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Hydrodynamic Pressures

The objective is to derived the dynamic pressures time series on a floater's hull once its motions are known.

To do so, the instant pressure is divided into several components:

  • The static pressure ( \(P_{stat}\) ) : Fluid pressure on the hull in still water which corresponds to the buoyancy of the floater after the static equilibirum,

  • The hydrostatic pressure (\(P_{hys}\)) : "Hydrostatic" pressure variation due to the floater's motions computed on the instant immersed volume,

  • The Froude-Krylov pressures ( \(P_{fkr}\) ): "Hydrostatic" pressure variation due to the wave elevation computed on the instant immersed volume,

  • The diffraction and radiation pressures ( \(P_{dif}\),\(P_{rad}\) ): Pressure variations due to the modification of the incident wave by the floater derived from the pressure transfer functions previously computed by Diodore.

At that stage, pressure due to current flow is disregarded. It is to be noted that the hydrostatic ( \(P_{hys}\)) and Froude-krylov ( \(P_{fkr}\)) pressures are computed at every time step with the instant immersed volume. That's the reason why this post-processing is associated with the non-linear Froude- Krylov option in the floater motion type. These terms are then fully non- linear.

On the contrary, diffraction and radiation pressures are derived from the hydrodynamic pressures transfer functions (RAOs) previously computed by Diodore at a selected draft and by consequence they are linearized terms. Recent comparison with model tests (see JH2018) exhibited a good correlation between the total pressure computed numercially and the recorded pressure on a selected pontoons of a windturbine floater.

The procedure to obtain diffracted and radiated pressures time series from the hydrodyanmic transfer functions are described here below:

Diffraction pressure

The diffraction pressure RAOs write:

\[ H_D(\omega) = \frac{P_D(\omega)}{\eta(\omega)} \]

and the Impulse Response Function (IRF) is derived as such:

\[ h_D(\tau)=\frac{1}{\pi} \int_0^{\infty}Re(H_D(\omega)e^{-i\omega t})d\omega \]

Note that the function h(t) is close to zero outside the range [d,d].

The integration shall be properly done especially at low frequencies aup to the asymptotic value \(H_D(0) = 0\)

Once computed, h d(t) allows calculating the dynamic pressure from any wave elevation time series as such:

\[ P_D(t) = \int_{-d}^d h_D({\tau})\eta(t-\tau)d\tau = \int_{-d}^dh(-\tau)\eta(t+\tau)d\tau \]

Radiation pressure

For every degree of freedom, the radiation pressure RAO writes :

\[ H_{R_j}(\omega)=\frac{P_d(\omega)}{\dot x_j(\omega)} \]

Radiation is computed as such:

\[ P_{R_j}(t) = P_{R_j}(\infty)\dot x_j(t) + \int_0^d R_j(\tau)\dot x_j(t-\tau)d\tau \]

with \(P_{R_j}()\) the impulse radiation pressure and \(R_j(t)\) the IRF of the radiation pressure RAO:

\[ R_j(t)= \frac {1}{\pi} \int_0^{\infty} Re([P_{R_j}(\omega) - P_{R_j}(\infty)]e^{-i \omega t})d\omega \]

At very high frequencies, the hull mesh used for the hydrodynamic calculations may raise problems and to overcome this issue \(R_j(t)\) is written as follows:

\[ R_j(t)= \frac{1}{\pi} \int_{\omega_{min}}^{\omega_{max}} Re([P_{R_j}(\omega)-P_{R_j}(\omega_{max})]e^{-i\omega t})d\omega \]

with

\[ P_{R_j}(\infty)=<{P_{R_j}(\omega) - \int _0^d Im(R_j(t)e^{+i\omega t})dt}>_{\omega \in WF} \]