湍流模型系数不确定度对翼型绕流模拟的影响
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摘要
使用非嵌入式多项式混沌方法研究了湍流模型系数的不确定度对RAE2822跨声速翼型绕流模拟的影响。计算中关注了数值模拟的积分量(升力系数、阻力系数)和局部量(壁面压力、摩擦系数和空间马赫数分布)的不确定度量化结果。首先,从单输入变量入手,研究卡门常数的不确定度对数值模拟的影响。然后,同时考虑Spalart-Allmaras模型中9个参数的不确定度带来的影响。通过多项式混沌展开,得到系统输出对不确定输入变量的响应,由此可以得到输出的统计特性,包括平均值、方差和极值等信息。最后,在多变量不确定度量化过程中,通过Sobol指标来量化每个输入变量的不确定度对输出不确定度的贡献程度。本文计算只考虑了RAE2822跨声速翼型模拟的单一计算状态,影响规律是否可以推及其他工况和算例需要进一步检验。
The effects of uncertainty in the turbulence model closure coefficients on the simulation of flow over RAE2822 airfoil are investigated in this paper using a non-intrusive polynomial chaos method.The integral values(including lift and drag coefficients)and local flow field variables(e.g.pressure coefficient,skin friction coefficient,and Mach number)are considered as the output quantities of interest.The investigation begins with a single stochastic input variable,the von Karman constant.Then the uncertainty in nine closure coefficients of the Spalart-Allmaras model is taken into account.The system response is thus obtained,including the mean value,variance and extreme values.Finally,Sobol indices are used to evaluate the relative contributions of each closure coefficient to the variation of the output quantities.Since the conclusion in this paper is drawn from one single test case of the RAE2822 airfoil,further verification through the simulations of other cases is still needed.
The effects of uncertainty in the turbulence model closure coefficients on the simulation of flow over RAE2822 airfoil are investigated in this paper using a non-intrusive polynomial chaos method.The integral values(including lift and drag coefficients)and local flow field variables(e.g.pressure coefficient,skin friction coefficient,and Mach number)are considered as the output quantities of interest.The investigation begins with a single stochastic input variable,the von Karman constant.Then the uncertainty in nine closure coefficients of the Spalart-Allmaras model is taken into account.The system response is thus obtained,including the mean value,variance and extreme values.Finally,Sobol indices are used to evaluate the relative contributions of each closure coefficient to the variation of the output quantities.Since the conclusion in this paper is drawn from one single test case of the RAE2822 airfoil,further verification through the simulations of other cases is still needed.
引文
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[13]张伟,王小永,于剑,等.来流导致的高超声速气动热不确定度量化分析[J].北京航空航天大学学报,2018,44(5):1102-1109.ZHANG W,WANG X Y,YU J,et al.Uncertainty quantification analysis in hypersonic aerothermodynamics due to freestream[J].Journal of Beijing University of Aeronautics and Astronautics,2018,44(5):1102-1109(in Chinese).
[14]刘全,王瑞利,林忠,等.爆轰计算JWL状态方程参数不确定度研究[J].爆炸与冲击,2013,33(6):647-654.LIU Q,WANG R L,LIN Z,et al.Uncertainty quantification for JWL EOS parameters in explosive numerical simulation[J].Explosion and Shock Waves,2013,33(6):647-654(in Chinese).
[15] WIENER N.The homogeneous chaos[J].American Journal of Mathematics,1938,60:897-936.
[16] GHANEM R G,SPANOS P D.Stochastic finite elements:A spectral approach[M].revised edition.New York:Dover Publications,2003.
[17] HOSDER S, WALTERS R W,BALCH M.Efficient sampling for non-intrusive polynomial chaos applications with multiple input uncertain variables:AIAA-2007-1939[R].Reston,VA:AIAA,2007.
[18] SOBOL I.Sensitivity estimates for nonlinear mathematical models[J].Math Modeling&Computing Experiment,1993,1:407-414.
[19]胡军,张树道.基于多项式混沌的全局敏感度分析[J].计算物理,2016,33(1):1-13.HU J,ZHANG S D.Global sensitivity analysis based on polynomial chaos[J].Chinese Journal of Computational Phsics,2016,33(1):1-13(in Chinese).
[20] COOK P H,MCDONALD M A,FIRMIN M C P.Aerofoil RAE2822pressure distributions,and boundary layer and wake measurements,experimental data base for computer program assessment:AGARD-1979-0138[R].Paris:AGARD,1979.
[21] CHEN J T,ZHANG Y B,ZHOU N C et al.Numerical investigations of the high-lift configuration with mflow solver[J].Journal of Aircraft,2015,52(4):1051-1062.
[22] DISKIN B,THOMAS J L.Comparison of node-centered and cell-centered unstructured finite volume discretizations:Inviscid fluxes[J].AIAA Journal,2011,49(4):836-854.
[23] VENKATAKRISHNAN V.On the accuracy of limiters and convergence to steady-state solutions:AIAA-1993-0880[R].Reston,VA:AIAA,1993.
[24] SPALART P R,ALLMARAS S R.A one-equation turbulence model for aerodynamic flows:AIAA-1992-0439[R].Reston,VA:AIAA,1992.
[25] BARDINA J E,HUANG P G,COAKLEY T J.Turbulence modeling validation testing and development[R].Washington,D.C.:NASA,1997.
[26]张志雄.基于结构系综的湍流复杂系统研究[D].北京:北京大学,2009:80-81.ZHANG Z X.Complex system study of turbulence based on sructural ensemble[D].Beijing:Peking University,2009:80-81(in Chinese).
[27] SHIVSAI A D,RAMESH O N.Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers[J].Journal of Fluid Mechanics,2008,615:445-475.
[28] NAGIB H M,CHAUHAN K A.Variations of von Karman coefficient in canonical flows[J].Physics of Fluids,2008,20:101518.
[29] FRENZEN P,VOGEL C A.On the magnitude and apparent range of variation of the von Karman constant in the atmospheric surface layer[J].Boundary-Layer Meteorology,1995,72(4):371-392.
[30] ANDREAS E L,CLAFFEY K J,JORDAN R E,et al.Evaluations of the von Karman constant in the atmospheric surface layer[J].Journal of Fluid Mechanics,2006,559:117-149.
[31] SHE Z S,WU Y,CHEN X,et al.A multi-state description of roughness effects in turbulent pipe flow[J].New Journal Physics,2012,14:54-93.
[32] WU Y,CHEN X,SHE Z S,et al.On the Karman constant in turbulent channel flow[J].Physics Scripta,2013,21(3):9-14.
[33] SCHAEFER J,WEST T,HOSDER S,et al.Uncertainty quantification of turbulence model closure coefficients for transonic wall-bounded flows:AIAA-2015-2461[R].Reston,VA:AIAA,2015.
[2]刘智益,王晓东,康顺.多元多项式混沌法在随机方腔流动模拟中的应用[J].工程热物理学报,2012,33(3):419-422.LIU Z Y,WANG X D,KANG S.Application of multidimensional polynomial chaos on numerical simulations of stochastic cavity flow[J].Journal of Engineering Thermophysics,2012,33(3):419-422(in Chinese).
[3]张兆顺,崔桂香,许春晓.湍流理论与模拟[M].北京:清华大学出版社,2005:211-212.ZHANG Z S,CUI G X,XU C X.Theory and modeling of turbulence[M].Beijing:Tsinghua University Press,2005:211-212(in Chinese).
[4] FISHMAN G.Monte carlo:Concepts,algorithms and applications[M].New York:Springer,1996.
[5] GHANEM R,SPANOS P.Stochastic finite elements:A spectral approach[M].New York:Springer,1938.
[6] XIU D,KARNIADAKIS G.The Wiener-Askey polynomial chaos for stochastic differential equations[J].SIAM Journal on Scientific Computing,2002,24(2):619-644.
[7] LE MAITRE O P,KNIO O,HABIB N N,et al.A stochastic projection method for fluid flow I.Basic formulation[J].Journal of Computational Physics,2001,173:481-511.
[8] XIU D,KARNIADAKIS G E.Modeling uncertainty in flow simulations via generalized polynomial chaos[J].Journal of Computational Physics,2003,187:137-167.
[9] LACOR C,SMIRNOV S.Uncertainty propagation in the solution of compressible Navier-Stokes equations using polynomial chaos decomposition[C]∥NATO RTO AVF-147Symposium on Computational Uncertainty in Military Vehide Design,2007.
[10]王晓东,康顺.多项式混沌法求解随机Burgers方程[J].工程热物理学报,2010,31:393-398.WANG X D,KANG S.Solving stochastic Burgers equation using polynomial chaos decomposition[J].Journal of Engineering Thermophysics,2010,31:393-398(in Chinese).
[11] ELDRED M S.Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design:AIAA-2009-2274[R].Reston,VA:AIAA,2009.
[12] HOSDER S,WALTERS RW,BALCH M.Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics[J].AIAA Journal,2010,48(12):2721-2730.
[13]张伟,王小永,于剑,等.来流导致的高超声速气动热不确定度量化分析[J].北京航空航天大学学报,2018,44(5):1102-1109.ZHANG W,WANG X Y,YU J,et al.Uncertainty quantification analysis in hypersonic aerothermodynamics due to freestream[J].Journal of Beijing University of Aeronautics and Astronautics,2018,44(5):1102-1109(in Chinese).
[14]刘全,王瑞利,林忠,等.爆轰计算JWL状态方程参数不确定度研究[J].爆炸与冲击,2013,33(6):647-654.LIU Q,WANG R L,LIN Z,et al.Uncertainty quantification for JWL EOS parameters in explosive numerical simulation[J].Explosion and Shock Waves,2013,33(6):647-654(in Chinese).
[15] WIENER N.The homogeneous chaos[J].American Journal of Mathematics,1938,60:897-936.
[16] GHANEM R G,SPANOS P D.Stochastic finite elements:A spectral approach[M].revised edition.New York:Dover Publications,2003.
[17] HOSDER S, WALTERS R W,BALCH M.Efficient sampling for non-intrusive polynomial chaos applications with multiple input uncertain variables:AIAA-2007-1939[R].Reston,VA:AIAA,2007.
[18] SOBOL I.Sensitivity estimates for nonlinear mathematical models[J].Math Modeling&Computing Experiment,1993,1:407-414.
[19]胡军,张树道.基于多项式混沌的全局敏感度分析[J].计算物理,2016,33(1):1-13.HU J,ZHANG S D.Global sensitivity analysis based on polynomial chaos[J].Chinese Journal of Computational Phsics,2016,33(1):1-13(in Chinese).
[20] COOK P H,MCDONALD M A,FIRMIN M C P.Aerofoil RAE2822pressure distributions,and boundary layer and wake measurements,experimental data base for computer program assessment:AGARD-1979-0138[R].Paris:AGARD,1979.
[21] CHEN J T,ZHANG Y B,ZHOU N C et al.Numerical investigations of the high-lift configuration with mflow solver[J].Journal of Aircraft,2015,52(4):1051-1062.
[22] DISKIN B,THOMAS J L.Comparison of node-centered and cell-centered unstructured finite volume discretizations:Inviscid fluxes[J].AIAA Journal,2011,49(4):836-854.
[23] VENKATAKRISHNAN V.On the accuracy of limiters and convergence to steady-state solutions:AIAA-1993-0880[R].Reston,VA:AIAA,1993.
[24] SPALART P R,ALLMARAS S R.A one-equation turbulence model for aerodynamic flows:AIAA-1992-0439[R].Reston,VA:AIAA,1992.
[25] BARDINA J E,HUANG P G,COAKLEY T J.Turbulence modeling validation testing and development[R].Washington,D.C.:NASA,1997.
[26]张志雄.基于结构系综的湍流复杂系统研究[D].北京:北京大学,2009:80-81.ZHANG Z X.Complex system study of turbulence based on sructural ensemble[D].Beijing:Peking University,2009:80-81(in Chinese).
[27] SHIVSAI A D,RAMESH O N.Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers[J].Journal of Fluid Mechanics,2008,615:445-475.
[28] NAGIB H M,CHAUHAN K A.Variations of von Karman coefficient in canonical flows[J].Physics of Fluids,2008,20:101518.
[29] FRENZEN P,VOGEL C A.On the magnitude and apparent range of variation of the von Karman constant in the atmospheric surface layer[J].Boundary-Layer Meteorology,1995,72(4):371-392.
[30] ANDREAS E L,CLAFFEY K J,JORDAN R E,et al.Evaluations of the von Karman constant in the atmospheric surface layer[J].Journal of Fluid Mechanics,2006,559:117-149.
[31] SHE Z S,WU Y,CHEN X,et al.A multi-state description of roughness effects in turbulent pipe flow[J].New Journal Physics,2012,14:54-93.
[32] WU Y,CHEN X,SHE Z S,et al.On the Karman constant in turbulent channel flow[J].Physics Scripta,2013,21(3):9-14.
[33] SCHAEFER J,WEST T,HOSDER S,et al.Uncertainty quantification of turbulence model closure coefficients for transonic wall-bounded flows:AIAA-2015-2461[R].Reston,VA:AIAA,2015.