不可压缩流体运动能量耗散方法之研究

The understanding of incompressible fluid motion bridges mathematics and applied sciences such as engineering and physical disciplines. The research of the subject greatly enriches the development of mathematical theory of partial differential equations. In return, the mathematical theory is applied to solve practical engineering and physical problems through the study of fluid motions. The initial motivation for the present project stems from the applicant’s examination on the bifurcation analysis of the fluid motion in an electro-magnetic field under the influence of free surface energy dissipation, which actually is a good source stabilizing fluid motion. We recently thus thought of investigation of fluid motions involving energy dissipation, which is utilized to smoothen flows, to remove singularities and to create compactness. The present project is a continuation of the so called energy dissipation method applied to the Green function method in hydrodynamics and atmospheric blocking problem defined by an atmospheric circulation equation. Free surface Green function is a fundamental element of hydrodynamics, but the evaluation of free surface Green function is still arduous due to the presence of a singular wave integral. However, the singular wave integral is due to irrotational flow assumption, which is artificial in engineering practice. The singularity can be removed by adding an ingredient of the free surface energy dissipation. Therefore Green function can be evaluated in a simpler manner and more accurate hydrodynamic results can be obtained. On the other hand, the energy dissipation technique in the analysis of atmospheric circulation equation gives rise to the compactness which is necessary in equilibrium bifurcation and Hopf bifurcation for the understanding of atmospheric blocking. It should be mentioned that the atmospheric circulation equation has very strong nonlinearity, and thus the traditional Navier-Stokes equation theory is not applicable to this problem.

不可压缩流体运动问题作为数学和工程物理间的桥梁, 其研究极大地丰富和发展了偏微分方程等理论。反过来, 这些理论通过不可压缩流认识和解决了不断出现的工程物理问题。本项目源于申请人有关流体在电磁场作用下自由面能量耗散所产生的流体分叉问题的研究。由此联想到建立称之为能量耗散的方法，并用此来进一步完善水动力学的自由面Green函数方法和深入理解大气环流方程的大气阻塞问题。自由面Green函数方法是水动力学的经典方法, 但由于其积分奇性而在数值模拟时困难重重，我们则从能量耗散角度认识Green函数而消除了积分奇性的约束，从而得到简便且富有物理意义的逼近形式, 并相对一些流体物体相互作用问题给出更精确的数值模拟。而建立于大气环流方程的能量耗散方法使得我们能得到平衡解和周期轨道分叉所需要的紧性。由于方程有很强的非线性性, 这种紧性在经典的Navier-Stokes方程理论下是无法得到的。

不可压缩流体运动问题作为数学和工程物理间的桥梁, 其研究极大地丰富和发展了偏微分方程等理论。反过来, 这些理论通过不可压缩流认识和解决了不断出现的工程物理问题。本项目源于主持人有关流体在电磁场作用下自由面能量耗散所产生的流体分叉问题的研究。由此联想到建立称之为能量耗散的方法，并用此来进一步完善水动力学的自由面Green函数方法，研究二维不可压缩流的稳定性和失稳性态，深入理解大气环流方程的大气阻塞问题以及二维湍流的。自由面Green函数方法是水动力学的经典方法, 但由于其积分奇性而在数值模拟时困难重重，我们则从能量耗散角度认识Green函数而消除了积分奇性的约束，从而得到简便且富有物理意义的逼近形式, 并相对一些流体物体相互作用问题给出更精确的数值模拟。而建立于二维不可压缩流体运动方程特别是大气方程和电磁场相互作用下流体运动方程的能量耗散方法使得我们能得到平衡解和周期轨道分叉所需要的紧性。由于方程有很强的非线性性, 这种紧性在经典的Navier-Stokes方程理论下是无法得到的。