Computational simulation of detonation waves and model reduction for reacting flows

COMPUTATIONAL SIMULATION OF DETONATION WAVES AND MODEL REDUCTION FOR REACTING FLOWS NGUYEN VAN BO (B.Eng., Hanoi University of Technology, Vietnam M.Eng., Institute of Technology Bandung, Indonesia) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHISOLOPHY IN COMPUTATIONAL ENGINEERING (CE) SINGAPORE-MIT ALLIANCE NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgments It is a great pleasure to thank people who helped me make my dissertation has been possible, without their love, encouragement, support and guidance I would never have completed this dissertation. First, I would like to express my gratitude to Prof. Karen Willcox for her persistent guidance, encouragement and understanding. I am really happy and lucky to have a very nice advisor who has been willing to show and make me understand as well as forgive all my mistakes during the working time under her guidance. Her supports in academic life and real life are great and very important to me for this dissertation and future. Second, I also would like to show my appreciation and thank a very important person, Prof. Khoo Boo Cheong, for his guidance, insightful discussion and comments for this dissertation. I would also like to thank for his kindly helps and support since I applied for Ph.D candidate at SingaporeMIT Alliance programme. His constant guidance and support are also the keys for the completion of this research. A much gratitude to the thesis committee members, Prof. Jaime Peraire and Prof. Lim Kiang Meng, for spending time to read my thesis and very valuable comments and suggestions. I also thank to their kindly help and support during the time I have been studying at NUS and MIT. A special thank to Dr. Marcelo Buffoni for his guidance, suggestion, discussion and support during two years working together. He plays a very important role not only like an advisor but a really good friend. A great appreciation is not enough to express my gratitude to what he has done for me. I would also like to thank to Dr. Dou Huashu for insight discussion and suggestion for this research. A lot of thanks to Dr. Ngoc Cuong Nguyen for very interesting and helpful discussion. This dissertation is dedicated to my parents, my wife and my son who give me their love, encouragement, and firmly support. To my father: I still remember the day he told me, a little years old boy, that “when you are going up, just earn a Ph.D degree for me” when we were repairing the roof of our house together. At ii that time, I didn’t understand what his meaning was, however, I only understood when i had been studying at the Hanoi University of Technology for my Bachelor degree. What his meaning was to study for myself for my family and special for his longing-study dream that he could not pursue because of some reasons. To my mom who spends her life for taking care of me, encouraging me, and supporting me in any situation. She has kept her eyes on me through all steps of my life. To my wife who has always been being beside me and encouraging me to pass all obstacles and difficulties on my way of life. She shares with me from the badness to the goodness. Specially, she takes care of my son as the both roles of a father as well as a mother. A thousand of words might not enough to thank to you - my lovely wife, but i can not find any word from deep inside of my heart better than simple word of “thank-you”. To my son who are all my life, my happiness, and motivations for not only this dissertation but all my future aiming targets. To all friends - ACDLers, SMAers, NUSers, and apartment mates, i would like to thank for supporting, encouraging, discussing, and sharing all information. A special thank to Mr. Thang and his wife for their delicious food and talk every month. I would also like to thank to Mr. Ha Nguyen, Mr. Xuan Sang Nguyen, Mr. Khac Chi Hoang, Mr. Cong Tinh Bui, and Mr. Duc Viet Nguyen, and Ms. Van Thanh for discussing, sharing, boosting me morally and providing me great information resources. I would also like to thank to all staff members at SMA office and specially are Mr. Michael, Ms. Nora, Ms. Hong Yanling for very kindly helps. This work was supported by the Singapore-MIT Alliance (SMA) Computational Engineering Programme, National University of Singapore. iii Contents Thesis Summary ix List of Tables xi List of Figures xiii List of Symbols xxi 0.1 Nomenclature with English symbols . . . . . . . . . . . . . . . . . . xxi 0.2 Nomenclature with Greek symbols . . . . . . . . . . . . . . . . . . . xxiii Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Review of Detonation Physics . . . . . . . . . . . . . . . . . . 1.2.2 Numerical simulation of reacting flows . . . . . . . . . . . . . 1.2.3 Numerical simulation of detonation waves . . . . . . . . . . . 1.2.4 Model order reduction for reacting flow applications . . . . . 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Governing Equations and Numerical Method for Reacting Problems 15 2.1 Conservative Navier-Stokes equations for reacting flows . . . . . . . 16 2.2 Combustion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Equation of state for a perfect gas and thermodynamic polynomial fits 22 iv 2.4 Thermal and transport properties . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Transport properties . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Viscosity Coefficient . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . 26 2.4.4 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . 26 Boundary conditions for reacting flow problems . . . . . . . . . . . . 27 2.5.1 Reacting Navier-Stokes equations near a boundary . . . . . . 28 2.5.2 Local One Dimensional Inviscid Relation (LODI) . . . . . . . 30 2.5.3 Characteristic boundary conditions for reacting flow problems 31 2.6 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Numerical methods for spatial discretization . . . . . . . . . . . . . . 35 2.7.1 Domain discretization . . . . . . . . . . . . . . . . . . . . . . 35 2.7.2 The fifth order WENO-LLF scheme . . . . . . . . . . . . . . 36 2.7.3 The fourth-order central differencing scheme for viscous terms 38 Numerical method for thermo-chemical kinetics of reacting flows . . 40 2.8.1 Numerical method for chemical kinetics of reacting flows . . . 40 2.8.2 Temperature evaluation . . . . . . . . . . . . . . . . . . . . . 41 The numerical implementation of boundary conditions . . . . . . . . 42 2.9.1 The fourth-order one-sided finite difference . . . . . . . . . . 42 2.9.2 Solid wall boundary conditions . . . . . . . . . . . . . . . . . 43 2.9.3 Inlet and Outlet boundary conditions . . . . . . . . . . . . . 43 2.5 2.8 2.9 Validation and Comparison of Computer Code using Benchmark Problems 47 3.1 Validation of the computer code using benchmark problems . . . . . 48 3.2 Validation of the code for transport properties . . . . . . . . . . . . 50 3.3 Poiseuille flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 Non-reacting Poiseuille flow . . . . . . . . . . . . . . . . . . . 53 3.3.2 Poiseuille Reacting flows . . . . . . . . . . . . . . . . . . . . . 56 Gaussian flame propagation . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 v 3.5 Code validation for one dimensional ZND detonation waves . . . . . 63 Computational simulation of detonation waves in viscous reacting flows 65 4.1 Simulation of one-dimensional detonation waves . . . . . . . . . . . . 65 4.1.1 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1.2 One dimensional detonation wave structure . . . . . . . . . . 66 4.1.3 Comparison of detonation waves between viscous and inviscid reacting flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 69 Numerical Simulation of two-dimensional detonation waves in viscous reacting flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.1 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.2 Detonation wave propagation mechanism in 2D straight chamber 73 4.2.3 Role of wave components in the onset of detonation waves . . 78 4.2.4 Two-dimensional detonation cellular structure . . . . . . . . . 79 Computational simulation of detonation waves in inviscid reacting flows 5.1 5.2 82 Computational simulation of detonation waves in an abrupt detonation chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.2 Transition and propagation mechanism of the detonation waves 84 5.1.3 Critical ratio of the widths . . . . . . . . . . . . . . . . . . . 91 5.1.4 Quenched and successfully transition of detonation waves . . 93 5.1.5 Evolution of detonation cellular structure . . . . . . . . . . . 97 Simulation of detonation waves in axi-symmetric diverging detonation chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.2 Propagation mechanism of detonation waves in transition region of diverging chamber . . . . . . . . . . . . . . . . . . . . 100 vi 5.2.3 Relation between oblique angle and transition length in a diverging chamber . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2.4 Evolution of detonation cellular structure inside diverging chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Simulation of detonation waves in axi-symmetric converging detonation chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.1 Propagation mechanism of detonation waves in transition region of converging chamber . . . . . . . . . . . . . . . . . . . 104 5.3.2 Relation between oblique angle and transition length in converging chamber . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3.3 Evolution of detonation cellular structure inside converging chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.4 Critical radius for axi-symmetric detonation chamber . . . . . . . . . 109 Model Order Reduction for Reacting Flow Applications 112 6.1 Reduced model construction . . . . . . . . . . . . . . . . . . . . . . . 112 6.2 Proper Orthogonal Decomposition technique . . . . . . . . . . . . . 114 6.3 Discrete Empirical Interpolation Method . . . . . . . . . . . . . . . . 116 6.4 Solution of the reacting flow problem using the POD-DEIM reducedorder model. 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Two-species one-dimensional stiff nonlinear diffusion-reaction problem.119 6.5.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.5.2 Fixed parameter 6.5.3 Comparison with the computational singular perturbation method123 6.5.4 Impact of changes in over the average concentration of species . . . . . . . . . . . . . . . . . . . . . . . . 121 y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.6 Example 2: Premixed Gaussian flame problem . . . . . . . . . . . . 129 6.6.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.6.2 Fixed parameters and inputs . . . . . . . . . . . . . . . . . . 130 6.6.3 Varying Prandtl number: P r ∈ [0.5, 1.0] . . . . . . . . . . . . 139 vii 6.6.4 Analysis of the impact of input parameters on the total heat released and the average value of species HO2 . . . . . . . . . 142 Conclusions and Recommendations for Future Work 148 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 151 Bibliography 153 viii Thesis Summary In this study, numerical simulations are performed for different detonation chambers to evaluate and analyze the influence of geometry on the detonation process. An efficient reduced-order model, obtained by systematic reduction of the original high-order full model, is performed to overcome the computationally expensive of the reacting flows. Here, a numerical simulation code has been developed for one and two-dimensional reacting flows. The numerical code is validated through comparisons to benchmark problems. The numerical results show that the detonation wave characteristics are in good agreement with the ZND model and experimental data. The physical and chemical characteristics of the detonation waves, the role of transverse waves, and detonation wave propagation mechanisms are investigated. For a two-dimensional abrupt detonation chamber, the propagation mechanism of detonation waves from the small chamber to larger chamber is investigated. Our findings indicate that there exists a critical value of ratio d2 /d1 = 1.8. Beyond this value, the detonation sustenance fails in the transition from the small to larger chamber, otherwise, it is ensured. The reasons of the failure and successful transition of detonation are founded. For an axi-symmetric converging/diverging detonation chamber, the behavior and mechanism of detonation wave propagation inside the chambers are investigated. For convergence case, two distinct cellular structure regions, separated by the triple point trajectory, are founded. There is no reflection region observed when the oblique angle is beyond 56o . For divergence case, all the detonation cells of the original detonation have disappeared before the new ones are created for an oblique angle greater than 45o , while the original detonation cells are somewhat maintained for an oblique angle smaller than 45o . The transition length is a function of both the oblique angle and the ratio d2 /d1 . Our findings reveal that the transition length reaches the minimum value when the oblique angle is about 45o . For a successful transition of all case, the evolution of detonation cellular structure inside the chamber is investigated, and the regular detonation cells in new stable state are reconstructed with size similar to those in the original stable region. ix The reduced-order model is obtained using the POD-DEIM method for chemical kinetics part of chemical reacting flows. The POD technique is employed to extract a low-dimensional basis that represents the dominant characteristics of the system trajectory in state-space. The DEIM algorithm is then applied to improve the efficiency in computing the projected nonlinear terms in the POD reduced system. To demonstrate the model order reduction method, the stiff diffusion-reaction model (1) and the multi-step reacting flow model (2) are considered. The reduced model of different dimensions is obtained to compute and analysis the relative accuracy and the computational time. The results show that the reduced model can accurately produce and predict the solution of the original full model over a wide range of parameters with some factors of reduction in the computational time (about 5.0 for (1) and 10.0 for (2)). Monte-Carlo simulations are performed for the reduced model to estimate variability in the outputs of interest of reacting flow simulations. The obtained results show that the reduced model can speed up computations by factors of about 5.0 for (1) and 10.0 for (2) compared to the original full model, and yet retain reasonable accuracy. x Chapter Conclusions and Recommendations for Future Work 7.1 Conclusions The design of a viable detonation engine requires detailed knowledge of the det- onation process. Of particular importance is the effect of geometry on the chemical and physical dynamics of detonation waves. Since experiments for pulse detonation engines can be very costly, numerical simulation provides an alternative means for the analysis of the detailed detonation process. In this study, numerical simulations are performed for different detonation chambers to evaluate and analyze the influence of geometry on the detonation process. Such numerical simulations of reacting flow can be computationally expensive, due to the need to resolve the many different time and length scales associated with the many species and chemical reactions. An efficient reduced-order model, obtained by systematic reduction of the original high-order full model, can overcome this computational burden. Here, a numerical simulation code has been developed for one and two-dimensional reacting flows. The numerical code is validated through comparisons to benchmark problems of non-reacting and reacting flows. One-dimensional and two-dimensional 148 straight detonation chamber models are considered for simulating the detonation waves in viscous reacting flows. The results show that the detonation wave characteristics (both one-dimensional and two-dimensional) are in good agreement with the ZND model and experimental data. The physical and chemical characteristics of the detonation waves, the role of transverse waves, and detonation wave propagation mechanisms are investigated. The two-dimensional abrupt detonation chamber and axi-symmetric converging/diverging chamber are also simulated to study the dynamics of detonation waves for inviscid reacting flows. For a two-dimensional abrupt detonation chamber, the propagation mechanism of detonation waves from the small chamber to the larger chamber is investigated. Our findings indicate that there exists a critical value of ratio d2 /d1 = 1.8, which is determined as the head of expansion line reaches the axis of the detonation chamber. When this ratio is larger than 1.8, detonation sustenance fails in the transition from the small to large chamber. Otherwise, the detonation sustenance is ensured in the transit from the small chamber to the larger chamber. This value is in good agreement with previous numerical and experimental results. The detonation waves successfully transit from the small chamber to the large chamber via three mechanisms: (1) the maintenance of triple points (hotspots), (2) creation of new hotspots from the intersection of the reflection waves and expansion waves, and (3) new bubbles or hotspots are generated from instability in the transition region (local explosion). Conversely, the detonation failure to transit from small channel to larger channel happen when all the original triple points and hotspots disappear, and there are no hotspots and triple point created in the domain. In order to reach the new stable state, the successful transition of detonation waves passes through five regions. These are the original stable region, expansion region, reflection region, transition region and new stable detonation region. The original stable region is narrowed with a loss of detonation cells as the detonation front moves forward in the streamwise direction. In the expansion region, all the detonation cells disappear as the pressure and temperature decrease. In the reflection region, new hotspots are created via the interaction of reflection waves and expansion waves. In the transition region, 149 the un-burnt mixture is re-ignited, and detonation waves are formed with irregular detonation cells. In the new stable detonation region, the regular detonation cells are reconstructed with size similar to those in the original stable region. For the converging/diverging detonation chamber, the behavior and mechanism of detonation wave propagation inside the axi-symmetric convergence/divergence chambers are investigated. The detonation wave is first compressed in the converging chamber and then expanded, while the detonation wave is first expanded and then compressed in the diverging chamber. Two distinct cellular structure regions are created by Mach-reflected detonation, and separated by the triple point trajectory. There is no reflection region observed when the oblique angle is beyond 56◦ . All the detonation cells of the original detonation have disappeared before the new ones are created for an oblique angle greater than 45o , while the original detonation cells are somewhat maintained for an oblique angle smaller than 45◦ . The transition length is a function of both the oblique angle (θ) and the ratio d2 /d1 . For d2 /d1 = 0.5 (convergent case) and d2 /d1 = 1.5 (divergent case), our findings reveal that the transition length reaches the minimum value when the oblique angle is about 45◦ . In order to reach the new stable state, the detonation waves in the axi-symmetric converging chamber also pass through five regions, which are the original stable region, the compression region, the expansion region, the transition region, and the new stable state region. When there is no compression region in the diverging chamber, the cell size in the downstream region is also similar to that in the original upstream region. The reduced-order model is obtained using the POD-DEIM method for chemical reacting flows. The POD technique is employed to extract a low-dimensional basis that represents the dominant characteristics of the system trajectory in state-space. The DEIM algorithm is then applied to improve the efficiency in computing the projected nonlinear terms in the POD reduced system. To demonstrate the model order reduction method, two examples are considered. The first is a stiff diffusionreaction model and the second is a more complex multi-step reacting flow model. In both cases, reduction is performed for the chemical kinetics. For the stiff diffusion- 150 reaction model, the reduced model has a dimension of 30 which is much smaller than 200 for the original full model. The results show that the reduced model can accurately produce and predict the solution of the original full model with factor of about 5.0 reduction in computational time. This factor might be expected to be larger as the dimension of the full model becomes larger. The results also show that the reduced model can accurately predict the solutions of the full model over a wide range of parameters. For more complex multi-step reacting flows, the dimensions of the reduced models are less than 60, which is much smaller than the dimension of 91809 for the original full model. Monte-Carlo simulations with 500 samples are performed for the reduced model to estimate variability in the outputs of interest of reacting flow simulations. The obtained results show that the reduced model can speed up computations by a factor of about 10.0 compared to the original full model, and yet retain reasonable accuracy. 7.2 Recommendations for Future Work The computer code has been developed for two-dimensional viscous reacting flows, but it should be fairly straightforward to extend it to three dimensional simulation. A three-dimensional analysis would allow the simulation of a three-dimensional multi-head detonation front, a spinning detonation front, and three-dimensional cellular structure. Furthermore, a rotating detonation wave engine model could be analyzed and studied. In terms of the numerical method for the chemical kinetics, the CHEMEQ package used in this research is a serial source code, which was modified from the (original) Fortran version. There still exists some limitations such as slow convergence. Hence, it is possible to use other solvers instead of the CHEMEQ, such as the CVODE solver. This is an optimization solver for stiff nonlinear ODEs system. Some simple comparisons between CHEMEQ and CVODE have been done for zerodimensional chemical kinetic models. We observed that the solution obtained from both methods achieve the same level of accuracy; however, the CVODE solver runs 151 much faster than the CHEMEQ package. Therefore, it is possible to use the CVODE instead of CHEMEQ for saving computational time. A limitation of the model reduction approach for typical reacting flow problems is that the method requires saving too many snapshots of both the solutions and the nonlinear term to get reasonable accurate solution. In some cases, the simulations need to run for a long time, resulting in too many snapshots to store for computing the POD basis and interpolation points. Therefore, the offline computation is a significant challenge, especially with limited computer resources. In this study, the POD-DEIM model reduction method is applied only to the chemical kinetics, while the fluid dynamics are still solved using the original full model. This is why the total simulation time only has a speed up factor of 10, in spite of several orders of magnitude in time saving for the chemical kinetics part. Therefore, the application of the model reduction method for the fluid dynamics part is another avenue for future work to obtain overall speedup in the computational simulations. The projection-based reduced model techniques may not accurately approximate a discontinuous solution. In particular, the POD-DEIM technique can not accurately approximate the solution of the detonation problem at the detonation front, due to the large discontinuity of the solution in this region. However, the POD-DEIM can approximate well the solution of the detonation problem in smooth regions. Another area of future work is to apply the POD-DEIM method in the smooth regions and solve using the full model over the thin detonation front region. However, this work requires an algorithm to detect the location of detonation front. In the context of the combustion process, besides the present input parameters employed in this current work, there are many other input parameters needed to gain better understanding of the physical process. For example the optimal ratio of fuel and air in the initial mixture for having maximum total heat released, reaction parameters (pre-coefficients A, the exponential parameter β, the activation energy E0 , etc.). Therefore Monte Carlo simulation over a larger input space using the reduced model may provide further opportunity to analyze these input parameters. 152 Bibliography [1] Kuo, K.K., Principle of combustion, John Wiley, 1986. [2] William, F.A., Combustion theory, Benjamin Cumming, 1985. [3] Poisot, T., Theoretical and numerical combustion, Edwards, Inc, 2001. [4] Wilson, G.J. and MacCormack, R.W., Modelling supersonic combustion using a full-implicit numerical method. AIAA 90-2307,1990. [5] CHEMKIN Release 4.1.1, Reaction Design, San Diego, CA, USA, 2007. Available at: http://www.reactiondesign.com/products/open/chemkin.html [6] Stull, D.R., JANAF Thermochemical Tables, National Standard Reference Data Series. U. S. National, Bureau of Standards No. 37. 2nd Ed. Gaithersberg, Maryland, 1971. [7] Gordon, S. and McBride, B.J., Computer program for calculation of complex chemical equilibrium compositions and application I, Analysis, Tech. Rep. NASA RP-1311, 1976. [Online] Available at: http://www.lerc.nasa.gov/WWW/CEAWEB. [8] Chase, M.W., Davis, C.A., Downey, J.R., Frurip, D.J., McDonal, R.A., and Syverud, A.N, JANAF thermochemical tables, Journal of Physical Chemistry, 1985. [9] Yi, T.H., Numerical study of chemically reacting viscous flow relevant to pulsed detonation engines, Ph.D. dissertation, The University of Texas at Arlington, USA, 2005. [10] Smith, G.P., Golden, D.M., Frenklach, M., Moriarty, N.W., Eiteneer, B., ant et al. GRI-Mech 3.1, [Online]. Available at: http://www.me.berkeley.edu/grimech. [11] Kee, R.J., Coltrin, M., and Glarborg P., Chemical reacting flows. New Jersey, Wiley-Interscience, 2003. [12] Kee, R. J., Warnatz, J., and Miller, J. A., A Fortran Computer Code Package for the Evaluation of Gas-Phase Viscosities, Conductivities, and Diffusion Coefficients, Sandia National Laboratories Report SAND83-8209, 1983. [13] Turns, S.R., An introduction to Combustion: McGraw-Hill, 1996. Concepts and Application, 153 [14] Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B., Molecular Theory of Gases and Liquids. New York, John Wiley, 1954. [15] Dixon-Lewis, G., Proceedings of the Royal Society A. 304:111 (1968). [16] Warnatz, J, Numerical Methods in Flame Propagation, edited by Peters, N., and Warnatz, J., Friedr. Vieweg and Sohn, Wiesbaden, 1982. [17] Monchick, L. and Mason, E. A., Transport properties of polar gases, J. Chem. Phys. 35:1676-97, 1961. [18] Wilke,C. R., , Journal of Chemical Physics 18:517, 1950. [19] Bird, R. B., Stewart, W. E., and Lightfoot, E. N., , Transport Phenomena, John Wiley and Sons, New York, 1960. [20] Mathur, S., Tondon, P. K., and Saxena, S. C., , Molecular Physics 12:569, 1967. [21] Poinsot, T., Lele, s., Boundary conditions for direct simulations of compressible viscous flows, Journal of Computational Physics 101:104129, 1992. [22] Yoo, C., Wang, Y., Trouve, A., Im, H., Characteristic boundary conditions for direct simulations of turbulent counterflow flames, Combustion Theory and Modeling 9(4):617646, 2005. [23] Yoo, C.S. and Im, H.G., Charateristic boundary conditions for simulation of compressible reacting flows with multi-dimensional, viscous and reaction effects, Combustion theory and modeling, 11(2), 259-286, 2007. [24] Baum, M., Poinsot, T., Thevenin, D., Accurate boundary conditions for multicomponent reactive flows, Journal of Computational Physics 116(2):247261, 1995. [25] Nicoud, F., Defining the waves amplitude in characteristic boundary conditions, Journal of Computational Physics 149: 418-422, 1999. [26] Okongo, N., Bellan, J., Consistent boundary conditions for multicomponent real gas mixtures based on characteristic waves, Journal of Computational Physics 176 (2): 330344, 2002. [27] Polifke, W., Wall, C., Moin, P., Partially reflecting and non-reflecting boundary conditions for simulation of compressible viscous flow, Journal of Computational Physics 213 (1): 437449, 2006. [28] Sutherland,J. and Kennedy, C., Improved boundary conditions for viscous, reacting, compressible flows, Journal of Computational Physics 191 (2): 502524, 2003. [29] Thompson, K.W., Time dependent boundary conditions for hyperbolic systems, Journal of Computational Physics 68: 124, 1987. [30] Lodato, G., Domingo, P., Vervisch, L., Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows, Journal of Computational Physics 227: 51055143, 2008. 154 [31] Harten, A., High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics 49: 357-393, 1983. [32] Shu, C.W., Essentially Non-Oscillatory and Weighted Essentially NonOscillatory schemes for Hyperbolic Conservation Laws, NASA/CR-97-206253, ICASE Report No. 97-65, 1997. [33] Jiang, G.S. and Shu, C.W., Efficient Implementation of Weighted ENO Scheme, Journal of Computational Physics 126:202-228, 1996. [34] Shu, C.W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes Journal of Computational Physics 77: 439-471, 1988. [35] Shu, C.W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II Journal of Computational Physics 83: 32-78, 1989. [36] Henrick, A.K., Aslam, T.D. and Powers, J.M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points Journal of Computational Physics 207: 542-567, 2005. [37] Roe, P.L., Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes, Journal of Computational Physics 43: 357-372, 1981. [38] Fedkiw, R.P., Merriman, B. and Osher, S., High accuracy numerical methods for thermal perfect gas flows with chemistry, Journal of Computational Physics 132:175-190, 1997. [39] Dou, H.S., Tsai, H.M, Khoo, B.C, and Qiu, J., Simulations of detonation wave propagation in rectangular ducts using a three-dimensional WENO scheme, Combustion and Flame 154: 644659, 2008. [40] Qu, Q., Khoo, B.C, Dou, H.S., and Tsai, H.M., The evolution of a detonation wave in a variable cross-sectional chamber, Shock Waves Journal, DOI 10.1007/s00193-008-0157-7, 2008. [41] Shen, Y.Q., Zha, G.C., and Chen, X., High order conservative differencing for viscous terms and the application to vortex-induced vibration flows, Journal of Computational Physics 228:82838300, 2009. [42] Shen, Y.Q., Wang, B.Y., and Zha, G.C., Implicit WENO scheme and high order viscous formulas for compressible flows, AIAA-paper 2007-4431, June 2007, to appear in AIAA Journal. [43] Shen, Y.Q., Zha, G.C., Simulation of Flows at All Speeds with Implicit HighOrder WENO Schemes, 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition - January 2009, Orlando, Florida. [44] Young, T.R, Boris, J.P., A numerical technique for solving stiff ordinary differential equations associated with the chemical kinetics of reactive-flow problems, J. Phys. Chem., 81(25), pp 24242427, 1977. [45] Young, T.R., CHEMEQ-Subroutine for solving stiff ordinary differential equations, Huntsville, AL. AIAA 2003-4511, AD-A0835545, 1979. 155 [46] Deiterding, R., Parallel adaptive simulation of multi-dimensional detonation structures, Ph.D. dissertation, Brandenburgische Technische University, Cottbus, Germany, 2003 [47] Anderson, J. D., Computational Fluid Dynamics: The Basics With Applications, Science/Engineering/Math, McGraw-Hill Science, ISBN 0070016852, 1995. [48] Oran, E.S. and Boris, J.P, Numerical simulation of reactive flows, Second edition, Cambrigde University Press, 2001. [49] Fedkiw, R.P., A survey of Chemically Reacting Compressible Flows, Ph.D. dissertation, UCLA, USA, 1996. [50] Wada, Y., Ogawa, S., Ishiguro, T. and Kubota, H., A Generalized Roe s Approximate Riemann Solver for Chemically Reaction Flows, AIAA-89-0202, 1989. [51] Liu, Y. and Vinokur, M., Upwind algrithm for general thermo-chemical nonequilibrium flows, AIAA-89-0201, 1989. [52] Rahul, K., Bhattacharyya, S. N., One-sided finite-difference approximations suitable for use with Richardson extrapolation, Journal of Computational Physics 219: 1320, 2006. [53] Lax, P.D., Weak solution of nonlinear hyperbolic equations and their numerical compuational, Comm. Pure. Math. 7, 159-193, 1954. [54] Goodwin, D.G., Cantera code package. [Online]. http://navier.engr.colostate.edu/tools/diffus.html. Available at: [55] Joseph, M.P. and Samuel, P., Accurate spatial resolution estimates for reactive supersonic flows with detailed chemistry, AIAA J. 43(5), 1088-1099, 2005. [56] Oran, E.S., Weber, J.E., Stefaniw, E.I, Lefebvre, M.H. and Aderson, J.D., A numerical study of two-dimensional H2-O2-Ar detonation using a detailed chemical reaction model Combustion and Flame 113:147-163, 1998. [57] Lefebvre M.H. and Oran E.S., Analysis of shock structures in regular detonation, Shock Waves 4:277-283, 1995. [58] Zeldovich, Ya B., Journal of Experiment and Theory in Physics, 10: 542-568, 1940. [59] Von Neumann, Theory of Detonation Waves, J. OSRD Report: 549, 1942. [60] Doering, W., On Detonation Processes in Gases, Annals of Physics, 43 (5): 421-436, 1943. [61] Chapman, D.L., On the rate of explosion in gases, Philosophical Magazine, Vol. 47, pp.90-104, 1899. [62] Jouguet, E.J., On the propagation of chemical reactions in gases, Math Pures Appl, Series 6, Vol. 1: pp.347-425, 1905. 156 [63] Tarver, C.M., Chemical energy release in one-dimensional detonation waves in gaseous exploisive, Combustion and Flame 46: 111-133, 1982. [64] Fickett, W. and Davis, W.C., Detonation, University of California Press, Berkeley, 1979. [65] Zhuravskaya, T.A., Propagation of detonation waves in plane channels with obstacles, Fluid Dynamics, 48 (6) 987-994, 2007. [66] Levin, V., Markov, V., Zhuravskaya, T.A., and Osinkin, S., Propagation of cellular detonation in the plane channels with obtacles, Shock Waves, Part IV, 347-351, 2009. [67] Moen, I.O., Funk, J.K., Ward, S.A., Rude,G.M., and Thibault, P.A., Detonation length scales for fuel air explosives, Dynamics of Shock Waves, Explo. and Deto., Progress in Astro. Aero., 94: 55-79, 1984. [68] Li, J., Lai, W.H., and Chung, K., Tube diameter effect on deflagration to detonation transition of propane-oxygen mixtures, Shock Waves, 16: 109-117, 2006. [69] Thomas, G.O. and Williams, R.L., Detonation interaction with wedges and bends, Shock Waves 11:481-492, 2002. [70] Guo, C.M., Zhang, D.L. and Xie, W., The Mach reflection of a detonation based on soot track measurements Combustion and Flame 127:2051-2058, 2001. [71] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., MCKenney, A., and Sorensen, D.C., ARPACK User’s Guide Third Edition, SIAM, 1999. [72] Daniel, J.W., Gragg, W.B., Kaufman, L., and Stewart, G.W, Reorthogonalization and Stable Algorithms for Updating the Gram-Schmidt QR factorization, Mathematics of Computation, 30: 772-795, 1976. [73] Barrault, M., Maday, Y., Nguyen, N.C, and Patera, A.T., An ‘Empirical Interpolation’ Method: Application to Efficient Reduced-Basis Discretization of Partial Differential Equations., Comptes Rendus Mathematique, 339(9): 667672, 2004. [74] Grepl, M.A., Maday, Y., Nguyen, N.C., and Patera, A.T., Efficients ReducedBasis Treatment of Nonaffine and Nonlinear Partial Differential Equation, Mathematical Modelling and Numerical Analysis, 41(3):575-605, 2007. [75] Nguyen, N.C., and J. Peraire, An efficient Reduced-order modelling approach for a nonlinear parameterized partial differential equations, Internaltional journal for Numerical Methods in Engineering, 76: 27-55, 2008. [76] Bui-Thanh, T., Damodaran, M., and Willcox, K., Aerodyanmics Data Reconstruction and Inverse Design using Proper Othorgonal Decomposition, AIAA Journal, 42: 1505-1516, 2004. [77] Bui-Thanh, T., Willcox, K., and Ghattas, O., Model Reduction for a LargeScale Systems with High-Dimensional Parametric Input Space, SIAM J. Sci. Comput, Vol. 30, No. 6, 3270-3288, 2008. 157 [78] Chatturantabut, S., and Sorensen, D., Discrete Empirical Interpolation for Nonlinear Model Reduction, Technique Report TR09-05, Department of Computational and Applied Mathematics, Rice University, 2009. [79] Chatturantabut, S., and Sorensen, D., Application of POD and DEIM on Dimension Reduction of Nonlinear Miscible Viscous Fingering in Porous Media, Technique Report TR09-25, Department of Computational and Applied Mathematics, Rice University, 2009. [80] Sorensen, D., and Antoulas, A., The Sylvester Equation and Approximate Balanced Reduction, Linear Algebra and Its Applications, Vol. 351352, pp. 671700, Aug. 2002. [81] Antoulas, A,. and Sorensen D., and Gugercin, S., A survey of model reduction methods for large-scale systems, Structured Matrices in Operator Theory, Numerical Analysis, Control,Signal and Image Processing, Chapter 280, American Mathematical Society: Providence, RI, pp. 193-219, 2001. [82] Lo`eve, M., Probability Theory, Van Nostrand, New York, 1955. [83] Feldmann, P., and Freund, R., Efficient Linear Circuit Analysis by Pad Approximation via the Lanczos Process, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 14, No. 5, pp. 639649, 1955. [84] Sirovich, L., Turbulence and the dynamics of coherent structures. Part {I,II,III.}, Q. Appl. Math., Vol. 45, No. 3, pp. 561-590, 1987. [85] Bai, Z., Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Applied Numerical Mathematics, Vol. 43, no. 1-2, pp. 9-44, 2002. [86] Gugercin, S., and Antoulas, A Survey of Model Reduction by Balanced Truncation and Some New Results, International Journal of Control, Vol. 77, No. 8,pp. 748766, 2004. [87] Glover, K., All Optimal Hankel-norm Approximations of Linear Multivariable Systems and Their L∞ Error Bounds, International Journal of Control, vol. 39, no. 6, pp. 1115-1193, 1984. [88] Holmes, P., and Lumley, J.L., and Berkooz, G., Turbulence, coherent structures, dynamical systems and symmetry, Cambridge University Press, 1998. [89] Willcox, K., Paduano, J.D., Peraire, J., and Hall, K. C., Low order aerodynamic models for aeroelastic control of turbomachines, in Proceedings of the American Institute of Aeronautics and Astronautics Structures, Structural Dynamics, and Materials Conference, vol. 3, 1999, pp. 2204-14. [90] Willcox, K., and Peraire, J., Balanced Model Reduction via the Proper Orthogonal Decomposition, AIAA Journal, vol. 40, no. 11, pp. 2323-30, 2002. [91] Willcox, K., Biegler, L., Ghattas, O., Heinkenschloss, M., Keyes, D., and van Bloemen Waanders, B. (Eds.), Model Reduction for Large-Scale Applications in Computational Fluid Dynamics, in Real-Time PDE-Constrained Optimization, SIAM Book Series, pp. 217-233, 2007. 158 [92] Willcox, K. and Peraire, J., Application of Reduced-Order Aerodynamic Modeling to the Analysis of Structural Uncertainty in Bladed Disks. ASME Paper GT-30680, presented at the ASME International Gas Turbine and Aeroengine Technical Conference, Amsterdam, The Netherlands, June 2002. [93] Astrid, P., Weiland, S., Willcox, K., and Backx, T., Missing Point Estimation in Models Described by Proper Orthogonal Decomposition, IEEE Transactions on Automatic Control, Vol. 53, Issue 10, pp. 2237-2251, 2008. [94] Everson, R, and Sirovich, L., The {Karhunen-Lo´eve} Procedure for Gappy Data, Journal of the Optical Society of America, Vol. 12, no. 8, pp. 1657-1664, 1995. [95] Bos, R., and Bombois, X., and van den Hof, P., Accelerating large-scale nonlinear models for monitoring and control using spatial and temporal correlations, Proceedings of American Control Conference, Boston, USA, 2004. [96] Lucia, D., King, P., and Beran, P., Reduced order modelling of the twodimensional flow with moving shock, Computers and Fluids, Vol. 32, pp. 917938, 2003. [97] Bufifoni, M., Camarri, S., Iollo, A., and Salvetti, M, Low-dimensional modelling of a confined three-dimensional wake flows, Journal of Fluids Mechanics, Vol.569, pp. 141-150, 2006. [98] Buffoni, M. Telib, H. and Iollo, A.,Iterative Methods for Model Reduction by Domain Decomposition, Computers & Fluids, Volume 38, Issue 6, pp. 11601167, 2009. [99] Buffoni, M. and Willcox, K., Projection-Based Model Reduction for Reacting Flows, AIAA-2010-5008, presented at 40th Fluid Dynamics Conference and Exhibit, Chicago, IL, June 28-July 1, 2010. [100] Amabili, M., Sarka, A., and Padousis, M.P., Chaotic vibrations of circular sylinddrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method, Journal of Sound and Vibration, Vol. 290, No. 3-5, pp. 736-762, 2006. [101] Kerschen, G., Golinval, J.C., Vakakis, A.F., and Bergman, L.A., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical system: an overview, Nonlinear dynamics, Vol. 41, pp. 147-169, 2005. [102] Graham, W.R., Perraire, J., and Tang, K.Y., Optimal control of vortex shedding using low-order models. Part I, Internaltional Journal for Numerical Method in Engineering, Vol. 44, pp. 945-972, 1998. [103] Lall, S., Marseden, J., and Glavaski, S., Empirical model reduction of controlled nonlinear system, Proceeding of the IFAC World Congress, pp.473-478, 1999. 159 [104] Rewienski, M., and White, J., A Trajectory Piecewise-Linear Approach to Model Order Reduction and Fast Simulation of Nonlinear Circuits and Micromachined Devices, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 22, no. 2, pp. 155-70, 2003. [105] Lam S.H., Using CSP to understand complex chemical kinetics, Combust. Sci. Tech., 89: 375-404, 1993. [106] Lam S.H., and Goussis, D.A., The CSP method for simplifying kinetics, Internat. J. Chem. Knetics, 26: 461-486, 1994. [107] Hadjinicolaou, M., and Goussis D.A., Asymptotic solution of stiff PDEs with the CSP method: The reaction-diffusion equation, SIAM J. Sci. Comput. Vol. 20, No. 3, pp. 781-810, 1999. [108] Valorani, M., Creta, F., and Goussis, D.A., Local and global manifolds in stiff reaction-diffusion systems, Computational Fluid and Solid Mechanics Journal, 1548-1551, 2003. [109] Ramshaw, J.D., Partial chemical equilibrium in fluid dynamics, Phys. Fluids, Vol. 23, pp. 675, 1980. [110] Peters, N. and Williams, F.A., The assymtotic structure of methane flames, Complex chemical reaction systems, Springer series in Chemical Physics, Vol. 47, pp.310-317, 1987. [111] Brown, N.J., Li, G.P., and Koszykowshi, M.L., Mechanism reduction via principal component analysis, Int. J. Chem. Kinet, Vol. 29 (6), pp. 393-414, 1997. [112] Maas, U. and Pope, S.B., Implementation of simplified chemical kinetics based on iinstrinsic low-dimensional manifolds, Proc. Combust. Inst. Vol. 24, 103-112, 1992. [113] Maas, U. and Pope, S.B., Simplifying chemical kinetics: Instrinsic low dimensional manifolds in composition space. Combust. Flame. Vol. 88, pp. 239-264, 1992 [114] Maas, U. and Pope, S.B., Laminar flame calculations using simplified chemical kinetics based on instrinsic low-dimensional manifolds, Proc. Combust. Inst, 25, pp. 1349-1356, 1994. [115] Pope, S.B., Computtionally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combustion Theory and Modelling, 1. 41-63, 1997. [116] Tones, S.R., Moriarty, N.W., Brown, N.J., and Frenklach, M., PRISM: Piecewise reusable implementation of solution mapping, Isael Journal of Chemistry, 39:97-106,1999. [117] Douglas, A., Schwer, Pisi Lu, and William, H.G., An adaptive chemistry approach to modelling complex kinetics in reacting flows, Combustion and Flame, Vol. 133, pp. 451-465, 2003. 160 [118] Banerjee, I. and Ierapetritou, M.G., An adaptive reduction scheme to model reactive flows, Combustion and Flame, Vol. 144, pp. 619-633, 2006. [119] Binita, B., Douglas, A., Schwer, Barton, P.I., and William, H.G., Optimallyreduced kinetic models: Reaction elimination in large-scale kinetic mechanism, Combustion and Flame, Vol. 135, pp. 191-208, 2003. [120] Mitsos, A., Oxberry, G.M., Barton, P.I., and William, H.G., Optimal automatic reaction and species elimination in kinetics mechanism, Combustion and Flame, Vol. 155, pp. 118-132, 2008. [121] Roy, G.D., Frolov, S.M., Borisov, A.A., and Netzer, D.W., Pulse detonation propulsion: challenges, current status, and future perspective, Progress in Energy and combustion Science, Vol. 30, pp. 545-672, 2004. [122] Kailasanath, K., Review of Propulsion Applications of Detonation Waves, AIAA Journal, Vol. 39, No. 9, pp. 1698-1708, 2000. [123] Bussing, T., and Murman, E. M., Finite volume method for the calculation of the compressible chemically reacting flows, AAIA journal, vol. 26, pp. 10701078, 1988. [124] Kim, H. W., Lu, F. K., Anderson, A. A., and Wilson, D. R., Numerical simulation of detonation process in a tube, Computational Fluid Dynamics Journal, vol. 12, no. 2, pp. 227-241, 2003. [125] Rogers, R.C., and Chinitz, W., Using a global hydrogen-air combustion model in turbulent reacting flow calculations, AIAA Journal, vol. 21, no. 4, pp. 586592, 1983. [126] Park, C., Assessment of two-temperature kinetic model for ionizing air, AIAA, 87-1574, 1987. [127] Deiterding, R., Parallel a daptive simulation of multidimensional detonation strucutures, Ph.D. dissertation, Brandenburgische Technische University, Cottbus, Germany, 2003. [128] Greenberg, J.B., Operator splitting methods for the computation of reacting flows, Journal of Computers & Fluids, Vol. 11, Issue 2, pp. 95-105, 1983. [129] Gregory, J. M., William, R. G.,, and John, H. S., Numerical solution of the atmospheric diffusion equation for chemically reacting flows, Journal of Computational Physics, Vol. 45, No. 1, pp.1-42, 1982. [130] Singer, M. A., Pope, S. B., and Najm, H. N., Operator-splitting with ISAT to model reacting flow with detailed chemistry, Journal of Combustion Theory and Modelling, Vol. 10, No. 2, pp.199217, 2006. [131] Rankine, W.J.M., On the thermodyamic theory of waves of finite longitudinal disturbance, Philos Trans R Soc London 1870: 277-88. [132] Hugoniot, H., Propagation des Mouvements dans les Corps et sp´ecialement dans les Gaz Parfaits, Journal de l’Ecole Polyt Cahier (1887) 57:1-97. 161 [133] Mikhelson, V.A., Ph.D Dissertation, Imperial Moscow University Publication: 1890. [134] Voinov, A.N., On the mechnism of formation of the spinning detonation, Doklady of the Academic of Sciences, SSSR, Vol. 73, 1950, p.125-128. [135] Voitsekhovsky, V.B., Study of the structure of the front of the spinning detonation, Research on Physics Technical Institute, State Edition of Defense Industry, Moscow, 1958, pp.81-91. [136] Denisov YuN, Troshin Yak, Doklady USSR Acad Sci (1959) 125:110-113. [137] Brown, P.N., Byrne, G.D., and Hindmarsh, A.C., VODE: A variable coefficient ODE solver, SIAM Journal on Scientific and Statistic Computing, Vol. 10, pp.1038-1051, 1989. [Online]. Available: http://www.llnl.gov/CASC/ [138] Cambier, J.L. and Tegner, J.K. Strategies for Pulse detonation engines performance oftimization, Journal of Propulsion and Power, 14(4):489-498, 1998. [139] Benedik, WB., Guirao, C.M., Knystautas, R. and Lee, J.H., Crtitical charge for direct initiation of detonation in gaseous fuel-air mixture, Progress in Aeronautics and Astronautics, 106: 181-202, 1986. [140] Cooper, M., Jackson, S., Austin, J., Wintenberger, E. and Shepherd, J.E., Direct Experimental Impulse Measurements for Detonations and Deflagrations, Journal of Propulsion and Power, 18-5: 10331041, 2002. [141] New, T.H., Panicker, P.K., Lu, F.K. and H. M. Tsai, H.M., Experimental Investigations on DDT Enhancements by Schelkin Spirals in a PDE AIAA 2006552, 2006. [142] Gamezo, V.N., Khokhlov, A.M., and Oran, E.S., Effect of the wakes on shockflame interaction and deflagration to detonation transition, Proceeding of the Combustion Institute, Volume 29, Issue 2, pp. 2803-2808, 2002. [143] Parra-Santos, M.T., Castro-Ruiz, F., and Mendez-Bueno, C., Numerical simulation of the deflagration to detonation transition, Journal of Combustion, Explosion, and Shockwaves, Volume 41 (2), pp. 215-222, 2005. [144] Sergey, M., Frolov, Ilya, V., Semenov, Pavel, S., Utkin, Pavel, V., Komissarov, Vladimir, V., and Markov, Enhancement of shock-to-detonation transition in channels with regular shaped obstacles, Proc. 21st, ICDERS, Poitiers, France, July 23-27, 2007, pp. 215.s., [145] Levin, V., Markov, V., Zhuravskaya, T., and Osinkin, Propagation of cellular detonation in plane channels with obstacles, Journal of Fluid Dynamics, vol. 42 (6), pp. 987-994, 2008. [146] Mitrofanov, V.V. and Soloukhin, R. I., The diffraction of multi-front detonation waves, Journal Soviet Physics Doklady, Vol. 9, pp.1055-1058, 1965. [147] Edward, D.H., Thomas, G.O., and Nettleton, M.A., The diffraction of planar detonation waves at an abrupt area change, J. Fluid. Mech., Vol. 95. pp. 79-96, 1979. 162 [148] Edward, D.H., Thomas, G.O., and Nettleton, M.A., Diffraction of planar detonations in various fuel-air mixture at an area change, In Gasdynamics of Detonation and Explosion, Vol. 75 of Progress in Astronautics and Aeronautics, pp. 341-357, 1981. [149] Lee, J.H.S., Dynamic parameters of gaseous detonations, Annu. Rev. Fluid Mech., 16:311-336, 1984. [150] Li, C., and Kailasanath, K., Detonation transmission and transition in channel of different size, Proceeding of the Combustion Institute, Vol. 28, pp. 603-609, 2000. [151] Viswanath, K., Collin, T., John, H., and Fredic, C., Initiation of detonation in a large tube, 19th International Colloquium on the Dynamics of Explosions and Reactive systems, Hakone, Japan, August, 2003. [152] Fan, H.Y. and Lu, F.K, Numerical simulation of detonation processes in a variable cross-section chamber, Proc. IMechE, Part G: J. Aerospace Engineering, Vol.222, pp. 673-686, 2007. [153] Ohyagi, S., Obara, T., Nakata, F., and Hoshi, S., A numerical simulation of reflection processes of detonation waves on a wedge, Shock Waves Journal, Vol.10 (3): pp.185-190, 2000. [154] Guo, C. M., Zhang, D.L., and Xie, W., The Mach reflection of a detonation based on soot track measurements, Combustion and Flame, 127:2051-2058, 2001 [155] Thomas, G.O. and Williams, R.L.,Detonation interaction with wedges and bends, Shock Waves, Vol. 11: 481-492, 2002. [156] Deng, B., Hu, Z.M., Teng, H.H., and Jiang, Z.L., Numerical investigation on detonation cell evolution in a channel with area-changing cross section, Sci China-Phys Mech Astro, Vol. 50 (6), 797-808, 2007. [157] Gropp, W., Lusk, E., and Skjellum, A.,Using MPI portable Parallel Programming with the Message-Passing Interface, Second Edition, The MIT Press, Cambridge, Massachusetts, USA, 1999. [158] Wilkinson, B. and Allen, M., Parallel Programming: Techniques and Applications Using Networked Workstations and Parallel computers, Second Edition, Pearson Education Inc, USA, 2005. 163 [...]... Comparison of solutions of the pressures evolution at three sensor locations between reduced model of size 40 and full model of size 91809.133 6-14 Comparison of solutions of the density evolution at three sensor locations between reduced model of size 40 and full model of size 91809 133 6-15 Comparison of solutions of the temperature evolution at three sensor locations between reduced model of size 40 and. .. transmission of detonation waves from small tube to larger tube with the presence of the downstream solid wall 6 For example, Li and Kailasanath [150] in their simulation study of the detonation waves propagation and transmission inside the channels of different sizes, concluded that a local region of high pressure and temperature can be created by collision of reflected waves and detonation waves at the... objectives of this thesis are: 1 To develop a computer code for the numerical simulation of chemically reacting viscous detonation 2 To use the developed code to gain insight into the physical and chemical phenomena associated with the detonation waves and into the effects on detonation of the viscous and diffusion terms, and to capture the evolution of the detonation cell for different geometries of the detonation. .. Numerical simulation of detonation waves In this study, we make use of pre -detonation initiation in the simulation to gain a better understanding of the physical phenomena and propagation mechanism of detonation waves as they emerge from the small to larger channel in the detonation chamber, as well as to determine the critical value of the ratio of widths of small to large channel (d2 /d1 ) for successful... relative error and online computational time for different numbers of POD basis vectors 139 6.5 Comparison between full model and reduced-order model; MCS results are shown for the average value of species HO2 and total heat released for 500 randomly sampled values of the peak temperature of the initial conditions 145 6.6 Comparison between full model and reduced-order... Mitrofanov and Soloukhin [146] proposed a minimum value of a diameter of the detonation chamber, which is required for successful detonation transmission, however, they did not discuss the downstream dimension A correlated relation of the detonation cell size and critical value of diameter of the detonation chamber is studied and analyzed by Edwards et al.[147, 148] Besides simulation of the same mixture for. .. Comparison of computational time and relative error between the POD model (using 30 PODmode), the POD-DEIM model (using 30 POD modes and 30 interpolation points), the CSP method, and the full model 125 6.3 Comparison between full model and reduced-order model; Results of MCS using 1000 randomly normal distributed values of reaction time scale are shown for the results of species... 2: Comparison of species HO2 between the full model and reduced-order model MCS results are shown for 500 randomly sampled values of the width of the initial conditions The dashed line shows the sample mean 146 6-39 Example 2: Comparison of species HO2 between the full model and reduced-order model MCS results are show for 500 randomly sampled values of the width of the initial... 3 To perform numerical simulations in one and two dimensions to determine the detonation wave structure, the detonation cellular structure, the propagation mechanism of the waves inside the detonation chambers and the role of wave components in sustaining the detonation waves 4 To measure the effect of the geometry of the combustion chamber on the detonation in order to find the critical value of the... the ratio between the diameters of the detonation chamber and the ignition chamber that enable successful transmission of detonation waves, to find the causes of failure and/ or successful transmission, to obtain a relationship between deflagration -detonation transition (DDT) length and the oblique angle of the detonation chamber, and to assess quenching of the detonation waves inside a small chamber 5 . COMPUTATIONAL SIMULATION OF DETONATION WAVES AND MODEL REDUCTION FOR REACTING FLOWS NGUYEN VAN BO (B.Eng., Hanoi University of Technology, Vietnam M.Eng., Institute of Technology Bandung,. factors of reduction in the computational time (about 5.0 for (1) and 10.0 for (2)). Monte-Carlo simulations are performed for the reduced model to estimate variability in the outputs of interest of. reduced model of size 40 and full model of size 91809.133 6-14 Comparison of solutions of the density evolution at three sensor lo- cations between reduced model of size 40 and full model of size