Computation of force in curvilinear co-ordinates

FORCE is the wave-driven stress, i.e. the force per unit surface driving the wave-driven current, expressed in N/m$^2$, is defined as the derivative of the radiation stresses

  $$S_{xx} = \rho g \int \lfloor n \cos^2\theta + n - \frac{1}{2} \rfloor E d\sigma d\theta$$ (3.102)

  $$S_{xy} = S_{yx} = \rho g \int n \sin\theta \cos\theta E d\sigma d\theta$$ (3.103)

  $$S_{yy} = \rho g \int \lfloor n \sin^2\theta + n - \frac{1}{2} \rfloor E d\sigma d\theta$$ (3.104)

Here, $n$ is the ratio of group velocity and phase velocity, that is,

  $$n = \frac{c_g k}{\omega}$$ (3.105)

The force is then

  $$F_x = -\frac{\partial S_{xx}}{\partial x} - \frac{\partial S_{xy}}{\partial y}$$ (3.106)

and

  $$F_y = -\frac{\partial S_{yx}}{\partial x} - \frac{\partial S_{yy}}{\partial y}$$ (3.107)

In order to compute the force, the derivative of the radiation stress tensor has to be taken. Let $f$ be one of the components of the tensor. We have to derive expressions for $\partial f/\partial x$ and $\partial f/\partial y$. Derivatives with respect to the computational grid co-ordinates $\xi$ and $\eta$ can easily be found. The transformation is based on

  $$\frac{\partial f}{\partial \xi} = \frac{\partial f}{\partial x}\f... ...{\partial \xi} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial \xi}$$ (3.108)

and

  $$\frac{\partial f}{\partial \eta} = \frac{\partial f}{\partial x}\... ...partial \eta} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial \eta}$$ (3.109)

Hence,

  $$\frac{\partial f}{\partial x} = \frac{\frac{\partial f}{\partial\... ...\partial x} + \frac{\partial f}{\partial \eta}\frac{\partial \eta}{\partial x}$$ (3.110)

and

  $$\frac{\partial f}{\partial y} = \frac{\frac{\partial f}{\partial\... ...\partial y} + \frac{\partial f}{\partial \eta}\frac{\partial \eta}{\partial y}$$ (3.111)

Numerical approximations are quite simple:

  $$\frac{\partial f}{\partial \xi} \approx \frac{f_{\xi+1,\eta} - f_... ...c{\partial f}{\partial \eta} \approx \frac{f_{\xi,\eta+1} - f_{\xi,\eta-1}}{2}$$ (3.112)

These expressions are also used for derivatives of $x$ and $y$. On the boundaries of the computational region a one-sided approximation can be used.