非定常流动模拟的时间离散方法

张伟伟; 贡伊明; 刘溢浪

doi:10.6052/1000-0992-17-018

非定常流动模拟的时间离散方法

doi: 10.6052/1000-0992-17-018

西北工业大学翼型叶栅重点实验室, 西安 710072

基金项目: 国家自然科学基金项目资助(11622220,11572252).

详细信息

作者简介:
null

作者简介：张伟伟, 西北工业大学教授, 流体系主任.主要从事复杂流固耦合力学、非定常空气动力学和流动控制研究.中国空气动力学会理事, 陕西省力学学会理事.主持国家重大专项、国家自然科学基金(3项)、863项目、教育部博士点基金、国防预研基金、航空基金等省部级以上基金13项.在国内外刊物和学术会议发表论文100余篇, 被SCI检索22篇, 其中JCRQ1区14篇, 被EI检索67篇.

中图分类号: V211.3;
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出版历程
- 收稿日期: 2017-09-20
- 刊出日期: 2019-02-08

Time discretization methods in the computation of unsteady flow

School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China

摘要

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.
- 非定常流动 /
- 时间离散方法 /
- 频域谐波法 /
- 时域配点法 /
- 时间谱方法
Abstract: 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.
- unsteady flow /
- time discretization method /
- frequency domain harmonic method /
- time domain collocation method /
- time spectral method

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参考文献(89)

[1]	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).
[2]	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).
[3]	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).
[4]	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).
[5]	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).
[6]	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).
[7]	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).
[8]	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.
[9]	Beam R M, Warming R F.1976. An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. Journal of Computational Physics, 22: 87-110.
[10]	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.
[11]	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.
[12]	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.
[13]	Brunton S L, Rowley C W.2013. Empirical state-space representations for Theodorsen's lift model. Journal of Fluids & Structures, 38: 174-186.
[14]	Butsuntorn N, Jameson A.2008a. Time spectral method for rotorcraft flow. [PhD Thesis]. Stanford CA: Stanford University.
[15]	Butsuntorn N, Jameson A.2008b. Time spectral method for rotorcraft flow with vorticity confinement//26th AIAA Applied Aerodynamics Conference.
[16]	Butsuntorn N.2008. Time spectral method for rotorcraft flow with vorticity confinement. [PhD Thesis]. California: Stanford University.
[17]	Butcher J C.2016. Numerical Methods for Ordinary Differential Equations. (First Edition). John Wiley & Sons.
[18]	Chen T, Vasanthakumar P, He L.2000. Analysis of unsteady blade row interaction using nonlinear harmonic approach.Journal of Propulsion & Power, 17: 651-658.
[19]	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.
[20]	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.
[21]	Choi S, Datta A.2008. CFD prediction of rotor loads using time-spectral method and exact fluid-structure interface//26th AIAA Applied Aerodynamics Conference.
[22]	Choi S, Lee K, Potsdam M M, Alonso J J.2008. Helicopter rotor design using a time-spectral and adjoint-based method//Aiaa/issmo Multidisciplinary Analysis and Optimization Conference, 412-423.
[23]	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.
[24]	Dinu A D, Botez R, Cotoi I.2006. Chebyshev polynomials for unsteady aerodynamic calculations in aeroservoelasticity.Journal of Aircraft, 43: 165-171.
[25]	Dong K I, Choi S, Mcclure J E, Skiles F.2015. Mapped Chebyshev pseudospectral method for unsteady flow analysis. AIAA Journal, 53: 1-16.
[26]	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.
[27]	Dufour G, Gourdain N, Sicot F.2012. A time-domain harmonic balance method for rotor/stator interactions. Journal of Turbomachinery, 134: 11001-11001.
[28]	Ekici K, Hall K C.2007. Nonlinear analysis of unsteady flows in multistage turbomachines using harmonic balance. AIAA Journal, 45: 1047-1057.
[29]	Ekici K, Hall K C.2008. Nonlinear frequency-domain analysis of unsteady flows in turbomachinery with multiple excitation frequencies. AIAA Journal, 46: 1912.
[30]	Ekici K, Beran P.2014. Adjoint sensitivity analysis of low-speed flows using an efficient harmonic balance technique. AIAA Journal, 52: 1330-1336.
[31]	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.
[32]	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.
[33]	Gopinath A K, Beran P S, Jameson A.2006. Comparative analysis of computational methods for limit-cycle oscillations. AIAA Paper, 2076: 1-4.
[34]	Gopinath A K, Jameson A.2005. Time spectral method for periodic unsteady computations over two-and three-dimensional bodies. AIAA Paper, 1220: 10-13.
[35]	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.
[36]	Guédeney 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.
[37]	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.
[38]	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.
[39]	Hall K C, Crawley E F.1989. Calculation of unsteady flows in turbomachinery using the linearized Euler equations. AIAA Journal, 27: 777-787.
[40]	Hall K C, Crawley E F.1994. A linearized Euler analysis of unsteady flows in turbomachinery. Massachusetts Inst.of Tech.Report, 116: V001T03A035.
[41]	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.
[42]	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.
[43]	He L.2008. Harmonic solution of unsteady flow around blades with separation. AIAA Journal, 46: 1299-1307.
[44]	He L.2010. Fourier methods for turbomachinery applications. Progress in Aerospace Sciences, 46: 329-341.
[45]	He L, Ning W.1998. Efficient approach for analysis of unsteady viscous flows in turbomachines. AIAA Journal, 36: 2005-2012.
[46]	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.
[47]	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.
[48]	Jameson A.1991. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings//Computational Fluid Dynamics Conference.
[49]	Jameson A, Shankaran S.2009. An assessment of dual-time stepping, time spectral and artificial compressibility based numerical algorithms for unsteady flow with applications to flapping wings//19th AIAA Computational Fluid Dynamics, 4273.
[50]	Kurdi M H, Beran P S.2008. Spectral element method in time for rapidly actuated systems. Journal of Computational Physics, 227: 1809-1835.
[51]	Lau S L, Cheung Y K.1981. Amplitude incremental variational principle for nonlinear vibration of elastic systems. Journal of Applied Mechanics, 48: 959.
[52]	Lee B H K, Liu L, Chung K W.2005. Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces. Journal of Sound & Vibration, 281: 699-717.
[53]	Leffell J, Sitaraman J, Lakshminarayan V, Wissink A.2016. Towards efficient parallel-in-time simulation of periodic flows//54th AIAA Aerospace Sciences Meeting (AIAA Paper 2016-0066), San Diego, CA, January 4--8.
[54]	Liu L, Dowell E H.2004. The secondary bifurcation of an aeroelastic airfoil motion: Effect of high harmonics. Nonlinear Dynamics, 37: 31-49.
[55]	Liu L, Dowell E H, Thomas J P.2007. A high dimensional harmonic balance approach for an aeroelastic airfoil with cubic restoring forces. Journal of Fluids & Structures, 23: 351-363.
[56]	Mavriplis D J, Yang Z, Mundis N.2012. Extensions of time spectral methods for practical rotorcraft problems//50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition paper AIAA, 423: 24.
[57]	McMullen M S.2003. The application of non-linear frequency domain methods to the Euler and Navier-Stokes equations.[PhD Thesis]. California: Stanford University.
[58]	Mcmullen M S, Jameson A.2006. The computational efficiency of non-linear frequency domain methods. Journal of Computational Physics, 212: 637-661.
[59]	Mcmullen M S, Jameson A, Alonso J J.2001. Acceleration of convergence to a periodic steady state in turbomachinery flows. AIAA Journal, 28: 100-152.
[60]	Mcmullen M, Jameson A, Alonso J J.2002. Application of a non-linear frerquency domain solver to the Euler and Navier-stokes equations//40th AIAA Aerospace Sciences Meeting & Exhibit.
[61]	Mcmullen M, Jameson A, Alonso J J.2006. Demonstration of nonlinear frequency domain methods. AIAA Journal, 44: 1428-1435.
[62]	Mundis N L, Mavriplis D J.2013a. GMRES applied to the time-spectral and quasi-periodic time-spectral methods. AIAA Paper, 3084: 24-27.
[63]	Mundis N L, Mavriplis D J.2013b. Quasi-periodic time spectral method for aeroelastic flutter analysis//AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2013-0638.
[64]	Mundis N L, Mavriplis D J.2014. An efficient flexible GMRES solver for the fully-coupled time-spectral aeroelastic system. AIAA Paper, 1427: 13-17.
[65]	Mundis N L, Mavriplis D J.2015a. Wave number independent preconditioning for GMRES time-spectral solvers//53rd AIAA Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, Kissimmee, Florida, 1-21.
[66]	Mundis N L, Mavriplis D J.2015b. Finite-element time discretizations for the unsteady Euler equations//53rd AIAA Aerospace Sciences Meeting, 0569.
[67]	Mundis N L, Mavriplis D J.2016. Toward an optimal solver for time-spectral solutions on unstructured meshes//54th AIAA Aerospace Sciences Meeting, 0069.
[68]	Mundis N L, Mavriplis D J, Sitaraman J.2013. Quasi-periodic time-spectral methods for flutter and gust response//69th Forum of the American Helicopter Society, AHS International, Alexandria.
[69]	Murman S M.2012. Reduced-frequency approach for calculating dynamic derivatives. AIAA Journal, 45: 2005-2840.
[70]	Murman S M, Aftosmis M J, Berger M J.2004. Numerical simulation of rolling airframes using a multilevel Cartesian method. Journal of Spacecraft and Rockets, 41: 426-435.
[71]	Nadarajah S, Jameson A.2007. Optimum shape design for unsteady three-dimensional viscous flows using a nonlinear frequency-domain method. Journal of Aircraft, 44: 1513-1527.
[72]	Ning W, He L.1997. Computation of unsteady flows around oscillating blades using linear and nonlinear harmonic Euler methods//ASME 1997 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, V004T14A039-V004T14A039.
[73]	Pechloff A N, Laschka B.2006. Small disturbance Navier-Stokes method: Efficient tool for predicting unsteady air loads. Journal of Aircraft, 43: 17-29.
[74]	Ronch A D, Ghoreyshi M, Badcock K J, Gortz S, Widhalm M, Dwight R P, Campobasso M S.2010. Linear frequency domain and harmonic balance predictions of dynamic derivatives//AIAA Applied Aerodynamics Conference.
[75]	Ronch A D, Mccracken A J, Badcock K J, Widhalm M, Campobasso M S.2013. Linear frequency domain and harmonic balance predictions of dynamic derivatives. Journal of Aircraft, 50: 694-707.
[76]	Rahmati M T, He L, Wang D X, Li Y S, Wells R G, Krishnababu S K.2014. Nonlinear time and frequency domain methods for multirow aeromechanical analysis. Journal of Turbomachinery, 136: 041010.
[77]	Su X, Yuan X.Implicit solution of time spectral method for periodic unsteady flows. Journal of Engineering Thermophysics, 63: 860-876.
[78]	Salles L, Blanc L, Gouskov A, Jean P.2014. Dual time stepping algorithms with the high order harmonic balance method for contact interfaces with fretting-wear. Journal of Engineering for Gas Turbines & Power, 134: 913-921.
[79]	Thomas J P, Dowell E H, Hall K C.2002. Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations. AIAA Journal, 40: 638-646.
[80]	Thomas J P, Dowell E H, Hall K C.2004. Modeling viscous transonic limit cycle oscillation behavior using a harmonic balance approach. Journal of Aircraft, 41: 1266-1274.
[81]	Thomas J P, Hall K C, Dowell E H.2005. Discrete adjoint approach for modeling unsteady aerodynamic design sensitivities. AIAA Journal, 43: 1931.
[82]	Van Der Weide E, Gopinath A, Jameson A.2005. Turbomachinery applications with the time spectral method. AIAA Paper, 4905: 2005.
[83]	Vilmin S, Lorrain E, Hirsch C, Swoboda M.2006. Unsteady flow modeling across the rotor/stator interface using the nonlinear harmonic method//ASME Turbo Expo 2006: Power for Land, Sea, and Air.
[84]	Widhalm M, Dwight R, Thormann R, Hubner A.2010. Efficient computation of dynamic stability data with a linearized frequency domain solver//Eccomas Cfd. DLR.
[85]	Woodgate M A, Barakos G N.2012. Implicit computational fluid dynamics methods for fast analysis of rotor flows. AIAA Journal, 50: 1217-1244.
[86]	Yang Z, Mavriplis D.2010. Time spectral method for periodic and quasi-periodic unsteady computations on unstructured meshes//40th AIAA Fluid Dynamics Conference, 28-1.
[87]	Yang Z, Mavriplis D, Sitaraman J. Yang Z, Mavriplis D, Sitaraman J.2011. Prediction of helicopter maneuver loads using BDF/time spectral method on unstructured meshes//49th AIAA Aerospace Sciences Meeting, 4-7.
[88]	Yi S, Im D, Choi S, Lee D.2015. An efficient fluid-structure interaction analysis based on time-spectral approaches//AIAA 53rd Aerospace Science Meeting, 5-9.
[89]	Zeiler T A.2000. Results of Theodorsen and Garrick revisited. Journal of Aircraft, 37: 918-920.