CMS-Wave:Wave Radiation Stresses

Wave Radiation Stresses

The wave radiation stresses $(S_{ij})$ are calculated using linear wave theory as (Longuet-Higgins and Stewart 1961; Dean and Dalrymple 1984)

$S_{ij}=\int \int E_{w}(f,\theta )\left[n_{g}\omega _{i}\omega _{j}+\delta _{ij}\left(n_{g}-{\frac {1}{2}}\right)\right]dfd\theta$

(1)

where:

f = the wave frequency [1/s]

$\theta$ = the wave direction [rad]

$E_{w}$ = wave energy = $1/16\ \rho gH_{s}^{2}$ [N/m]

$H_{s}$ = significant wave height [m]

$w_{i}$ = wave unit vector = $(cos\ \theta ,sin\ \theta )$[-]

$\delta _{ij}=\left\{{\begin{aligned}&1\ for\ i=j\\&0\ for\ i\neq j\end{aligned}}\right.$

$n_{g}={\frac {c_{g}}{c}}={\frac {1}{2}}\left(1+{\frac {2kh}{sinh\ 2kn}}\right)[-]$

$c_{g}$ = wave group velocity [m/s]

$c$ = wave celerity [m/s]

$k$ = wave number [rad/s]

The wave radiation stresses and their gradients are computed within the wave model and interpolated in space and time in the flow model.

References

• Dean, R. G., and R. A. Dalrymple. 1984. Water wave mechanics for engineers and scientists. Englewood Cliffs, NJ: Prentice-Hall.
• Longuet-Higgins, M. S., and R. W. Stewart. 1961. The changes in amplitude of short gravity waves on steady non-uniform currents. Journal of Fluid Mechanics 10(4):529–549.

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