Book contents
- Frontmatter
- Contents
- Contributors
- Prologue
- Part I Paradigms and Tools
- 1 Numerical Simulations and Coarse-Graining
- 2 An Overview of Scale-Resolving Simulation Models for Practical Flows
- 3 Filtering Approaches and Coarse Graining
- 4 Filtered Density Function
- 5 Symmetries, Subgrid-Scale Modeling, and Coarse Graining
- 6 Coarse-Graining Turbulence Using the Mori–Zwanzig Formalism
- 7 Data-Driven Modeling for Coarse Graining
- 8 Verification, Validation, Uncertainty Quantification, and Coarse-Graining
- Part II Challenges
- Epilogue
- Abbreviations
- Index
- References
6 - Coarse-Graining Turbulence Using the Mori–Zwanzig Formalism
from Part I - Paradigms and Tools
Published online by Cambridge University Press: 31 January 2025
Book contents
- Frontmatter
- Contents
- Contributors
- Prologue
- Part I Paradigms and Tools
- 1 Numerical Simulations and Coarse-Graining
- 2 An Overview of Scale-Resolving Simulation Models for Practical Flows
- 3 Filtering Approaches and Coarse Graining
- 4 Filtered Density Function
- 5 Symmetries, Subgrid-Scale Modeling, and Coarse Graining
- 6 Coarse-Graining Turbulence Using the Mori–Zwanzig Formalism
- 7 Data-Driven Modeling for Coarse Graining
- 8 Verification, Validation, Uncertainty Quantification, and Coarse-Graining
- Part II Challenges
- Epilogue
- Abbreviations
- Index
- References
Summary
Originating from irreversible statistical mechanics, the Mori–Zwanzig (M–Z) formalism provides a mathematical procedure for the development of coarse-grained models of complex systems, such as turbulence, that lack scale separation. The M–Z formalism begins with the application of a specialized class of projectors to the governing equations. By leveraging these projectors, the M–Z procedure results in a reduced system, commonly referred to as the generalized Langevin equation (GLE). The GLE encapsulates the system’s behavior on a macroscopic (resolved) scale. The influence of the microscopic (unresolved) scales on resolved scales appears as a convolution integral – often referred to as memory – and an additional noise term. In essence, fully resolved Markovian dynamics is transformed into coarse grained non-Markovian dynamics. The appearance of the memory term in the GLE demonstrates that the coarse-graining procedure leads to nonlocal memory effects, which have to be modeled. This chapter introduces the mathematics behind the projection approach and the derivation of the GLE. Beyond the theoretical developments, the practical application of the M–Z procedure in the construction of subgrid-scale models for large eddy simulations is also presented.
- Type
- Chapter
- Information
- Coarse Graining TurbulenceModeling and Data-Driven Approaches and their Applications, pp. 177 - 201Publisher: Cambridge University PressPrint publication year: 2025
References
Bazilevs, Y., Calo, V. M., Cottrell, J. A., Hughes, T. J. R., Reali, A., and Scovazzi, G. 2007. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Computer Methods in Applied Mechanics and Engineering, 197(1–4), 173–201.CrossRefGoogle Scholar
Bernstein, David. 2007. Optimal prediction of Burgers’s equation. Multiscale Modeling and Simulation, 6(1), 27–52.CrossRefGoogle Scholar
Castillo, E., and Codina, R. 2019. Dynamic term-by-term stabilized finite element formulation using orthogonal subgrid-scales for the incompressible Navier–Stokes problem. Computer Methods in Applied Mechanics and Engineering, 349, 701–721.CrossRefGoogle Scholar
Chorin, A., Hald, O., and Kupferman, R. 2002. Optimal prediction with memory. Journal of Physics D, 166(3–4), 239–257.Google Scholar
Codina, Ramon. 2000. Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Computer Methods in Applied Mechanics and Engineering, 190(13), 1579–1599.CrossRefGoogle Scholar
Codina, Ramon. 2002. Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Computer Methods in Applied Mechanics and Engineering, 191(39), 4295–4321.CrossRefGoogle Scholar
Codina, Ramon. 2008. Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Applied Numerical Mathematics, 58(3), 264–283.CrossRefGoogle Scholar
Codina, Ramon, and Blasco, Jordi. 2002. Analysis of a stabilized finite element approximation of the transient convection–diffusion–reaction equation using orthogonal subscales. Computing and Visualization in Science, 4(3), 167–174.CrossRefGoogle Scholar
Codina, Ramon, Principe, Javier, Guasch, Oriol, and Badia, Santiago. 2007. Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Computer Methods in Applied Mechanics and Engineering, 196(21), 2413–2430.CrossRefGoogle Scholar
Franca, L. P., Frey, S. L., and Hughes, T. J. R. 1992. Stabilized finite element methods: 1. Application to the advective–diffusive model. Computer Methods in Applied Mechanics and Engineering, 95(2), 253–276.CrossRefGoogle Scholar
Germano, M. 1992. Turbulence: the filtering approach. Journal of Fluid Mechanics, 238, 325–336.CrossRefGoogle Scholar
Ginard, Margarida Moragues. 2015. Variational multiscale stabilization and local preconditioning for compressible flow. Ph.D. thesis, Universitat Politécnica de Catalunya.Google Scholar
Gouasmi, Ayoub, Parish, Eric J., and Duraisamy, Karthik. 2017. A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori–Zwanzig formalism. Proceedings of the Royal Society of London A, 473(20170385).CrossRefGoogle Scholar
Hald, Ole H., and Stinis, P. 2007. Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions. Proceedings of the National Academy of Sciences, 104(16), 6527–6532.CrossRefGoogle ScholarPubMed
Heppe, Burkhard MO. 1998. Generalized Langevin equation for relative turbulent dispersion. Journal of Fluid Mechanics, 357, 167–198.CrossRefGoogle Scholar
Hughes, Thomas J. R. 1995. Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgridscale models, bubbles and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering, 127, 387–401.CrossRefGoogle Scholar
Hughes, Thomas J. R., Feijoo, GR, Mazzei, L, and Qunicy, J. B. 1998. The variational multiscale method – a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 166, 173–189.CrossRefGoogle Scholar
Hughes, T. J. R., and Sangalli, G. 2007. Variational multiscale analysis: The fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM Journal on Numerical Analysis, 45(2), 539–557.CrossRefGoogle Scholar
Lee, Wonjung. 2021. Quasi-Markovian property of strong wave turbulence. Physical Review E, 103(5), 052101.CrossRefGoogle ScholarPubMed
Lin, Yen Ting, Tian, Yifeng, Livescu, Daniel, and Anghel, Marian. 2021. Data-driven learning for the Mori–Zwanzig formalism: A generalization of the Koopman learning framework. SIAM Journal on Applied Dynamical Systems, 20(4), 2558–2601.CrossRefGoogle Scholar
Moin, P., and Kim, J. 1980. On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries. Journal of Computational Physics, 381–392.CrossRefGoogle Scholar
Moser, R., Kim, J., and Mansour, N. 1999. Direct numerical simulation of channel flow up to Reτ = 590. Physics of Fluids, 11(4), 943–945.CrossRefGoogle Scholar
Parish, Eric J. 2018. Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations. Ph.D. thesis, University of Michigan.Google Scholar
Parish, Eric, and Duraisamy, Karthik. 2016. Reduced order modeling of turbulent flows using statistical coarse-graining. In: 46th AIAA Fluid Dynamics Conference, AIAA AVIATION Forum.CrossRefGoogle Scholar
Parish, Eric J., and Duraisamy, Karthik. 2017a. A dynamic subgrid scale model for Large Eddy Simulations based on the Mori–Zwanzig formalism. Journal of Computational Physics, 349, 154–175.CrossRefGoogle Scholar
Parish, Eric J., and Duraisamy, Karthik. 2017b. Non-Markovian closure models for large eddy simulation using the Mori–Zwanzig Formalism. Physical Review Fluids, 2(1), 014604.CrossRefGoogle Scholar
Parish, Eric J., and Duraisamy, Karthik. 2017c. A Unified Framework for Multiscale Modeling using the Mori–Zwanzig Formalism and the Variational Multiscale Method. ArXiv:1712.09669.Google Scholar
Pradhan, Aniruddhe, and Duraisamy, Karthik. 2020. Variational multiscale closures for finite element discretizations using the Mori–Zwanzig approach. Computer Methods in Applied Mechanics and Engineering, 368, 113152.CrossRefGoogle Scholar
Price, Jacob, and Stinis, Panos. 2017. Renormalized reduced order models with memory for long time prediction. ArXiv:1707.01955v1.Google Scholar
Price, Jacob R. 2018. Multiscale Techniques for Nonlinear Dynamical Systems: Applications and Theory. Ph.D. thesis, University of Washington.Google Scholar
Rogallo, Robert S. 1981 (September). Numerical Experiments in Homogeneous Turbulence. Technical Memorandum 81315. NASA.Google Scholar
Sondak, David, and Oberai, Assad A. 2012. Large eddy simulation models for incompressible magnetohydrodynamics derived from the variational multiscale formulation. Physics of Plasmas, 19(10), 102308.CrossRefGoogle Scholar
Stinis, Panos. 2012. Mori–Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty. ArXiv:1211.4285.Google Scholar
Stinis, Panos. 2015. Renormalized Mori–Zwanzig reduced models for systems without scale separation. Proceedings of the Royal Society of London A, 471(20140446).CrossRefGoogle Scholar
Taylor, G. I., and Green, A. E. 1937. Mechanism of the production of small eddies from large ones. Proceedings of the Royal Society of London A, 158, 499–521.Google Scholar
Tian, Yifeng, Lin, Yen Ting, Anghel, Marian, and Livescu, Daniel. 2021. Data-driven learning of Mori–Zwanzig operators for isotropic turbulence. Physics of Fluids, 33(12).CrossRefGoogle Scholar
Wang, Z., and Oberai, A. A. 2010. Spectral analysis of the dissipation of the residual-based variational multiscale method. Computer Methods in Applied Mechanics and Engineering, 199(13–16).CrossRefGoogle Scholar
Woodward, Michael, Tian, Yifeng, Mohan, Arvind T, Lin, Yen Ting, Hader, Christoph, Livescu, Daniel, Fasel, Hermann F, and Chertkov, Misha. 2023. Data-driven Mori–Zwanzig: Approaching a reduced order model for hypersonic boundary layer transition. Page 1624 of: AIAA SCITECH 2023 Forum.CrossRefGoogle Scholar
Zhu, Yuanran. 2021. Nonequilibrium statistical mechanics for stationary turbulent dispersion. ArXiv:2106.07120.Google Scholar
Zhu, Yuanran, and Venturi, Daniele. 2017. Faber approximation to the Mori–Zwanzig equation. ArXiv:1708.03806v2.Google Scholar