Results: Quadratic loads

These quantities are reported on the mean drift loads (PI), mean drift loads (CS), mean drift loads (MC), quadratic loads (PI) and quadratic loads (CS) sheets of the results tables.

The quadratic load is one of the contributions to the second-order load on a body. It is the contribution that arises from products of two first-order quantities, and therefore can be evaluated from the results of a first-order analysis. We use the terminology time-varying quadratic loads to refer to the complete set of quadratic loads, including both zero and non-zero frequencies, whereas mean drift loads are the special case of zero-frequency quadratic loads.

OrcaWave can evaluate quadratic loads using three different calculation methods. Full details on each method are given below. The following table gives a summary:

Quadratic loads are often slow to converge, requiring a relatively fine mesh in order to obtain accurate results. Given that, the existence of three different calculation methods can be helpful. Their results differ only because of discretisation error, i.e. the error that can be reduced by using a finer mesh.

Mean drift loads

Mean drift loads, also known as Newman QTFs, are the special case of quadratic loads corresponding to zero difference frequency.

OrcaWave gives results for a complex quantity $\vec{F}_q$, following the complex notation used in the second-order theory. If the incoming wave at a given frequency includes contributions with complex amplitude $A_i$ for multiple wave headings $\beta_i$, the physical load is given by \begin{equation} \sum_{i,j} \Re\left\{A_i A_j^* \vec{F}_q(\beta_i, \beta_j)\right\} = \sum_i |A_i|^2 \vec{F}_q(\beta_i, \beta_i) + 2\sum_{j>i} \Re\left\{ A_i A_j^* \vec{F}_q(\beta_i, \beta_j) \right\} \end{equation}

Time-varying quadratic loads

Time-varying quadratic loads to refer to the complete set of quadratic loads, including both zero and non-zero frequencies. OrcaWave only reports complete quadratic load results if the solve type is full QTF. Each quadratic load can be added to the corresponding potential load to obtain a full QTF.

OrcaWave gives results for a complex quantity $\vec{F}_q^{\pm}$, following the complex notation used in the second-order theory. Assuming a wave spectrum with complex amplitudes $A_i$, the physical load is given by \begin{equation} \begin{aligned} \sum_{ij} \Re\left\{A_i A_j \vec{F}^+_{q,ij}\textrm{e}^{\textrm{i}(\omega_i+\omega_j) t}\right\} & \;\;\;\textrm{ for a sum frequency} \\ \sum_{ij} \Re\left\{A_i A_j^* \vec{F}^-_{q,ij}\textrm{e}^{\textrm{i}(\omega_i-\omega_j) t} \right\} & \;\;\;\textrm{ for a difference frequency} \end{aligned} \end{equation}

Pressure integration method

The pressure integration method, also known as the near-field method, obtains the quadratic load by direct integration of the hydrodynamic pressure. It is only available if the solve type includes the source formulation, because fluid velocity $\nabla\phi$ is required on the body surface $\SB$.

Results include mean drift loads and, if the solve type is full QTF, time-varying quadratic loads. Results are reported in body coordinates $\Bxyz$.

For this calculation method, the quadratic load is most conveniently expressed in terms of real quantities. In the formulae below, this is emphasised by using $\Phi$ for the potential, rather than the complex potential, $\phi$. In the translation degrees of freedom, the force on a displacement body is \begin{align} \vec{f}_q = &\; \frac{\rho g}{2}\oint_{\CWL} \left(\eta - d_3 \right)^2 \frac{\vec{n}}{\sqrt{1-n_z^2}} \ud l - \rho \int_{\SB}\left\{ \frac{1}{2}\left(\nabla\Phi\right)^2 + \vec{d} \cdot \nabla\PDt{\Phi} \right\}\vec{n} \ud S \nonumber \\ &\; + \vec{\alpha}\times\vec{f}\fo - \rho g |\SI| (\vec{\alpha} \cdot \vec{x}_f) \alpha_3 \vec{e}_z \label{eqPressureIntegrationForce} \\ \textrm{where}\;\;\; \eta(x,y,t) = &\; -\frac{1}{g}\PDt{\Phi}(x,y,0) \\ \vec{d}(\vec{x},t) = &\; \vec{\xi}(t) + \vec{\alpha}(t)\times\vec{x} \\ \vec{f}\fo(t) = &\; -\rho\int_{\SB} \PDt{\Phi} \vec{n}\ud S - \rho g |\SI| d_3(\vec{x}_f, t) \vec{e}_z \end{align} Here

Each quantity is first order. The corresponding superscripts, e.g. $\Phi\fo$, are omitted for visual clarity.

$\vec{\xi}(t)$ is the translational displacement of the body, obtained from the surge, sway and heave displacement RAOs

$\vec{\alpha}(t)$ is the rotational displacement of the body, obtained from the roll, pitch and yaw displacement RAOs

$\vec{x}_f$ is the centre of floatation

$|\SI|$ is the water plane area

In the line integral, $\vec{n}$ is the three dimensional normal to the body surface at each point along the waterline, $\CWL$.

In the rotation degrees of freedom, the moment on a displacement body is \begin{align} \vec{m}_q = &\; \frac{\rho g}{2}\oint_{\CWL} \left(\eta -d_3 \right)^2 \frac{\vec{x}\times\vec{n}}{\sqrt{1-n_z^2}} \ud l - \rho \int_{\SB}\left\{ \frac{1}{2} \left(\nabla\Phi\right)^2 + \vec{d}\cdot\nabla\PDt{\Phi} \right\}(\vec{x}\times\vec{n}) \ud S \nonumber \\ &\; + \vec{\xi}\times\vec{f}\fo + \vec{\alpha}\times\vec{m}\fo + \rho g |\VInt| \Big\{(\vec{\alpha}\cdot\vec{\xi})\vec{e}_z - \alpha_3 \vec{d}(\vec{x}_b, t)\Big\} \nonumber \\ &\; + \rho g\left\{ \half |\VInt|\left(2\alpha_1\alpha_2 x_b - \alpha_1^2 y_b - \alpha_3^2 y_b\right) - \alpha_1\alpha_3 L_{xy} -\alpha_2\alpha_3 L_{yy} \right\}\vec{e}_x \nonumber \\ &\; + \rho g\left\{ \half |\VInt| \left({\alpha_2^2 + \alpha_3^2}\right)x_b + \alpha_1\alpha_3 L_{xx} + \alpha_2\alpha_3 L_{xy} \right\}\vec{e}_y \label{eqPressureIntegrationMoment} \\ \textrm{where}\;\;\; \vec{m}\fo(t) = &\; -\rho\int_{\SB} \vec{x}\times\vec{n}\, \PDt{\Phi} \ud S + \rho g |\VInt| \vec{d}(\vec{x}_b, t) \times \vec{e}_z \nonumber \\ &\; - \rho g \left(|\SI|y_f\xi_3 + L_{yy}\alpha_1 - L_{xy}\alpha_2\right) \vec{e}_x + \rho g \left(|\SI|x_f\xi_3 + L_{xy}\alpha_1 - L_{xx}\alpha_2\right) \vec{e}_y \end{align} Here

$\vec{x}_b$ is the centre of buoyancy

$|\VInt|$ is the volume of the body

$L_{xx}$, $L_{xy}$ and $L_{yy}$ are the water plane moments of the body

Control surface integration method

The control surface integration method, also known as the middle-field method, obtains the quadratic load from integrals over both the body surface and a control surface. Equations (\ref{eqPressureIntegrationForce}) and (\ref{eqPressureIntegrationMoment}) are transformed (Chen 2004) into this alternative form by repeated applications of the divergence theorem. It is available in all models but is not applicable to sectional bodies.

Control surface integration often achieves more accurate (converged) results than the pressure integration method for a given mesh. Alternatively, a coarser mesh may be sufficient to achieve a given accuracy in the results.

Results include mean drift loads and, if the solve type is full QTF, time-varying quadratic loads. Results are reported in body coordinates $\Bxyz$.

As for the pressure integration method, the quadratic load is most conveniently expressed in terms of real quantities. In the translation degrees of freedom, the force is \begin{align} \vec{f}_q = &\; \rho g \vec{e}_z \oint_{\CWL} \left\{ -\eta (\vec{d}\cdot\vec{n}^h) + \frac{\left(\eta - d_3\right)^2 n_z}{2\sqrt{1-n_z^2}} \right\} \ud l - \frac{\rho g}{2}\oint_{\CCL}\vec{n}^h \eta^2 \ud l \nonumber \\ &\; + \rho \int_{\SCsub}\left\{ \nabla\Phi\PD{\Phi}{n} -\frac{\vec{n}}{2} \left(\nabla\Phi\right)^2 \right\} \ud S \nonumber \\ &\; + \rho \int_{\SCfre}\left\{ \nabla\Phi\PD{\Phi}{n} -\frac{\vec{n}}{2} \left(\nabla\Phi\right)^2 + g \eta \nabla^h\eta \right\} \ud S \nonumber \\ &\; - \rho\int_{\SB} \left\{ \nabla\Phi \left(\PDt{\vec{d}}\cdot\vec{n}\right) + \left(\vec{d}\cdot\vec{n}\right) \nabla\PDt{\Phi} \right\} \ud S \nonumber \\ &\; - \rho g |\SI| (\vec{\alpha} \cdot \vec{x}_f) \alpha_3 \vec{e}_z \label{eqControlSurfaceForce} \end{align} Here

In the line integrals, $\vec{n}^h$ is a two dimensional normal to the line (i.e. $\CWL$ or $\CCL$) pointing out of $\SCfre$.

All other quantities have the same definitions as for the pressure integration method.

In the rotation degrees of freedom, the moment is \begin{align} \vec{m}_q = &\; \rho g \oint_{\CWL} \left\{ -\eta \left(\vec{d}\cdot\vec{n}^h\right) + \frac{\left(\eta - d_3\right)^2 n_z}{2\sqrt{1-n_z^2}} \right\} \left(\vec{x}\times\vec{e}_z\right) \ud l - \frac{\rho g}{2}\oint_{\CCL} \left(\vec{x}\times\vec{n}^h\right) \eta^2 \ud l \nonumber \\ &\; + \rho \int_{\SCsub} \left\{ \vec{x}\times\nabla\Phi \PD{\Phi}{n} - \frac{\vec{x}\times\vec{n}}{2} \left(\nabla\Phi\right)^2 \right\} \ud S \nonumber \\ &\; + \rho \int_{\SCfre} \left\{ \vec{x}\times\nabla\Phi \PD{\Phi}{n} - \frac{\vec{x}\times\vec{n}}{2} \left(\nabla\Phi\right)^2 + g \eta \vec{x}\times\nabla^h \eta \right\} \ud S \nonumber \\ &\; - \rho\int_{\SB}\left\{ (\vec{x}\times\nabla\Phi) \left(\PDt{\vec{d}}\cdot\vec{n}\right) + (\vec{d}\cdot\vec{n}) \left(\vec{x} \times \nabla\PDt{\Phi}\right) \right\}\ud S \nonumber \\ &\; - \rho g |\SI| \left( d_3(\vec{x}_f, t) \vec{\xi}\times\vec{e}_z + \xi_3\alpha_3\vec{x}_f \right) \nonumber \\ &\; + \rho g\left\{ \half |\VInt|\left(2\alpha_1\alpha_2x_b - \alpha_1^2y_b - \alpha_3^2y_b\right) + \alpha_2\alpha_3 (L_{xx} - L_{yy}) -2\alpha_1\alpha_3 L_{xy} \right\}\vec{e}_x \nonumber \\ &\; + \rho g\left\{ \half |\VInt|\left(\alpha_2^2 + \alpha_3^2\right) x_b + \alpha_1\alpha_3 (L_{xx} - L_{yy}) + 2\alpha_2\alpha_3 L_{xy} \right\}\vec{e}_y \label{eqControlSurfaceMoment} \end{align}

Momentum conservation method

The momentum conservation method, also known as the far-field method, obtains the quadratic load from a far-field asymptotic analysis of the velocity potential. It is available in all models except those with a damping lid.

Momentum conservation is closely related to control surface integration and, like that method, it often achieves more accurate (converged) results than the pressure integration method for a given mesh.

In contrast to the other methods, results are reported in global coordinates $\GXYZ$. In addition, there are some significant limitations:

• Results are given for mean drift loads only. This method cannot give time-varying quadratic loads.
• Only horizontal (i.e. surge, sway and yaw) load components can be computed. This method gives no information about vertical components (i.e. heave, roll and pitch), for which zero values are returned.
• The results give the surge, sway and yaw load on the ensemble of all bodies included in the model. In a multibody model it is not possible to get the mean drift load on an individual body.

For this calculation method, the quadratic load is most conveniently expressed in complex notation. First we define $H_{\beta}(\theta)$, the Kochin function for an incident wave of heading $\beta$ \begin{equation} \begin{aligned} H_{\beta}(\theta) = &\; \int_{S_B} \left\{ \psi(\vec{\xi}, \theta) \PD{\phi}{n_\xi}(\vec{\xi}) - \phi(\vec{\xi})\PD{\psi}{n_\xi}(\vec{\xi}, \theta) \right\} \ud S_\xi \\ \psi(\vec{\xi}, \theta) = &\; f(k \zeta) \textrm{e}^{-\textrm{i}k (\xi\cos\theta + \eta\sin\theta)} \end{aligned} \label{eqKochinFunction} \end{equation} Here

$\phi$ is the first-order complex velocity potential associated with an incident wave of heading $\beta$ and unit amplitude

$\vec{\xi}=(\xi,\eta,\zeta)$, $k$ is the wavenumber, $f(k \zeta)$ is the function of depth in the incident wave

The surge component of the mean drift load is given in terms of the Kochin functions \begin{align} \left[\vec{F}_q(\beta_i, \beta_j)\right]_{\textrm{surge}} = &\; \frac{\rho k^3}{8\pi \nu} \left(\frac{\nu}{\nu + h(k^2 - \nu^2)}\right) \int_0^{2\pi} \cos\theta\, H_{\beta_i}(\theta) H^*_{\beta_j}(\theta) \ud \theta \nonumber \\ &\; + \frac{\rho k \omega}{4 \nu} \left\{ \cos\beta_j\, H_{\beta_i}(\pi + \beta_j) + \cos\beta_i\, H^*_{\beta_j}(\pi + \beta_i) \right\} \label{eqMomentumConservationSurge} \end{align} The formula for the sway component is the same as (\ref{eqMomentumConservationSurge}), but with $\cos\theta$, $\cos\beta_i$ and $\cos\beta_j$ replaced by $\sin\theta$, $\sin\beta_i$ and $\sin\beta_j$ respectively.

The yaw component of the mean drift load is given by \begin{align} \left[\vec{F}_q(\beta_i, \beta_j)\right]_{\textrm{yaw}} = &\; \frac{-\textrm{i} \rho k^2}{8\pi \nu} \left(\frac{\nu}{\nu + h(k^2 - \nu^2)}\right) \int_0^{2\pi} H^{\prime}_{\beta_i}(\theta) H^*_{\beta_j}(\theta) \ud \theta \nonumber \\ &\; + \frac{\textrm{i} \rho \omega}{4 \nu} \left\{ H^{\prime}_{\beta_i}(\pi + \beta_j) - H^{* \prime}_{\beta_j}(\pi + \beta_i) \right\} \label{eqMomentumConservationYaw} \end{align} In (\ref{eqMomentumConservationSurge}) and (\ref{eqMomentumConservationYaw}) the integrations with respect to $\theta$ are performed by numerical quadrature using the user-specified number of nodes.

Sectional bodies

The control surface method is not available for sectional bodies. The pressure integration method is available, using modified equations. Instead of (\ref{eqPressureIntegrationForce}), the force is \begin{align} \vec{f}_q = &\; \frac{\rho g}{2}\oint_{\CWL} \left(\eta - d_3 \right)^2 \frac{\vec{n}}{\sqrt{1-n_z^2}} \ud l - \rho \int_{\SB}\left\{ \frac{1}{2}\left(\nabla\Phi\right)^2 + \vec{d} \cdot \nabla\PDt{\Phi} \right\}\vec{n} \ud S \nonumber \\ &\; + \vec{\alpha} \times \left(\vec{f}\fo - \vec{\alpha} \times \vec{f}\zo\right) + H \vec{f}\zo - q^b \vec{h} \\ \textrm{where}\;\;\; \vec{f}\zo(t) = &\; -q^b_{i3} \\ \vec{f}\fo(t) = &\; -\rho\int_{\SB} \PDt{\Phi} \vec{n}\ud S - K_{i, 3} \xi_3 - K_{i, j+3}\alpha_{j} \\ H = &\; \frac{1}{2}\left( \begin{array}{ccc} -(\alpha_{2}^2 + \alpha_{3}^2) & 0 & 0 \\ 2\alpha_{1}\alpha_{2} & -(\alpha_{1}^2 + \alpha_{3}^2) & 0 \\ 2\alpha_{1}\alpha_{3} & 2\alpha_{2}\alpha_{3} & -(\alpha_{1}^2 + \alpha_{2}^2) \end{array} \right) \\ \vec{h} = &\; H_{3i} \end{align} Instead of (\ref{eqPressureIntegrationMoment}), the moment is \begin{align} \vec{m}_q = &\; \frac{\rho g}{2}\oint_{\CWL} \left(\eta -d_3 \right)^2 \frac{\vec{x}\times\vec{n}}{\sqrt{1-n_z^2}} \ud l - \rho \int_{\SB}\left\{ \frac{1}{2} \left(\nabla\Phi\right)^2 + \vec{d}\cdot\nabla\PDt{\Phi} \right\}(\vec{x}\times\vec{n}) \ud S \nonumber \\ &\; + \vec{\xi}\times\vec{f}\fo + \vec{\alpha} \times \left( \vec{m}\fo - \vec{\alpha} \times \vec{m}\zo - \vec{\xi} \times \vec{f}\zo \right) + H \vec{m}\zo - q^c \vec{h} \\ \textrm{where}\;\;\; \vec{m}\zo(t) = &\; -q^c_{i3} \\ \vec{m}\fo(t) = &\; -\rho\int_{\SB} \vec{x}\times\vec{n}\, \PDt{\Phi} \ud S -K_{i+3, 3}\xi_3 - K_{i+3, j+3}\alpha_j + M g (\alpha_3 \vec{x}_m - z_m \vec{\alpha}) + \vec{\xi} \times \vec{f}\zo \end{align} Here

$K_{ij}$ is the hydrostatic stiffness matrix

$q^b_{ij}$ and $q^c_{ij}$ are matrices which form part of the definition of $K_{ij}$