Kinematic stretching

The linear wave theory is based on the linearization of the free surface condition on the mean water level \(z=0\). In case of irregular wave profile, the superposition of regular wave components can lead to unrealistic velocity estimation particularly in the crest. Stretching method are applied on the scaling factor \(E(z)\) to overcome this issue.

\[E(z) = \frac{\cosh(k(z+H))}{\sinh(kH)}\]

The different methods are presented in the following.

Vertical stretching:

The vertical stretching method assumes that the scaling factor above the mean water level is constant and equals to its value at the mean water level \(E(0)\).

\[E(z) = \Biggl \lbrace { E(0) \text{ if } z \geq 0 \atop E(z) \text{ otherwise} }\]

This method results in an abrupt, non-physical, \(E(z)\) evolution when crossing the mean water level.

Extrapolation stretching:

To correct this non-physical evolution, the extrapolation stretching method introduces a correction to the vertical method.

\[E(z) = \Biggl \lbrace { E(0) + z \frac{\partial E}{\partial z}(0) \text{ if } z \geq 0 \atop E(z) \text{ otherwise} }\]

Wheeler stretching:

This method stretches the vertical scale factor, in order to obtain at \(z=\eta\), the kinematic given by the linear theory at \(z=0\). The corresponding transformation changes \(z\) into \(z'\) :

\[z' = H \frac{z-\eta}{H+\eta}\]

where \(H\) is the water depth and \(\eta\) is the instantaneous wave elevation given by the linear wave model.

The wheeler stretching is defined for : \(H < z < \eta\)

Chakrabarti stretching:

In this method, proposed by Chakrabarti [Chakrabarti1971], the water depth \(h\) in the denominator is corrected by the wave elevation \(\eta\) :

\[E(z) = \frac{\cosh(k(z+h))}{\sinh(k(h+\eta))}\]

Delta-stretching:

This method is a coupling between the Wheeler and the extrapolation stretching, proposed by Rodenbush and Forristall [Rodenbush1986]. While \(E(z)\) is still replaced by \(E(z')\), the transformation \(z \longrightarrow z'\) becomes

  • for \(z > -H_{\Delta}\) : \(z' = (z + H_{\Delta}) \frac{H_{\Delta} + \Delta.eta}{H_{\Delta} + \eta} - H_{\Delta}\)
  • for \(z < -H_{\Delta}\) : \(z' = z\)

\(\Delta\) is a parameter taken between 0 and 1, and \(H_{\Delta}\) is the water height on which the stretching is applied.

The Delta-stretching method can be downgraded to one of the two methods, in using specific values for \(H_{\Delta}\) and \(\Delta\):

  • for \(H_{\Delta} = H\) and \(\Delta = 0\), the method is equivalent to the Wheeler stretching.
  • for \(H_{\Delta} = H\) and \(\Delta = 1\), it is equivalent to the extrapolation stretching.

Rodenbush and Forristall recommend using \(\Delta = 0.3\) and \(H_{\Delta} = H_S/2\). However Molin [Molin2002] suggests that \(H_{\Delta} = H\), the water depth, is usually chosen.

References:

[Chakrabarti1971]Chakrabarti, SK, Discussion on “dynamics of single point mooring in deep water”. J Waterwayss, Harbour and Coastal Eng Div ASCE, Vol 97, 588-590, 1971
[Rodenbush1986]Rodenbusch, G, Forristall, GZ, An empirical model for random wave kinematics near the free surface Proc Offshore Technology Conf, Paper 5098, 1986
[Molin2002]Molin, B., Hydrodynamique des Structures Offshore, Editions Technip, 2002