computational models for turbulent reacting flows cambridge series in chemical engineering

This page intentionally left blank Computational Models for Turbulent Reacting Flows This book presents the current state of the art in computational models for turbulent reacting flows, and analyzes carefully the strengths and weaknesses of the various techniques described The focus is on formulation of practical models as opposed to numerical issues arising from their solution A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness The book is aimed at chemical, mechanical, and aerospace engineers in academia and industry, as well as developers of computational fluid dynamics codes for reacting flows r o d n e y o f o x received his Ph.D from Kansas State University, and is currently the Herbert L Stiles Professor in the Chemical Engineering Department at Iowa State University He has held visiting positions at Stanford University and at the CNRS Laboratory in Rouen, France, and has been an invited professor at ENSIC in Nancy, France; Politecnico di Torino, Italy; and Aalborg University, Denmark He is the recipient of a National Science Foundation Presidential Young Investigator Award, and has published over 70 scientific papers C AMB R IDGE S E RIE S IN CHE M IC AL ENG INEERING Series Editor: Arvind Varma, University of Notre Dame Editorial Board: Alexis T Bell, University of California, Berkeley John Bridgwater, University of Cambridge L Gary Leal, University of California, Santa Barbara Massimo Morbidelli, ETH, Zurich Stanley I Sandler, University of Delaware Michael L Schuler, Cornell University Arthur W Westerberg, Carnegie-Mellon University Titles in the Series: Diffusion: Mass Transfer in Fluid Systems, Second Edtion, E L Cussler Principles of Gas-Solid Flows, Liang-Shih Fan and Chao Zhu Modeling Vapor-Liquid Equilibria: Cubic Equations of State and their Mixing Rules, Hasan Orbey and Stanley I Sandler Advanced Transport Phenomena, John C Slattery Parametric Sensitivity in Chemical Systems, Arvind Varma, Massimo Morbidelli and Hua Wu Chemical Engineering Design and Analysis, T Michael Duncan and Jeffrey A Reimer Chemical Product Design, E L Cussler and G D Moggridge Catalyst Design: Optimal Distribution of Catalyst in Pellets, Reactors, and Membranes, Massimo Morbidelli, Asterios Gavriilidis and Arvind Varma Process Control: A First Course with MATLAB, Pao C Chau Computational Models for Turbulent Reacting Flows, Rodney O Fox Computational Models for Turbulent Reacting Flows Rodney O Fox Herbert L Stiles Professor of Chemical Engineering Iowa State University    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521650496 © Cambridge University Press 2003 This book is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2003 - - ---- eBook (EBL) --- eBook (EBL) - - ---- hardback --- hardback - - ---- paperback --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate ` a Roberte Contents Preface Turbulent reacting flows 1.1 Introduction page xiii 1 1.2 Chemical-reaction-engineering approach 1.2.1 PFR and CSTR models 1.2.2 RTD theory 1.2.3 Zone models 1.2.4 Micromixing models 1.2.5 Micromixing time 10 12 14 1.3 Fluid-mechanical approach 1.3.1 Fundamental transport equations 1.3.2 Turbulence models 1.3.3 Chemical source term 1.3.4 Molecular mixing 15 16 17 18 23 1.4 Relationship between approaches 24 1.5 A road map to Chapters 2–7 25 Statistical description of turbulent flow 2.1 Homogeneous turbulence 2.1.1 One-point probability density function 2.1.2 Spatial correlation functions 2.1.3 Temporal correlation functions 2.1.4 Turbulent energy spectrum 2.1.5 Model velocity spectrum 2.1.6 Spectral transport 27 27 29 32 34 36 39 41 vii viii Contents 2.2 Inhomogeneous turbulence 2.2.1 Expected values of derivatives 2.2.2 Mean velocity 2.2.3 Reynolds stresses 2.2.4 Turbulent dissipation rate Statistical description of turbulent mixing 44 45 47 48 51 56 3.1 Phenomenology of turbulent mixing 3.1.1 Length scales of turbulent mixing 3.1.2 Phenomenological model for turbulent mixing 56 57 58 3.2 Homogeneous turbulent mixing 3.2.1 One-point velocity, composition PDF 3.2.2 Conditional velocity and scalar statistics 3.2.3 Spatial correlation functions 3.2.4 Scalar energy spectrum 3.2.5 Model scalar spectrum 3.2.6 Scalar spectral transport 62 62 67 69 71 73 78 3.3 Inhomogeneous turbulent mixing 3.3.1 Scalar mean 3.3.2 Scalar flux 3.3.3 Scalar variance 3.3.4 Scalar dissipation rate 3.3.5 Scalar covariance 3.3.6 Joint scalar dissipation rate 80 81 82 84 86 90 92 3.4 Differential diffusion 3.4.1 Homogeneous turbulence 3.4.2 Mean scalar gradients 3.4.3 Decaying scalars 96 97 98 98 Models for turbulent transport 100 4.1 Direct numerical simulation 4.1.1 Homogeneous turbulence 4.1.2 Reacting flow 100 101 102 4.2 Large-eddy simulation 4.2.1 Filtered Navier–Stokes equation 4.2.2 LES velocity PDF 4.2.3 Scalar transport 4.2.4 Reacting flow 104 104 106 108 109 4.3 Linear-eddy model 4.3.1 Homogeneous flows 4.3.2 Inhomogeneous flows 110 111 113 405 References Torrest, R S and W E Ranz (1970) Concentration fluctuations and chemical conversion associated with mixing in some turbulent flows AIChE Journal 16, 930–942 Tsai, K and R O Fox (1993) PDF modeling of free-radical polymerization in an axisymmetric reactor EES Report 254, Kansas State University, Manhattan, Kansas (1994a) Modeling the effect of turbulent mixing on a series-parallel reaction in a tubular reactor ICRES Report 9403, Kansas State University, Manhattan, Kansas (1994b) PDF simulation of a turbulent series-parallel reaction in an axisymmetic reactor Chemical Engineering Science 49, 5141–5158 (1995a) Modeling multiple reactive scalar mixing with the generalized IEM model Physics of Fluids 7, 2820–2830 (1995b) PDF modeling of turbulent mixing and chemical reactions in a tubular jet reactor In G B Tatterson (ed.), Process Mixing: Industrial Mixing Fundamentals, pp 31–38 New York: AIChE (1996a) Modeling the scalar dissipation rate for a series-parallel reaction Chemical Engineering Science 51, 1929–1938 (1996b) PDF modeling of turbulent mixing effects on initiator efficiency in a tubular LDPE reactor AIChE Journal 42, 2926–2940 (1998) The BMC/GIEM model for micromixing in non-premixed turbulent reacting flows Industrial and Engineering Chemistry Research 37, 2131–2141 Tsai, K and E E O’Brien (1993) A hybrid one- and two-point approach for isothermal reacting flows in homogeneous turbulence Physics of Fluids A: Fluid Dynamics 5, 2901–2910 Vali˜ o, L and C Dopazo (1990) A binomial sampling model for scalar turbulent mixing n Physics of Fluids A: Fluid Dynamics 2, 1204–1212 (1991) A binomial Langevin model for turbulent mixing Physics of Fluids A: Fluid Dynamics 3, 3034–3037 Van Slooten, P R and S B Pope (1997) PDF modeling of inhomogeneous turbulence with exact representation of rapid distortions Physics of Fluids 9, 1085–1105 (1999) Application of PDF modeling to swirling and non-swirling turbulent jets Flow, Turbulence and Combustion 62, 295–333 Van Slooten, P R., Jayesh, and S B Pope (1998) Advances in PDF modeling for inhomogeneous flows Physics of Fluids 10, 246–265 Vassilatos, G and H L Toor (1965) Second-order chemical reaction in a nonhomogeneous turbulent fluid AIChE Journal 11, 666–673 Vedula, P (2001) Study of Scalar Transport in Turbulent Flows Using Direct Numerical Simulations Ph D thesis, Georgia Institute of Technology, Atlanta Vedula, P., P K Yeung, and R O Fox (2001) Dynamics of scalar dissipation in isotropic turbulence: A numerical and modeling study Journal of Fluid Mechanics 433, 29–60 Verman, B., B Geurts, and H Kuertan (1994) Realizability conditions for the turbulent stress tensor in large-eddy simulations Journal of Fluid Mechanics 278, 351–362 Vervisch, L (1991) Prise en compte d’effets de cin´ tique chimique dans les flammes de e diffusion turbulente par l’approche fonction densit´ de probabilit´ Ph D thesis, e e Universit´ de Rouen, France e Vervisch, L and T Poinsot (1998) Direct numerical simulation of non-premixed turbulent flames Annual Reviews of Fluid Mechanics 30, 655–691 Veynante, D and L Vervisch (2002) Turbulent combustion modeling Progress in Energy and Combustion Science 28, 193–266 Villermaux, J (1991) Mixing effects on complex chemical reactions in a stirred reactor Reviews in Chemical Engineering 7, 51–108 406 References Villermaux, J and J C Devillon (1972) Repr´ sentation de la coalescence et de la e redispersion des domaines de s´ gr´ gation dans un fluide par un mod` le d’interaction e e e ph´ nom´ nologique In Proceedings of the 2nd International Symposium on Chemical e e Reaction Engineering, pp 1–13 New York: Elsevier Villermaux, J and L Falk (1994) A generalized mixing model for initial contacting of reactive fluids Chemical Engineering Science 49, 5127–5140 Wall, C., B J Boersma, and P Moin (2000) An evaluation of the assumed beta probability density function subgrid-scale model for large eddy simulation of nonpremixed, turbulent combustion with heat release Physics of Fluids 12, 2522–2529 Wand, M P and M C Jones (1995) Kernel Smoothing London: Chapman & Hall Wang, D and C Tong (2002) Conditionally filtered scalar dissipation, scalar diffusion, and velocity in a turbulent jet Physics of Fluids 14, 2170–2185 Warhaft, Z (2000) Passive scalars in turbulent flows Annual Reviews of Fluid Mechanics 32, 203–240 Warnatz, J., U Maas, and R W Dibble (1996) Combustion Berlin: Springer-Verlag Weinstein, H and R J Adler (1967) Micromixing effects in continuous chemical reactors Chemical Engineering Science 22, 65–75 Welton, W C and S B Pope (1997) PDF model calculations of compressible turbulent flows using smoothed particle hydrodynamics Journal of Computational Physics 134, 150–168 Wen, C Y and L T Fan (1975) Models for Flow Systems and Chemical Reactors New York: Marcel Dekker Wilcox, D C (1993) Turbulence Modeling for CFD La Ca˜ ada, California: DCW n Industries Inc Wouters, H A (1998) Lagrangian Models for Turbulent Reacting Flows Ph D thesis, Technische Universiteit Delft, The Netherlands Wouters, H A., T W J Peters, and D Roekaerts (1996) On the existence of a generalized Langevin model representation for second-moment closures Physics of Fluids 8, 1702–1704 Wright, D L., R McGraw, and D E Rosner (2001) Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering particle populations Journal of Colloid and Interface Science 236, 242–251 Xu, J and S B Pope (1999) Assessment of numerical accuracy of PDF/Monte Carlo methods for turbulent reactive flows Journal of Computational Physics 152, 192–230 (2000) PDF calculations of turbulent nonpremixed flames with local extinction Combustion and Flame 123, 281–307 Yeung, P K (1994) Spectral transport of self-similar passive scalar fields in isotropic turbulence Physics of Fluids 6, 2245–2247 (1996) Multi-scalar triadic interactions in differential diffusion with and without mean scalar gradients Journal of Fluid Mechanics 321, 235–278 (1997) One- and two-particle Lagrangian acceleration correlations in numerically simulated homogeneous turbulence Physics of Fluids 9, 2981–2990 (1998a) Correlations and conditional statistics in differential diffusion: Scalars with uniform mean gradients Physics of Fluids 10, 2621–2635 (1998b) Multi-scalar mixing and Lagrangian approaches In Proceedings of the Second Monte Verita Colloquium on Fundamental Problematic Issues in Turbulence, Ascona, Switzerland (2001) Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations Journal of Fluid Mechanics 427, 241–274 407 References (2002) Lagrangian investigations of turbulence Annual Reviews of Fluid Mechanics 34, 115–142 Yeung, P K and C A Moseley (1995) Effects of mean scalar gradients on differential diffusion in isotropic turbulence Paper 95-0866, AIAA Yeung, P K and S B Pope (1989) Lagrangian statistics from direct numerical simulations of isotropic turbulence Journal of Fluid Mechanics 207, 531–586 (1993) Differential diffusion of passive scalars in isotropic turbulence Physics of Fluids A: Fluid Dynamics 5, 2467–2478 Yeung, P K., M C Sykes, and P Vedula (2000) Direct numerical simulation of differential diffusion with Schmidt numbers up to 4.0 Physics of Fluids 12, 1601–1604 Yeung, P K., S Xu, and K R Sreenivasan (2002) Schmidt number effects on turbulent transport with uniform mean scalar gradient Physics of Fluids 14, 4178–4191 Zwietering, T N (1959) The degree of mixing in continuous flow systems Chemical Engineering Science 11, 1–15 (1984) A backmixing model describing micromixing in single-phase continuous-flow systems Chemical Engineering Science 39, 1765–1788 Index arithmetic-to-geometric-diffusivity ratio, 91 batch reactor, 3, 4, 9, 10, 188 Batchelor scale, 57–60, 73, 74, 85, 103, 111, 112, 121, 128, 198, 199, 272 beta PDF, 65, 110, 175, 176, 206, 210, 214, 215, 217, 218, 232, 236, 239, 265, 276, 283, 286, 325 bi-variate, 176 conditional scalar dissipation rate, 276, 283, 286, 325 Fokker–Planck equation, 276, 283, 286 multi-variate, 176, 283 uni-variate, 175, 176, 286, 325 binary mixing, 64, 65, 161, 175, 179, 182, 189, 267, 284, 286, 379 boundary conditions, 57, 114, 126, 146, 157–160, 202, 204, 206, 212, 228, 236, 255, 260, 264, 346–348 composition PDF, 267, 344 conditional moments, 286 Fokker–Planck equation, 280 inflow, 346, 347 non-zero-flux wall, 348 outflow, 158, 346 periodic, 101, 112, 199 symmetry, 346, 347 use in Monte-Carlo simulations, 344–346 velocity PDF, 260 zero-flux wall, 158, 346 CD model, 264, 268, 269, 273–275, 297 constraints and desirable properties, 273, 274 CFL condition, 341, 343, 346, 353 Chapman–Kolmogorov equation, 298 chemical reactions, 2–4, 7, 9, 11, 16, 18, 20, 24, 58, 64, 91, 95, 104, 110, 113, 120, 125, 250, 264, 266, 269, 384 conserved scalars, 147 element conservation, 144, 145 elementary, 141, 142, 144–146, 152, 177 408 fast, 4, 7, 14, 15, 95, 127, 134, 152–154, 207, 221, 310, 346 finite-rate, 7, 153–156 isothermal, 151 liquid-phase, sensitivity to micromixing, 207 non-elementary, 141, 143, 146–150, 152 non-isothermal, 150 one-step, 154, 155 simple, 149, 180–193 slow, 4, 7, 152, 153 chemical source term, 5, 9, 11, 13, 16–18, 21–23, 26, 58, 62, 68, 69, 91, 110, 113, 125, 140–144, 149, 150, 152, 156, 157, 177, 179, 180, 193, 194, 203, 209, 211, 216–219, 221, 232, 234, 235, 239, 241–244, 249–251, 266, 270, 272, 282, 285, 293, 308–321, 329, 330, 337, 383 characteristic length scale, 125 conditional, 68, 207, 209, 211, 234 correlation with velocity, 84, 124, 125 decoupling from transport, 178, 309, 345 definition, 18, 141–143 Arrhenius form, 144 chemical species, 142 composition vector, 143 element matrix, 144 equilibrium constant, 144 Jacobian matrix, 151, 313 rate constants, 142, 144 reaction coefficient matrix, 143, 146 reaction rate functions, 143 reaction rate vector, 143 specific enthalpies, 143 stoichiometric coefficients, 142 time scales, 152 effect on joint scalar dissipation rate, 271 equilibrium-chemistry limit, 177 filtered, 110, 238 fluctuating, 90 for covariance dissipation range, 95 for joint scalar dissipation rate, 95, 327 for scalar covariance, 91, 95, 156, 326 409 Index in LSR model, 326 in PDF transport equation, 249, 261 Jacobian, 92, 95, 141, 143, 327 lookup table, 178–180, 240, 310–321, 330, 346 moment closures, 84, 90, 91, 125, 150, 151, 153–156 first-order, 153–155 higher-order, 155–156 reaction-progress variables, 181, 182 Reynolds-averaged, 18, 21, 22, 67, 81, 114, 141, 150–151, 155, 157, 207, 209–211, 219, 221, 222, 233, 243, 244 SGS closures, 109, 110, 180, 237, 238 stiff kinetics, 180, 308, 346 volume-averaged, 6, 22 chemical time scales, 2, 7, 11, 110, 141, 142, 151–154, 157, 177, 180, 308, 310, 312, 315 definition, 151, 152 in definition of Da, 152 local, 314 competitive-consecutive reactions, 147, 151, 155, 181, 184–188, 193, 210, 273 infinite-rate, 185 limiting cases, 185, 187, 210 mixing line, 186–188 mixture fraction, 185 reaction-progress variables, 185 in a semi-batch reactor, 188 sensitivity to micromixing, 188, 193 composition PDF, 9, 10, 18, 19, 21, 22, 25, 62, 64–66, 68, 109, 141, 142, 151, 153, 156, 157, 194, 219, 221, 232, 241–244, 249–251, 262, 270, 273, 275, 276, 281, 285, 287, 293, 300, 330, 331, 334, 335, 337, 372–374 allowable region, 264, 266–267, 284 binary mixing, 65 conditional, 68, 141, 157, 208, 233, 235, 237 definition, 62 derivation of transport equation, 249–251 in DQMOM, 373, 374, 379, 383, 384 effect of molecular diffusion, 65 empirical PDF, 307 Gaussian limit, 264, 265 histogram, 18, 21, 22 Lagrangian, 294, 298 LES, 109, 237 linearity, 264, 266–268 local mixing, 264, 268–271 models for conditional diffusion, 262, 263, 273–287 moment closures, 153 presumed, 25, 142, 156, 219, 221, 222, 227, 233, 237 Re, Sc, and Da dependence, 264, 272–273 relationship to CRE approach, 66, 194, 195, 251 relationship to mixture-fraction PDF, 68 relationship to Reynolds-averaged chemical source term, 67 relationship to velocity, composition PDF, 63, 244 scalar length-scale distribution, 66, 264, 265, 271–272 scalar-conditioned velocity fluctuations, 251 two-point, 70 composition PDF codes, 329–339, 361 advantages and disadvantages, 339, 353–354 Eulerian, 329–339, 343 numerical diffusion, 352 Lagrangian, 329, 330, 340–354 numerical diffusion, 352 LES, 329 Monte-Carlo simulation, 344–346 notional-particle representation, 340–344 particle-field estimation, 340, 348–352 computational fluid dynamics, 2, 4, 8, 11, 13–15, 17–19, 21, 25–27, 29, 32, 44, 67, 88, 113, 114, 120–122, 140, 154, 161, 175, 176, 179, 180, 197, 200, 223, 232, 240, 259, 260, 330, 383 conditional acceleration, 26, 248, 254–261, 294, 295, 321 decoupling from scalar field, 255 definition, 254 extension to velocity, composition PDF, 258 models, 255–258 GLM, 257–258 SLM, 257 conditional diffusion, 26, 248, 254, 259, 261–287, 296, 308 constraints and properties, 262–273 definition, 261 models, 261–265, 273–287 CD, 273–274 EMST, 269 FP, 275–286 GIEM, 267 IEM, 274–275 VCIEM, 275, 297 conditional fluxes, 248–249 acceleration, 248 reaction/diffusion, 248 conditional gradient-correlation matrix, 279 relationship to FP model, 278 conditional moment closures, 69, 207–216, 218, 236, 239, 277, 279, 285 conditional scalar dissipation rate, 69, 211–214 presumed forms, 214 relationship to mixture-fraction PDF, 211 formulation, 207–209 conditional fluctuations, 208 conditional means, 208 higher-order conditional moments, 208 unconditional means, 208 homogeneous flow, 211–212 inhomogeneous flow, 214–216 gradient-diffusion model, 215 limitations, 215 presumed conditional moments, 209–211 treatment of chemical source term, 209 conditional PDF, 31, 67–69, 107–109, 196, 204, 216, 217, 233–237, 250, 288 conditional expected values, 31 definition, 31 Lagrangian, 300 conditional scalar dissipation rate, 23, 24, 212, 213, 234, 236, 279, 325 boundary conditions in composition space, 280 closed form for mixture-fraction vector, 283 410 Index conditional scalar (cont.) definition, 69 effect of chemistry, 286 joint, 211 relationship to beta PDF, 214, 276, 285 relationship to CMC, 211 relationship to FP model, 275 conditional statistics, 67–69, 287 definition, 67 use in PDF methods, 69 correlation functions, 32, 62 spatial, 32–34, 36, 69–71 longitudinal, 33 scalar, 70, 71 scalar cross-correlation, 70 scalar-velocity, 70 transverse, 33 velocity, 33, 36 temporal, 34–35 Lagrangian, 34, 288, 292, 293, 297 velocity, 35 CRE approach, 2–17, 25, 194, 195, 251 micromixing models, 12–14 micromixing time, 14–15 PFR and CSTR models, 5–7 relationship to composition PDF description, 66, 195 relationship to FM approach, 16, 23, 24 RTD theory, 8–10 zone models, 10–11 CSTR model, 3–10, 13, 14, 66, 127, 128, 194, 195, 198 Damkă hler number, 103, 152 o definition, 152 effect on molecular mixing, 264, 272–273 differential diffusion, 56, 78, 80, 91, 96–99, 122, 135, 149, 215, 244, 264, 272, 274, 275, 278, 280, 284, 365 decaying scalars, 98–99, 138–139 dissipation-range correlation function, 138 homogeneous turbulence, 97 mean scalar gradients, 98, 137–138 multi-variate LSR model, 325–326 multi-variate SR model, 135–139, 325 Re dependence, 96, 98, 99, 139 scalar correlation function, 96 scalar-gradient correlation function, 96 spectral model, 135 dissipation range, 40, 53, 54, 131, 136, 367 scalar, 75, 79, 88, 89, 94, 113, 131, 137–139, 326, 365, 369, 370 velocity, 39, 43, 54, 88, 104 DNS, 18, 100–104, 110, 239, 245 computational cost, 18, 101, 102 homogeneous turbulence, 28, 29, 63, 64, 101–102 use for model validation, 20, 23, 24, 74, 75, 97–99, 102, 104, 127, 131, 132, 135, 139, 201, 207, 242, 249, 258, 263, 265, 267, 270, 276, 289, 292, 293, 322, 364, 368, 370, 371 reacting flow, 102–104, 178 relationship to LEM, 110, 113 relationship to LES, 104, 106, 110 relationship to RANS, 121 DQMOM, 373–386 bi-variate case, 379–382 linear system for source terms, 380 presumed PDF, 379 relationship to moments, 380 transport equation, 380 multi-variate case, 382–384 linear system for source terms, 382 transport equation, 382 relationship to multi-environment PDF models, 383 relationship to QMOM, 373 uni-variate case, 374–379 linear system for source terms, 374 presumed PDF, 374 relationship to moments, 375 transport equation, 374 DQMOM–IEM, 384–386 realizability, 384 relationship to DQMOM, 384 relationship to multi-environment PDF models, 386 Vandermonde matrix, 385 ellipsoid of accuracy, 316–318 definition, 316 use in ISAT, 317, 318 empirical PDF, 300–302, 307, 331, 332, 337 definition, 301 dependence on grid and particle numbers, 301 Glivenko–Cantelli theorem, 302 PDF estimation, 307 relationship to notional particles, 330, 337 relationship to particle-field estimation, 340 EMST model, 269–271 flame-sheet example, 270 energy spectrum coherency, 365 energy dissipation, 38, 41, 197 LES, 105 scalar, 56, 57, 59, 62, 66, 69, 71–75, 77–79, 87, 88, 94, 97, 113, 127, 264, 265, 270, 363, 364, 368 compensated, 76 definition, 71 energy-containing range, 73 inertial-convective sub-range, 73 inertial-diffusive sub-range, 73 model, 72–78 Re and Sc dependence, 75 viscous-convective sub-range, 75 viscous-diffusive sub-range, 74 scalar dissipation, 72, 75, 369 scalar-covariance, 72, 94, 363–365 scalar-flux, 72, 78, 128, 130, 363, 366 transfer, 78, 363, 364, 367, 368 turbulent, 36–42, 51, 53, 54, 73, 75, 76, 78, 79, 85, 104, 127, 132, 364, 368 compensated, 76 definition, 37 dissipation range, 39 energy-containing range, 39 inertial range, 39 model, 39–41, 54, 73–75 411 Index Re dependence, 40 universal equilibrium range, 40 energy-containing range, 73, 129, 365 scalar, 73 velocity, 39, 40, 57, 85 ensemble average, 20, 108, 200, 245, 260, 289, 293, 298, 302, 309, 323, 328, 348 cloud-in-cell, 349 relationship to empirical PDF, 332 statistical error, 332 equilibrium-chemistry limit, 156, 157, 177–180, 182, 205, 207, 211 chemical time scales, 177 LES, 180 lookup table, 178 reacting scalars, 177–178, 180 covariances, 179 means, 179 relationship to mixture-fraction PDF, 179–180 turbulent flow, 178–180 estimation errors, 26, 299, 302, 304, 305, 309, 328, 342, 352 deterministic, 300–302, 305–307, 341 bias, 298, 300, 301, 306, 328, 352, 359 discretization, 300, 301, 305, 306, 328, 329, 359 in mean-field estimation, 300–307 in particle-field estimation, 298 in PDF estimation, 308 statistical, 295, 298, 300–302, 306, 328, 329, 332, 337–339, 341, 342, 352, 353, 359 Eulerian composition PDF codes, 329–339, 343, 352, 353, 360 advantages and disadvantages, 339 local time stepping, 338 numerical diffusion, 336–337 particle transport processes, 332–335 relationship to empirical PDF, 332 time averaging, 338 Eulerian correspondence, 293 fundamental transport equation pressure field, 17 reacting scalars, 16 velocity, 16 fundamental transport equations, 16–17 molecular mixing, 23–24 relationship to CRE approach, 16, 24 turbulence models, 17–18 Fokker–Planck equation, 251, 288 boundary conditions, 280 FP model, 275, 276 Lagrangian PDF, 288, 294 stationary solution, 285 velocity PDF, 256 velocity, composition PDF, 291 FP model, 264, 266, 267, 270, 271, 273, 275–287, 325–327, 377 conditional gradient-correlation matrix, 279 conditional joint scalar dissipation rate, 266, 277, 279 conditional scalar dissipation rate, 276 constraints and desirable properties, 280–281 differential diffusion, 280, 326 Lagrangian, 297, 325, 345 limiting cases, 271, 279, 284–285 for mixture-fraction vector, 282–283, 286 multiple scalars, 276 relationship to CMC, 285, 286 relationship to GIEM, 268 relationship to IEM, 277 relationship to mapping closure, 281–282 scalar covariance, 277 shape matrix, 266, 272, 281, 327 single scalar, 276 stochastic differential equation, 278 fractional time stepping, 309, 310, 332, 340, 345, 346 full PDF methods, 241, see transported PDF methods 387 flame-sheet example, 268–270 flamelet model, 142, 201–207, 264, 271, 272, 285 conditional scalar dissipation rate, 204 definition, 201–204 differential diffusion, 135 flamelet library, 206 inhomogeneous flow, 206–207 limitations, 206, 207 mixture-fraction, dissipation rate PDF, 205–206 relationship to FP model, 277 stationary, 204–205 transport equation stationary, 204 unstationary, 204 fluid particles, 5, 9, 13, 27, 63, 113, 121, 188, 194, 195, 197–201, 215, 216, 269, 270, 273, 285, 289–290, 323, 331, 335 relationship to Eulerian PDF, 290 relationship to Lagrangian PDF, 289 relationship to notional particles, 289 FM approach, 2, 3, 15–24, 197 chemical source term, 18–23 composition PDF, 22 Gaussian PDF, 23, 24, 30, 63–65, 108, 175, 176, 214, 219–221, 227, 251, 256, 261, 262, 264–267, 275–277, 281, 295, 378 conditional expected values, 32 correlation function, 32 covariance matrix, 30 eigenvector transformation, 220 functional form, 30 mean vector, 30 properties of, 31 GIEM model, 267, 273 GLM, 257–258, 292 definition, 257 extension to velocity, composition PDF, 258–259 Lagrangian, 295, 296 relationship to Reynolds-stress models, 258 relationship to scalar-flux model, 258 higher-order PDF models, 321–327 differential diffusion, 325–326 LSR model, 322–327 reacting scalars, 326–327 turbulence frequency, 321–322 412 Index hybrid PDF codes, see velocity, composition PDF codes IEM model, 13, 14, 23, 24, 194–197, 264, 267, 269, 270, 274–276, 281, 296, 340, 373, 377, 378, 383, 384, 386 for conditional diffusion, 273, 274 constraints and desirable properties, 275 for CSTR, 13, 14, 194–195 flame-sheet example, 269 Lagrangian, 296, 297, 308, 345, 348, 350 for PFR, 13 relationship to FP model, 277 relationship to VCIEM model, 275 inertial range, 38–42, 51, 53–55, 58, 59, 73–75, 79, 104, 106, 129, 198 inertial-convective sub-range, 73, 79, 87, 126–129, 365, 368 inertial-diffusive sub-range, 73, 74 integral scale, 20, 60, 76, 102, 121, 131, 259, 266, 271 scalar, 20, 57, 70, 72, 77, 95, 127, 134, 197 velocity, 33–35, 37, 38, 40, 54, 57, 59–61, 99, 113, 120, 121, 127, 197, 305, 308 intensity of segregation, 65, 66, 72 ISAT, 312–321 binary-tree tabulation, 318–319 ellipsoid of accuracy, 316–318 flow diagram, 318 further improvements, 319–321 dimension reduction, 320 parallelization, 321 linear interpolation, 314–316 reaction mapping, 313–314 joint scalar dissipation rate, 80, 91–96, 251, 262, 265–267, 270, 280, 281, 285 conditional, 211, 212, 270, 276 Da dependence, 152 definition, 90 effect of chemical reactions, 91, 95, 271, 286 fluctuating, 93, 271 non-equilibrium models, 135–137 Re dependence, 95 relationship to FP model, 275, 277 relationship to mixing models, 262 role in differential diffusion, 91, 97 spectral model, 95 in SR model, 135 transport equation, 93 derivation, 92–93 model, 136, 327 kernel function, 301, 348, 349 constant, 301, 302, 349 use in empirical PDF, 301 grid cell, 301, 307, 341, 349 Kolmogorov scale, 15, 34, 35, 43, 50, 57, 59, 60, 73, 74, 76, 89, 101, 102, 105, 125, 129, 153, 197–199, 201, 202, 327, 365 Lagrangian composition PDF codes, 329, 339–354 advantages and disadvantages, 353–354 boundary conditions, 346–348 coupling to FV codes, 340 flow diagram, 343, 344 local time stepping, 353 Monte-Carlo simulation, 344–346 notional-particle representation, 340–344 particle weights, 342–343 particle-field estimation, 348–352 Lagrangian correspondence, 293 Lagrangian micromixing models, 4, 10, 12–14, 66, 193–201, 221, 226, 239, 240, 242, 271 age-based, 13, 195–196 IEM, 13–14 for CSTR, 14, 194–195 for PFR, 13 inhomogeneous flows, 200–201 mechanistic, 198–200 micromixing rate, 197–198 relationship to composition PDF, 25, 195, 251 Lagrangian PDF methods, 63, 100, 135, 241, 256, 287–298, 329, 340, 368 Eulerian correspondence, 293 fluid particles, 289–290 Lagrangian correspondence, 293 mixing models, 296–298 notional particles, 287–289 spatial distribution, 290 relationship to Eulerian PDF methods, 290–292 stochastic differential equations, 292–294 velocity PDF closures, 294–296 laminar diffusion flamelets, 201–207 length scales scalar, 69–71 Batchelor, 57; see Batchelor scale dependence on initial conditions, 57 integral, 70, 72; see integral scale, scalar phenomenological model, 58–62 Sc dependence, 57, 71 Taylor microscale, 70; see Taylor microscales, scalar scalar-to-velocity length-scale ratio, 61, 77 turbulence, 33–34, 36 integral, 34, 35; see integral scale, velocity integral, longitudinal, 33 integral, transverse, 33 Kolmogorov, 34, 35; see Kolmogorov scale Re dependence, 34 Taylor microscales, 33, 35; see Taylor microscales, velocity turbulent mixing, 57–58 LES, 104–110 CMC, 214 composition PDF, 109 equilibrium-chemistry limit, 180 filtered chemical source term, 110 density function, 108 Navier–Stokes equation, 104–106 scalar field, 108 strain rate, 106 velocity field, 105 filtering function, 104, 105 effect on energy spectrum, 105 kernel, 349 413 Index positive, 108 sharp-spectral, 104 gradient-diffusion model, 109 mixture fraction, 179 mixture-fraction PDF, 109 modified filtered pressure, 106 multi-environment PDF models, 110 reacting flow, 109–110, 113 relationship to DNS, 104 relationship to RANS, 104, 114 residual kinetic energy, 107 mixture-fraction variance, 110, 238 mixture-fraction-variance dissipation rate, 238 Reynolds stresses, 107 scalar field, 109 scalar flux, 108, 109, 238 scalar variance, 109 stress tensor, 105, 106 velocity field, 106 scalar transport, 108–109 SGS turbulent diffusivity, 109, 238 Smagorinsky model, 106 velocity PDF, 106–108 velocity, composition PDF, 109 LES–PDF methods, 106–110, 260–261 composition PDF, 109 composition PDF codes, 329, 340, 341, 354 computational cost, 261 DQMOM, 384 relationship to composition PDF models, 110 relationship to FDF approach, 108 relationship to multi-environment PDF models, 237–239 velocity, composition PDF, 109 velocity PDF, 106–108 linear-eddy model, 110–114, 199, 239, 287 homogeneous flows, 111–112 inhomogeneous flows, 113–114 Lagrangian LEM, 113 relationship to DNS, 111, 113 relationship to PDF methods, 113 relationship to RANS, 112 triplet map, 111 location-conditioned statistics, 308, 329, 358, 359 estimation errors, 302–307 estimation methods, 299–302 use in Lagrangian PDF methods, 288–292 in LFP model, 297 in LGLM, 295–296 in LIEM model, 297 in LVCIEM model, 297 in mixing models, 296–297 in particle-pressure-field equation, 295 in velocity PDF closures, 294–296 lookup tables, 240, 310–321, 330, 346 binary-tree tabulation, 318–319 use in equilibrium-chemistry limit, 178–180 flamelet library, 206 ILDM, 312 ISAT, 312–321 pre-computed, 310–312 LSR model, 322–327 conditional joint scalar dissipation rate, 325 conditional scalar covariances, 325 conditional scalar dissipation rate, 322 conditional scalar variances, 323 coupling to turbulence frequency model, 323 differential diffusion, 325–326 joint-scalar-dissipation chemical source term, 327 reacting scalars, 326–327 relationship to FP model, 282, 327 relationship to SR model, 135 spectral transfer rates, 323, 324 mean pressure field, 48, 295, 296, 355 coupling with PDF transport equation, 259–260 relationship to particle pressure field, 295 mean velocity field, 11, 17, 27, 28, 30, 37, 40, 41, 47–49, 51, 53, 55, 83, 85, 87, 101, 106, 116, 120–123, 126, 132, 195, 250, 251, 254, 259, 294, 296, 298, 300, 329, 331–337, 340, 341, 345–347, 352, 356, 361 transport equation, 47, 252–253, 259, 295, 296 mean-field estimation errors, 300–307 relationship to empirical PDF, 300 mechanical-to-scalar time-scale ratio, 77, 78, 127, 130, 368 definition, 76 Re and Sc dependence, 77 relationship to Obukhov–Corrsin constant, 73 time dependence, 128, 131 mesomixing, 57, 197, 198 micromixing models, 4, 10, 12–14, 16, 23, 25, 66, 239, 240, 242, 271 age-based, 195–196 coalescence-redispersion, 12 DQMOM, 383 DQMOM–IEM, 386 IEM, 13, 14, 194–195 inhomogeneous flows, 200–201 Lagrangian, 193–201, 251 maximum-mixedness, 12 mechanistic, 198–200 micromixing rate, 197–198 minimum-mixedness, 12 multi-environment, 12 micromixing time, 4, 13–15, 25, 153, 308, 338 in definition of Da, 152 dependence on initial conditions, 15 in IEM model, 13, 194 relationship to scalar dissipation rate, 14–15 mixture-fraction PDF, 68, 142, 156, 161, 166, 172, 174–177, 179, 180, 184, 193, 197, 200, 205, 210, 212–216, 233, 234, 285 conditional scalar dissipation rate, 211, 213 from FP model, 282–283 LES, 109 mixture-fraction covariances, 174 mixture-fraction means, 174 presumed, 110, 174–177, 207, 209, 211, 216 bi-variate beta, 176 limitations, 175 multi-variate, 176 uni-variate beta, 175 relationship to CMC, 209 relationship to composition PDF, 68 414 Index mixture-fraction (cont.) relationship to equilibrium-chemistry limit, 179–180 relationship to flamelet model, 205, 206 transport equation in homogeneous flow, 212 mixture-fraction vector, 109, 141, 156–179, 181, 183, 193, 201, 211, 212, 216, 221, 222, 232–234, 282–284 allowable region, 282 in CMC, 207 definition, 161–168 initial/inlet conditions, 167 linear mixture, 161, 162 linear-mixture basis, 161, 166 mixture-fraction basis, 164 transformation matrix, 167 example flows, 168–174 filtered, 239 general formulation, 157–161 linear mixture, 163 PDF, 174–177 principle of linear superposition, 157 relationship to initial/inlet conditions, 157 relationship to linear mixture, 167 relationship to reaction-progress vector, 166 residual, 239 molecular diffusion, 10, 13, 23, 47, 56–58, 65, 72, 78, 109–111, 113, 125, 136, 149, 198, 199, 239, 262, 268, 269, 287, 363 coefficients, 90, 111, 146, 149, 264, 267, 272, 276 differential diffusion, 135, 267 effect on composition PDF, 65, 104, 240, 242, 249 molecular mixing models, 24, 114, 250, 262, 264–266, 272, 273, 275, 281, 293, 351 constraints, 262–263 on correlation with velocity field, 262 on joint scalar dissipation rate, 262 on scalar means, 262 desirable properties, 263–265 boundedness, 264 dependence on scalar length scales, 264 Gaussian limit, 264 linearity, 264 local mixing, 264 Re, Sc, and Da dependence, 264 prospects for further improvements, 286–287 moment closures, 156 chemical source term, 90, 91, 125, 150, 151, 153–156 first-order, 153–155 higher-order, 155–156 conditional, 207–216 scalar fields, 123–127, 193 velocity field, 114–120 Monte-Carlo simulation, 26, 259, 287, 298, 328, 344–348, 357–358 boundary conditions, 346–348 fractional time stepping, 345 relationship to DQMOM, 373, 383, 385 stochastic differential equations, 344, 345 use in transported PDF codes, 240 multi-environment conditional PDF models, 233–236 one-step reaction, 235 relationship to CMC, 236 multi-environment presumed PDF models, 221–239 conditioned on mixture fraction, 233–236 extension to LES, 237–239 general formulation, 222–226 inhomogeneous flows, 226–233 micromixing terms, 222 presumed PDF, 222 relationship to DQMOM, 383–384 relationship to micromixing models, 221 spurious dissipation rate, 226 ternary mixing, 232 Navier–Stokes equation, 16, 27, 40, 41, 67, 74, 82, 100, 111, 244, 292 definition, 101 filtered, 104–107 Fourier-transformed, 101 Reynolds-averaged, 17, 102, 114, 253 non-equilibrium models, 54, 62, 71, 78, 85, 118, 127, 201 differential diffusion, 135–139, 325–326 reacting scalars, 326–327 scalar dissipation rate, 127–135, 322–325 non-premixed reactants, 2–5, 13, 14, 23, 24, 58, 125, 126, 141, 168, 170, 172, 194, 206, 212, 229, 271, 284, 286, 310, 326 binary mixing, 161 relationship to mixture fraction, 156 ternary mixing, 161 notional particles, 240, 287–289, 298–300, 309, 310, 323, 328–330, 332, 335–339 in Eulerian PDF codes, 331–332 initial distribution, 290 in Lagrangian PDF codes, 340–344, 356–357 in Lagrangian PDF methods, 287 particle-field estimation, 298–300, 348–352 relationship to computational cost, 339, 353, 360 relationship to empirical PDF, 300–302, 330 relationship to Eulerian PDF, 290–292 spatial distribution, 290, 339 stochastic differential equations, 288 weights in PDF codes, 330, 339, 342–343 Obukhov–Corrsin constant, 73, 75 relationship to mechanical-to-scalar time-scale ratio, 73 one-point PDF, 22, 26, 29–32, 69, 107, 205, 240, 246, 271 composition, 18, 22, 62, 64–67, 242, 243, 262 mixture-fraction, 174–177 velocity, 29–32, 107 velocity, composition, 44–45, 62–64, 67, 81, 84, 140, 240, 241, 245, 246 one-step reaction, 103, 110, 154, 181–184, 202, 235, 272, 273, 326 allowable region, 266 fast-chemistry limit, 154 finite-rate, 210 flamelet model, 201 infinite-rate, 154, 184, 271 limiting cases, 183 mixture fraction, 183 415 Index moment closure, 155 non-isothermal, 184, 201, 203, 233, 235, 268 presumed CMC, 209 reaction-progress variable, 183 parallel reactions, 181, 189–193, 210 limiting cases, 190, 191 mixing line, 191, 192 mixture fraction, 189 reaction-progress variables, 189 sensitivity to micromixing, 192, 193 particle transport processes, 332–335, 344–348, 357–358 Eulerian PDF codes, 332–335 inter-cell, 332–335 intra-cell, 332 numerical diffusion, 336–337 Lagrangian PDF codes, 344–348, 357–358 boundary conditions, 346–348 Monte-Carlo simulation, 344–346, 357–358 particle-field estimation, 298–308, 332, 340, 341, 343–346, 348–352, 358–359 bi-linear basis functions, 350 consistency, 358–359 empirical PDF, 300–302 estimation errors, 302–307 global estimators, 350, 351 GLME, 350 in Lagrangian PDF codes, 348–352, 358–359 local estimators, 349 LCME, 349 LLME, 351 notional particles, 298–300 PDF estimation, 307–308 PDF beta, 175–177; see beta PDF bi-variate, 176 uni-variate, 175 composition, 9, 244; see composition PDF conditional, 31, 67; see conditional PDF in DQMOM, 373 empirical, 300–302; see empirical PDF estimation, 307–308 bi-variate, 20 uni-variate, 19 Eulerian, 29, 34 Gaussian, 30; see Gaussian PDF joint, Lagrangian, 288 LES composition, 109 LES velocity, 106–108 LES velocity, composition, 109 marginal, 30 mixture-fraction, 174–177; see mixture-fraction PDF mixture-fraction, dissipation rate, 205–206 multi-point, 32 multi-variate, 30 one-point, 22, 29–32, 62–67; see one-point PDF in QMOM, 372 transport equation composition, 249–251, 374 velocity, 255 velocity, composition, 244–249 turbulence frequency, 322 two-point, 29, 32 two-time, 45 uni-variate, 29 velocity, 30; see velocity PDF velocity, composition, 16, 44, 62–67, 242; see velocity, composition PDF PDF simulation codes, 328–362 Eulerian, 331–339 hybrid, 354–361; see velocity, composition PDF codes Lagrangian, 340–354 overview, 329–331 PFR model, 3, 5–7, 9, 10, 13, 14, 194, 197 premixed reactants, 2, 3, 125, 161, 207, 216, 269–272, 286 presumed PDF methods, 141, 174, 175, 205, 216–221 multi-environment models, 221–239 conditional, 233–236 in DQMOM, 383 formulation, 222–226 inhomogeneous flows, 226–233 LES, 237–239 in QMOM, 372 multiple scalars, 218–221 Gaussian PDF, 220 limitations, 221 single scalar, 216–218 conditional PDF, 217 limitations, 217 QMOM, 372–373 presumed PDF, 372 product-difference algorithm, 373 relationship to DQMOM, 373 relationship to scalar moments, 372 random field, 27, 44, 69, 242, 245, 246, 288 scalar, 62, 63 velocity, 27–29, 32, 62, 107, 108 random process, 27, 28, 64, 249, 297 Eulerian, 63 Lagrangian, 63 random variables, 29, 108, 241, 249 Gaussian, 31 relationship to sample space, 29 RANS models, 17, 106, 110, 114–127 mean velocity, 114 relationship to LEM, 111, 112 relationship to LES, 104 relationship to PDF methods, 243, 249, 252–254, 354, 360 mean velocity, derivation of transport equation, 252–253 Reynolds stresses, derivation of transport equation, 254, 354 Reynolds stresses, 117–120 scalar dissipation rate, 126–127 equilibrium model, 126 non-equilibrium models, 127–135 scalar flux, 123–125 consistency with Reynolds-stress model, 124 effect of chemical reactions, 125 416 Index RANS models, (cont.) gradient-diffusion, 122, 123 modified gradient-diffusion, 122 scalar mean, 120 scalar variance, 125–126 turbulence frequency, 116 turbulent dissipation rate, 116 turbulent kinetic energy, 115 turbulent viscosity, 116 turbulent-diffusivity-based, 121–122 turbulent diffusivity, 122 turbulent-viscosity-based, 114–116 k–ω model, 116 k–ε model, 115 one-equation, 115 two-equation, 115 reaction mapping, 313–314 direct integration, 314 Jacobian, 312, 313 linearized, 314–316 conserved manifold, 315 fast manifold, 315 use in ISAT, 316 sensitivity matrix, 315 slow manifold, 315 reaction-progress variables, 169, 171, 173, 181–183, 205, 207, 209, 210, 216–221, 235, 236, 266, 268, 272, 286 competitive-consecutive reactions, 185 conditional, 209, 239 filtered, 239 flamelet model, 203 initial/inlet conditions, 183, 204 one-step reaction, 183 parallel reactions, 189 relationship to reaction-progress vector, 166 simple chemistry, 181 transport equation, 203 reaction-progress vector, 156, 166, 167, 171, 209, 212, 221, 233, 284, 286 conditional, 214, 233 equilibrium, 178, 207 example flows, 168–174 initial/inlet conditions, 166, 167 relationship to mixture-fraction vector, 166 transformation matrix, 166 transport equation, 177 Reynolds number Kolmogorov, 35 Taylor-scale, 34, 35 turbulence, 34–36 turbulent, 365 Reynolds stresses, 30, 37, 48–51, 55, 82, 83, 90, 104, 106, 107, 114, 123, 243, 253, 259, 261, 354–356, 358 anisotropy tensor, 118 dissipation rate tensor, 50 effect on mean pressure, 48 effect on mean velocity, 47 normal stresses, 48 pressure-diffusion term, 50 pressure-rate-of-strain tensor, 50 production term, 49 relationship to energy spectrum, 36 shear stresses, 48 transport equation, 83 consistency with scalar flux, 123, 259 derivation, 48–51 models, 117–120 relationship to PDF methods, 254, 256–258 turbulent-viscosity model, 115 velocity-pressure-gradient term, 50 RTD theory, 3, 4, 8–10, 12, 25, 194 internal-age distribution in CSTR, 13 in PFR, 13 relationship to micromixing models, 10, 12 relationship to PDF methods, RTD function, in CSTR, in PFR, sample space variables, 9, 29, 308 composition, 9, 62 velocity, 27, 29, 44, 63, 80 scalar correlation, 96, 98, 99, 136–139, 218, 221, 267, 277–279, 282, 284, 297 differential diffusion, 96, 137 in FP model, 277 scalar covariance, 70, 71, 90–92, 94, 96–98, 135, 136, 151, 154–156, 174, 176, 179, 180, 217, 219–221, 243, 251, 264, 265, 267, 276–278, 280, 283, 285, 325, 326, 364, 381–383 chemical source term, 90, 95, 156 conditional, 208 covariance-production term, 90, 137 eigenvector transformation, 220 joint scalar dissipation rate, 90 spatial-transport terms, 90 spectrum, 72, 363–365 transport equation, 90, 156 derivation, 90–91 model, 135–139 scalar dissipation energy spectrum, see energy spectrum, scalar dissipation scalar dissipation rate, 86–89, 225, 228 in CMC, 212 conditioned on scalar, 23, 24, 69, 204, 212–214, 236, 285 conditioned on turbulence frequency, 287, 322 definition, 71, 85, 86 equilibrium model, 88, 126 in flamelet model, 201 fluctuating, 86, 205, 242 in LSR model, 322–325 in multi-environment presumed PDF models, 223 non-equilibrium models, 127–135 Re, Sc, and Da dependence, 272 relationship to conditional diffusion, 275 relationship to micromixing time, 14–15, 62, 200 relationship to scalar energy spectrum, 72 relationship to scalar length-scale distribution, 240, 264, 265, 271 relationship to scalar spectral energy transfer rate, 15, 62, 79, 80, 89 relationship to scalar variance, 23 spectral model, 79, 88, 89 417 Index in SR model, 367–371 joint, 90; see joint scalar dissipation rate transport equation, 86, 126–127 derivation, 86 model, 126–127 scalar energy spectrum, see energy spectrum, scalar scalar flux, 18, 46, 47, 71, 82–85, 89, 90, 96, 97, 104, 121, 125, 215, 219, 226, 243, 244, 251, 252, 258, 275, 325, 340, 354, 360 conditioning on velocity, 263 consistent models, 254, 258, 259 Da dependence, 152 definition, 44, 81 gradient-diffusion model, 85, 121–122 molecular transport term, 83 pressure-diffusion term, 83 pressure-scalar-gradient term, 83 pressure-scrambling term, 83, 259 production term, 83 reacting scalars, 84 residual, 108 in scalar mean transport equation, 81 scalar-flux dissipation, 84 transport equation, 83, 123–125 derivation, 82–84 model, 123–125 scalar length scales, see length scales, scalar scalar mean, 7, 9, 14, 18, 57, 65, 66, 81–82, 89, 96, 125, 151, 179, 180, 208, 213, 214, 262, 265, 267, 379, 382 conditional, 207, 208, 211–213, 377 estimated, 301, 331 location-conditioned, 297 transport equation, 81, 120 derivation, 81–82 model, 120 scalar spectral transport, see spectral transport, scalar scalar variance, 7, 14, 18, 65, 67, 69, 71, 84–86, 89, 155, 197, 200, 223, 225–227, 242, 265, 271, 378, 379 conditional, 324, 325 definition, 15, 65 intensity of segregation, 66 in LSR model, 325 relationship to micromixing time, 15, 62 relationship to scalar energy spectrum, 71, 72 relationship to Taylor microscale, 70 residual, 109 in scalar dissipation range, 88 scalar dissipation rate, 85 scalar variance flux, 85 scalar-variance-production term, 85 in SR model, 127, 130, 131, 134, 136, 364 transport equation, 85, 125–126 derivation, 84–86 model, 125 scalar vectors conserved-constant, 159, 160, 171 conserved-scalar, 148, 149, 158 conserved-variable, 159–162, 166, 168, 171, 172 mixture-fraction, 141, 156, 157, 161–179, 181, 183, 193, 201, 207, 211, 212, 216, 221, 222, 232–234, 282–284 reacting-scalar, 148, 149, 156 reaction-progress, 156, 166, 167, 171, 207, 209, 212, 214, 221, 233, 284, 286 scalar-conditioned velocity fluctuations, 250–251 scalar-to-velocity length-scale ratio, 61, 77 Schmidt number molecular, 57 SGS, 109 turbulent, 122 two scalars, 94, 135 SGS fields, see LES, residual simple chemistry, 148, 180–193, 209, 216 competitive-consecutive reactions, 184–188 one-step reaction, 182–184 parallel reactions, 189–193 reaction-progress variables, 181–182 singular value decomposition, 145, 147, 148, 150, 151, 157, 159, 161, 169, 170, 172, 315 definition, 147 orthogonal matrices, 147 singular values, 147 SLM, see conditional acceleration, models spatial correlation function, see correlation functions, spatial spectral transport, 41–43, 78–80, 118, 272, 367 scalar, 78–80 cospectral transfer function, 78 relationship to scalar dissipation rate, 79 spectral energy transfer rates, 79 spectral transfer function, 78 transport equation, 363–365 velocity, 41–43 characteristic time scale, 42 relationship to turbulent dissipation rate, 42 spectral energy transfer rate, 42 spectral transfer function, 41 SR model, 127–135 derivation, 365–371 differential diffusion, 135 multi-variate, 135–139 covariance, 136 covariance-dissipation-production term, 136 decaying scalars, 138–139 dissipation transfer rate, 136 joint scalar dissipation rate, 136 mean scalar gradients, 137–138 relationship to LSR model, 322 scalar dissipation rate, 131 scalar energies, 129 scalar variance, 131 scalar-variance production, 130 spectral transfer rates, 132–134 wavenumber bands, 129 stochastic differential equations, 251, 276, 278, 288, 298, 301–304, 308, 329, 340, 355 Eulerian correspondence, 290, 293 FP model, 279 Lagrangian correspondence, 293 stochastic differential (cont.) in Lagrangian PDF methods, 287 for notional particles, 288, 292–294 numerical simulation, 344 Euler, 344 multi-step, 345 Wiener process, 279 418 Index Taylor microscales, 33 scalar, 70 velocity, 33 longitudinal, 33 transverse, 33 temporal correlation function, see correlation functions, temporal ternary mixing, 161 see non-premixed reactants, ternary mixing time scales chemical, see chemical time scales mechanical-to-scalar time-scale ratio, 76 scalar micromixing, 7, 194 mixing, 7, 58, 61, 62, 67, 70, 72, 76, 149, 156, 240 recirculation, turbulence, 2, 36 eddy turnover, 38 integral, 7, 11, 35 Kolmogorov, 15, 35 Taylor, 35 transport equation composition PDF, 250 joint scalar dissipation rate, 93 mean pressure, 48 mean velocity, 47 pressure, 17 Reynolds stresses, 49 scalar covariance, 90 scalar dissipation rate, inert, 86 scalar flux inert, 83 reacting, 84 scalar mean inert, 82 reacting, 81 scalar variance, inert, 85 scalar, reacting, 16 turbulent dissipation rate, 52 turbulent kinetic energy, 51 velocity, 16 velocity, composition PDF, 248 velocity PDF, 255 transported PDF methods, 14, 26, 44, 65, 67, 81, 83, 84, 91, 109, 125, 140, 233, 236, 239–327 chemical source term, 308–321; see lookup tables composition PDF, 244; see composition PDF composition PDF transport equation, 249–251 higher-order models, 321–327; see higher-order PDF models Lagrangian, 287–298; see Lagrangian PDF methods models for conditional acceleration, 254–261; see conditional acceleration, models models for conditional diffusion, 261–287; see conditional diffusion, models overview, 241–244 particle-field estimation, 298–308; see particle-field estimation relationship to RANS models, 252–254 simulation codes, 328–362; see PDF simulation codes velocity, composition PDF, 242–243; see velocity, composition PDF velocity, composition PDF transport equation, 244–249 transported PDF simulations, see PDF simulation codes turbulence frequency, 329, 331, 341, 346, 355, 356, 359, 361 definition, 116 fluctuating, 321 PDF models, 321–322 gamma-distribution, 321 log-normal, 321 stretched-exponential, 322 RANS model, 116 relationship to LSR model, 322 relationship to turbulence time scale, 321 turbulence length scales, see length scales, turbulence turbulence time scales, see time scales, turbulence turbulent diffusivity, 11, 61, 85, 201, 251, 331, 332, 337, 340, 361 boundary condition, 347 definition, 122 in LEM, 112 in model for scalar-conditioned velocity fluctuations, 251 relationship to molecular diffusion, 122 in SDE for notional-particle position, 294 SGS, 109, 238 turbulent Schmidt number, 122 turbulent dissipation rate, 7, 15, 17, 18, 34, 35, 38, 41, 43, 51–55, 70, 89, 197, 251, 272, 321 definition, 38, 52 fluctuating, 52 relationship to spectral energy transfer rate, 38, 42, 43 in Reynolds-stress models, 117 transport equation, 52 in two-equation RANS models, 115 derivation, 51–53 model, 116 turbulent energy dissipation spectrum, see energy spectrum, energy dissipation turbulent energy spectrum, see energy spectrum, turbulent turbulent flow, statistical description, 27–55 homogeneous, 27–43 correlation functions, 32–36 energy spectra, 36–43 one-point PDF, 29–32 inhomogeneous transport equations, 44–55 turbulent kinetic energy, 7, 17, 28, 34, 35, 37, 40–42, 50, 55, 251, 304, 321, 361 definition, 37 dissipation rate, 51; see turbulent dissipation rate production term, 51 residual, 107 spatial transport, 51 transport equation derivation, 51 model, 115 in two-equation models, 115 turbulent mixing, statistical description, 56–99 differential diffusion, 96–99 419 Index homogeneous, 62–80 correlation functions, 69–71 energy spectra, 71–80 one-point PDF, 62–69 inhomogeneous transport equations, 80–96 phenomenological model, 56–62 turbulent transport, models, 100–140 differential diffusion, 135–139 DNS, 100–104 LEM, 110–114 LES, 104–110 non-equilibrium, 127–135 RANS scalars, 120–127 velocity, 114–120 transported PDF, 140 turbulent-diffusivity-based models, see RANS models, turbulent-diffusivity-based turbulent-viscosity-based models, see RANS models, turbulent-viscosity-based universal equilibrium range, 40 VCIEM model, 275; see conditional diffusion, models Lagrangian, 297 velocity, composition PDF, 44, 62–67, 81, 84, 140, 240–244, 258, 290, 321, 328, 330 conditional acceleration, 258; see conditional acceleration conditional diffusion, see conditional diffusion definition, 62 in DQMOM, 384 LES, 109 relationship to composition PDF, 242, 244 transport equation, 244–249 velocity, composition PDF codes, 354–361 advantages and disadvantages, 360 mean conservation equations, 355–356 Monte-Carlo simulation, 357–358 notional-particle representation, 356–357 particle-field estimation and consistency, 358–359 velocity field, 16 filtered, 105 fluctuating, 27 Fourier-transformed, 101 Lagrangian, 28 mean, 17, 47 random, 27 residual, 106 velocity PDF, 29, 254–256 boundary conditions, 260 conditional, 31 definition, 29 Gaussian, 30 Lagrangian models, 294–296 LES, 106–108 models for conditional acceleration, 256; see conditional acceleration, models Reynolds stresses, 30 transport equation, 255 two-point, 32 velocity, wavenumber PDF models, 261 viscous-convective sub-range, 75, 79, 80, 94, 127, 128 viscous-diffusive sub-range, 74, 80, 94 zone models, 10–11 ... chemical reaction engineering (most notably in the area of multiphase turbulent reacting flows), one is certainly justified in pointing to computational models for turbulent reacting flows as a highly... Gavriilidis and Arvind Varma Process Control: A First Course with MATLAB, Pao C Chau Computational Models for Turbulent Reacting Flows, Rodney O Fox Computational Models for Turbulent Reacting Flows Rodney...This page intentionally left blank Computational Models for Turbulent Reacting Flows This book presents the current state of the art in computational models for turbulent reacting flows, and