WANG Jincheng, QI Jin, WU Chuijie. Analysis and Modelling of Optimal Dynamical Systems of Incompressible NavierStokes Equations[J]. Applied Mathematics and Mechanics, 2020, 41(1): 1-15. doi: 10.21656/1000-0887.400279
 Citation: WANG Jincheng, QI Jin, WU Chuijie. Analysis and Modelling of Optimal Dynamical Systems of Incompressible NavierStokes Equations[J]. Applied Mathematics and Mechanics, 2020, 41(1): 1-15.

# Analysis and Modelling of Optimal Dynamical Systems of Incompressible NavierStokes Equations

##### doi: 10.21656/1000-0887.400279
Funds:  The National Natural Science Foundation of China(11601033; 11372068); The National Basic Research Program of China(973 Program)(2014CB744104)
• Rev Recd Date: 2019-11-26
• Publish Date: 2020-01-01
• The modeling method for the optimal dynamical systems of NavierStokes equations satisfying both arbitrary velocity boundary conditions and velocity incompressible conditions was studied. Through the modeling and analysis of the optimal dynamical systems of the flow around the square column, it is found that the dynamics characteristics of the optimal dynamical systems are limit cycles. At the same time, the optimal dynamical system with only 3 optimal basis functions could well describe all the main flow field characteristics and the dynamics characteristics of the problem, so the proposed method is applicable to complex flow problems and their dynamics with minimal basis functions.
•  [1] POPE S B. Turbulent Flows [M]. Cambridge: Cambridge University Press, 2000. [2] HOLMES P, LUMLEY J L, BERKOOZ G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry [M]. Cambridge: Cambridge University Press, 1996. [3] 是勋刚. 湍流[M]. 天津: 天津大学出版社, 1994.(SHI Xungang. Turbulence [M] Tianjin: Tianjin University Press, 1994.(in Chinese)) [4] SHRIER H N. Nonlinear Hydrodynamic Modeling: a Mathematical Introduction [M]. Berlin: Springer-Verlag, 1987. [5] DANIELSON T J, OTTINO J M. Structural stability in two-dimensional model flows: Lagrangian and Eulerian turbulence[J]. Physics of Fluids A: Fluid Dynamics,1990,2(11): 2024-2035. [6] POJE C, LUMLEY J L. A model for large scale structures in turbulence shear flows[J]. Journal of Fluid Mechanics,1995,〖STHZ〗 285: 349-369. [7] BERKOOZ G, HOLMES P, LUMLEY J L. The proper orthogonal decomposition in analysis of turbulent flows[J]. Annual Review of Fluid Mechanics,1993,25(1): 539-575. [8] KALASHNIKOVA I, WAANDERS B V, ARUNAJATESAN S, et al. Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment[J]. Computer Methods in Applied Mechanics and Engineering,2014,272: 251-270. [9] LUO Z D, YANG X Z, ZHOU Y J. A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation[J]. Journal of Computational and Applied Mathematics,2009,229(1): 97-107. [10] ALONSO D, VELAZQUEZ A, VEGA J M. A method to generate computationally efficient reduced order models[J]. Computer Methods in Applied Mechanics and Engineering,2009,198(33/36): 2683-2691. [11] WANG X L, JIANG Y L. Model order reduction methods for coupled systems in the time domain using Laguerre polynomials[J]. Computers and Mathematics With Applications,2011,62(8): 3241-3250. [12] AMMAR A, MOKDAD B, CHINESTA F, et al. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[J]. Journal of Non-Newtonian Fluid Mechanics,2006,139(3): 153-176. [13] WU C J. Large optimal truncated low-dimensional dynamical systems[J]. Discrete and Continuous Dynamical Systems (Series A),1996,2(4): 559-583. [14] WU C J, SHI H S. An optimal theory for an expansion of flow quantities to capture the flow structures[J]. Fluid Dynamics Research,1995,17(2): 67-85. [15] 吴锤结, 赵红亮. 不依赖数据库的最优动力系统建模理论及其应用[J]. 力学学报, 2001,33(3): 289-300.(WU Chuijie, ZHAO Hongliang. A new database-free method of constructing optimal low-dimensional dynamical systems and its application[J] Acta Mechanica Sinica,2001,33(3): 289-300.(in Chinese)) [16] WU C J, WANG L. A method of constructing a database-free optimal dynamical system and a global optimal dynamical system[J]. Science in China(Series G): Physics, Mechanics & Astronomy,2008,51(7): 905-915. [17] PENG N F, GUAN H, WU C J. Research on the optimal dynamical systems of three-dimensional Navier-Stokes equations based on weighted residual[J]. Science China: Physics, Mechanics & Astronomy,2016,59(4): 644-701. [18] PENG N F, GUAN H, WU C J. Optimal dynamical systems of Navier-Stokes equations based on generalized helical-wave bases and the fundamental elements of turbulence[J]. Science China: Physics, Mechanics & Astronomy,2016,59(11): 114713. DOI: 10.1007/s11433-016-0247-3. [19] TORN A, ZILINSKAS A. Global Optimization [M]. Berlin: Springer-Verlag, 1989. [20] 叶庆凯, 王肇明. 优化与最优控制中的计算方法[M]. 北京: 科学出版社, 1986.(YE Qingkai, WANG Zhaoming. Computational Methods of Optimization and Optimum Control [M]. Beijing: Science Press, 1986.(in Chinese)) [21] 黄克智, 薛明德, 陆明万. 张量分析[M]. 北京: 清华大学出版社, 2003.(HUANG Kezhi, XUE Mingde, LU Mingwan. Tensor Analysis [M]. Beijing: Tsinghua University Press, 2003.(in Chinese)) [22] TORREY M D, MJOLSNESS R C, STEIN L R. NASA-VOF3D: a three-dimensional computer program for incompressible flows with free surfaces[R]. Nasa Sti/Recon Technical Report N, 1987. [23] SIROVICH L. Turbulence and the dynamics of coherent structures, part Ⅲ: dynamics and scaling[J]. Quarterly of Applied Mathematics,1987,45(3): 561-571. [24] CARBONE F, AUBRY N. Hierarchical order in wall-bounded shear turbulence[J]. Physics of Fluids,1996, 8(4): 1061-1075. [25] KAUFMAN E H, LEEMING D J, TAYLOR G D. An ODE-based approach to nonlinearly constrained minimax problems[J]. Numerical Algorithms,1995,9(1): 25-37.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142