微可压缩模型预处理求解方法研究
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摘要
研究表明,邓小刚等提出的微可压缩模型(SCM)求解低速流动很有效。然而由于采用了与完整Navier-Stokes方程相同的连续方程和动量方程,其特征系统在来流马赫数很低的时候表现出较强的刚性,难以求解极低马赫数的流动。针对这一问题发展的预处理形式微可压模型(SCM-P)本质上也存在着方程不相容的缺陷,在定常领域内取得较大成功的同时,却在非定常流场和临界状态计算中很可能由于误差的积累而得到非物理解。针对这些问题,本文发展了一种新的代数预处理形式微可压模型(SCM-AP),通过一系列典型算例的模拟,验证了代数预处理后的微可压模型算法,取得了良好的效果。
本文共分为六章,各章内容如下:
第一章为引言,简要介绍了数值模拟低马赫数流动研究背景和目前比较流行的算法,简述了微可压缩模型(SCM)和预处理形式的微可压模型(SCM-P)在低马赫数和极低马赫数流动计算中的模拟特性以及面临的问题。最后介绍了本文的工作。
第二章简要介绍了微可压模型(SCM)及其在极低马赫数条件下采用特征分裂方法时的刚性问题,并通过代数预处理方法提出了四种解决方案,确保了离散方程左端隐式矩阵主对角占优。
第三章对第二章提出的四种方案进行了稳定性分析,选择了两种满足稳定性要求的方案作进一步的发展,并分析了两种方案实施过程中所作的近似处理对稳定性的影响。
第四章考察了本文发展的两种新算法:SCM-AP1、SCM-AP2,把它们应用于模拟方腔流、定常/非定常粘性圆柱绕流、定常/非定常翼型绕流等二维算例,并与SCM和SCM-P在计算精度和效率上进行了比较,发现SCM-AP2有良好的计算精度和收敛速度。
第五章将SCM-AP2应用到三角翼流场、椭球流场的模拟中,展示了SCM-AP2对大攻角下尖前缘外形分离涡流场的良好模拟能力和计算精度。
第六章为结束语,对本文工作进行了概括总结,指出了仍需进一步研究的一些工作。
最后是致谢及本文的参考文献。
The Slightly Compressible ModeI(SCM)efficient for lOW Mach number flow calculationderived by Deng Xiaogang is veryBut the eigenvalue system ofthe SCM equations shows strong stiffness when the freesteam Mach number is very low,because the same governing equations with the Navier-Stokes equations are introducedin SCM．A preconditioned method proposed for SCM(SCM-P)is not very effective in solving the unsteady flow calculations and critical state，since the continuity equationis only satisfied in the steady state．In this thesis based on SCM．a new preconditioned method has been proposed．
This thesis iS composed of six chapters：
The first chapter is the preface of this thesis．The background of the simulation forlOW Mach number flows iS introduced jn brief SO as to the popular arithmetic．The characteristic and problems confronted in using SCM and SCM．P to calculate the low and very low Mach number flows are included in the next part．The last part is theresearch work ofthis thesis．
In the second chapter,the stiffneSS of SCM iS introduced．Four schemes areproposed based on algebraic preconditioned methods．the results indicate an importantimprovement．
In the third chapter，the stability and the convergence characteristics of methodsproposed in chapter 2 are investigated．SCM-APl and SCM-AP2 which exhibitfeasible results are further developed in this thesis．The effects of the approximationsintroduced in the implementations on the stability are analysed．
In the fourth chapter,some two．dimensionaJ results calculated by SCM．APl andSCM-AP2 are presented and compared with the results calculated by SCM and SCM-PIn the present work，SCM·AP2 is more excellent in the precision and convergence．
In the fifth chapter，flow fields around a delta wing and an ellipsoid are simulatedby the SCM-AP2．Results show that SCM．AP2 is effective in simulating unsteadyseparation flows at high angle of attack．
The concluding remarks are given in the sixth chapter,where the summary ofcurrent study work is presented and the job tO be carried on in future is pointed out asweIl
In the end，the references and acknowledgements are presented．
本文共分为六章,各章内容如下:
第一章为引言,简要介绍了数值模拟低马赫数流动研究背景和目前比较流行的算法,简述了微可压缩模型(SCM)和预处理形式的微可压模型(SCM-P)在低马赫数和极低马赫数流动计算中的模拟特性以及面临的问题。最后介绍了本文的工作。
第二章简要介绍了微可压模型(SCM)及其在极低马赫数条件下采用特征分裂方法时的刚性问题,并通过代数预处理方法提出了四种解决方案,确保了离散方程左端隐式矩阵主对角占优。
第三章对第二章提出的四种方案进行了稳定性分析,选择了两种满足稳定性要求的方案作进一步的发展,并分析了两种方案实施过程中所作的近似处理对稳定性的影响。
第四章考察了本文发展的两种新算法:SCM-AP1、SCM-AP2,把它们应用于模拟方腔流、定常/非定常粘性圆柱绕流、定常/非定常翼型绕流等二维算例,并与SCM和SCM-P在计算精度和效率上进行了比较,发现SCM-AP2有良好的计算精度和收敛速度。
第五章将SCM-AP2应用到三角翼流场、椭球流场的模拟中,展示了SCM-AP2对大攻角下尖前缘外形分离涡流场的良好模拟能力和计算精度。
第六章为结束语,对本文工作进行了概括总结,指出了仍需进一步研究的一些工作。
最后是致谢及本文的参考文献。
The Slightly Compressible ModeI(SCM)efficient for lOW Mach number flow calculationderived by Deng Xiaogang is veryBut the eigenvalue system ofthe SCM equations shows strong stiffness when the freesteam Mach number is very low,because the same governing equations with the Navier-Stokes equations are introducedin SCM．A preconditioned method proposed for SCM(SCM-P)is not very effective in solving the unsteady flow calculations and critical state，since the continuity equationis only satisfied in the steady state．In this thesis based on SCM．a new preconditioned method has been proposed．
This thesis iS composed of six chapters：
The first chapter is the preface of this thesis．The background of the simulation forlOW Mach number flows iS introduced jn brief SO as to the popular arithmetic．The characteristic and problems confronted in using SCM and SCM．P to calculate the low and very low Mach number flows are included in the next part．The last part is theresearch work ofthis thesis．
In the second chapter,the stiffneSS of SCM iS introduced．Four schemes areproposed based on algebraic preconditioned methods．the results indicate an importantimprovement．
In the third chapter，the stability and the convergence characteristics of methodsproposed in chapter 2 are investigated．SCM-APl and SCM-AP2 which exhibitfeasible results are further developed in this thesis．The effects of the approximationsintroduced in the implementations on the stability are analysed．
In the fourth chapter,some two．dimensionaJ results calculated by SCM．APl andSCM-AP2 are presented and compared with the results calculated by SCM and SCM-PIn the present work，SCM·AP2 is more excellent in the precision and convergence．
In the fifth chapter，flow fields around a delta wing and an ellipsoid are simulatedby the SCM-AP2．Results show that SCM．AP2 is effective in simulating unsteadyseparation flows at high angle of attack．
The concluding remarks are given in the sixth chapter,where the summary ofcurrent study work is presented and the job tO be carried on in future is pointed out asweIl
In the end，the references and acknowledgements are presented．
引文
[1] Xiaogang Deng, Fenggan Zhuang and Meiliang Mao,“On Low Mach Number Perfect Gas Flow Calculations”AIAA Paper 99-3317, 1999.
[2] Deng Xiaogang, Zhuang Fenggan, A Novel Slightly Compressible Model for Low Mach Number Perfect Gas Flow Calculations, ACTA MECHANICA SINICA,Vol.18, No.3, June 2002。
[3]邓小刚,向大平,毛枚良,微可压缩模型数值模拟研究,The Fourth Cross-Strait CFD Workshop. Yunnan, April 12, 2003。
[4]向大平,邓小刚,毛枚良,低马赫数流动数值模拟方法的研究,空气动力学学报,第20卷,第4期:373～378,2002。
[5]陶文铨,计算传热学的近代进展,科学出版社,2000。
[6] Deng G B, Piquet J, Queutey P Visonneau M, A new fully coupled solution of the Navier-Stokes equations, Int J Number Methods Fluids, 1994.19:605~639。
[7] Min B K , Chang K S, A momentum coupling method for the unsteady incompressible Navier-Stokes equations on the staggered grid, Int J Number Methods Fluids, 1998.28:443~460。
[8] Galpin P F, Raithby G D, Treatment of non-linearities in the numerical solution of the incompressible Navier-Stokes equations, Int J Number Methods Fluids,1986.6:409~426。
[9]郭鸿志,张欣欣,刘向军,李杰。传输过程数值模拟。北京:冶金工业出版社,1998.100-130.
[10] Speziale C.G.On the advantages of the vorticity of the equations of fluid dynamics.J.Comput Phys,1987.73:476-480.
[11] Karnadiaki G E, Tsraeli M, Orszag S A, High-order splitting methods for the incompressible Navier-Stokes equations, J Comput Phys, 1991.97:414~443。
[12] Jordan S A ,Ragab S A.An efficient fractional-step technique for unsteady incompressible flows using a semistaggered grid strategy.J Comput Phys,1996,127:218-225.
[13] Abdallah S.Numerical solutions for the incompressible Navier-Stokes equations in primitive variables using a non-staggered grid ,J Comput Phy,1987,70:183-202.
[14] Sotiropoulos F,Abdallah S.A primitive variable method for the solution of three-dimensional incompressible viscous flows,J Comput Phys,1992,103:336-349.
[15] Kausbik S,Rubin S G.Incompressible Navier-Stokes solutions with a newprimitive variable solver.Comput Fluids,1995.24(1):27-40.
[16]伍亚舟,黄兰洁。非均匀交错网格上的Temam方法及驱动方腔流动的数值模拟。计算物理,1994,11(2):141-148.
[17] Kwak D,Chang J L C,Shanks S P,Chakravathy S R.A three-dimentional incompressible Navier-Stokes flow solver using primitive variables.AIAA J,1986.24:390-396.
[18] Lee S L,Tzong R Y.Arificial pressure for pressure-linked equarion.Int J Heat Mass Transfer,1992.35(10:) 2705-2716.
[19] Shuen J S,Chen K H,Choi Y A.A coupled implicit method for chemical non-equilibrium flow of all speeds.J Comput Phys,1993.106:306-869.
[20] Chorin A J, A numerical method for solving incompressible viscous flow problems,J Compt Phys, 1967.2:12~26。
[21] Shih T M, Hayes L J, Minkowycz W J , Yang K T, Aung W. Parallel computations in heat transfer .Numer Heat Trnasfer,1986.9:639-662.
[22] Lewis A J, Brent A D. A comparison of coarse and fine grain parallelization strategies for the SIMPLE pressure correction algorithm. Int J Numer Methods Fluids, 1993.16:891-914.
[23] Blosch E L, Shyy W. Sequential pressure-based Navier-Stokes algorithms on SIMD computers :computational issues. Numer Heat Transfer ,Part B, 1994.26:115-132.
[24]向大平,新微可压缩模型数值模拟方法研究,硕士学位论文,中国空气动力研究与发展中心,2000.6
[25]丛成华,微可压缩模型预处理技术研究,硕士学位论文,中国空气动力研究与发展中心,2005.3。
[26] John M. Hsu, Antony Jameson. An Implicit-Explicit Hybrid Scheme forCalculating Complex Unsteady Flows, AIAA 2002-0714.
[27] J. Hsu, A. Jameson. An Implicit-Explicit Scheme for Calculating Complex Unsteady Flows, Industrial Affiliates Meeting 2002.
[28]邓小兵.不可压缩流动的计算研究,硕士学位论文,中国空气动力研究与发展中心,2001.6
[29]闵耀兵,微可压缩模型预处理求解技术研究,硕士学位论文,中国空气动力研究与发展中心,2008.6
[30]陈亮中,双三角翼非定常分离流动的数值模拟研究,博士学位论文,中国空气动力研究与发展中心,2009.
[31] Yu jiang and Andrzej J.Przekwas CFD Research Corporation Huntsville,Al 35805.Implicit,Pressure-based Incompressible Navier-Stokes Equations Solver For Unstructured Meshes.AIAA-94-0305.
[32] S.M.H.K arimian.Numerical Solution of Two-Dimensional IncompressibleNavier-Stokes Equations Treatment of Velocity-Pressure Coupling,AIAA-94-2359.
[33] Cord-Christian Rossow,Efficient computation of compressible and incompressible flows, Journal of Computational Physics 220 (2007) 879–899。
[34] K.C.Karki and S.V.Patankar,Pressure Based Calculation Procedure for Viscous Flows at All Speeds in Arbitrary Configurations.AIAA,VOL.27,No.9.
[35] G. B. Deng, J. Piquet, P. Queutey AND M. Visonneau, A NEW FULLY COUPLED SOLUTION OF THE NAVIER-STOKES EQUATIONS, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS. VOL. 19,605-639 (1994).
[36] P.F.Galpin,J.P.Van Doormaal and G.D.Raithry,Solution of the Incompressible Mass and Momentum Equations by Application of a Coupled Equation Line Solver, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS. VOL. 5,615-625 (1985).
[37] M.L.Mansour and A.Hamed, Implicit Solution of the Incompressible Navier-Stokes Equations on a Non-staggered Grid, Journal Computational Physics, Vol.86, 147-167(1990)
[38]吴子牛,计算流体力学基本原理,科学出版社,2001。
[39] O.Coutier-Delgosha,R.Fortes-Patella.Stability of preconditioned Navier-Stokes equations associated with a cavitation model,Computers & Fluids 34(2005)319-349.
[40] Tang L Q ,Cheng T and Tsang T T H. Transient solutions for three-dimensional lid-driven cavity flows by aleast-squares finite element methood [J] International Journal for Numerical Methods in Fluids,1995,21:413- 432.
[41] R.Schreiber.Driven cavity flows by efficient numerical techniques.J Comput Phys,49,310-33(1983).
[42] Ghia U, Ghia K N, Shin C T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method[J].J.Computational Physics,1982,48:387-411.
[43]康钦军,圆柱绕流的数值模拟,博士学位论文,清华大学研究生院,1999。
[44] Rogers S E, Kwak D, Upwind differencing scheme for time-accurate incomgpressible Navier-Stokes equations,AIAA J, 1990,28:253~262。
[45] Roshko A, On the development of turbulent wakes from vortex streets,NACA Rep,1191,1954。
[46] S.A.Morton,R.B.Melville,M.R.Visbal,Accuracy and Coupling Issues of Aeroelastic Navier-Stokes Solution on Deforming Meshes,Journal of Aircraft,Vol.35,No.5,September-October 1998.
[47]袁先旭,非定常流动数值模拟及飞行器动态特性分析研究,博士学位论文,中国空气动力研究与发展中心,2002
[48] Werner Haase, Eric Chaput, etc. Notes on Numerical Fluid Mechanics, Volume 58,ECARP-European Computational Aerodynamics Research Project: Validation of CFD Codes and Assessment of Turbulence Models, 1997
[49] Wadcock A J, Two-Dimensional Stalled Airfoil, AFOSR-HTTM, Vol.1:234~252,1981
[50] Hummel D. On the Vortex Formation over a Slender Wing at Large Angle of Incidence, High Angle of Attack Aerodynamics, AGARD-CP-247, Paper 15, Jan ,1979.
[51]刘伟,细长机翼摇滚机理的非线性动力学分析及数值模拟方法研究,博士学位论文,国防科学技术大学研究生院,2004.
[2] Deng Xiaogang, Zhuang Fenggan, A Novel Slightly Compressible Model for Low Mach Number Perfect Gas Flow Calculations, ACTA MECHANICA SINICA,Vol.18, No.3, June 2002。
[3]邓小刚,向大平,毛枚良,微可压缩模型数值模拟研究,The Fourth Cross-Strait CFD Workshop. Yunnan, April 12, 2003。
[4]向大平,邓小刚,毛枚良,低马赫数流动数值模拟方法的研究,空气动力学学报,第20卷,第4期:373～378,2002。
[5]陶文铨,计算传热学的近代进展,科学出版社,2000。
[6] Deng G B, Piquet J, Queutey P Visonneau M, A new fully coupled solution of the Navier-Stokes equations, Int J Number Methods Fluids, 1994.19:605~639。
[7] Min B K , Chang K S, A momentum coupling method for the unsteady incompressible Navier-Stokes equations on the staggered grid, Int J Number Methods Fluids, 1998.28:443~460。
[8] Galpin P F, Raithby G D, Treatment of non-linearities in the numerical solution of the incompressible Navier-Stokes equations, Int J Number Methods Fluids,1986.6:409~426。
[9]郭鸿志,张欣欣,刘向军,李杰。传输过程数值模拟。北京:冶金工业出版社,1998.100-130.
[10] Speziale C.G.On the advantages of the vorticity of the equations of fluid dynamics.J.Comput Phys,1987.73:476-480.
[11] Karnadiaki G E, Tsraeli M, Orszag S A, High-order splitting methods for the incompressible Navier-Stokes equations, J Comput Phys, 1991.97:414~443。
[12] Jordan S A ,Ragab S A.An efficient fractional-step technique for unsteady incompressible flows using a semistaggered grid strategy.J Comput Phys,1996,127:218-225.
[13] Abdallah S.Numerical solutions for the incompressible Navier-Stokes equations in primitive variables using a non-staggered grid ,J Comput Phy,1987,70:183-202.
[14] Sotiropoulos F,Abdallah S.A primitive variable method for the solution of three-dimensional incompressible viscous flows,J Comput Phys,1992,103:336-349.
[15] Kausbik S,Rubin S G.Incompressible Navier-Stokes solutions with a newprimitive variable solver.Comput Fluids,1995.24(1):27-40.
[16]伍亚舟,黄兰洁。非均匀交错网格上的Temam方法及驱动方腔流动的数值模拟。计算物理,1994,11(2):141-148.
[17] Kwak D,Chang J L C,Shanks S P,Chakravathy S R.A three-dimentional incompressible Navier-Stokes flow solver using primitive variables.AIAA J,1986.24:390-396.
[18] Lee S L,Tzong R Y.Arificial pressure for pressure-linked equarion.Int J Heat Mass Transfer,1992.35(10:) 2705-2716.
[19] Shuen J S,Chen K H,Choi Y A.A coupled implicit method for chemical non-equilibrium flow of all speeds.J Comput Phys,1993.106:306-869.
[20] Chorin A J, A numerical method for solving incompressible viscous flow problems,J Compt Phys, 1967.2:12~26。
[21] Shih T M, Hayes L J, Minkowycz W J , Yang K T, Aung W. Parallel computations in heat transfer .Numer Heat Trnasfer,1986.9:639-662.
[22] Lewis A J, Brent A D. A comparison of coarse and fine grain parallelization strategies for the SIMPLE pressure correction algorithm. Int J Numer Methods Fluids, 1993.16:891-914.
[23] Blosch E L, Shyy W. Sequential pressure-based Navier-Stokes algorithms on SIMD computers :computational issues. Numer Heat Transfer ,Part B, 1994.26:115-132.
[24]向大平,新微可压缩模型数值模拟方法研究,硕士学位论文,中国空气动力研究与发展中心,2000.6
[25]丛成华,微可压缩模型预处理技术研究,硕士学位论文,中国空气动力研究与发展中心,2005.3。
[26] John M. Hsu, Antony Jameson. An Implicit-Explicit Hybrid Scheme forCalculating Complex Unsteady Flows, AIAA 2002-0714.
[27] J. Hsu, A. Jameson. An Implicit-Explicit Scheme for Calculating Complex Unsteady Flows, Industrial Affiliates Meeting 2002.
[28]邓小兵.不可压缩流动的计算研究,硕士学位论文,中国空气动力研究与发展中心,2001.6
[29]闵耀兵,微可压缩模型预处理求解技术研究,硕士学位论文,中国空气动力研究与发展中心,2008.6
[30]陈亮中,双三角翼非定常分离流动的数值模拟研究,博士学位论文,中国空气动力研究与发展中心,2009.
[31] Yu jiang and Andrzej J.Przekwas CFD Research Corporation Huntsville,Al 35805.Implicit,Pressure-based Incompressible Navier-Stokes Equations Solver For Unstructured Meshes.AIAA-94-0305.
[32] S.M.H.K arimian.Numerical Solution of Two-Dimensional IncompressibleNavier-Stokes Equations Treatment of Velocity-Pressure Coupling,AIAA-94-2359.
[33] Cord-Christian Rossow,Efficient computation of compressible and incompressible flows, Journal of Computational Physics 220 (2007) 879–899。
[34] K.C.Karki and S.V.Patankar,Pressure Based Calculation Procedure for Viscous Flows at All Speeds in Arbitrary Configurations.AIAA,VOL.27,No.9.
[35] G. B. Deng, J. Piquet, P. Queutey AND M. Visonneau, A NEW FULLY COUPLED SOLUTION OF THE NAVIER-STOKES EQUATIONS, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS. VOL. 19,605-639 (1994).
[36] P.F.Galpin,J.P.Van Doormaal and G.D.Raithry,Solution of the Incompressible Mass and Momentum Equations by Application of a Coupled Equation Line Solver, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS. VOL. 5,615-625 (1985).
[37] M.L.Mansour and A.Hamed, Implicit Solution of the Incompressible Navier-Stokes Equations on a Non-staggered Grid, Journal Computational Physics, Vol.86, 147-167(1990)
[38]吴子牛,计算流体力学基本原理,科学出版社,2001。
[39] O.Coutier-Delgosha,R.Fortes-Patella.Stability of preconditioned Navier-Stokes equations associated with a cavitation model,Computers & Fluids 34(2005)319-349.
[40] Tang L Q ,Cheng T and Tsang T T H. Transient solutions for three-dimensional lid-driven cavity flows by aleast-squares finite element methood [J] International Journal for Numerical Methods in Fluids,1995,21:413- 432.
[41] R.Schreiber.Driven cavity flows by efficient numerical techniques.J Comput Phys,49,310-33(1983).
[42] Ghia U, Ghia K N, Shin C T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method[J].J.Computational Physics,1982,48:387-411.
[43]康钦军,圆柱绕流的数值模拟,博士学位论文,清华大学研究生院,1999。
[44] Rogers S E, Kwak D, Upwind differencing scheme for time-accurate incomgpressible Navier-Stokes equations,AIAA J, 1990,28:253~262。
[45] Roshko A, On the development of turbulent wakes from vortex streets,NACA Rep,1191,1954。
[46] S.A.Morton,R.B.Melville,M.R.Visbal,Accuracy and Coupling Issues of Aeroelastic Navier-Stokes Solution on Deforming Meshes,Journal of Aircraft,Vol.35,No.5,September-October 1998.
[47]袁先旭,非定常流动数值模拟及飞行器动态特性分析研究,博士学位论文,中国空气动力研究与发展中心,2002
[48] Werner Haase, Eric Chaput, etc. Notes on Numerical Fluid Mechanics, Volume 58,ECARP-European Computational Aerodynamics Research Project: Validation of CFD Codes and Assessment of Turbulence Models, 1997
[49] Wadcock A J, Two-Dimensional Stalled Airfoil, AFOSR-HTTM, Vol.1:234~252,1981
[50] Hummel D. On the Vortex Formation over a Slender Wing at Large Angle of Incidence, High Angle of Attack Aerodynamics, AGARD-CP-247, Paper 15, Jan ,1979.
[51]刘伟,细长机翼摇滚机理的非线性动力学分析及数值模拟方法研究,博士学位论文,国防科学技术大学研究生院,2004.