Wave Drift Forces

Nonlinear Effects

Some of the most important nonlinear effects arising in connection with wave-body interactions are

• Drift forces. They are the mean forces exerted on floating or submerged bodies by ambient waves may be treated very well by perturbation theory

• Slamming. These are highly nonlinear effects arising when a ship section acts upon the water surface or when a steep or breaking wave impinges upon a floating structure. May be modeled by fully or partially nonlinear potential flow models of analytical or numerical nature

• Forces due to viscous flow separation around floating structures and their subsystems, e.g. risers, mooring lines etc. Vortex induced vibrations (VIV) is an important example. Such effects can be treated experimentally and computationally by solving the Navier-Stokes equations.

• Nonlinear ship motions in steep waves. These effects are mostly of potential flow nature and are being treated by nonlinear Rankine panel methods. The primary nonlinearity is the variable wetness of the ship hull, the nonlinearity of the kinematics of ambient waves and the numerical solution of the equations of motion in the time domain.

• Nonlinear responses of deep water offshore platforms in certain flexural modes of their tethers. These effects are known as springing & ringing and are treated by a combination of perturbation and nonlinear methods and experiments.

Drift Forces

• Drift forces will first be considered in regular waves. The main results will then be extended in random waves

• One very important property of drift forces other than their practical significance is that they depend only on the linear solution.

Mean Drift Force on a Vertical Wall

[math]\displaystyle{ \phi_I = \mathbb{R}\mathbf{e} \left\{ \frac{i g A}{\omega} e^{KZ-iKX+i\omega t} \right\} \, }[/math] [math]\displaystyle{ \phi_D = \mathbb{R}\mathbf{e} \left\{ \frac{i g A}{\omega} e^{KZ+iKX+i\omega t} \right\} \, }[/math] [math]\displaystyle{ \phi = \phi_I + \phi_D; \quad \phi_X = 0, X = 0 \, }[/math]

• Nonlinear hydrodynamic pressure:

[math]\displaystyle{ P = - \rho \left( \frac{\partial\phi}{\partial t} + \frac{1}{2} \nabla\phi \cdot \nabla\phi + g Z \right) }[/math]

• Nonlinear horizontal force on the wall:

[math]\displaystyle{ F_X = \int_{-\infty}^\zeta P dZ, \quad \zeta = - \frac{1}{g} \frac{\partial\phi}{\partial t} \int_{Z=0} + O \left( Z^2 \right) }[/math]

• We need to evaluate [math]\displaystyle{ F_X\, }[/math] correct to [math]\displaystyle{ O \left( A^2 \right) \, }[/math]. Noting that the mean time value of effects of [math]\displaystyle{ O(A) \, }[/math] which are linear, is zero.

Assume a perturbation expansion for [math]\displaystyle{ \phi\, }[/math]:

[math]\displaystyle{ \phi = \phi_1 + \phi_2 + \phi_3 + \cdots \, }[/math]

where the potential derived earlier is the linear term denoted above by [math]\displaystyle{ \phi_1 \, }[/math].

• In evaluating the leading order effect in the mean horizontal force we drop terms with zero mean values or of order [math]\displaystyle{ A^3\, }[/math] and higher. We note without proof that

[math]\displaystyle{ {\overline{\phi_2(t)}}^t = 0 \, }[/math]

The total horizontal force [math]\displaystyle{ F_X \, }[/math] also accepts the expansion:

[math]\displaystyle{ F_X = F_1 + F_2 + F_3 + \cdots \, }[/math] [math]\displaystyle{ F_1 = \int_{-\infty}^0 P_1 dZ = - \rho \int_{-\infty}^0 \frac{\partial\phi}{\partial t} dZ = O (A) }[/math] [math]\displaystyle{ {\overline{F_1(t)}}^t = 0 \, }[/math] [math]\displaystyle{ F_2 = \int_{-\infty}^0 P_2 dZ + \int_0^\zeta P_1 dZ = O \left(A^2\right) }[/math] [math]\displaystyle{ = - \rho \int_{-\infty}^0 \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 \right) dZ }[/math] [math]\displaystyle{ - \rho \int_0^\zeta \left( \frac{\partial\phi_1}{\partial t} + g Z \right) dZ + O \left( A^3 \right) \, }[/math]

with errors of [math]\displaystyle{ O \left( A^3 \right) \, }[/math], the last integral may be approximated by Taylor expanding about [math]\displaystyle{ Z=0\, }[/math] the first term and by direct integration of the second. It follows that:

[math]\displaystyle{ \int_0^\zeta \left( \frac{\partial\phi_1}{\partial t} + g Z \right) dZ = \zeta \left. \frac{\partial\phi_1}{\partial t} \right|_{Z=0} + \frac{1}{2} g \zeta^2 + O \left( A^3 \right) }[/math] [math]\displaystyle{ = - g \zeta^2 + \frac{1}{2} g \zeta^2 = - \frac{1}{2} g \zeta^2 \ , }[/math]

Collecting terms:

[math]\displaystyle{ F_2(t) = - \rho \int_{-\infty}^0 \frac{\partial\phi_2}{\partial t} dZ - \frac{1}{2} \rho \int_{-\infty}^0 \nabla\phi_1 \cdot \nabla\phi_1 dZ + \frac{1}{2} \rho g \zeta^2(t) + O \left( A^3 \right) }[/math]

The mean time value of [math]\displaystyle{ {\overline{\frac{\partial\phi_2}{\partial t}}}^t = 0 \, }[/math] for any stationary signal [math]\displaystyle{ \phi_2(t)\, }[/math]. So the second-order potential does not contribute to the mean drift force as stated above!

• Of the remaining two terms the quadratic Bernoulli term contributes a suction force which is "pulling" the wall into the wave (Counterintuitive but true!) while the last term is always positive pushing the wall in the direction of the wave as expected.

• It follows that all of the mean drift force arises from the pressure integration over the surf-zone which is more than enough to overcome the suction force.

Upon substitution of the linear velocity potential derived above and use of the familiar identity:

[math]\displaystyle{ \overline{\mathbb{R}\mathbf{e} \left(A_1 e^{i\omega t} \right) \mathbb{R}\mathbf{e} \left(A_2 e^{i\omega t} \right)} = \frac{1}{2} \mathbb{R}\mathbf{e} \left\{ A, A_2^* \right\} }[/math]

It is easy to verify:

[math]\displaystyle{ {\overline{F_X}}^t = \frac{1}{2} \rho g A^2 + O \left( A^3 \right) \, }[/math]

So the mean horizontal force on a vertical wall of infinite draft by a plane progressive wave has a finite mean value [math]\displaystyle{ \sim A^2 \, }[/math].

If the plane regular wave is incident at an angle the mean horizontal force may be shown to take the value:

[math]\displaystyle{ \overline{F_X} = \frac{1}{2} \rho g A^2 \cos^2\beta \, }[/math]

In spite of their simplicity the above results have a number of useful applications in practice when waves that are sufficiently short interact with floating structures.

Some examples follow:

1. [math]\displaystyle{ \frac{\lambda}{2} \lt T \Rightarrow \lambda \lt \frac{T}{2} \, }[/math]: Ship with wall sided geometry acts like a wall

• For a barge like ship with length [math]\displaystyle{ L\, }[/math] the mean sway drift force in reasonably short waves is approximately given by:

[math]\displaystyle{ \overline{F_Y} \simeq \frac{1}{2} \rho g A^2 L \, }[/math]

1. Extend the above result when the ship hull section has flare with slope [math]\displaystyle{ \alpha\, }[/math].
2. * Short waves incident upon a ship at an angle are locally reflected as if they encounter a continuum of vertical walls inclined at varying angles. We can thus apply the result derived above.

• Only part of the ship waterline will encounter waves. This region is boldfaced above and can be determined by a simple geometrical argument. Denote this portion of the ship waterline by [math]\displaystyle{ \Gamma\, }[/math]. It is easy to show that the mean drift force in the [math]\displaystyle{ (x,y)\, }[/math] directions is given by

[math]\displaystyle{ \vec{F} = \left( \begin{matrix} \overline{F_x} \\ \overline{F_y} \end{matrix} \right) = \frac{1}{2} \rho g A^2 \int_{\Gamma} \left| \vec{n} \cdot \left( \begin{matrix} \cos\beta \\ \sin\beta \end{matrix} \right) \right|^2 \vec{n} dl }[/math]

This result is known as Ray Theorem.