- #1
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- TL;DR
- How do surface waves change in the presence of vorticity?
The classical free surface profile for the solitary wave for irrotational and incompressible fluids for small amplitude and long wavelength is the classical Korteweg-deVries(KdV) equation given by:\frac{\partial\eta}{\partial t}+\frac{\partial \eta}{\partial x}+\eta\frac{\partial\eta}{\partial x}+\frac{\partial^{3}\eta}{\partial x^{3}}=0
I'm interested in removing the irrrotational aspect of the initial assumptions. The vorticity equation in 2D can be written as:\frac{D\omega}{Dt}=0,\quad\omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}, ``Combining'' these two equations yields:\frac{D}{Dt}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=0
The full equation is:\frac{\partial}{\partial t}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+u\frac{\partial}{\partial x}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+v\frac{\partial}{\partial y}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=0
From here I can do the usual approximations to get a similar equation to the KdV? Where am I going wrong here?
I'm interested in removing the irrrotational aspect of the initial assumptions. The vorticity equation in 2D can be written as:\frac{D\omega}{Dt}=0,\quad\omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}, ``Combining'' these two equations yields:\frac{D}{Dt}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=0
The full equation is:\frac{\partial}{\partial t}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+u\frac{\partial}{\partial x}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+v\frac{\partial}{\partial y}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=0
From here I can do the usual approximations to get a similar equation to the KdV? Where am I going wrong here?