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4 - Filtered Density Function

A Stochastic Closure for Coarse-Grained Simulation

from Part I - Paradigms and Tools

Published online by Cambridge University Press:  31 January 2025

Fernando F. Grinstein
Affiliation:
Los Alamos National Laboratory
Filipe S. Pereira
Affiliation:
Los Alamos National Laboratory
Massimo Germano
Affiliation:
Duke University, North Carolina
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Summary

An overview is presented of the filtered density function (FDF) methodology as a closure for large eddy simulation (LES) of turbulent reacting flows. The theoretical basis and the solution strategy of LES/FDF are briefly discussed, with the focus on some of the closure issues. Some of the recent applications of LES/FDF are reviewed, along with some speculations about future prospects for such simulations.

Type
Chapter
Information
Coarse Graining Turbulence
Modeling and Data-Driven Approaches and their Applications
, pp. 127 - 150
Publisher: Cambridge University Press
Print publication year: 2025

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References

Aitzhan, A., Sammak, S., Givi, P., and Nouri, A. G. 2022a. PeleLM-FDF large eddy simulator of turbulent reacting flows. Combustion Theory and Modelling, 26(6), 1–18.Google Scholar
Aitzhan, A., Nouri, A. G., Givi, P., and Babaee, H. 2022b. Reduced order modeling of turbulence-chemistry interactions using dynamically bi-orthonormal decomposition. ArXiv:2201.02097.CrossRefGoogle Scholar
Aliramezani, M., Koch, C. R., and Shahbakhti, M. 2022. Modeling, diagnostics, optimization, and control of internal combustion engines via modern machine learning techniques: A review and future directions. Progress in Energy and Combustion Science, 88, 100967.CrossRefGoogle Scholar
Ansari, N., Jaberi, F. A., Sheikhi, M. R. H., and Givi, P. 2011a. Filtered density function as a modern CFD tool. Engineering Applications of Computational Fluid Dynamics, 1, 1–22.Google Scholar
Ansari, N., Goldin, G. M., Sheikhi, M. R. H., and Givi, P. 2011b. Filtered density function simulator on unstructured meshes. Journal of Computational Physics, 230(19), 7132–7150.CrossRefGoogle Scholar
Ansari, N., Pisciuneri, P. H., Strakey, P. A., and Givi, P. 2012. Scalar-filtered mass-density-function simulation of swirling reacting flows on unstructured grids. AIAA Journal, 50(11), 2476–2482.CrossRefGoogle Scholar
Ansari, N., Strakey, P. A., Goldin, G. M., and Givi, P. 2015. Filtered density function simulation of a realistic swirled combustor. Proceedings of the Combustion Institute, 35, 1433–1442.CrossRefGoogle Scholar
Bao, T., Chen, S., Johnson, T. T., Givi, P., Sammak, S., and Jia, X. 2022. Physics guided neural networks for spatio-temporal super-resolution of turbulent flows. Pages 118–128 in Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. PMLR.Google Scholar
Beishuizen, N. A. 2008. PDF modelling and particle-turbulence interaction of turbulent spray flames. Thesis, Delft University of Technology.Google Scholar
Bhaya, R., De, A., and Yadav, R. 2014. Large eddy simulation of mild combustion using PDF-based turbulence-chemistry interaction models. Combustion Science and Technology, 186(9), 1138–1165.CrossRefGoogle Scholar
Bode, M., Gauding, M., Lian, Z., Denker, D., Davidovic, M., Kleinheinz, K., Jitsev, J., and Pitsch, H. 2019. Using physics-informed super-resolution generative adversarial networks for subgrid modeling in turbulent reactive flows. ArXiv:1911.11380.Google Scholar
Borghi, R. 2021. Ted O’Brien and turbulent combustion. Physics of Fluids, 33(8), 080401.CrossRefGoogle Scholar
Bray, K. N. C., Champion, M., Libby, P. A., and Swaminathan, N. 2006. Finite rate chemistry and presumed PDF models for premixed turbulent combustion. Combustion and Flame, 146(4), 665–673.CrossRefGoogle Scholar
Brunton, S. L., Noack, B. R., and Koumoutsakos, P. 2020. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 52, 477–508.CrossRefGoogle Scholar
Cary, A., Chawner, E. D., Gropp, W., Kleb, B., Kolonay, R., Nielsen, E., and Smith, B. 2022. Realizing the vision of CFD in 2030. Computing in Science & Engineering, 24, 64–70.CrossRefGoogle Scholar
Chen, S., Sammak, S., Givi, P., Yurko, J. P., and Jia, X. 2021a. Reconstructing high-resolution turbulent flows using physics-guided neural networks. In: 2021 IEEE International Conference on Big Data. IEEE.Google Scholar
Chen, Z. X., Iavarone, S., Ghiasi, G., Kannan, V., D’Alessio, G., Parente, A., and Swaminathan, N. 2021b. Application of machine learning for filtered density function closure in MILD combustion. Combustion and Flame, 225, 160–179.CrossRefGoogle Scholar
Cleary, M. J., and Klimenko, A. Y. 2011. A detailed quantitative analysis of sparse-Lagrangian filtered density function simulations in constant and variable density reacting jet flows. Physics of Fluids, 23(11), 115102.CrossRefGoogle Scholar
Cleary, M. J., Klimenko, A. Y., Janicka, J., and Pfitzner, M. 2009. A sparse-Lagrangian multiple mapping conditioning model for turbulent diffusion flames. Proceedings of the Combustion Institute, 32 I(1), 1499–1507.Google Scholar
Colucci, P. J., Jaberi, Farhad A., Givi, P., and Pope, S. B. 1998. Filtered density function for large eddy simulation of turbulent reacting flows. Physics of Fluids, 10(2), 499–515.CrossRefGoogle Scholar
Curl, R. L. 1963. Dispersed phase mixing: I. Theory and effects in simple reactors. AIChE Journal, 9(2), 175–181.CrossRefGoogle Scholar
Damasceno, M. M. R., de Freitas Santos, J. G., and Vedovoto, J. M. 2018. Simulation of turbulent reactive flows using a FDF methodology: Advances in particle density control for normalized variables. Computers & Fluids, 170, 128–140.CrossRefGoogle Scholar
Dopazo, C. 1973. Non-isothermal turbulent reactive flows: Stochastic approaches. Thesis, State University of New York at Stony Brook.Google Scholar
Dopazo, C. 1976. A probabilistic approach to turbulent flame theory. Acta Astronautica, 3(9-10), 853–878.CrossRefGoogle Scholar
Dopazo, C. 1994. Recent developments in PDF methods. In: Turbulent Reacting Flows. Academic Press.Google Scholar
Dopazo, C., and O’Brien, E. E. 1974a. An approach to the autoignition of a turbulent mixture. Acta Astronautica, 1(9), 1239–1266.CrossRefGoogle Scholar
Dopazo, C., and O’Brien, E. E. 1974b. Functional formulation of nonisothermal turbulent reactive flows. Physics of Fluids, 17(11), 1968–1975.Google Scholar
Drozda, T. G., Sheikhi, M. R. H., Madnia, C. K., and Givi, P. 2007. Developments in formulation and application of the filtered density function. Flow, Turbulence and Combustion, 78(1), 35–67.Google Scholar
Duraisamy, K. 2021. Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence. Physical Review Fluids, 6(5), 050504.CrossRefGoogle Scholar
Durbin, P. A., and Speziale, C. G. 1994. Realizability of second-moment closure via stochastic analysis. Journal of Fluid Mechanics, 280, 395–407.CrossRefGoogle Scholar
Fox, R. O. 2003. Computational Models for Turbulent Reacting Flows. Cambridge University Press.CrossRefGoogle Scholar
Frankel, S. H., Jiang, T. L., and Givi, P. 1992. Modeling of isotropic reacting turbulence by a hybrid mapping-EDQNM closure. AIChE Journal, 38(4), 535–543.CrossRefGoogle Scholar
Frankel, S. H., Adumitroaie, V., Madnia, C. K., and Givi, P. 1993. Large eddy simulations of turbulent reacting flows by assumed PDF methods. In: Engineering Applications of Large Eddy Simulations. ASME.Google Scholar
Fukami, K., Fukagata, K., and Taira, K. 2019. Super-resolution reconstruction of turbulent flows with machine learning. Journal of Fluid Mechanics, 870(May), 106–120.CrossRefGoogle Scholar
Galindo-Lopez, S., Salehi, F., Cleary, M. J., Masri, A. R., Neuber, G., Stein, O. T., Kronenburg, A., Varna, A., Hawkes, E. R., Sundaram, B., et al. 2018. A stochastic multiple mapping conditioning computational model in OpenFOAM for turbulent combustion. Computers & Fluids, 172, 410–425.CrossRefGoogle Scholar
Gao, F., and O’Brien, E. E. 1993. A large-eddy simulation scheme for turbulent reacting flows. Physics of Fluids, 5(6), 1282–1284.Google Scholar
Garnier, E., Adams, N., and Sagaut, P. 2009. Large Eddy Simulation for Compressible Flows. Springer.CrossRefGoogle Scholar
Gatski, T. B., and Jongen, T. 2000. Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Progress in Aerospace Sciences, 36(8), 655–682.CrossRefGoogle Scholar
Gatski, Thomas B, Sarkar, Sutanu, and Speziale, Charles G. 2012. Studies in Turbulence. Springer.Google Scholar
Ge, H. 2006. Probability density function modeling of turbulent non-reactive and reactive spray flows. Thesis, Ruprecht-Karls-Universität Heidelberg.Google Scholar
Ge, H., and Gutheil, E. 2006. Probability density function (PDF) simulation of turbulent spray flows. Atomization and Sprays, 16(5), 531–542.CrossRefGoogle Scholar
Ge, H., and Gutheil, E. 2008. Simulation of a turbulent spray flame using coupled PDF gas phase and spray flamelet modeling. Combustion and Flame, 153(1-2), 173–185.CrossRefGoogle Scholar
Ge, H., Düwel, I., Kronemayer, H., Dibble, R. W., Gutheil, E., Schulz, C., and Wolfrum, J. 2008. Laser-based experimental and Monte Carlo PDF numerical investigation of an ethanol/air spray flame. Combustion Science and Technology, 180(8), 1529–1547.CrossRefGoogle Scholar
Ge, Y., Cleary, M. J., and Klimenko, A. Y. 2013. A comparative study of Sandia flame series (D-F) using sparse-Lagrangian MMC modelling. Proceedings of the Combustion Institute, 34(1), 1325–1332.CrossRefGoogle Scholar
Georgiadis, N. J., Rizzetta, D. P., and Fureby, C. 2010. Large-eddy simulation: Current capabilities, recommended practices, and future research. AIAA Journal, 48(8), 1772–1784.CrossRefGoogle Scholar
Germano, M. 1992. Turbulence: The filtering approach. Journal of Fluid Mechanics, 238, 325–336.CrossRefGoogle Scholar
Germano, M. 1996. A statistical formulation of dynamic model. Physics of Fluids, 8(2), 565–570.CrossRefGoogle Scholar
Germano, M., Piomelli, U., Moin, P., and Cabot, W. H. 1991. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A: Fluid Dynamics, 3(7), 1760–1765.CrossRefGoogle Scholar
Gicquel, L. Y. M., Givi, P., Jaberi, F. A., and Pope, S. B. 2002. Velocity filtered density function for large eddy simulation of turbulent flows. Physics of Fluids, 14(3), 1196–1213.CrossRefGoogle Scholar
Givi, P. 1989. Model-free simulations of turbulent reactive flows. Progress in Energy and Combustion Science, 15(1), 1–107.Google Scholar
Givi, P. 2006. Filtered density function for subgrid scale modeling of turbulent combustion. AIAA Journal, 44(1), 16–23.CrossRefGoogle Scholar
Givi, P. 2021. Perspective: Machine learning and quantum computing for reactive turbulence modeling and simulation. Mechanics Research Communications, 116, 103759.CrossRefGoogle Scholar
Givi, P., Daley, A. J., Mavriplis, D., and Malik, M. 2020. Quantum speedup for aeroscience and engineering. AIAA Journal, 58(8), 3715–3727.CrossRefGoogle Scholar
Gounder, J. D., Kourmatzis, A., and Masri, A. R. 2012. Turbulent piloted dilute spray flames: Flow fields and droplet dynamics. Combustion and flame, 159(11), 3372–3397.CrossRefGoogle Scholar
Gourianov, N. 2022. Exploiting the structure of turbulence with tensor networks. Thesis, University of Oxford.Google Scholar
Gourianov, N., Lubasch, M., Dolgov, S., van den Berg, Q. Y., Babaee, H., Givi, P., Kiffner, M., and Jaksch, D. 2022. A quantum-inspired approach to exploit turbulence structures. Nature Computational Science, 2, 30–37.CrossRefGoogle ScholarPubMed
Grigoriu, M. 1995. Applied Non-Gaussian Processes. Prentice-Hall.Google Scholar
Han, W., Raman, V., and Chen, Z. 2016. LES/PDF modeling of autoignition in a lifted turbulent flame: Analysis of flame sensitivity to differential diffusion and scalar mixing time-scale. Combustion and Flame, 171, 69–86.CrossRefGoogle Scholar
Haworth, D. C., and Pope, S. B. 1986. A second-order Monte Carlo method for the solution of the Ito stochastic differential equation. Stochastic Analysis and Applications, 151–186.CrossRefGoogle Scholar
Haworth, D. C., and Pope, S. B. 1987a. Monte Carlo solutions of a joint PDF equation for turbulent flows in general orthogonal coordinates. Journal of Computational Physics, 72(2), 311–346.CrossRefGoogle Scholar
Haworth, D. C, and Pope, S. B. 1987b. A PDF modeling study of self-similar turbulent free shear flows. Physics of Fluids, 30(4), 1026–1044.Google Scholar
Haworth, D. C. 2010. Progress in probability density function methods for turbulent reacting flows. Progress in Energy and Combustion Science, 36(2), 168–259.CrossRefGoogle Scholar
Heinz, S. 2007. Unified turbulence models for LES and RANS, FDF and PDF simulations. Theoretical and Computational Fluid Dynamics, 21(2), 99–118.CrossRefGoogle Scholar
Hopf, E. 1952. Statistical hydromechanics and functional calculus. Journal National Mechanics and Analysis, 1, 87–123.Google Scholar
Ievlev, V. M. 1973. Equations for the finite-dimensional probability distributions of pulsating variables in a turbulent flow. Dokl. Akad. Nauk SSSR, 208(5), 1044–1047. English translation: Soviet Physics Doklady 18, 117–119.Google Scholar
Ihme, M., Chung, W. T., and Mishra, A. A. 2022. Combustion machine learning: Principles, progress and prospects. Progress in Energy and Combustion Science, 91, 101010.CrossRefGoogle Scholar
Inkarbekov, M., Aitzhan, A., Kaltayev, A., and Sammak, S. 2020. A GPU-accelerated filtered density function simulator of turbulent reacting flows. International Journal of Computational Fluid Dynamics, 34(6), 381–396.CrossRefGoogle Scholar
Jaberi, F. A., Colucci, P. J., James, S., Givi, P., and Pope, S. B. 1999. Filtered mass density function for large-eddy simulation of turbulent reacting flows. Journal of Fluid Mechanics, 401, 85–121.CrossRefGoogle Scholar
Janicka, J., Kolbe, W., and Kollmann, W. 1979. Closure of the transport equation for the probability density function of turbulent scalar fields. Journal of Non-Equilibrium Thermodynamics, 4(1), 47–66.CrossRefGoogle Scholar
Jenny, P., Roekaerts, D., and Beishuizen, N. 2012. Modeling of turbulent dilute spray combustion. Progress in Energy and Combustion Science, 38(6), 846–887.CrossRefGoogle Scholar
Jiang, T.-L., and O’Brien, E. E. 1991. Simulation of scalar mixing by stationary isotropic turbulence. Physics of Fluids, 3(6), 1612–1624.Google Scholar
Klimenko, A. Y., and Pope, S. B. 2003. The modeling of turbulent reactive flows based on multiple mapping conditioning. Physics of Fluids, 15(7), 1907–1925.CrossRefGoogle Scholar
Kloeden, P. E., Platen, E., and Schurz, H. 1997. Numerical Solution of Stochastic Differential Equations through Computer Experiments. 2nd ed. Springer.Google Scholar
Kolla, H., Rogerson, J. W., Chakraborty, N., and Swaminathan, N. 2009. Scalar dissipation rate modeling and its validation. Combustion Science and Technology, 181(3), 518–535.CrossRefGoogle Scholar
Kolla, H. 2010. Scalar dissipation rate based flamelet modelling of turbulent premixed flames. Thesis, University of Cambridge.Google Scholar
Kuo, K. K., and Acharya, R. 2012. Fundamentals of Turbulent and Multiphase Combustion. Wiley.CrossRefGoogle Scholar
Kuo, Y.-Y., and O’Brien, E. E. 1981. Two‐point probability density function closure applied to a diffusive‐reactive system. Physics of Fluids, 24(2), 194–201.Google Scholar
Kuron, M., Ren, Z., Hawkes, E. R., Zhou, H., Kolla, H., Chen, J. H., and Lu, T. 2017. A mixing timescale model for TPDF simulations of turbulent premixed flames. Combustion and Flame, 177, 171–183.CrossRefGoogle Scholar
Ladeinde, F., Givi, P., and Dopazo, C. 2021. Preface to special issue: In memory of Edward E. (Ted) O’Brien. Physics of Fluids, 33(8), 080402.CrossRefGoogle Scholar
Launder, B. E., and Spalding, D. B. 1972. Lectures in Mathematical Modeling of Turbulence. Academic Press.Google Scholar
Lindstedt, R. P., Louloudi, S. A., and Váos, E. M. 2000. Joint scalar probability density function modeling of pollutant formation in piloted turbulent jet diffusion flames with comprehensive chemistry. Proceedings of the Combustion Institute, 28(1), 149–156.CrossRefGoogle Scholar
Liu, J., and Wang, H. 2022. Machine learning assisted modeling of mixing timescale for LES/PDF of high-Karlovitz turbulent premixed combustion. Combustion and Flame, 238, 111895.CrossRefGoogle Scholar
Lu, T., and Law, C. K. 2009. Toward accommodating realistic fuel chemistry in large-scale computations. Progress in Energy and Combustion Science, 35(2), 192–215.CrossRefGoogle Scholar
Lundgren, T. S. 1967. Distribution functions in the statistical theory of turbulence. Physics of Fluids, 10(5), 969–975.Google Scholar
Lundgren, T. S. 1969. Model equation for nonhomogeneous turbulence. Physics of Fluids, 12(3), 485–497.Google Scholar
Lundgren, T. S. 1972. A closure hypothesis for the hierarchy of equations for turbulent probability distribution functions. In: Statistical Models and Turbulence. Springer.Google Scholar
Madnia, C. K., and Givi, P. 1993. Direct numerical simulation and large eddy simulation of reacting homogeneous turbulence. In: Large Eddy Simulations of Complex Engineering and Geophysical Flows. Cambridge University Press.Google Scholar
Madnia, C. K., Jaberi, F. A., and Givi, P. 2006. Large eddy simulation of heat and mass transport in turbulent flows. In: Handbook of Numerical Heat Transfer, 2nd ed. Wiley.Google Scholar
Magnussen, B. 1981. On the structure of turbulence and a generalized eddy dissipation concept for chemical reaction in turbulent flow. In: 19th Aerospace Sciences Meeting.CrossRefGoogle Scholar
McGraw, R. 1997. Description of aerosol dynamics by the quadrature method of moments. Aerosol Science and Technology, 27(2), 255–265.CrossRefGoogle Scholar
Meester, R. 2012. Analysis of scalar mixing in hybrid RANS-PDF calculations of turbulent gas and spray flames. Thesis, Ghent University.Google Scholar
Meier, W., Weigand, P., Duan, X. R., and Giezendanner-Thoben, R. 2007. Detailed characterization of the dynamics of thermoacoustic pulsations in a lean premixed swirl flame. Combustion and Flame, 150(1–2), 2–26.CrossRefGoogle Scholar
Miller, R. S., and Foster, J. W. 2016. Survey of turbulent combustion models for large-eddy simulations of propulsive flowfields. AIAA Journal, 54(10), 2930–2946.CrossRefGoogle Scholar
Mira, D., Pérez Sánchez, E. J., Borrell, R., and Houzeaux, G. 2022. HPC-enabling technologies for high-fidelity combustion simulations. Proceedings of the Combustion Institute.CrossRefGoogle Scholar
Mohammadi, K., Immonen, J., Blackburn, L. D., Tuttle, J. F., Andersson, K., and Powell, K. M. 2022. A review on the application of machine learning for combustion in power generation applications. Reviews in Chemical Engineering.Google Scholar
Mokhtarpoor, R., Turkeri, H., and Muradoglu, M. 2014. A new robust consistent hybrid finite-volume/particle method for solving the PDF model equations of turbulent reactive flows. Computers & Fluids, 105, 39–57.CrossRefGoogle Scholar
Monin, A. S., and Yaglom, A. M. 1975. Statistical Fluid Mechanics. MIT Press.Google Scholar
Muradoglu, M., Jenny, P., Pope, S. B., and Caughey, D. A. 1999. A consistent hybrid finite-volume/particle method for the PDF equations of turbulent reactive flows. Journal of Computational Physics, 154(2), 342–371.CrossRefGoogle Scholar
Naderi, M. H., and Babaee, H. 2023. Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases. Computer Methods in Applied Mechanics and Engineering, 405, 115813.CrossRefGoogle Scholar
Naud, B. 2003. PDF modeling of turbulent sprays and flames using a particle stochastic approach. Thesis, Technische Universiteit Delft.Google Scholar
Nik, M., Yilmaz, S., Sheikhi, M. R. H., and Givi, P. 2010. Grid resolution effects on VSFMDF/LES. Flow, Turbulence and Combustion, 85(3–4), 677–688.CrossRefGoogle Scholar
Nonaka, A., Day, M. S., and Bell, J. B. 2018. A conservative, thermodynamically consistent numerical approach for low Mach number combustion. Part I: Single-level integration. Combustion Theory and Modelling, 22(1), 156–184.CrossRefGoogle Scholar
Nouri, A. G., Nik, M. B., Givi, P., Livescu, D., and Pope, S. B. 2017. Self-contained filtered density function. Physical Review Fluids, 2(9), 094603.CrossRefGoogle Scholar
Nouri, A. G., Babaee, H., Givi, P., Chelliah, H. K., and Livescu, D. 2022. Skeletal model reduction with forced optimally time dependent modes. Combustion and Flame, 235, 111684.CrossRefGoogle Scholar
O’Brien, E. E. 1980. The probability density function (PDF) approach to reacting turbulent flows. In: Turbulent Reacting Flows. Springer.Google Scholar
Owens, J. D., Houston, M., Luebke, D., Green, S., Stone, J. E., and Phillips, J. C. 2008. GPU computing. Proceedings of the IEEE, 96(5), 879–899.CrossRefGoogle Scholar
Pisciuneri, P. H., Yilmaz, S. L., Strakey, P. A., and Givi, P. 2013. An irregularly portioned FDF simulator. SIAM Journal of Scientific Computing, 35(4), C438C452.CrossRefGoogle Scholar
Pitsch, H., and Attili, A. 2020. Data Analysis for Direct Numerical Simulations of Turbulent Combustion. Springer.CrossRefGoogle Scholar
Pope, S. B. 1976. The probability approach to the modelling of turbulent reacting flows. Combustion and Flame, 27, 299–312.CrossRefGoogle Scholar
Pope, S. B. 1977. The implications of the probability equations for turbulent combustion models. Combustion and Flame, 29, 235–246.CrossRefGoogle Scholar
Pope, S. B. 1979a. A rational method of determining probability distributions in turbulent reacting flows. Journal of Non-Equilibrium Thermodynamics, 4, 309–320.CrossRefGoogle Scholar
Pope, S. B. 1979b. The relationship between the probability approach and particle models for reaction in homogeneous turbulence. Combustion and Flame, 35, 41–45.CrossRefGoogle Scholar
Pope, S. B. 1981a. A Monte Carlo method for the PDF equations of turbulent reactive flow. Combustion Science and Technology, 25, 159–174.CrossRefGoogle Scholar
Pope, S. B. 1981b. Transport equation for the joint probability density function of velocity and scalars in turbulent flow. Physics of Fluids, 24(4), 588–596.Google Scholar
Pope, S. B. 1982. An improved turbulent mixing model. Combustion Science and Technology, 28, 131–145.CrossRefGoogle Scholar
Pope, S. B. 1985. PDF methods for turbulent reactive flows. Progress in Energy and Combustion Science, 11(2), 119–192.CrossRefGoogle Scholar
Pope, S. B. 1991. Computations of turbulent combustion: Progress and challenges. Symposium (International) on Combustion, 23(1), 591–612.CrossRefGoogle Scholar
Pope, S. B. 1994a. Lagrangian PDF methods for turbulent flows. Annual Review of Fluid Mechanics, 23, 63.Google Scholar
Pope, S. B. 1994b. On the relationship between stochastic Lagrangian models of turbulence and second-moment closures. Physics of Fluids, 6(2), 973–985.CrossRefGoogle Scholar
Pope, S. B. 1995. Particle method for turbulent flows: Integration of stochastic model equations. Journal of Computational Physics, 117(2), 332–349.CrossRefGoogle Scholar
Pope, S. B. 2000. Turbulent Flows. Cambridge University Press.Google Scholar
Pope, S. B. 2004. Ten questions concerning the large-eddy simulation of turbulent flows. New Journal of Physics, 6(1), 35.CrossRefGoogle Scholar
Pope, S. B. 2013a. A model for turbulent mixing based on shadow-position conditioning. Physics of Fluids, 25(11).CrossRefGoogle Scholar
Pope, S. B. 2013b. Small scales, many species and the manifold challenges of turbulent combustion. Proceedings of the Combustion Institute, 34(1), 1–31.CrossRefGoogle Scholar
Popov, P. P., and Pope, S. B. 2014. Large eddy simulation/probability density function simulations of bluff body stabilized flames. Combustion and Flame, 161(12), 3100–3133.CrossRefGoogle Scholar
Popov, P. P., Wang, H., and Pope, S. B. 2015. Specific volume coupling and convergence properties in hybrid particle/finite volume algorithms for turbulent reactive flows. Journal of Computational Physics, 294, 110–126.CrossRefGoogle Scholar
Raissi, M., Babaee, H., and Givi, P. 2019. Deep learning of turbulent scalar mixing. Physical Review Fluids, 4(Dec), 124501.CrossRefGoogle Scholar
Raju, M. S. 2002. On the importance of chemistry/turbulence interactions in spray computations. Numerical Heat Transfer: Part B: Fundamentals, 41(5), 409–432.CrossRefGoogle Scholar
Raman, V., and Pitsch, H. 2007. A consistent LES/filtered-density function formulation for the simulation of turbulent flames with detailed chemistry. Proceedings of the Combustion Institute, 31, 1711–1719.CrossRefGoogle Scholar
Raman, V., Fox, R. O., and Harvey, A. D. 2004. Hybrid finite-volume/transported PDF simulations of a partially premixed methane–air flame. Combustion and Flame, 136(3), 327–350.CrossRefGoogle Scholar
Raman, V., Pitsch, H., and Fox, R. O. 2006. Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion. Combustion Theory and Modelling, 10(3), 439–458.CrossRefGoogle Scholar
Ramezanian, D., Nouri, A. G., and Babaee, H. 2021. On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds. Computer Methods in Applied Mechanics and Engineering, 382, 113882.CrossRefGoogle Scholar
Ren, J., Wang, H., Wang, C., Luo, K., and Fan, J. 2023. A-priori and a-posterior studies of filtered probability density function models and NO formation prediction in turbulent stratified premixed combustion using machine learning. Fuel, 333, 126358.CrossRefGoogle Scholar
Ren, Z., and Goldin, G. M. 2011. An efficient time scale model with tabulation of chemical equilibrium. Combustion and Flame, 158(10), 1977–1979.CrossRefGoogle Scholar
Ren, Z., Lu, Z., Hou, L., and Lu, L. 2014. Numerical simulation of turbulent combustion: Scientific challenges. Science China Physics, Mechanics & Astronomy, 57(8), 1495–1503.CrossRefGoogle Scholar
Reveillon, J., and Vervisch, L. 2000. Accounting for spray vaporization in non-premixed turbulent combustion modeling: A single droplet model (SDM). Combustion and Flame, 121(1–2), 75–90.CrossRefGoogle Scholar
Rieth, M., Chen, J. Y., Menon, S., and Kempf, A. M. 2019. A hybrid flamelet finite-rate chemistry approach for efficient LES with a transported FDF. Combustion and Flame, 199, 183–193.CrossRefGoogle Scholar
Sabelnikov, V., and Soulard, O. 2005. Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker–Planck equations for probability density functions of turbulent reactive scalars. Physical Review E, 72(1), 1–22.Google ScholarPubMed
Sagaut, P. 2010. Large Eddy Simulation for Incompressible Flows. 3rd ed. Springer.Google Scholar
Sammak, S., Brazell, M. J., Givi, P., and Mavriplis, D. J. 2016. A hybrid DG-Monte Carlo FDF simulator. Computers & Fluids, 140(13), 158–166.CrossRefGoogle Scholar
Sammak, S., Aitzhan, A., Givi, P., and Madnia, C. K. 2020a. High fidelity spectral-FDF-LES of turbulent scalar mixing. Combustion Science and Technology, 192(7), 1219–1232.CrossRefGoogle Scholar
Sammak, S., Ren, Z., and Givi, P. 2020b. Modern developments in filtered density function. In: Modeling and Simulation of Turbulent Mixing and Reaction: For Power, Energy and Flight. Springer.Google Scholar
Sheikhi, M. R. H. 2005. Joint velocity-scalar filtered density function for large eddy simulation of turbulent reacting flows. Thesis, University of Pittsburgh.Google Scholar
Sheikhi, M. R. H., Drozda, T. G., Givi, P., and Pope, S. B. 2003. Velocity-scalar filtered density function for large eddy simulation of turbulent flows. Physics of Fluids, 15(8), 2321–2337.CrossRefGoogle Scholar
Sheikhi, M. R. H., Givi, P., and Pope, S. B. 2007. Velocity-scalar filtered mass density function for large eddy simulation of turbulent reacting flows. Physics of Fluids, 19(9), 095106.CrossRefGoogle Scholar
Sheikhi, M. R. H., Givi, P., and Pope, S. B. 2009. Frequency-velocity-scalar filtered mass density function for large eddy simulation of turbulent flows. Physics of Fluids, 21(7), 075102.CrossRefGoogle Scholar
Shi, Y., GreenJr, W. H.,Wong, H.-W., and Oluwole, O. O. 2011. Redesigning combustion modeling algorithms for the Graphics Processing Unit (GPU): Chemical kinetic rate evaluation and ordinary differential equation integration. Combustion and Flame, 158(5), 836–847.CrossRefGoogle Scholar
Sitaraman, H., Yellapantula, S., Henry de Frahan, M. T., Perry, B., Rood, J., Grout, R., and Day, M. 2021. Adaptive mesh based combustion simulations of direct fuel injection effects in a supersonic cavity flame-holder. Combustion and Flame, 232, 111531.CrossRefGoogle Scholar
Slotnick, J. P., Khodadoust, A., Alonso, J., Darmofal, D., Gropp, W., Lurie, E., and Mavriplis, D. J. 2014. CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences. Report. NASA: Washington, DC, USA.Google Scholar
Smagorinsky, J. 1963. General circulation experiments with the primitive equations. I. The basic experiment. Monthly Weather Review, 91(3), 99–164.2.3.CO;2>CrossRefGoogle Scholar
So, R. M. C., and Speziale, C. G. 1999. A review of turbulent heat transfer modeling. Annual Review of Heat Transfer, 10, 177–220.CrossRefGoogle Scholar
Spafford, K., Meredith, J., Vetter, J., Chen, J. H., Grout, R., and Sankaran, R. 2009. Accelerating S3D: A GPGPU case study. In: European Conference on Parallel Processing. Springer.Google Scholar
Speziale, C. G. 1991. Analytical methods for the development of Reynolds-stress closures in turbulence. Annual Review of Fluid Mechanics, 23(1), 107–157.CrossRefGoogle Scholar
Speziale, C. G. 1998. Turbulence modeling for time-dependent RANS and VLES: A review. AIAA Journal, 36(2), 173–184.CrossRefGoogle Scholar
Stein, O. T., Böhm, B., Dreizler, A., and Kempf, A. M. 2011. Highly-resolved LES and PIV analysis of isothermal turbulent opposed jets for combustion applications. Flow, Turbulence and Combustion, 2(87), 425–447.Google Scholar
Stratonovich, R. L. 1963. Introduction to the Theory of Random Noise. Gordon and Breach.Google Scholar
Su, X., Wei, J., Wang, X., Zhou, H., Hawkes, E. R., and Ren, Z. 2023. A pairwise mixing model with kernel constraint and its appraisal in transported PDF simulations of ethylene flames. Combustion and Flame, 255, 112916.CrossRefGoogle Scholar
Subramaniam, A., Wong, M. L., Borker, R., Nimmagadda, S., and Lele, S. K. 2020. Turbulence enrichment using physics-informed generative adversarial networks. ArXiv:2003.01907.Google Scholar
Subramaniam, S., and Pope, S. B. 1998. A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees. Combustion and Flame, 115(4), 487–514.CrossRefGoogle Scholar
Tang, J. C. K., Wang, H., Bolla, M., Wehrfritz, A., and Hawkes, E. R. 2019. A DNS evaluation of mixing and evaporation models for TPDF modelling of nonpremixed spray flames. Proceedings of the Combustion Institute, 37(3), 3363–3372.Google Scholar
Tang, J. C. K. 2018. Modelling of multiphase flames using direct numerical simulation and transported PDF methods. Thesis, University of New South Wales.Google Scholar
Tang, Q., Zhao, W., Bockelie, M., and Fox, R. O. 2007. Multi-environment probability density function method for modelling turbulent combustion using realistic chemical kinetics. Combustion Theory and Modelling, 11(6), 889–907.CrossRefGoogle Scholar
Taulbee, D. B. 1989. Engineering turbulence models. In: Advances in Turbulence. Hemisphere Publishing Co.Google Scholar
Tirunagari, R. R., and Pope, S. B. 2016. LES/PDF for premixed combustion in the DNS limit. Combustion Theory and Modelling, 20(5), 834–865.CrossRefGoogle Scholar
Turkeri, H., Zhao, X., Pope, S. B., and Muradoglu, M. 2019a. Large eddy simulation/probability density function simulations of the Cambridge turbulent stratified flame series. Combustion and Flame, 199, 24–45.CrossRefGoogle Scholar
Turkeri, H., Pope, S. B., and Muradoglu, M. 2019b. A LES/PDF simulator on block-structured meshes. Combustion Theory and Modelling, 23(1), 1–41.CrossRefGoogle Scholar
Valino, L. 1998. A field Monte Carlo formulation for calculating the probability density function of a single scalar in a turbulent flow. Flow, Turbulence and Combustion, 60(2), 157–172.CrossRefGoogle Scholar
Villermaux, J., and Devillon, J. C. 1972. Représentation de la coalescence et de la redispersion des domaines de ségrégation dans un fluide par un modele d’interaction phénoménologique. In: Proceedings of the 2nd International Symposium on Chemical Teaction Engineering. Elsevier.Google Scholar
Vreman, A. W. 2004. An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications. Physics of Fluids, 16(10), 3670–3681.CrossRefGoogle Scholar
Wang, H., and Zhang, P. 2017. A unified view of pilot stabilized turbulent jet flames for model assessment across different combustion regimes. Proceedings of the Combustion Institute, 36(2), 1693–1703.CrossRefGoogle Scholar
Wang, H., Pant, T., and Zhang, P. 2018. LES/PDF modeling of turbulent premixed flames with locally enhanced mixing by reaction. Flow, Turbulence and Combustion, 100(1), 147–175.CrossRefGoogle Scholar
Wang, M. 2012. Combustion simulations using graphic processing units. Thesis, University of Connecticut.Google Scholar
Wilcox, D. C. 2006. Turbulence Modeling for CFD. 3rd ed. DCW Industries, Inc.Google Scholar
Wouters, H. A., Peeters, T. W., and Roekaerts, D. J. 1996. On the existence of a generalized Langevin model representation for second-moment closures. Physics of Fluids, 8(7), 1702–1704.CrossRefGoogle Scholar
Xie, W., Xie, Q., Zhou, H., and Ren, Z. 2019. An exponential distribution scheme for the two-way coupling in transported PDF method for dilute spray combustion. Combustion Theory and Modelling, 24(1), 105–128.Google Scholar
Xu, G., Daley, A. J., Givi, P., and Somma, R. D. 2018. Turbulent mixing simulation via a quantum algorithm. AIAA Journal, 56(2), 687–699.CrossRefGoogle Scholar
Xu, G., Daley, A. J., Givi, P., and Somma, R. D. 2019. Quantum algorithm for the computation of the reactant conversion rate in homogeneous turbulence. Combustion Theory and Modelling, 23(6), 1090–1104.CrossRefGoogle Scholar
Xu, J., and Pope, S. B. 1999. Assessment of numerical accuracy of PDF/Monte Carlo methods for turbulent reacting flows. Journal of Computational Physics, 152, 192–230.CrossRefGoogle Scholar
Yang, Y., Wang, H., Pope, S. B., and Chen, J. H. 2013. Large-eddy simulation/probability density function modeling of a non-premixed CO/H2 temporally evolving jet flame. Proceedings of the Combustion Institute, 34(1), 1241–1249.CrossRefGoogle Scholar
Yang, T., Yin, Y., Zhou, H., and Ren, Z. 2021. Review of Lagrangian stochastic models for turbulent combustion. Acta Mechanica Sinica, 37(9), 1349.CrossRefGoogle Scholar
Yang, T., Xie, Q., Zhou, H., and Ren, Z. 2020. On the modeling of scalar mixing timescale in filtered density function simulation of turbulent premixed flames. Physics of Fluids, 32(11), 115130.CrossRefGoogle Scholar
Yang, T., Yin, Y., Zhou, H., Mo, Y., Chen, Y., and Ren, Z. 2022. Consistent submodel coupling in hybrid particle/finite volume algorithms for zone-adaptive modelling of turbulent reactive flows. Combustion Theory and Modelling, 26(7), 1159–1184.CrossRefGoogle Scholar
Yang, T., Zhou, H., Yin, Y., and Ren, Z. 2023. Zone-adaptive modeling of turbulent flames with multiple chemical mechanisms. Proceedings of the Combustion Institute, 39(2), 2409–2418.CrossRefGoogle Scholar
Yilmaz, S. L., Nik, M. B., Sheikhi, M. R. H., Strakey, P. A., and Givi, P. 2011. An irregularly portioned Lagrangian Monte Carlo method for turbulent flow simulation. Journal of Scientific Computing, 47(1), 109–125.CrossRefGoogle Scholar
Yilmaz, S. L., Ansari, N., Pisciuneri, P. H., Nik, M. B., Otis, C. C., and Givi, P. 2013. Applied filtered density function. Journal of Applied Fluid Mechanics, 6(3), 311–320.Google Scholar
Yilmaz, L., Nik, M. B., Givi, P., and Strakey, P. A. 2010. Scalar filtered density function for large eddy simulation of a Bunsen burner. Journal of Propulsion and Power, 26(1), 84–93.CrossRefGoogle Scholar
Yin, Y., Yang, T., Zhou, H., and Ren, Z. 2022. Assessment of finite-rate chemistry effects in a turbulent dilute ethanol spray flame. Journal of Propulsion and Power, 38(4), 607–622.CrossRefGoogle Scholar
Yoshizawa, A., and Horiuti, K. 1985. A statistically-derived subgrid-scale kinetic energy model for the large-eddy simulation of turbulent flows. Journal of the Physical Society of Japan, 54(8), 2834–2839.CrossRefGoogle Scholar
Zhang, P., Masri, A. R., and Wang, H. 2017. Studies of the flow and turbulence fields in a turbulent pulsed jet flame using LES/PDF. Combustion Theory and Modelling, 21(5), 897–924.CrossRefGoogle Scholar
Zhang, Y. Z., and Haworth, D. C. 2004. A general mass consistency algorithm for hybrid particle/finite-volume PDF methods. Journal of Computational Physics, 194(1), 156–193.CrossRefGoogle Scholar
Zhao, X., Haworth, D. C., and Huckaby, E. D. 2012. Transported PDF modeling of nonpremixed turbulent CO/H2/N2 jet flames. Combustion Science and Technology, 184(5), 676–693.CrossRefGoogle Scholar
Zhou, H., Ren, Z., Rowinski, D. H., and Pope, S. B. 2020. Filtered density function simulations of a near-limit turbulent lean premixed flame. Journal of Propulsion and Power, 36(3), 381–399.CrossRefGoogle Scholar
Zhou, H., Li, Z., Yang, T., Hawkes, E. R., Ren, Z., Wang, H., and Wehrfritz, A. 2021. An evaluation of gas-phase micro-mixing models with differential mixing timescales in transported PDF simulations of sooting flame DNS. Proceedings of the Combustion Institute, 38(2), 2731–2739.CrossRefGoogle Scholar
Zhou, L., Song, Y., Ji, W., and Wei, H. 2022. Machine learning for combustion. Energy and AI, 7, 100128.CrossRefGoogle Scholar
Zhu, M., Bray, K. N. C., Rumberg, O., and Rogg, B. 2000. PDF transport equations for two-phase reactive flows and sprays. Combustion and Flame, 122(3), 327–338.CrossRefGoogle Scholar

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