Hydrostatic force

Three levels of approximation can be defined for the hydrostatic restoring force:

  • a linear model;
  • a weakly nonlinear model;
  • a fully nonlinear model.

Linear model

The generalized hydrostatic static force expression for the linear approximation is given by:

\[\mathbf{f}_H(t) = -\mathbf{K}_h \mathbf{x}(t)\]

where

  • \(\mathbf{x}\) is the generalized position vector, with respect to the equilibrium frame;
  • \(\mathbf{K}_h\) is the hydrostatic stiffness matrix.

The stiffness coefficients for the horizontal degrees of freedom (surge, sway and yaw) are null. The coefficients for the heave, roll and pitch degrees of freedom are specified with respect to the equilibrium frame.

Note

The hydrostatic stiffness matrix is an input of the numerical model, considered constant during the simulation.

Weakly nonlinear model

On contrary to the linear model, the exact position of the body is now considered. However, the incident wave is still supposed to be of small curvature; the free surface can then be represented by the plane \(z = 0\).

As a consequence to these two approximations, the hydrostatic load is now computed by integrating the hydrostatic pressure over the wetted body surface \(S_0\), defined by the exact position of the body mesh, clipped by the plane \(z = 0\):

\[\mathbf{f}_H(t)= \iint_{S_0} \rho gz \mathbf{n} dS\]

where

  • \(\rho\) is the water density;
  • \(g\) denotes the gravity constant;
  • \(\mathbf{n}\) is normal vector, pointing outward the body surface and inward the fluid.

Note

The hydrostatic force is computed at the center of buoyancy and then transported to the center of gravity, to yield the hydrostatic torque.

Fully nonlinear model

The assumption of small incident wave curvature is no longer considered; the free surface wave elevation is then \(z = \eta_I\). The wetted body surface is now the exact wetted body surface, \(S_I\), defined by the exact position of the body mesh, and delimited by the incident wave field. The expression of the hydrostatic force is:

\[\mathbf{f}_H(t) = \iint_{S_I} \rho gz \mathbf{n} dS\]

And for the hydrostatic torque:

\[\mathbf{f}_H(t) = \iint_{S_I} \rho gz (\mathbf{x}\times\mathbf{n}) dS\]

Warning

In this approach, the hydrostatic loads do not match the weight of the water column as there is an incident wave. Neverthess, by definition, the hydrostatic loads are the integration of the term \(-\rho gz\) over the wetted body surface. If the integration of the weight of the water column is considered, that is to say \(-\rho g(z - \eta^I)\), then it is necessary to update the expression of the Froude-Krylov loads accordingly. Basically, the nonlinear hydrostatic loads should always be applied with the nonlinear Froude-Krylov loads to be used with consistency.