NUMERICALLY ACCURATE RANS/PDF AND LES/PDF CALCULATIONS OF TURBULENT FLAMES ADissertation Presented to the Faculty of the Graduate School ofCornell University in Partial Fulfillment of the Requirementsfor the Degree of Doctor ofPhilosophy by HaifengWang May 2010 (cid:13)c 2010HaifengWang ALL RIGHTSRESERVED NUMERICALLYACCURATERANS/PDFANDLES/PDF CALCULATIONS OF TURBULENTFLAMES HaifengWang, Ph.D. Cornell University 2010 Numerically accurate probability density function (PDF) calculations of turbu- lent jet flames are performed in the Reynolds-averaged Navier-Stokes (RANS) context. First, the effect is investigated of the time-averaging of the mean feed- back quantities from the particle solver to the RANS solver on the bias errors thatarecausedbyfeedingthenoisymeanquantitesextractedfromafinitenum- ber of particles back into the calculations. The time-averaging of the feedback quantities leads to approximately the same convergent results as those with- out time-averaging, while it reduces the bias errors significantly for the same number of particles per cell. Second, the particle time-series from the PDF cal- culations are analyzed, for the first time, to investigate the local extinction and re-ignition in the Sandia piloted flameE, and the auto-ignition in the Cabrahy- drogen/nitrogen lifted jet flame. The particle time-series provide deep insight intothecomplicatedcombustion processesintheseflamesanddemonstratethe capability ofthe modelsto represent these processes. Next, different types of weak second-order splitting schemes applicable to the stochastic differential equations from the composition PDF method are de- veloped and validated, which, for the first time, makes the composition PDF calculations second-order accurate in time in contrast to first-order accuracy in all previous composition PDFpractices. Finally,thecurrentRANS/PDFcapabilityisadvancedtothelargeeddysim- ulations (LES) with the composition PDF method. A new high-performance PDF code, called HPDF, is developed with the following attributes: second- order accuracy in space and time; scalable up to at least 4096 cores; supporting Cartesian and polar cylindrical coordinate systems; parallelizable by domain decomposition in two dimensions; and it has a general interface to facilitate coupling to different existing LES (or RANS) codes etc. The new HPDF code is combined with an existing LES code, and the first set of LES/PDF calculations based on the new code is performed. The numerical convergence of the HPDF code is verified. The overall good agreement of the LES/PDF results with the experimentaldataisobserved. ThenewLES/PDFcapabilityestablishestheba- sis for the future LES/PDF work to consider more advanced models, realistic chemistry, differential diffusion etc. BIOGRAPHICALSKETCH Haifeng Wang was born in April 1977, in Yancheng, China. He received his Bachelor degree in Engineering Thermophysics from the University of Science andTechnology ofChina(USTC)in 2000. InAugust 2002,hemarried Haixia Ji. He and his wife came to the Unite States in 2005. He enrolled in the Ph.D. pro- gram at Cornell University in August 2005, majoring in Mechanical Engineer- ing, and received his master degree in June 2008. He has accepted a postdoc position with Professor Stephen B.Pope atCornell University aftergraduation. iii Tomy wifeHaixiaJi iv ACKNOWLEDGEMENTS I am most grateful to my Ph.D. advisor, Professor Stephen B. Pope, for his pa- tience, outstanding guidance and constant help. He directs me to develop crit- ical thinking, judging and decision making toward a rigorous researcher. I am also grateful to Professors David A. Caughey and Alexander B. Vladimirsky for their suggestions and help to my Ph.D. study and for serving in my special committee. I want to thank all professors at Cornell who taught me courses, especially Professor Franklin K. Moore who provided a lot of advice about my course project which ledto a journal publication. I appreciate my colleagues for their discussions and suggestions. They are the current membersandaffiliates ofthe Turbulence andCombustion Group at Cornell, Ms. Sharadha Viswanathan, Mr. Pavel P. Popov, Mr. David H. Rowin- ski, Dr. Konstantin A.Kemenov,Mr. VarunHiremath, Mr. Parvez Sukheswalla, Dr. Steven R. Lantz, and the previous group members, Dr. Renfeng R. Cao, Dr. Zhuyin Ren, Dr. Liuyan Lu, and Dr. Randall J. McDermott. I want to thank the consulting group from Cornell Center for Advanced Computing, especiallyDr. Steven R. Lantz, fortheir tremendous helpin high-performance computing. I am sincerely grateful to mywife, our parents, brothers and sisters fortheir endless love, encouragement and support. I wish to remember my father who passed away during my study at Cornell. My father has the deepest influence on my growing. His leaving is the greatest loss of my life. His spirit, however, will live forever in myheart. Ifinallywishtothankallmyfriendsfortheirfriendshipandenormoushelp. v TABLE OFCONTENTS Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table ofContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List ofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List ofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Study ofturbulent combustion . . . . . . . . . . . . . . . . . . . . 3 1.3 Overviewof chapters . . . . . . . . . . . . . . . . . . . . . . . . . . 7 References 12 2 Time-averaging strategies in the finite-volume/particle hybrid algo- rithm for thejoint PDFequation of turbulentreactive flows 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Time-averagingin HYB2D . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Summary of HYB2D . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Time-averagingtechnique . . . . . . . . . . . . . . . . . . . 18 2.2.3 Biasdue to the velocity correction . . . . . . . . . . . . . . 22 2.3 Influenceof time-averaging on bias . . . . . . . . . . . . . . . . . . 26 2.3.1 Cabra lifted H /N jetflame . . . . . . . . . . . . . . . . . 26 2 2 2.3.2 Sandia piloted flameE . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Influenceof the model constantC . . . . . . . . . . . . . . . . . . 34 ω1 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References 42 3 Lagrangian investigation of local extinction, re-ignition and auto- ignition in turbulentflames 44 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Particle calculations and Eulerian scatter plots . . . . . . . . . . . 49 3.3 Lagrangian particle tracking . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Particle trajectories in Sandia flameE . . . . . . . . . . . . . . . . . 58 3.4.1 Trajectories ofcontinuously burningparticles . . . . . . . 63 3.4.2 Trajectories oflocally extinguished particles . . . . . . . . 65 3.5 Particle trajectories in Cabra H /N lifted flame . . . . . . . . . . 73 2 2 3.6 Rolesof mixing andreaction during re-ignition and auto-ignition 80 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 References 86 vi 4 Weaksecond-ordersplittingschemesforLagrangianMonteCarlopar- ticle methodsfor thecomposition PDF/FDF transportequations 90 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 PDFmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3 Numericalsolutions of SDEs . . . . . . . . . . . . . . . . . . . . . 101 4.3.1 Ito SDEsand weakconvergence . . . . . . . . . . . . . . . 101 4.3.2 SDE system with frozen coefficients . . . . . . . . . . . . . 102 4.3.3 Weaksecond-order ItoSDE schemes . . . . . . . . . . . . . 103 4.4 Method of manufactured solutions (MMS) for Monte Carlo par- ticle methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.1 Augmented SDE system for MMS . . . . . . . . . . . . . . 108 4.4.2 Error analysisfor weakconvergence . . . . . . . . . . . . . 110 4.4.3 Computational cost of a Monte Carloconvergence study . 113 4.5 Weaksecond-order splitting schemes . . . . . . . . . . . . . . . . . 114 4.5.1 Sub-stepping of scalarevolution . . . . . . . . . . . . . . . 115 4.5.2 Splitting schemesofthe coupled SDE system . . . . . . . . 120 4.6 Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.6.1 Order of weakconvergence ofdifferent splitting schemes 129 4.6.2 Comparison ofdifferent splitting schemes . . . . . . . . . 136 4.6.3 Convergence ofhigh moments . . . . . . . . . . . . . . . . 138 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 References 149 5 Large Eddy Simulation/Probability Density Function Modeling of a TurbulentCH /H /N JetFlame 154 4 2 2 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.2.1 LESsolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.2.2 PDF methods andparticle methods . . . . . . . . . . . . . 161 5.3 Verification ofthe HPDF code . . . . . . . . . . . . . . . . . . . . . 163 5.4 LES/PDF calculations ofDLR FlameA . . . . . . . . . . . . . . . . 165 5.4.1 Effect of LESgrid resolution . . . . . . . . . . . . . . . . . . 165 5.4.2 Consistency betweenLESand PDF . . . . . . . . . . . . . . 167 5.4.3 Composition fields . . . . . . . . . . . . . . . . . . . . . . . 170 5.4.4 Effect of time integration schemes . . . . . . . . . . . . . . 173 5.4.5 Computational cost . . . . . . . . . . . . . . . . . . . . . . . 174 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 References 177 vii 6 Conclusions andfuturework 180 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.2 Future work on LES/PDF . . . . . . . . . . . . . . . . . . . . . . . 185 6.2.1 LEScode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.2.2 LES/PDF code . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2.3 LES/PDF applications . . . . . . . . . . . . . . . . . . . . . 187 References 189 A Particle trajectories in physicalspace 190 B Proof of theSDE system with frozen coefficients 195 B.1 Proof of the frozen-coefficient ItoSDE . . . . . . . . . . . . . . . . 195 B.2 Proof of the frozen-coefficient scalarequation . . . . . . . . . . . . 197 C Measurementof the globalerror forweak convergence 199 D Manufacturedsolutions to one-dimensionaltestcase 201 References 204 viii