厄塞尔数
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厄塞尔数(Ursell number)是流体动力学中的无量纲,表示流体层中长的表面重力波的非线性程度,得名自1953年发现此重要性的弗里茨·厄塞尔[1]。
厄塞尔数是推导自史托克波,一个针对非线性周期波的摄动序列,在浅水的长波极限-其波长远大于水深时,Ursell数U可以定义如下:
若不考虑常数3 / (32 π2)的话,上述公式就是自由表面提升振幅中,二次项和一次项的比例,[2] 有用到的参数有
因此厄塞尔数U是相对波高H / h乘以相对波长的平方。
针对厄塞尔数小(U ≪ 32 π2 / 3 ≈ 100)的长波(λ ≫ h)[3],可以用线性的波理论求解。否则(多半是通常)若针对比较长的波(λ > 7 h)[4],需使用像KdV方程或博欣内斯克方程等非线性的理论。此参数(经过不同的正规化)已由乔治·斯托克斯写在他1847年的表面重力波论文中[5]。
脚注
- ^ Ursell, F. The long-wave paradox in the theory of gravity waves. Proceedings of the Cambridge Philosophical Society. 1953, 49 (4): 685–694. Bibcode:1953PCPS...49..685U. doi:10.1017/S0305004100028887.
- ^ Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
- ^ This factor is due to the neglected constant in the amplitude ratio of the second-order to first-order terms in the Stokes' wave expansion. See Dingemans (1997), p. 179 & 182.
- ^ Dingemans (1997), Part 2, pp. 473 & 516.
- ^ Stokes, G. G. On the theory of oscillatory waves. Transactions of the Cambridge Philosophical Society. 1847, 8: 441–455.
Reprinted in: Stokes, G. G. Mathematical and Physical Papers, Volume I. Cambridge University Press. 1880: 197–229.
参考资料
- Dingemans, M. W. Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering 13. Singapore: World Scientific. 1997. ISBN 981-02-0427-2. In 2 parts, 967 pages.
- Svendsen, I. A. Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering 24. Singapore: World Scientific. 2006. ISBN 981-256-142-0. 722 pages.
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