基于Helmholtz-Hodge分解的纳维-斯托克斯方程的一种改进
On a reformulation of Navier-Stokes equations based on Helmholtz-Hodge decomposition
摘要：
新的纳维-斯托克斯方程公式的建议是基于Helmholtz-Hodge分解，其中对应于物理现象的所有项被写为一个无发散项和另一个无卷曲项的和。这些变换是建立在离散力学的基础上的，离散力学是连续介质力学的一种替代方法，其中一个部分上的加速度守恒取代了一个体积上的动量守恒。这样，运动方程就变成了单位体积总机械能守恒定律，而质量守恒不再是一个附加定律。新的纳维-斯托克斯方程公式在不改变其初始形式的情况下恢复了离散方法的性质;经典形式的解也是所提出的公式的解。将由Helmholtz-Hodge分解得到的惯性项写成两个分量，当微分算子直接应用于运动方程时，给出了新的性质。
英文摘要：
The proposal for a new formulation of the Navier-Stokes equations is based on a Helmholtz-Hodge decomposition where all the terms corresponding to the physical phenomena are written as the sum of a divergence-free term and another curl-free term. These transformations are founded on the bases of discrete mechanics, an alternative approach to the mechanics of continuous media, where conservation of the acceleration on a segment replaces that of the momentum on a volume. The equation of motion thus becomes a law of conservation of total mechanical energy per volume unit where the conservation of mass is no longer necessarily an additional law. The new formulation of the Navier-Stokes equations recovers the properties of the discrete approach without altering those of its initial form; the solutions of the classical form are also those of the proposed formulation. Writing inertial terms in two components resulting from the Helmholtz-Hodge decomposition gives the equation of motion new properties when differential operators are applied to it directly.
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