非定常流动模拟的时间离散方法
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摘要
对于不同非定常流动问题,采用合适的时间离散方法,可有效提高数值精度和计算效率.本文在总结传统时间离散方法的基础上,对近些年发展的非线性频域法、谐波平衡法、经典时间谱方法、时间谱元法、时间有限差分法等进行了系统地总结.根据离散形式的不同,将上述方法分为时域推进法、频域谐波法、时域配点法和混合方法 4大类.首先简要介绍了各类方法的数学思想以及研究进展,并重点比较了(准)周期性非定常流动计算中各方法的精度、效率以及适用范围.然后,对各种时间离散格式的特点进行总结,并就不同的非定常流动问题如何选择合适的时间离散方法给予了建议.最后,对这些新型时间离散格式在工程中的应用进行了简要介绍,并对其发展方向进行展望.
For the numerical computation of unsteady flow, the computational accuracy and efficiency would have a significant difference with different time discretization methods. This paper based on the summarize of the development situation of time discretization methods at present, briefly introduces the time discretization methods developed in recent years like the nonlinear frequency domain method, harmonic balance method, time spectral method, time spectral element method, time finite difference method and so on. Based on the difference between discrete versions, the time discretization methods can be divided into four types: time domain marching method, frequency domain harmonic method, time domain collocation method and hybrid method. This paper briefly introduces the mathematical thought and study progress of each discretization method, and selective compare the accuracy, efficiency, and scope of application of each time discretization method in the computation of unsteady flow. Then we systematically summarize the characteristic of each time discretization method and advise how to choose appropriate time discretization methods in different unsteady flow problems. Finally, briefly introduce the application of current time discretization methods in the projects and discuss the development directions of the time discretization method in the future.
For the numerical computation of unsteady flow, the computational accuracy and efficiency would have a significant difference with different time discretization methods. This paper based on the summarize of the development situation of time discretization methods at present, briefly introduces the time discretization methods developed in recent years like the nonlinear frequency domain method, harmonic balance method, time spectral method, time spectral element method, time finite difference method and so on. Based on the difference between discrete versions, the time discretization methods can be divided into four types: time domain marching method, frequency domain harmonic method, time domain collocation method and hybrid method. This paper briefly introduces the mathematical thought and study progress of each discretization method, and selective compare the accuracy, efficiency, and scope of application of each time discretization method in the computation of unsteady flow. Then we systematically summarize the characteristic of each time discretization method and advise how to choose appropriate time discretization methods in different unsteady flow problems. Finally, briefly introduce the application of current time discretization methods in the projects and discuss the development directions of the time discretization method in the future.
引文
陈琦,陈坚强,谢昱飞,袁先旭.2014a.谐波平衡法在非定常流场中的应用.航空学报,35:736-743(Chen Q,Chen J Q,Xie Y F,et al.2014a.Application of harmonic balance method to unsteady flow field.Acta Aeronautica et Astronautica Sinica,35:736-743).
陈琦,陈坚强,袁先旭,谢昱飞.2014b.谐波平衡法在动导数快速预测中的应用研究.力学学报,46:183-190(Chen Q,Chen J Q,Yuan X X,et al.2014b.Application of a harmonic balance method in rapid predictions of dynamic stability derivatives.Chinese Journal of Theoretical and Applied Mechanics,46:183-190).
刘南,白俊强,刘艳,华俊.2016.基于谐波平衡法和V-g法的高效颤振预测分析(英文).空气动力学学报,34:631-637(Liu N,Bai J Q,Liu Y,et al.2016.Efficient flutter prediction based on harmonic balance and V-g methods.Acta Aerodynamica Sinica,34:631-637).
刘南.2016.机翼跨声速非线性颤振及高校分析方法研究.[博士论文].西安:西北工业大学(Liu N.2016.Investigation of transonic nonlinear flutter and efficient analysis approach.[PhD Thesis].Xi’an:Northwestern Polytechnical University).
谢立军,杨云军,刘周,周伟江.2015.基于时间谱方法的飞行器动导数高效计算技术.航空学报,36:2016-2026(Xie L J,Yang Y J,Liu Z,Zhou W J.2015.A high efficient method for computing dynamic derivatives of aircraft based on time spectral method.Acta Aeronautica et Astronautica Sinica,36:2016-2026).
袁新,苏欣荣.2009.谱方法用于非定常流动计算的隐式求解.工程热物理学报,30:2010-2012(Yuan X,Su X Y.2009.Implicit solution of time spectral method for periodic unsteady flows.Journal of Engineering Thermophysics,30:2010-2012).
赵军,刘宝杰.2008.非线性谐波法的进一步校验分析.航空动力学报,23:680-686(Zhao J,Liu B J.2008.Further verification of nonlinear harmonic method.Journal of Aerospace Power,23:680-686).
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Beam R M,Warming R F.1976.An implicit finite-difference algorithm for hyperbolic systems in conservationlaw form.Journal of Computational Physics,22:87-110.
Besem F M,Thomas J P,Kielb R E,Dowell E H.2016.An aeroelastic model for vortex-induced vibrating cylinders subject to frequency lock-in.Journal of Fluids&Structures,61:42-59.
Bonfiglioli A,Carpentieri B,Campobasso S.2009.Parallel unstructured three-dimensional turbulent flow analyses using efficiently preconditioned Newton-Krylov Solvers//19th AIAA Computational Fluid Dynamics,San Antonio.
Briley W R,Mcdonald H.1975.Solution of the three-dimensional compressible Navier Stokes equations by an implicit technique.Lecture Notes in Physics,35:105-110.
Brunton S L,Rowley C W.2013.Empirical state-space representations for Theodorsen’s lift model.Journal of Fluids&Structures,38:174-186.
Butsuntorn N,Jameson A.2008a.Time spectral method for rotorcraft flow.[PhD Thesis].Stanford CA:Stanford University.
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Choi S,Lee K H,Alonso J J,Datta A.2008.Preliminary study on time-spectral and adjoint-based design optimization of helicopter rotors//AHS specialist meeting,San Francisco,CA.
Choi S,Datta A.2008.CFD prediction of rotor loads using time-spectral method and exact fluid-structure interface//26th AIAA Applied Aerodynamics Conference.
Choi S,Lee K,Potsdam M M,Alonso J J.2008.Helicopter rotor design using a time-spectral and adjointbased method//Aiaa/issmo Multidisciplinary Analysis and Optimization Conference,412-423.
Custer C H,Weiss J M,Subramanian V,Clark W S,Hall K C.2012.Unsteady simulation of a 1.5 stage turbine using an implicitly coupled nonlinear harmonic balance method//ASME Turbo Expo:Turbine Technical Conference and Exposition,2303-2317.
Dinu A D,Botez R,Cotoi I.2006.Chebyshev polynomials for unsteady aerodynamic calculations in aeroservoelasticity.Journal of Aircraft,43:165-171.
Dong K I,Choi S,Mcclure J E,Skiles F.2015.Mapped Chebyshev pseudospectral method for unsteady flow analysis.AIAA Journal,53:1-16.
Du P,Ning F.2012.The development and application of a time-domain harmonic balance flow solver//ASME 2012 Fluids Engineering Division Summer Meeting Collocated with the ASME 2012 Heat Transfer Summer Conference and the ASME 2012,International Conference on Nanochannels,Microchannels,and Minichannels.
Dufour G,Gourdain N,Sicot F.2012.A time-domain harmonic balance method for rotor/stator interactions.Journal of Turbomachinery,134:11001-11001.
Ekici K,Hall K C.2007.Nonlinear analysis of unsteady flows in multistage turbomachines using harmonic balance.AIAA Journal,45:1047-1057.
Ekici K,Hall K C.2008.Nonlinear frequency-domain analysis of unsteady flows in turbomachinery with multiple excitation frequencies.AIAA Journal,46:1912.
Ekici K,Beran P.2014.Adjoint sensitivity analysis of low-speed flows using an efficient harmonic balance technique.AIAA Journal,52:1330-1336.
Farhat C,Lesoinne M.2000.Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems.Computer Methods in Applied Mechanics and Engineering,182:499-515.
Frey C,Ashcroft G,Kersken H P.2015.Simulations of unsteady blade row interactions using linear and non-linear frequency domain methods//ASME Turbo Expo 2015:Turbine Technical Conference and Exposition,V02BT39A037.
Gopinath A K,Beran P S,Jameson A.2006.Comparative analysis of computational methods for limit-cycle oscillations.AIAA Paper,2076:1-4.
Gopinath A K,Jameson A.2005.Time spectral method for periodic unsteady computations over two-and three-dimensional bodies.AIAA Paper,1220:10-13.
Gopinath A K,Van Der Weide E,Alonso J J,Jameson A.2007.Three-dimensional unsteady multi-stage turbomachinery simulations using the harmonic balance technique//45th AIAA Aerospace Sciences Meeting and Exhibit,892.
Gu′edeney T,Gomar A,Gallard F,Sicot F.2013.Non-uniform time sampling for multiple-frequency harmonic balance computations.Journal of Computational Physics,236:317-345.
Guo Y,Keller J,Parker R G.2012.Dynamic analysis of wind turbine planetary gears using an extended harmonic balance approach:Preprint.Office of Scientific&Technical Information Technical Reports,8:4329-4343.
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Hall K C,Crawley E F.1994.A linearized Euler analysis of unsteady flows in turbomachinery.Massachusetts Inst.of Tech.Report,116:V001T03A035.
Hall K C,Thomas J P,Clark W S.2002.Computation of unsteady nonlinear flows in cascades using a harmonic balance technique.AIAA Journal,40:879-886.
Hassan D,Sicot F.2011.A time-domain harmonic balance method for dynamic derivatives predictions.//49th AIAA Aerospace Sciences Meeting including the New Horizons and Aerospace Exposition,4-7.
He L.2008.Harmonic solution of unsteady flow around blades with separation.AIAA Journal,46:1299-1307.
He L.2010.Fourier methods for turbomachinery applications.Progress in Aerospace Sciences,46:329-341.
He L,Ning W.1998.Efficient approach for analysis of unsteady viscous flows in turbomachines.AIAAJournal,36:2005-2012.
Huang X,Wang D X.2016.Stabilizing and accelerating solution of harmonic balance equation system using the LU-SGS and block Jacobi methods//ASME Turbo Expo 2016:Turbomachinery Technical Conference and Exposition,V02DT44A026.
Hull T E,Enright W H,Fellen B M,Sedgwick A E.1972.Comparing numerical methods for ordinary differential equations.SIAM Journal on Numerical Analysis,9:603-637.
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陈琦,陈坚强,袁先旭,谢昱飞.2014b.谐波平衡法在动导数快速预测中的应用研究.力学学报,46:183-190(Chen Q,Chen J Q,Yuan X X,et al.2014b.Application of a harmonic balance method in rapid predictions of dynamic stability derivatives.Chinese Journal of Theoretical and Applied Mechanics,46:183-190).
刘南,白俊强,刘艳,华俊.2016.基于谐波平衡法和V-g法的高效颤振预测分析(英文).空气动力学学报,34:631-637(Liu N,Bai J Q,Liu Y,et al.2016.Efficient flutter prediction based on harmonic balance and V-g methods.Acta Aerodynamica Sinica,34:631-637).
刘南.2016.机翼跨声速非线性颤振及高校分析方法研究.[博士论文].西安:西北工业大学(Liu N.2016.Investigation of transonic nonlinear flutter and efficient analysis approach.[PhD Thesis].Xi’an:Northwestern Polytechnical University).
谢立军,杨云军,刘周,周伟江.2015.基于时间谱方法的飞行器动导数高效计算技术.航空学报,36:2016-2026(Xie L J,Yang Y J,Liu Z,Zhou W J.2015.A high efficient method for computing dynamic derivatives of aircraft based on time spectral method.Acta Aeronautica et Astronautica Sinica,36:2016-2026).
袁新,苏欣荣.2009.谱方法用于非定常流动计算的隐式求解.工程热物理学报,30:2010-2012(Yuan X,Su X Y.2009.Implicit solution of time spectral method for periodic unsteady flows.Journal of Engineering Thermophysics,30:2010-2012).
赵军,刘宝杰.2008.非线性谐波法的进一步校验分析.航空动力学报,23:680-686(Zhao J,Liu B J.2008.Further verification of nonlinear harmonic method.Journal of Aerospace Power,23:680-686).
Badcock K J,Richards B E,Woodgate M A.2000.Elements of computational fluid dynamics on block structured grids using implicit solvers.Progress in Aerospace Sciences,36:351-392.
Beam R M,Warming R F.1976.An implicit finite-difference algorithm for hyperbolic systems in conservationlaw form.Journal of Computational Physics,22:87-110.
Besem F M,Thomas J P,Kielb R E,Dowell E H.2016.An aeroelastic model for vortex-induced vibrating cylinders subject to frequency lock-in.Journal of Fluids&Structures,61:42-59.
Bonfiglioli A,Carpentieri B,Campobasso S.2009.Parallel unstructured three-dimensional turbulent flow analyses using efficiently preconditioned Newton-Krylov Solvers//19th AIAA Computational Fluid Dynamics,San Antonio.
Briley W R,Mcdonald H.1975.Solution of the three-dimensional compressible Navier Stokes equations by an implicit technique.Lecture Notes in Physics,35:105-110.
Brunton S L,Rowley C W.2013.Empirical state-space representations for Theodorsen’s lift model.Journal of Fluids&Structures,38:174-186.
Butsuntorn N,Jameson A.2008a.Time spectral method for rotorcraft flow.[PhD Thesis].Stanford CA:Stanford University.
Butsuntorn N,Jameson A.2008b.Time spectral method for rotorcraft flow with vorticity confinement//26th AIAA Applied Aerodynamics Conference.
Butsuntorn N.2008.Time spectral method for rotorcraft flow with vorticity confinement.[PhD Thesis].California:Stanford University.
Butcher J C.2016.Numerical Methods for Ordinary Differential Equations.(First Edition).John Wiley&Sons.
Chen T,Vasanthakumar P,He L.2000.Analysis of unsteady blade row interaction using nonlinear harmonic approach.Journal of Propulsion&Power,17:651-658.
Cherif M A,Emamirad H,Mnif M.2012.Derivatives for time-spectral computational fluid dynamics using an automatic differentiation adjoint.AIAA Journal,50:2809-2819.
Choi S,Lee K H,Alonso J J,Datta A.2008.Preliminary study on time-spectral and adjoint-based design optimization of helicopter rotors//AHS specialist meeting,San Francisco,CA.
Choi S,Datta A.2008.CFD prediction of rotor loads using time-spectral method and exact fluid-structure interface//26th AIAA Applied Aerodynamics Conference.
Choi S,Lee K,Potsdam M M,Alonso J J.2008.Helicopter rotor design using a time-spectral and adjointbased method//Aiaa/issmo Multidisciplinary Analysis and Optimization Conference,412-423.
Custer C H,Weiss J M,Subramanian V,Clark W S,Hall K C.2012.Unsteady simulation of a 1.5 stage turbine using an implicitly coupled nonlinear harmonic balance method//ASME Turbo Expo:Turbine Technical Conference and Exposition,2303-2317.
Dinu A D,Botez R,Cotoi I.2006.Chebyshev polynomials for unsteady aerodynamic calculations in aeroservoelasticity.Journal of Aircraft,43:165-171.
Dong K I,Choi S,Mcclure J E,Skiles F.2015.Mapped Chebyshev pseudospectral method for unsteady flow analysis.AIAA Journal,53:1-16.
Du P,Ning F.2012.The development and application of a time-domain harmonic balance flow solver//ASME 2012 Fluids Engineering Division Summer Meeting Collocated with the ASME 2012 Heat Transfer Summer Conference and the ASME 2012,International Conference on Nanochannels,Microchannels,and Minichannels.
Dufour G,Gourdain N,Sicot F.2012.A time-domain harmonic balance method for rotor/stator interactions.Journal of Turbomachinery,134:11001-11001.
Ekici K,Hall K C.2007.Nonlinear analysis of unsteady flows in multistage turbomachines using harmonic balance.AIAA Journal,45:1047-1057.
Ekici K,Hall K C.2008.Nonlinear frequency-domain analysis of unsteady flows in turbomachinery with multiple excitation frequencies.AIAA Journal,46:1912.
Ekici K,Beran P.2014.Adjoint sensitivity analysis of low-speed flows using an efficient harmonic balance technique.AIAA Journal,52:1330-1336.
Farhat C,Lesoinne M.2000.Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems.Computer Methods in Applied Mechanics and Engineering,182:499-515.
Frey C,Ashcroft G,Kersken H P.2015.Simulations of unsteady blade row interactions using linear and non-linear frequency domain methods//ASME Turbo Expo 2015:Turbine Technical Conference and Exposition,V02BT39A037.
Gopinath A K,Beran P S,Jameson A.2006.Comparative analysis of computational methods for limit-cycle oscillations.AIAA Paper,2076:1-4.
Gopinath A K,Jameson A.2005.Time spectral method for periodic unsteady computations over two-and three-dimensional bodies.AIAA Paper,1220:10-13.
Gopinath A K,Van Der Weide E,Alonso J J,Jameson A.2007.Three-dimensional unsteady multi-stage turbomachinery simulations using the harmonic balance technique//45th AIAA Aerospace Sciences Meeting and Exhibit,892.
Gu′edeney T,Gomar A,Gallard F,Sicot F.2013.Non-uniform time sampling for multiple-frequency harmonic balance computations.Journal of Computational Physics,236:317-345.
Guo Y,Keller J,Parker R G.2012.Dynamic analysis of wind turbine planetary gears using an extended harmonic balance approach:Preprint.Office of Scientific&Technical Information Technical Reports,8:4329-4343.
Hall K C,Clark W S.1991.Prediction of unsteady aerodynamic loads in cascades using the linearized Euler equations on deforming grids//27th Joint Propulsion Conference.
Hall K C,Crawley E F.1989.Calculation of unsteady flows in turbomachinery using the linearized Euler equations.AIAA Journal,27:777-787.
Hall K C,Crawley E F.1994.A linearized Euler analysis of unsteady flows in turbomachinery.Massachusetts Inst.of Tech.Report,116:V001T03A035.
Hall K C,Thomas J P,Clark W S.2002.Computation of unsteady nonlinear flows in cascades using a harmonic balance technique.AIAA Journal,40:879-886.
Hassan D,Sicot F.2011.A time-domain harmonic balance method for dynamic derivatives predictions.//49th AIAA Aerospace Sciences Meeting including the New Horizons and Aerospace Exposition,4-7.
He L.2008.Harmonic solution of unsteady flow around blades with separation.AIAA Journal,46:1299-1307.
He L.2010.Fourier methods for turbomachinery applications.Progress in Aerospace Sciences,46:329-341.
He L,Ning W.1998.Efficient approach for analysis of unsteady viscous flows in turbomachines.AIAAJournal,36:2005-2012.
Huang X,Wang D X.2016.Stabilizing and accelerating solution of harmonic balance equation system using the LU-SGS and block Jacobi methods//ASME Turbo Expo 2016:Turbomachinery Technical Conference and Exposition,V02DT44A026.
Hull T E,Enright W H,Fellen B M,Sedgwick A E.1972.Comparing numerical methods for ordinary differential equations.SIAM Journal on Numerical Analysis,9:603-637.
Jameson A.1991.Time dependent calculations using multigrid,with applications to unsteady flows past airfoils and wings//Computational Fluid Dynamics Conference.
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