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. 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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