Gravity-capillary waves modulated by linear shear flow in arbitrary water depth

doi:10.1088/1674-1056/abb3e4

Abstract Considering that the fluid is inviscid and incompressible and the flow is irrotational in a fixed frame of reference and using the multiple scale analysis method, we derive a nonlinear Schrödinger equation (NLSE) describing the evolution dynamics of gravity-capillary wavetrains in arbitrary constant depth. The gravity-capillary waves (GCWs) are influenced by a linear shear flow (LSF) which consists of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of GCWs with the LSF is analyzed using the NLSE. The MI is effectively modified by the LSF. In infinite depth, there are four asymptotes which are the boundaries between MI and modulational stability (MS) in the instability diagram. In addition, the dimensionless free surface elevation as a function of time for different dimensionless water depth, surface tension, uniform flow and vorticity is exhibited. It is found that the decay of free surface elevation and the steepness of free surface amplitude change over time, which are greatly affected by the water depth, surface tension, uniform flow and vorticity.

Keywords: gravity-capillary waves nonlinear Schrödinger equation linear shear flow modulational instability

Received: 27 May 2020
Revised: 13 July 2020
Accepted manuscript online: 01 September 2020

PACS:	47.35.Fg	(Solitary waves)
	47.35.Pq	(Capillary waves)
	92.10.Hm	(Ocean waves and oscillations)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 41830533) and the National Key Research and Development Program of China (Grant Nos. 2016YFC1401404 and 2017YFA0604102).

Corresponding Authors: ^†Corresponding author. E-mail: songjb@zju.edu.cn ^‡Corresponding author. E-mail: caoanzhou@zju.edu.cn

Cite this article:

Shaofeng Li(李少峰), Jinbao Song(宋金宝), and Anzhou Cao(曹安州) Gravity-capillary waves modulated by linear shear flow in arbitrary water depth 2020 Chin. Phys. B 29 124702

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Gravity-capillary waves modulated by linear shear flow in arbitrary water depth

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