<span style="white-space: normal;">Construction of Spatiotemporal-Coupling Optimal Low-Dimensional Dynamical Systems for Compressible Navier-Stokes Equations</span>

齐进; 吴锤结

doi:10.21656/1000-0887.430220

Construction of Spatiotemporal-Coupling Optimal Low-Dimensional Dynamical Systems for Compressible Navier-Stokes Equations

doi: 10.21656/1000-0887.430220

QI Jin^1,,
WU Chuijie^2, ,

1.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R.China
2.
School of Aeronautics and Astronautics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

详细信息

中图分类号: O35
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出版历程
- 收稿日期: 2022-07-01
- 修回日期: 2022-09-30
- 网络出版日期: 2022-10-20
- 刊出日期: 2022-10-31

可压缩Navier-Stokes方程的时空耦合优化低维动力系统建模方法

齐进^1
,,
吴锤结^{2
, ,}

1.
北京应用物理与计算数学研究所，北京 100088
2.
大连理工大学航空航天学院，辽宁大连 116024

摘要

摘要:
For the low-dimensional dynamical system model to study dynamics properties of Navier-Stokes equations, it is very important that the attraction domain of the low-dimensional model is the same as that of Navier-Stokes equations. However, to date, there is no universal approach to ensure this purpose for general problems. Herein, it is found that any low-dimensional model based on spatial bases, such as proper orthogonal decomposition bases, optimal spatial bases, and other classical spatial bases, is not predictable, i.e., the error increases with the time evolution of the flow field. With the theoretical framework for building optimal dynamical systems and the new concept of spatiotemporal-coupling spectrum expansion, the low-dimensional model for compressible Navier-Stokes equations was constructed to approximate the numerical solution to large-eddy simulation equations, and the numerical results and novel time evolution of spatiotemporal-coupling bases were given. The entire field error is typically below 10⁻²%, and the average error at each grid point is below 10⁻⁸%. The spatiotemporal-coupling optimal low-dimensional dynamical systems can ensure that the attraction domain of the low-dimensional model is the same as that of Navier-Stokes equations. Therefore, characteristic dynamics properties of spatiotemporal-coupling optimal low-dimensional dynamical systems are the same as those of real flow.
Abstract:
当采用低维动力系统模型研究Navier-Stokes方程的动力学性质时，保持低维模型的吸引域与Navier-Stokes方程的吸引域相同是非常重要的。然而，到目前为止，还没有一种普适的方法能确保对于一般问题都能达到这一目的。该文发现任何基于空间基的低维模型，如本征正交分解基、优化空间基和其他经典空间基，都不具有可预测性，即低维动力系统的误差随着流场的时间演化而增大。在构造优化动力系统的理论框架和时空耦合谱展开的新概念下，该文构造了可压缩Navier-Stokes方程的低维模型来逼近大涡模拟方程的数值解，给出了高精度的流场数值模拟结果和全新的时空耦合基时空演化数值结果。全场误差在10⁻²%以下，而每个网格点的平均误差在10⁻⁸%以下。时空耦合化化低维动力系统可以保证低维模型的吸引域与Navier-Stokes方程的吸引域相同。因此，保证了时空耦合优化低维动力系统的特征动力学性质与真实流场的特征动力学性质是一致的。
- 时空耦合谱展开 /
- 时空耦合优化低维动力系统 /
- 可压缩Navier-Stokes方程 /
- 大涡模拟 /
- 大涡模拟

HTML全文

Figure 1. The basic idea of the OLDDS theory

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Figure 2. Schematic diagram of the double-scale global optimization method

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Figure 3. The 3D sketch map of the compressible laminar backward-facing step flow

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Figure 4. Comparison of the evolution curves of the velocity errors with time in the whole field of low-dimensional dynamical system models constructed by POD bases, POD bases with a time interval, spatial optimal bases, SCOBs with random initial bases or POD initial bases

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Figure 5. Time evolution curves of errors of velocity, density and temperature

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Figure 6. Comparison of time evolution of flow fields of CFD and OLDDS, from left to right, $ t = 100\;{\rm{s}},300\;{\rm{s}},5\;000\;{\rm{s}},9\;990\;{\rm{s}} $: (a) velocity field u; (b) velocity field v; (c) velocity field w; (d) density field ρ; (e) temperature field T

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Figure 7. Time evolution curves of coefficients of velocity basis $ {\boldsymbol{\xi}} $, density basis $ \zeta $ and temperature basis $ \eta $

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Figure 8. Time evolution of spatiotemporal-coupling optimal velocity basis $ {{\boldsymbol{\xi}}} $, density basis $ \zeta $ and temperature basis $ \eta $: from left to right, $ t = 100\;{\rm{s}},5\;000\;{\rm{s}},9\;990\;{\rm{s}} $

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Figure 9. The 3D sketch map of the compressible turbulent straight jet flow

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Figure 10. Time evolution curves of errors of velocity, density and temperature

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Figure 11. Comparison of time evolution of turbulent flow fields between the LES and the SCOLDDS, from left to right, $ t = 100\;{\rm{s}},5\;000\;{\rm{s}},16\;670\;{\rm{s}} $: (a) velocity field u; (b) velocity field v; (c) velocity field w; (d) density field ρ; (e) temperature field T

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Figure 12. Time evolution curves of the proportion of each order of approximate flow variables: (a) the 1st-order; (b) the 2nd-order; (c) the 3rd-order

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Figure 13. Time evolution curves of coefficients of velocity basis $ {\boldsymbol{\xi}} $, density basis $ \zeta $ and temperature basis $ \eta $

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Figure 14. Time evolution of spatiotemporal-coupling optimal velocity basis $ {{\boldsymbol{\xi}}} $, density basis $ \zeta $ and temperature basis $ \eta $: from left to right, $ t = 100\;{\rm{s}},5\;000\;{\rm{s}},16\;670\;{\rm{s}} $

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Figure 15. Time evolution curves of errors of velocity, density and temperature

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Figure 16. Time evolution curves of the proportion of each order of approximate flow variables: (a) the 1st order; (b) the 2nd order; (c) the 3rd order; (d) the 4th order; (e) the 5th order

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Figure 17. Time evolution curves of coefficients of velocity basis $ {\boldsymbol{\xi}} $, density basis $ \zeta $ and temperature basis $ \eta $

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Figure 18. Time evolution of spatiotemporal-coupling optimal velocity basis $ {{\boldsymbol{\xi}}} $, density basis $ \zeta $ and temperature basis $ \eta $: from left to right, $ t = 100\;{\rm{s}},3\;000\;{\rm{s}},6\;000\;{\rm{s}} $

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