Waves: Kinematic stretching

Kinematic stretching is the process of extending linear Airy wave theory to provide predictions of fluid velocity and acceleration (kinematics) at points above the mean water level. It only applies to Airy waves and to random waves (which are made up of a number of Airy waves).

Linear wave theory applies, in principle, only to very small waves. It does not therefore predict kinematics for points above the mean water level, since they are not in the fluid, and the theory needs to be 'stretched' to cover such points. OrcaFlex offers a choice of methods: vertical, Wheeler, and extrapolation stretching methods.

Consider, for example, the horizontal particle velocity $u$. In Airy wave theory the formula for $u$ at position $(x,z)$ and time $t$ is \begin{equation} \label{Airy_u} u = E(z)\ a \ \omega\ \cos(\omega t - \phi - kx) \end{equation} where $a$, $\omega$, $\phi$ and $k$ are the wave amplitude, angular frequency, phase lag and wave number, respectively, $x$ is distance downstream from the wave origin and $z$ is measured positive upwards from the mean water level.

The term $E(z)$ is a scaling factor, given for Airy waves by \begin{equation} \label{Airy_Ez} E(z) = \frac{\cosh(k(d+z))}{\sinh(kd)} \end{equation} where $d$ is mean water depth. It is an exponential decay term that models the fact that the fluid velocity reduces with increasing depth. However for $z\!\gt\!0$, i.e. above the mean water level, $E(z)$ is greater than 1 so it amplifies the velocity in this region. This can give unrealistically large particle velocities, all the more so for higher frequency waves. The various stretching methods deal with this problem by replacing $E(z)$ with a more realistic expression.

Note that all the stretching methods apply not only to the scaling factor $E(z)$ in the horizontal velocity formula (\ref{Airy_u}), but also to the scaling factors in the corresponding Airy wave theory formulae for the vertical velocity and the horizontal and vertical acceleration.

Vertical stretching

This is the simplest of the available methods. Expression (\ref{Airy_Ez}) for $E(z)$ is left unchanged for $z\!\le\!0$, but for $z\!\gt\!0$, $E(z)$ takes the constant value $E(0)$. This has the effect of setting the kinematics above the mean water level to be identical to those at the mean water level.

Wheeler stretching

This method stretches (or compresses) the water column linearly into a height equivalent to the mean water depth. This is done by replacing $E(z)$ with $E(z')$, where \begin{equation} z' = d\ \frac{d+z}{d+\zeta} - d \end{equation} and $\zeta$ is the $z$-value at the instantaneous water surface. This formula for $z'$ essentially shifts $z$ linearly to be in the range $-\!d$ to 0.

Extrapolation stretching

This method extends $E(z)$ to points above the mean water level by linear extrapolation of the tangent to $E(z)$ at the mean water level. Expression (\ref{Airy_Ez}) for $E(z)$ is left unchanged for $z\!\le\!0$, but for $z\!\gt\!0$, $E(z)$ is replaced by $E(0) + z\ E'(0)$, where $E'$ is the rate of change of $E$ with $z$. Additionally, $z$ is capped by a maximum value of $\zeta$ when above the mean water level, meaning that it can never be above the instantaneous water surface.

Warning: This implementation of extrapolation stretching can lead to small discontinuities in fluid kinematics for objects that straddle the mean water level above the trough of a wave component. This can be mitigated by finer discretisation of the object in question, since fluid forces are applied in proportion to how much of the object is submerged.