非旋转性自由表面重力驻波的Euler与Lagrange方法解间的转换

The transformation between the Eulerian and Lagrangian solutions for irrotational standing gravity waves

对于等深水中的非旋转性重力驻波流场,本文用Euler与Lagrange两种方法求得其至三阶的解,根据同一粒流体质点在相同时间与位置处其流速值为唯一与质量守恒及在自由表面水位的Euler形式解与Lagrange形式解相同等特性,来推导其间互可转换.由一系列连续的Taylor级数展开,在考虑波动场中各流体质点的运动轨迹与运动周期条件下,将已知的Euler解转换成完全未知的Lagrange形式解.接着再将所得的Lagrange解转换成对应的Euler形式,均可得到完全相同的结果.由此可得知,在考虑波动场各流体质点运动的真实性下,对于非旋转性的重力驻波流场而言,此两种描述方法所得之解是互通可转换的.并证实了流场中所有流体质点的运动周期是完全相同的,且恰等于自由表面波形的脉动周期.

This study reports the transformations between the third-order Eulerian and Lagrangian solutions for the standing gravity waves on water of uniform depth. Regarding the motion of a marked fluid particle, the instantaneous velocity, the mass conservation and the free surface must be the same for either Eulerian or Lagrangian methods. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, we obtain the third-order asymptotic expressions for the Lagrangian particle trajectories and the period of particle motion directly in Lagrangian form. In particular, the Lagrangian mean level which differs from that in the Eulerian approach is also found as a part of the solutions. Therefore, the given Eulerian solutions can be transformed into the completely unknown Lagrangian solutions and the reversible process is also identified.