Abstract：[Significance] With the requirement of high efficiency and heavy thrust, fluid mixing and combustion under transcritical and supercritical conditions have been increasingly used in advanced propulsion systems. Transcritical and supercritical fluids exhibit numerous mixing and combustion properties different from those of subcritical (low-pressure) fluids because of thermodynamic nonidealities and transport anomalies. Therefore, elucidating the fundamental characteristics of transcritical and supercritical fluid flows and combustion is vital. Extreme operating conditions pose severe challenges to experimental measurements and optical diagnostics, while large eddy simulation (LES) can be used to detail flow structures and combustion dynamics in a relatively affordable manner. To capture abnormal variations in thermophysical properties, introducing a real-fluid equation of state (EOS) is vital, as the EOS is required for the closure of governing equations. [Progress] In high-fidelity simulations of transcritical and supercritical flows, cubic EOSs, including the Soave-Redlich-Kwong EOS (SRK EOS), the Peng-Robinson EOS (PR EOS), and volume-translation methods, have been frequently used owing to their relatively simple forms and high computation efficiency. In LES-based system equations, EOS filtering introduces an unclosed subgrid term, which is generally neglected in ideal-gas problems. However, this subgrid term plays a substantial role in transcritical and supercritical flows owing to the highly nonlinear changes in thermophysical properties. For density-based solvers, subgrid pressure is required, while subgrid density is required for pressure-based solvers. For demonstration, the relative magnitude of the subgrid density with respect to the filtered density is evaluated using the results of direct numerical simulation (DNS) of the transcritical liquid oxygen/methane mixing layer. The contribution of the subgrid density increases with the filtering size, with up to 60% of the DNS-filtered density in the transcritical mixing regions. To account for the subgrid density effect, various types of subgrid density models are considered, including the Reynolds-filtered model (RFM), gradient model (GM), scale similarity model (SSM), and filtered density function (FDF) model. In RFM, the EOS is evaluated using Reynolds-filtered primitive variables rather than Favre-filtered variables, which leads to an underestimation of the filtered density and limited improvement in accuracy. GM is analogous to the Smagorinsky model with both static and dynamic forms, while SSM is formulated upon the assumption of scale similarity between the grid-filter and test-filter levels. Both GM and SSM improve the accuracy of the filtered density to some extent. FDF shows the overall best performance, with the lowest modeling errors. The model assumes a beta distribution form of the Favre FDF to model the subgrid-scale fluctuations. In summary, GM, SSM, and FDF show evident improvements in the filtered density, while RFM needs further improvement to consistently achieve the required accuracy. The development of the subgrid pressure is briefly introduced, and RFM best approximates the subgrid pressure. [Conclusions and Prospects] Future research on the subgrid modeling of the real-fluid EOSs for transcritical and supercritical flows is needed in the following aspects. First, a more accurate and computationally efficient EOS for high-fidelity simulations needs to be developed. Second, more universal and effective subgrid models of EOSs and other thermophysical properties should be explored. The prevailing multidisciplinary methods, such as deep neural networks and random forests, show great potential to improve both the accuracy and efficiency of LES. Enhanced models would strengthen the predictive capability of current modeling and simulation tools and support the development of next-generation propulsion systems.
周明烁, 丁思宇, 王兴建. 跨/超临界流体大涡模拟状态方程亚格子模型综述[J]. 清华大学学报（自然科学版）, 2023, 63(4): 473-486.
ZHOU Mingshuo, DING Siyu, WANG Xingjian. Review of subgrid models of the equation of state in the large eddy simulation of transcritical and supercritical flows and combustion. Journal of Tsinghua University(Science and Technology), 2023, 63(4): 473-486.
[1] ULAS A, BOYSAN E. Numerical analysis of regenerative cooling in liquid propellant rocket engines[J]. Aerospace Science and Technology, 2013, 24(1):187-197. [2] BOYER R R, COTTON J D, MOHAGHEGH M, et al. Materials considerations for aerospace applications[J]. MRS Bulletin, 2015, 40(12):1055-1066. [3] DAVID S A, SIEFERT J A, FENG Z. Welding and weldability of candidate ferritic alloys for future advanced ultrasupercritical fossil power plants[J]. Science and Technology of Welding and Joining, 2013, 18(8):631-651. [4] ALLAM R, MARTIN S, FORREST B, et al. Demonstration of the Allam cycle:An update on the development status of a high efficiency supercritical carbon dioxide power process employing full carbon capture[J]. Energy Procedia, 2017, 114:5948-5966. [5] SINGLA G, SCOUFLAIRE P, ROLON C, et al. Transcritical oxygen/transcritical or supercritical methane combustion[J]. Proceedings of the Combustion Institute, 2005, 30(2):2921-2928. [6] HUO H F, WANG X J, YANG V. A general study of counterflow diffusion flames at subcritical and supercritical conditions:Oxygen/hydrogen mixtures[J]. Combustion and Flame, 2014, 161(12):3040-3050. [7] WANG X J, HUO H F, WANG Y X, et al. Comprehensive study of cryogenic fluid dynamics of swirl injectors at supercritical conditions[J]. AIAA Journal, 2017, 55(9):3109-3119. [8] WANG X J, LI Y X, WANG Y X, et al. Near-field flame dynamics of liquid oxygen/kerosene bi-swirl injectors at supercritical conditions[J]. Combustion and Flame, 2018, 190:1-11. [9] WANG X J, ZHANG L W, LI Y X, et al. Supercritical combustion of gas-centered liquid-swirl coaxial injectors for staged-combustion engines[J]. Combustion and Flame, 2018, 197:204-214. [10] ZHANG L W, WANG X J, LI Y X, et al. Supercritical fluid flow dynamics and mixing in gas-centered liquid-swirl coaxial injectors[J]. Physics of Fluids, 2018, 30(7):075106. [11] 谢良涛,杨建国,董飞.船用低速柴油机烟气余热S-CO2布雷顿循环系统优化[J].大连海事大学学报, 2021, 48(1):89-97. XIE L T, YANG J G, DONG F. Optimization of S-CO2 Brayton cycle system for flue gas waste heat of marine low-speed diesel engine[J]. Journal of Dalian Maritime University, 2021, 48(1):89-97.(in Chinese) [12] CANDEL S, SCHMITT T, DARABIHA N. Progress in transcritical combustion:Experimentation, modeling and simulation[C]//Proceedings of the 23rd ICDERS. Irvine, USA:IDERS Press, 2011:111. [13] 解茂昭.内燃机跨临界/超临界燃料喷雾混合过程的机理与模型[J].燃烧科学与技术, 2014, 20(1):1-9. XIE M Z. Mechanism and modeling of transcritical/supercritical fuel spray and mixture formation in internal combustion engines[J]. Journal of Combustion Science and Technology, 2014, 20(1):1-9.(in Chinese) [14] 李云清,何鹏.超临界环境下燃料液滴蒸发过程的计算研究[J].内燃机学报, 2008, 26(1):55-61. LI Y Q, HE P. Numerical investigation of fuel droplet evaporation in a supercritical environment[J]. Transactions of CSICE, 2008, 26(1):55-61.(in Chinese) [15] DAHMS R N, MANIN J, PICKETT L M, et al. Understanding high-pressure gas-liquid interface phenomena in diesel engines[J]. Proceedings of the Combustion Institute, 2013, 34(1):1667-1675. [16] DAHMS R N, OEFELEIN J C. On the transition between two-phase and single-phase interface dynamics in multicomponent fluids at supercritical pressures[J]. Physics of Fluids, 2013, 25(9):092103. [17] CRUA C, MANIN J, PICKETT L M. On the transcritical mixing of fuels at diesel engine conditions[J]. Fuel, 2017, 208:535-548. [18] 唐纯逸,王忠,李瑞娜,等.亚/超临界环境正十二烷液滴相变过程分析[J].车用发动机, 2021(1):63-68. TANG C Y, WANG Z, LI R N, et al. Phase transition process of N-dodecane droplets under sub/supercritical environment[J]. Vehicle Engine, 2021(1):63-68.(in Chinese) [19] OEFELEIN J C, YANG V. Modeling high-pressure mixing and combustion processes in liquid rocket engines[J]. Journal of Propulsion and Power, 1998, 14(5):843-857. [20] ZONG N, MENG H, HSIEH S Y, et al. A numerical study of cryogenic fluid injection and mixing under supercritical conditions[J]. Physics of Fluids, 2004, 16(12):4248-4261. [21] ZONG N, YANG V. Cryogenic fluid jets and mixing layers in transcritical and supercritical environments[J]. Combustion Science and Technology, 2006, 178(1-3):193-227. [22] MASQUELET M, MENON S. Large-eddy simulation of flame-turbulence interactions in a shear coaxial injector[J]. Journal of Propulsion and Power, 2010, 26(5):924-935. [23] SCHMITT T, RODRIGUEZ J, LEYVA I A, et al. Experiments and numerical simulation of mixing under supercritical conditions[J]. Physics of Fluids, 2012, 24(5):055104. [24] SMAGORINSKY J. General circulation experiments with the primitive equations:I. The basic experiment[J] Monthly Weather Review, 1963, 91(3):99-164. [25] MILLER R S, FOSTER J W. Survey of turbulent combustion models for large-eddy simulations of propulsive flowfields[J]. AIAA Journal, 2016, 54(10):2930-2946. [26] PIERCE C D, MOIN P. Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[J]. Journal of Fluid Mechanics, 2004, 504:73-97. [27] O'BRIEN J, URZAY J, IHME M, et al. Subgrid-scale backscatter in reacting and inert supersonic hydrogen-air turbulent mixing layers[J]. Journal of Fluid Mechanics, 2014, 743:554-584. [28] BARDINA J, FERZIGER J H, REYNOLDS W C. Improved turbulence models based on large eddy simulation of homogeneous, incompressible turbulent flows[D]. California:Stanford University, 1983. [29] GERMANO M, PIOMELLI U, MOIN P, et al. A dynamic subgrid-scale eddy viscosity model[J]. Physics of Fluids A:Fluid Dynamics, 1991, 3(7):1760-1765. [30] LILLY D K. A proposed modification of the Germano subgrid-scale closure method[J]. Physics of Fluids A:Fluid Dynamics, 1992, 4(3):633-635. [31] MAULIK R, SAN O, RASHEED A, et al. Data-driven deconvolution for large eddy simulations of kraichnan turbulence[J]. Physics of Fluids, 2018, 30(12):125109. [32] MAULIK R, SAN O, RASHEED A, et al. Subgrid modelling for two-dimensional turbulence using neural networks[J]. Journal of Fluid Mechanics, 2018, 858:122-144. [33] WANG Z, LUO K, LI D, et al. Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation[J]. Physics of Fluids, 2018, 30(12):125101. [34] XIE C Y, WANG J C, LI H, et al. A modified optimal LES model for highly compressible isotropic turbulence[J]. Physics of Fluids, 2018, 30(6):065108. [35] YUAN Z L, XIE C Y, WANG J C. Deconvolutional artificial neural network models for large eddy simulation of turbulence[J]. Physics of Fluids, 2020, 32(11):115106. [36] 吴磊,肖左利.基于人工神经网络的亚格子应力建模[J].力学学报, 2021, 53(10):2667-2681. WU L, XIAO Z L. Subgrid-scale stress modeling based on artificial neural network[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10):2667-2681.(in Chinese) [37] 谢晨月,袁泽龙,王建春,等.基于人工神经网络的湍流大涡模拟方法[J].力学学报, 2021, 53(1):1-16. XIE C Y, YUAN Z L, WANG J C, et al. Artificial neural network-based subgrid-scale models for large-eddy simulation of turbulence[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(1):1-16.(in Chinese) [38] CHUNG W T, MISHRA A A, IHME M. Interpretable data-driven methods for subgrid-scale closure in LES for transcritical LOx/GCH4 combustion[J]. Combustion and Flame, 2022, 239:111758. [39] MENG H, YANG V. A unified treatment of general fluid thermodynamics and its application to a preconditioning scheme[J]. Journal of Computational Physics, 2003, 189(1):277-304. [40] RIBERT G, TAIEB D, PETIT X, et al. Simulation of supercritical flows in rocket-motor engines:Application to cooling channel and injection system[J]. Progress in Propulsion Physics, 2013, 4:205-226. [41] SCHMITT T, SELLE L, CUENOT B, et al. Large-eddy simulation of transcritical flows[J]. Comptes Rendus Mécanique, 2009, 337(6-7):528-538. [42] 李云清,王宏楠,王德福.状态方程对高温高压条件下燃料液滴蒸发计算的影响[J].内燃机学报, 2009, 27(4):344-351. LI Y Q, WANG H N, WANG D F. Effect of state equation on numerical simulation of fuel droplet evaporation at high temperature and high pressure[J]. Transactions of CSICE, 2009, 27(4):344-351.(in Chinese) [43] SCHMITT T, SELLE L, RUIZ A, et al. Large-eddy simulation of supercritical-pressure round jets[J]. AIAA Journal, 2010, 48(9):2133-2144. [44] SELLE L, SCHMITT T. Large-eddy simulation of single-species flows under supercritical thermodynamic conditions[J]. Combustion Science and Technology, 2010, 182(4-6):392-404. [45] 解茂昭,杨康.超临界燃料喷射过程大涡模拟:状态方程的影响[J].热科学与技术, 2015, 14(5):409-417. XIE M Z, YANG K. Effect of equations of state on numerical simulation of supercritical fuel spray[J]. Journal of Thermal Science and Technology, 2015, 14(5):409-417.(in Chinese) [46] 熊先闯,刘正先,赵明,等.基于BWR状态方程的超临界二氧化碳主要热力参数推算[J].石油化工, 2021, 50(5):425-431. XIONG X C, LIU Z X, ZHAO M, et al. Calculation of main thermodynamic parameters of supercritical carbon dioxide based on BWR equation of state[J]. Petrochemical Technology, 2021, 50(5):425-431.(in Chinese) [47] OEFELEIN J, DAHMS R N U, PICKETT L M. A dense fluid approximation for the simulation of diesel engine fuel injection processes[R]. Livermore:Sandia National Lab., 2011. [48] RIBERT G, PETIT X, DOMINGO P. High-pressure methane-oxygen flames. Analysis of sub-grid scale contributions in filtered equations of state[J]. The Journal of Supercritical Fluids, 2017, 121:78-88. [49] SELLE L C, OKONG'O N A, BELLAN J, et al. Modelling of subgrid-scale phenomena in supercritical transitional mixing layers:An a priori study[J]. Journal of Fluid Mechanics, 2007, 593:57-91. [50] TASKINOGLU E S, BELLAN J. A posteriori study using a DNS database describing fluid disintegration and binary-species mixing under supercritical pressure:Heptane and nitrogen[J]. Journal of Fluid Mechanics, 2010, 645:211-254. [51] BORGHESI G, BELLAN J. A priori and a posteriori investigations for developing large eddy simulations of multi-species turbulent mixing under high-pressure conditions[J]. Physics of Fluids, 2015, 27(3):035117. [52] LAPENNA P E, CRETA F. Mixing under transcritical conditions:An a-priori study using direct numerical simulation[J]. The Journal of Supercritical Fluids, 2017, 128:263-278. [53] WANG X J, HUO H F, UNNIKRISHNAN U, et al. A systematic approach to high-fidelity modeling and efficient simulation of supercritical fluid mixing and combustion[J]. Combustion and Flame, 2018, 195:203-215. [54] UNNIKRISHNAN U, WANG X J, YANG S, et al. Subgrid scale modeling of the equation of state for turbulent flows under supercritical conditions[C]//Proceedings of the 53rd AIAA/SAE/ASEE Joint Propulsion Conference. Atlanta, USA:AIAA, 2017:2017-4855. [55] UNNIKRISHNAN U, OEFELEIN J C, YANG V. Subgrid modeling of the filtered equation of state with application to real-fluid turbulent mixing at supercritical pressures[J]. Physics of Fluids, 2022, 34(6):065112. [56] ZONG N, YANG V. An efficient preconditioning scheme for real-fluid mixtures using primitive pressure-temperature variables[J]. International Journal of Computational Fluid Dynamics, 2007, 21(5-6):217-230. [57] TAŞKINOGLU E S, BELLAN J. Subgrid-scale models and large-eddy simulation of oxygen stream disintegration and mixing with a hydrogen or helium stream at supercritical pressure[J]. Journal of Fluid Mechanics, 2011, 679:156-193. [58] GNANASKANDAN A, BELLAN J. Large eddy simulations of high pressure jets:Effect of subgrid scale modeling[C]//Proceedings of the 55th AIAA Aerospace Sciences Meeting. Grapevine, USA:AIAA, 2017. [59] 杨富方,刘航滔,杨震,等.超临界二氧化碳循环工质热物性研究进展[J].热力发电, 2020, 49(10):21-29. YANG F F, LIU H T, YANG Z, et al. Thermophysical properties of working fluid of supercritical carbon dioxide cycle:Research progress[J]. Thermal Power Generation, 2020, 49(10):21-29.(in Chinese) [60] YANG F F, LIU Q, DUAN Y Y, et al. Crossover multiparameter equation of state:General procedure and demonstration with carbon dioxide[J]. Fluid Phase Equilibria, 2019, 494:161-171. [61] LOPEZ-ECHEVERRY J S, REIF-ACHERMAN S, ARAUJO-LOPEZ E. Peng-robinson equation of state:40 years through cubics[J]. Fluid Phase Equilibria, 2017, 447:39-71. [62] MA P C, Lü Y, IHME M. An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows[J]. Journal of Computational Physics, 2017, 340:330-357. [63] ABUDOUR A M, MOHAMMAD S A, ROBINSON JR R L, et al. Volume-translated peng-robinson equation of state for saturated and single-phase liquid densities[J]. Fluid Phase Equilibria, 2012, 335:74-87. [64] BALED H, ENICK R M, WU Y, et al. Prediction of hydrocarbon densities at extreme conditions using volume-translated SRK and PR equations of state fit to high temperature, high pressure PVT data[J]. Fluid Phase Equilibria, 2012, 317:65-76. [65] MATHEIS J, MüLLER H, LENZ C, et al. Volume translation methods for real-gas computational fluid dynamics simulations[J]. The Journal of Supercritical Fluids, 2016, 107:422-432. [66] MüLLER H, NIEDERMEIER C A, MATHEIS J, et al. Large-eddy simulation of nitrogen injection at trans-and supercritical conditions[J]. Physics of Fluids, 2016, 28(1):015102. [67] MüLLER H, PFITZNER M. Large-eddy simulation of transcritical liquid oxygen/methane jet flames[J]. Progress in Propulsion Physics, 2019, 11:177-194. [68] MüLLER H, PFITZNER M, MATHEIS J, et al. Large-eddy simulation of coaxial LN2/GH2 injection at trans-and supercritical conditions[J]. Journal of Propulsion and Power, 2016, 32(1):46-56. [69] POLING B E, PRAUSNITZ J M, O'CONNELL J P. Properties of gases and liquids:5th ed[M]. New York:McGraw-Hill, 2001. [70] YANG V. Modeling of supercritical vaporization, mixing, and combustion processes in liquid-fueled propulsion systems[J]. Proceedings of the Combustion Institute, 2000, 28(1):925-942. [71] OEFELEIN J C. Thermophysical characteristics of shear-coaxial LOx-H2 flames at supercritical pressure[J]. Proceedings of the Combustion Institute, 2005, 30(2):2929-2937. [72] SCHMITT T. Large-eddy simulations of the mascotte test cases operating at supercritical pressure[J]. Flow, Turbulence and Combustion, 2020, 105(1):159-189. [73] HUO H F, YANG V. Large-eddy simulation of supercritical combustion:Model validation against gaseous H2-O2 injector[J]. Journal of Propulsion and Power, 2017, 33(5):1272-1284. [74] BANUTI D T, MA P C, HICKEY J P, et al. Thermodynamic structure of supercritical LOx-GH2 diffusion flames[J]. Combustion and Flame, 2018, 196:364-376. [75] MA P C, WU H, BANUTI D T, et al. On the numerical behavior of diffuse-interface methods for transcritical real-fluids simulations[J]. International Journal of Multiphase Flow, 2019, 113:231-249. [76] GUVEN U, RIBERT G. Impact of non-ideal transport modeling on supercritical flow simulation[J]. Proceedings of the Combustion Institute, 2019, 37(3):3255-3262. [77] MüLLER H, PFITZNER M. A flamelet model for transcritical LOX/GCH4 flames[J]. Journal of Physics:Conference Series, 2017, 821:012010. [78] MARTIN J J. Cubic equations of state-which?[J]. Industrial&Engineering Chemistry Fundamentals, 1979, 18(2):81-97. [79] HARSTAD K G, MILLER R S, BELLAN J. Efficient high-pressure state equations[J]. AIChE Journal, 1997, 43(6):1605-1610. [80] GOTTIPARTHI K C, SANKARAN R, RUIZ A M, et al. Large eddy simulation of a supercritical fuel jet-in-cross-flow using GPU-acceleration[C]//Proceedings of the 54th AIAA Aerospace Sciences Meeting. San Diego, USA:AIAA, 2016:2016-1939. [81] YANG S, LI Y X, WANG X J, et al. Comparison of tabulation and correlated dynamic evaluation of real fluid properties for supercritical mixing[C]//Proceedings of the 53rd AIAA/SAE/ASEE Joint Propulsion Conference. Atlanta, USA:AIAA, 2017:2017-4858. [82] KUTZ J N. Deep learning in fluid dynamics[J]. Journal of Fluid Mechanics, 2017, 814:1-4. [83] MILAN P J, HICKEY J P, WANG X J, et al. Deep-learning accelerated calculation of real-fluid properties in numerical simulation of complex flowfields[J]. Journal of Computational Physics, 2021, 444:110567. [84] MENEVEAU C, KATZ J. Scale-invariance and turbulence models for large-eddy simulation[J]. Annual Review of Fluid Mechanics, 2000, 32:1-32. [85] HUO H F, YANG V. Subgrid-scale models for large-eddy simulation of supercritical combustion[C]//Proceedings of the 51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Texas, USA:AIAA, 2013:2013-0706. [86] UNNIKRISHNAN U, HUO H F, WANG X J, et al. Subgrid scale modeling considerations for large eddy simulation of supercritical turbulent mixing and combustion[J]. Physics of Fluids, 2021, 33(7):075112. [87] RIBERT G, DOMINGO P, VERVISCH L. Analysis of sub-grid scale modeling of the ideal-gas equation of state in hydrogen-oxygen premixed flames[J]. Proceedings of the Combustion Institute, 2019, 37(2):2345-2351. [88] LAPENNA P E, INDELICATO G, LAMIONI R, et al. Modeling the equations of state using a flamelet approach in LRE-like conditions[J]. Acta Astronautica, 2019, 158:460-469. [89] RIBERT G, DOMINGO P, VERVISCH L. Sub-grid scale modeling of the equation of state for fully compressible combustion LES[C]//Proceedings of Summer Program 2016. California, USA:Center for Turbulence Research Press, 2016:123-132. [90] KIM S K, CHOI H-S, KIM Y. Thermodynamic modeling based on a generalized cubic equation of state for kerosene/LOx rocket combustion[J]. Combustion and Flame, 2012, 159(3):1351-1365. [91] FORERO G L A, VELÁSQUEZ J J A. A modified Patel-Teja cubic equation of state:Part I-Generalized model for gases and hydrocarbons[J]. Fluid Phase Equilibria, 2013, 342:8-22. [92] GHODERAO P N P, DALVI V H, NARAYAN M. A four-parameter cubic equation of state for pure compounds and mixtures[J]. Chemical Engineering Science, 2018, 190:173-189. [93] GHODERAO P N P, DALVI V H, NARAYAN M. A five-parameter cubic equation of state for pure fluids and mixtures[J]. Chemical Engineering Science:X, 2019, 3:100026. [94] ZHU H Y, BATTISTONI M, NINGEGOWDA M B, et al. Thermodynamic modeling of trans/supercritical fuel sprays in internal combustion engines based on a generalized cubic equation of state[J]. Fuel, 2022, 307:121894. [95] NI C X, WANG X J, LIU H T, et al. Gradient-harmonizing-based deep learning for thermophysical properties of carbon dioxide[J]. Journal of Thermophysics and Heat Transfer, 2022:1-12. [96] ZHOU M S, CHEN W, SU X Y, et al. Data-driven modeling of general fluid density under subcritical and supercritical conditions[J]. AIAA Journal, 2023:1-13. [97] CHANG Y H, WANG X J, ZHANG L W, et al. Reduced-order modeling for complex flow emulation by common kernel-smoothed proper orthogonal decomposition[J]. AIAA Journal, 2021, 59(9):3291-3303. [98] WANG X J, CHANG Y H, LI Y X, et al. Surrogate-based modeling for emulation of supercritical injector flow and combustion[J]. Proceedings of the Combustion Institute, 2021, 38(4):6393-6401.
2023, Vol. 63 
