二维非定常N-S方程及两方程湍流模型耦合求解技术研究
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摘要
现代飞行器设计迫切需要计算流体力学为其提供准确、高效和实用的气动力数据以及流场分析工具,数值求解Euler方程已无法满足现代飞行器设计过程中对气动力特性模拟的精度要求。从理论上讲,求解N-S方程可以比较真实地模拟流动的粘性效应,提供比较准确详实的流场信息。计算机软、硬件技术日新月异、计算流体力学数值方法日益发展完善,使数值求解N-S方程组成为可能。
本文主要发展了计算翼型跨音速定常及非定常粘性绕流的数值模拟方法。基于雷诺平均的思想,出现在RANS方程中的雷诺应力项一般采用统计意义下的湍流模型进行封闭。文中湍流模型分别采用B-L零方程模型及k-ε、k-ω两种线性涡粘两方程模型,对N-S方程的几种数值求解方法进行了探讨。
在空间离散上,基于有限体积法,采用迎风型通量矢量分裂类Van Leer格式,并引入三阶MUSCL插值方法和连续可微的van Albada通量限制器,使格式具有高阶精度,且具有良好的稳定性和收敛性;在时间推进上,分别采用显式四步Runge-Kutta和隐式LU-SGS推进方法,对主控方程与湍流模型方程进行耦合求解,虽然本文工作侧重于定常绕流的数值模拟,但从原理上说,本文所开发的LU-SGS定常计算程序只需略做修改即可实现非定常流动模拟;边界条件采用虚拟网格技术作为统一的边界处理方式,有效地提高了编程和计算效率。
飞行器定常、非定常气动力的准确数值模拟,特别是跨音速范围内的数值求解及研究,一直以来都是飞行器设计工作者面对的重要研究课题。本文主要针对NACA0012、RAE2822翼型的定常粘性流场进行数值模拟;在此基础上,以NACA0012翼型为例,对非定常粘性流场进行了数值求解。计算网格采用贴体结构粘性网格,其中非定常算例采用了刚性运动网恪的处理方法。文中对比分析了不同方法、不同湍流模型之间的差异,获得了比较理想的计算结果,验证了本文数值求解技术的可行性以及计算程序的正确性。
In modern aircraft design, there is a pressing demand for accurate, efficient and applied aerodynamic data and computational analysis tools. However, the Euler equations are not satisfied all the requirement for simulating the precision of aeroelastic problems. It's necessary to solve N-S equations to simulate viscous effects, offer more accurate and more detailed flow-fields information. It is possible to solve N-S equations with the improvement of the hardware and software in computer fields, and the development of numeric methods in computational fluid dynamics domains.The present work is mainly focused on developing airfoils numerical simulation methods for steady and unsteady transonic flow. With the numerical solution of RANS equations, the applications of B-L zero-equation 、 k-ε and k-ω two-equation linear eddy turbulence models are discussed.On the basis of a finite volume method, the spatial discrete scheme about the convection terms of N-S equations: the Van Leer scheme of flux vector splitting, is studied. To get higher order accuracy、well stability and astringency, the MUSCL interpolation function and the continuous differential van Albada limiter are applied for Van Leer. In this paper, two time stepping schemes, explicit four-stage Runge-Kutta scheme and implicit LU-SGS scheme, are employed to solve the governing equations. Boundary conditions are treated by adopting "ghost" cells, thecomputational efficiency is improved and the program's complexity is reduced, too. In aircraft design, accurate numerical simulation for steady and unsteady flows.especially for transonic flows, is a crucial problem for designers. To confirm the validity and reliability of developed flow solvers, test cases of steady viscous flows around NACA0012 and RAE2822 airfoils, are given in this paper. On the base of steady computation, unsteady viscous flows about NACA0012 airfoil, are simulated. The structured viscous grid is used. The unsteady viscous motorial grid is fixed relative to airfoils. Numerical results are in good agreement with the experimental data. The method and the procedure presented in this dissertation are verified.
本文主要发展了计算翼型跨音速定常及非定常粘性绕流的数值模拟方法。基于雷诺平均的思想,出现在RANS方程中的雷诺应力项一般采用统计意义下的湍流模型进行封闭。文中湍流模型分别采用B-L零方程模型及k-ε、k-ω两种线性涡粘两方程模型,对N-S方程的几种数值求解方法进行了探讨。
在空间离散上,基于有限体积法,采用迎风型通量矢量分裂类Van Leer格式,并引入三阶MUSCL插值方法和连续可微的van Albada通量限制器,使格式具有高阶精度,且具有良好的稳定性和收敛性;在时间推进上,分别采用显式四步Runge-Kutta和隐式LU-SGS推进方法,对主控方程与湍流模型方程进行耦合求解,虽然本文工作侧重于定常绕流的数值模拟,但从原理上说,本文所开发的LU-SGS定常计算程序只需略做修改即可实现非定常流动模拟;边界条件采用虚拟网格技术作为统一的边界处理方式,有效地提高了编程和计算效率。
飞行器定常、非定常气动力的准确数值模拟,特别是跨音速范围内的数值求解及研究,一直以来都是飞行器设计工作者面对的重要研究课题。本文主要针对NACA0012、RAE2822翼型的定常粘性流场进行数值模拟;在此基础上,以NACA0012翼型为例,对非定常粘性流场进行了数值求解。计算网格采用贴体结构粘性网格,其中非定常算例采用了刚性运动网恪的处理方法。文中对比分析了不同方法、不同湍流模型之间的差异,获得了比较理想的计算结果,验证了本文数值求解技术的可行性以及计算程序的正确性。
In modern aircraft design, there is a pressing demand for accurate, efficient and applied aerodynamic data and computational analysis tools. However, the Euler equations are not satisfied all the requirement for simulating the precision of aeroelastic problems. It's necessary to solve N-S equations to simulate viscous effects, offer more accurate and more detailed flow-fields information. It is possible to solve N-S equations with the improvement of the hardware and software in computer fields, and the development of numeric methods in computational fluid dynamics domains.The present work is mainly focused on developing airfoils numerical simulation methods for steady and unsteady transonic flow. With the numerical solution of RANS equations, the applications of B-L zero-equation 、 k-ε and k-ω two-equation linear eddy turbulence models are discussed.On the basis of a finite volume method, the spatial discrete scheme about the convection terms of N-S equations: the Van Leer scheme of flux vector splitting, is studied. To get higher order accuracy、well stability and astringency, the MUSCL interpolation function and the continuous differential van Albada limiter are applied for Van Leer. In this paper, two time stepping schemes, explicit four-stage Runge-Kutta scheme and implicit LU-SGS scheme, are employed to solve the governing equations. Boundary conditions are treated by adopting "ghost" cells, thecomputational efficiency is improved and the program's complexity is reduced, too. In aircraft design, accurate numerical simulation for steady and unsteady flows.especially for transonic flows, is a crucial problem for designers. To confirm the validity and reliability of developed flow solvers, test cases of steady viscous flows around NACA0012 and RAE2822 airfoils, are given in this paper. On the base of steady computation, unsteady viscous flows about NACA0012 airfoil, are simulated. The structured viscous grid is used. The unsteady viscous motorial grid is fixed relative to airfoils. Numerical results are in good agreement with the experimental data. The method and the procedure presented in this dissertation are verified.
引文
1.朱自强等,《应用计算流体力学》,北京航空航天大学出版社,1998
2. Hanel, D., "On the Accuracy of Upwind Scheme in the Solution of Navier-Stokes Equations", AIAA Paper 87-1105, 1987
3.梁强,非定常气动力的N-S方程解及其应用,西北工业大学硕士学位论文,2001年
4. Jameson A. et al. Numerical Simulation of the Euler Equations by Finite Volume Methods Using Methods Using Runge-Kutta Time-Marching Schemes. AIAA paper 81-1295.
5.李杰,“先进民用飞机复杂外形跨音速绕流数值分析”,西北工业大学博士学位论文,1999年
6. Jameson A. and Turkel E. Implicit Schemes and LU Decomposations. Mathematics of computation, Vol.37, No. 156, 1981, pp385-397.
7.肖志祥,“复杂流动Nayier-Stokes方程数值模拟及湍流模型研究”,西北工业大学博士学位论文,2003年
8. Menter, F.R., "Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows", AIAA Paper 93-2906, 1993
9.蒋胜矩,“基于N-S方程的大迎角分离流及翼型结冰数值模拟”,西北工业大学硕士学位论文,2004年
10.梁强,复杂流场的非定常气动力计算以及气动弹性研究,西北工业大学博士学位论文,2003
11. J.A.White, "Elliptic Grid Generation With Orthogonality And Spacing Control On An Arbitrary Number Of Boundafies",Analytical Services And Materials,Inc.Hampton,VA.AIAA 90-1568.
12.徐华舫,《空气动力学基础》,国防工业出版社,1979年
13.张强,“两方程湍流模型的应用研究”,西北工业大学硕士学位论文,2002 年
14.张兆顺著,《湍流》,国防工业出版社,2002.1
15. Christopher L. Rumsey and Veer N. Vatsa,"Comparison of the Predictive Capabilities of Several Turbulence Models", Journal of Aircraft, Vol.32, No.3, 1995
16. Soo HYung Park and Jang Hyuk Kwon, "Implementation of k-ω Turbulence Models in an Implicit Multigrid Method "AIAA Journal, Vol.42, No.7, 2004
17.是勋刚,《湍流》,天津:天津大学出版社,1994
18.郭广利,杨永年,叶正寅,“翼型跨音速定常及非定常粘性绕流的数值模拟”,西北工业大学学报,Vol.15,No.1,1997
19. L.Dubuc, et al. "Solution of the Unsteady Euler Equations Using an Implicit Dual-time Method", AIAA Journal 36(8): 1417-1424, August 1998.
20.蔡晋生,“跨音速大迎角欧拉方程数值模拟”,西北工业大学博士学位论文,1992年
21.北京空气动力研究所编,空气动力学发展论文集.北京:国防工业出版社,1997.
22. van Leer, B., "Flux Vector Spitting for Euler Equations", Lect. Notes in Phys., Vol.170, pp.507-512, 1982
23. Jameson, A., et al, "Numerical Solutions of Euler Equations by Finite Volume Methods with Runge-Kutta Time Stepping Schemes", AIAA Paper 81-1259,1981
24.傅德熏,《计算空气动力学》,宇航出版社,1994
25. Zheng, Xiaoqing, and Liu, Feng, "Staggered Upwind Method for Solving Navier-Stokes and k-ω Turbulence Model Equations", AIAA Journal, Vol.33, pp.991-998, 1995
26. Robert, F.K., et al, "Explicit Navier-Stokes Computation of Cascade Flows Using the k-ε Turbulence Model", AIAA Journal, Vol.30, pp.13-22, 1992
27. van Leer, B., "Flux Vector Spitting for Euler Equations", Lect. Notes in Phys., Vol.170, pp.507-512, 1982
28. F.R.Menter, "Two-equation eddy-viscosity turbulence models for engineering applications". AIAA Journal, 32(8):1598-1605, 1994.
29. Swanson, R.C. and Turkel, E. "On Central-Difference and Upwind Schemes", Journal of Computational Physics Vol 101,pp.292-306, 1992
30.张强,杨永,“EASM两方程湍流模型的应用研究”,航空学院第三届研究生学术年会论文集,2004.6
31. Antti Hellsten, "New Two-Equation Turbulence Model For Aerodynamics Applications", Report A-21
32.张仲寅,乔志德,粘性流体力学(修订本)[M].国防工业出版社,1989
33.桑为民,“基于自适应直角叉树切割网格的Euler及N-S方程数值模拟”,西北工业大学博士学位论文,2003年
34.罗时均,跨音速无粘有旋流和有限体积法[A],西北工业大学讲义,1984
35.聂铁军等,《数值计算方法》,西北工业大学出版社,1990
36.黄守智,“基于动态变形网格的非定常粘性流动数值分析”,西北工业大学硕士学位论文,2005.3
37. E,Q., Li,F.W., and Yang G.W. "Numerical Solution of the Euler Equations for Transonic Flow about the Complete Aircraft", AIAA-96-2406, 1996.
38.李凤蔚,鄂秦,“有限体积法的分析和改进”,空气动力学报,1994年第四期
39.王刚,三维非结构粘性网格生成与N-S方程求解,西北工业大学硕士学位论文,2001
40.叶均科,“基于拟压缩方法的二维低速流动数值模拟”,西北工业大学硕士学位论文,2004
41. Kallinderis, Y., Algebraic Turbulence Modeling for Adaptive Unstructured Grids, AIAA Journal, 30(3), 1992.
42. Rumsey, C.L., et al, "Efficiency and Accuracy of Time-Accurate Turbulent Navier-Stokes Computations", AIAA Paper 95-1835-CP, 1995
43.柳长安,“非结构网格生成及二维非定常流场及悬挂物投放数值模拟”,西北工业大学硕士学位论文,2000
44. Jie Li,Fengwei Li and Qing E, "A fully implict method for steady and unsteady viscous flow simulations", international journal numerical methods in fluilds, int.J.Numer.Meth.Fluids 2003;43:147-163
45. Kalitzin, G., et al, "Application of Two-Equation Turbulence Models in Aircraft Design", AIAA Paper 96-0327, 1996
46. Higenstock, A., "A Fast Method for the Elliptic Generation of Three Dimensional Grids With Full Boundary Control", Numerical Grid Generation In Computational Fluid Mechanics '88, Pineridge Press Limited, pp.137-146, 1988.
47.高正红,“振荡三角翼非定常气动特性的模拟”,西北工业大学学报,1997
48. Volpe, G. and Jameson, A., "An Efficient Solution for Computing Transonic and Supersonic Flows about Aircraft", ICAS-88-4.8.3, 1988.
2. Hanel, D., "On the Accuracy of Upwind Scheme in the Solution of Navier-Stokes Equations", AIAA Paper 87-1105, 1987
3.梁强,非定常气动力的N-S方程解及其应用,西北工业大学硕士学位论文,2001年
4. Jameson A. et al. Numerical Simulation of the Euler Equations by Finite Volume Methods Using Methods Using Runge-Kutta Time-Marching Schemes. AIAA paper 81-1295.
5.李杰,“先进民用飞机复杂外形跨音速绕流数值分析”,西北工业大学博士学位论文,1999年
6. Jameson A. and Turkel E. Implicit Schemes and LU Decomposations. Mathematics of computation, Vol.37, No. 156, 1981, pp385-397.
7.肖志祥,“复杂流动Nayier-Stokes方程数值模拟及湍流模型研究”,西北工业大学博士学位论文,2003年
8. Menter, F.R., "Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows", AIAA Paper 93-2906, 1993
9.蒋胜矩,“基于N-S方程的大迎角分离流及翼型结冰数值模拟”,西北工业大学硕士学位论文,2004年
10.梁强,复杂流场的非定常气动力计算以及气动弹性研究,西北工业大学博士学位论文,2003
11. J.A.White, "Elliptic Grid Generation With Orthogonality And Spacing Control On An Arbitrary Number Of Boundafies",Analytical Services And Materials,Inc.Hampton,VA.AIAA 90-1568.
12.徐华舫,《空气动力学基础》,国防工业出版社,1979年
13.张强,“两方程湍流模型的应用研究”,西北工业大学硕士学位论文,2002 年
14.张兆顺著,《湍流》,国防工业出版社,2002.1
15. Christopher L. Rumsey and Veer N. Vatsa,"Comparison of the Predictive Capabilities of Several Turbulence Models", Journal of Aircraft, Vol.32, No.3, 1995
16. Soo HYung Park and Jang Hyuk Kwon, "Implementation of k-ω Turbulence Models in an Implicit Multigrid Method "AIAA Journal, Vol.42, No.7, 2004
17.是勋刚,《湍流》,天津:天津大学出版社,1994
18.郭广利,杨永年,叶正寅,“翼型跨音速定常及非定常粘性绕流的数值模拟”,西北工业大学学报,Vol.15,No.1,1997
19. L.Dubuc, et al. "Solution of the Unsteady Euler Equations Using an Implicit Dual-time Method", AIAA Journal 36(8): 1417-1424, August 1998.
20.蔡晋生,“跨音速大迎角欧拉方程数值模拟”,西北工业大学博士学位论文,1992年
21.北京空气动力研究所编,空气动力学发展论文集.北京:国防工业出版社,1997.
22. van Leer, B., "Flux Vector Spitting for Euler Equations", Lect. Notes in Phys., Vol.170, pp.507-512, 1982
23. Jameson, A., et al, "Numerical Solutions of Euler Equations by Finite Volume Methods with Runge-Kutta Time Stepping Schemes", AIAA Paper 81-1259,1981
24.傅德熏,《计算空气动力学》,宇航出版社,1994
25. Zheng, Xiaoqing, and Liu, Feng, "Staggered Upwind Method for Solving Navier-Stokes and k-ω Turbulence Model Equations", AIAA Journal, Vol.33, pp.991-998, 1995
26. Robert, F.K., et al, "Explicit Navier-Stokes Computation of Cascade Flows Using the k-ε Turbulence Model", AIAA Journal, Vol.30, pp.13-22, 1992
27. van Leer, B., "Flux Vector Spitting for Euler Equations", Lect. Notes in Phys., Vol.170, pp.507-512, 1982
28. F.R.Menter, "Two-equation eddy-viscosity turbulence models for engineering applications". AIAA Journal, 32(8):1598-1605, 1994.
29. Swanson, R.C. and Turkel, E. "On Central-Difference and Upwind Schemes", Journal of Computational Physics Vol 101,pp.292-306, 1992
30.张强,杨永,“EASM两方程湍流模型的应用研究”,航空学院第三届研究生学术年会论文集,2004.6
31. Antti Hellsten, "New Two-Equation Turbulence Model For Aerodynamics Applications", Report A-21
32.张仲寅,乔志德,粘性流体力学(修订本)[M].国防工业出版社,1989
33.桑为民,“基于自适应直角叉树切割网格的Euler及N-S方程数值模拟”,西北工业大学博士学位论文,2003年
34.罗时均,跨音速无粘有旋流和有限体积法[A],西北工业大学讲义,1984
35.聂铁军等,《数值计算方法》,西北工业大学出版社,1990
36.黄守智,“基于动态变形网格的非定常粘性流动数值分析”,西北工业大学硕士学位论文,2005.3
37. E,Q., Li,F.W., and Yang G.W. "Numerical Solution of the Euler Equations for Transonic Flow about the Complete Aircraft", AIAA-96-2406, 1996.
38.李凤蔚,鄂秦,“有限体积法的分析和改进”,空气动力学报,1994年第四期
39.王刚,三维非结构粘性网格生成与N-S方程求解,西北工业大学硕士学位论文,2001
40.叶均科,“基于拟压缩方法的二维低速流动数值模拟”,西北工业大学硕士学位论文,2004
41. Kallinderis, Y., Algebraic Turbulence Modeling for Adaptive Unstructured Grids, AIAA Journal, 30(3), 1992.
42. Rumsey, C.L., et al, "Efficiency and Accuracy of Time-Accurate Turbulent Navier-Stokes Computations", AIAA Paper 95-1835-CP, 1995
43.柳长安,“非结构网格生成及二维非定常流场及悬挂物投放数值模拟”,西北工业大学硕士学位论文,2000
44. Jie Li,Fengwei Li and Qing E, "A fully implict method for steady and unsteady viscous flow simulations", international journal numerical methods in fluilds, int.J.Numer.Meth.Fluids 2003;43:147-163
45. Kalitzin, G., et al, "Application of Two-Equation Turbulence Models in Aircraft Design", AIAA Paper 96-0327, 1996
46. Higenstock, A., "A Fast Method for the Elliptic Generation of Three Dimensional Grids With Full Boundary Control", Numerical Grid Generation In Computational Fluid Mechanics '88, Pineridge Press Limited, pp.137-146, 1988.
47.高正红,“振荡三角翼非定常气动特性的模拟”,西北工业大学学报,1997
48. Volpe, G. and Jameson, A., "An Efficient Solution for Computing Transonic and Supersonic Flows about Aircraft", ICAS-88-4.8.3, 1988.