Construction of Spatiotemporal-Coupling Optimal Low-Dimensional Dynamical Systems for Compressible Navier-Stokes Equations
doi: 10.21656/1000-0887.430220
可压缩Navier-Stokes方程的时空耦合优化低维动力系统建模方法
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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。
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Key words:
- 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
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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
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
Figure 7. Time evolution curves of coefficients of velocity basis
$ {\boldsymbol{\xi}} $ , density basis$ \zeta $ and temperature basis$ \eta $ 
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}} $ 
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
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
Figure 13. Time evolution curves of coefficients of velocity basis
$ {\boldsymbol{\xi}} $ , density basis$ \zeta $ and temperature basis$ \eta $ 
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}} $ 
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
Figure 17. Time evolution curves of coefficients of velocity basis
$ {\boldsymbol{\xi}} $ , density basis$ \zeta $ and temperature basis$ \eta $ -
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