] | Quarteroni A.; Rozza G.; Manzoni A., Certified reduced basis approximation for parametrized partial differential equations and applications, J Math Ind, 1, 3 (2011) · Zbl 1273.65148 |
| [37] | Reynolds O., 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 (1895) · JFM 26.0872.02 |
| [38] | Rozza G., 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 (2014) · Zbl 1312.93027 |
| [39] | Sagaut P., Large Eddy Simulation for Incompressible Flows, Berlin: Springer-Verlag. (2001) · Zbl 0964.76002 |
| [40] | San O.; Maulik R., Neural network closures for nonlinear model order reduction, Adv Comput Math, 44, 1717-1750 (2018) · Zbl 1404.37101 |
| [41] | Wells D.; Wang Z.; Xie X.; et al., An evolve-then-filter regularized reduced order model for convection-dominated flows, Internat J. Numer Methods Fluids, 84, 598-615 (2017) |
| [42] | Xie X.; Wells D.; Wang Z.; et al., Approximate deconvolution reduced order modeling, Comput Methods Appl Mech Engrg, 313, 512-534 (2017) · Zbl 1439.76038 |
| [43] | Xie X.; Mohebujjaman M.; Rebholz LG; et al., Data-driven filtered reduced order modeling of fluid flows, SIAM J Sci Comput, 40, B834-B857 (2018) · Zbl 1412.65158 |
| [44] | Xie X.; Mohebujjaman M.; Rebholz LG; et al., Lagrangian data-driven reduced order modeling of finite time Lyapunov exponents, arXiv preprint arXiv:1808.05635. (2018) |
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