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Hydrodynamic Load Models for Submerged Elements

This section will give an introduction to hydrodynamic load models for submerged or surface-piercing normal slender structures (i.e. pipes, cables, risers, etc.).

1. Generalized Morison’s Equation

The velocity potential \(\mathrm {\phi ^I}\) of the undisturbed wave motion will be written as a superposition of linear waves of different circular frequencies \(\mathrm {\omega _\textrm{i}}\) and wave propagation angles \(\mathrm {\omega _\textrm{j}}\) ,

\[\phi ^I=\sum_{\textrm{j}=1}^{N_\beta }\,\sum_{\textrm{k}=1}^{N_\infty}\,\frac{gA_\textrm{jk}}{\omega _\textrm{k}}\frac{\textrm{cosh}[k_\textrm{k}(Z+D)]}{\textrm{cosh}(k_\textrm{k}D)}\cos(k_\textrm{k}X\cos\beta _\textrm{j}+k_\textrm{k}Y\sin\beta _\textrm{j}-\omega _\textrm{k}t-\phi _\textrm{jk})\]

Here \(\mathrm {g}\) is the acceleration due to gravity, \(\mathrm {t}\) is the time variable and \(\mathrm {(X,Y,Z)}\) is a Cartesian coordinate system with \(\mathrm {Z=0}\) in the mean free surface level.

Positive Z-axis is upwards. The mean water depth is \(\mathrm {D}\) . The wave numbers \(\mathrm {k_\textrm{k}}\) are related to the wave frequency \(\mathrm {\omega _\textrm{k}}\) by the dispersion relationship \(\mathrm {\omega _\textrm{k}^2/g=k_\textrm{k}\textrm{tanh}(k_\textrm{k}h)}\). \(\mathrm {\phi _\textrm{jk}}\) are random phase angles and \(\mathrm {A_\textrm{jk}}\) are determined by a wave spectrum for short crested sea.

When calculating the free surface elevation \(\mathrm {\eta }\) we will use that

\[\boldsymbol{\eta }=-\frac{1}{g}\frac{\partial \boldsymbol{\Phi}^I}{\partial t}\bigg|_{Z=0}\]

This value will be used to position the calculated velocity and acceleration profiles, which enables us in an easy way to calculate the loads on structural surface-piercing parts.

Figure 1. Flexible riser system

Figure 2. Riser cross sections

A schematic picture of a possible flexible riser system is shown in Figure 1. Cross-sections of typical flexible pipe configurations are shown in Figure 2.

The hydrodynamic forces are calculated according to a generalized Morison’s equation formulation, with the optional addition of a linear drag force term. A local coordinate system \(\mathrm {(x,y,z)}\) is defined. See figure Figure 1 where \(\mathrm {e(y,z)}\) is in the cross-sectional plane. Note that small letters indicate the local coordinate system while large letters \(\mathrm {(X,Y,Z)}\) indicate the global coordinate system.

1.1. Bi-symmetric cross section

The motion and the water velocity are decomposed into 3 components in the local element coordinate system, with \(\mathrm {x}\) in the axial direction, \(\mathrm {y}\) and \(\mathrm {z}\) normal to the axis, cfr. Figure 2.

It is assumed that both the \(\mathrm {y}\) - and \(\mathrm {z}\) -axis are symmetry-axes for the cross-sectional surface geometry. The longitudinal, \(\mathrm {F_x}\) , force components and transverse force components \(\mathrm {F_y}\) and \(\mathrm {F_z}\) , on an element of length \(\mathrm {dx}\) subjected to waves and current can then be written:

\[\scriptstyle{\mathrm {F_x=dx[(\rho A+m_{a}^x)\dot {u}_{x}^I-m_{a}^x\ddot {r}_x+C_D^x|U_x^c+u_x^I-\dot {r}_x|(U_x^c+u_x^I-\dot r_x)+C_{DL}^x(U_x^c+u_x^I-\dot r_x)}}\]

\[\scriptstyle{\mathrm {F_y=dx[(\rho A+m_{a}^y)\dot {u}_{y}^I-m_{a}^y\ddot {r}_y+C_D^y|U_y^c+u_y^I-\dot {r}_y|(U_y^c+u_y^I-\dot r_y)+C_{DL}^y(U_y^c+u_y^I-\dot r_y)]}}\]

\[\scriptstyle{\mathrm {F_z=dx[(\rho A+m_{a}^z)\dot {u}_{z}^I-m_{a}^z\ddot {r}_z+C_D^z|U_z^c+u_z^I-\dot {r}_z|(U_z^c+u_z^I-\dot r_z)+C_{DL}^z(U_z^c+u_z^I-\dot r_z)]}}\]

here dots denote time derivatives and

\(\mathrm {\rho \qquad\qquad\qquad\,}\) : mass density of water
\(\mathrm {A\qquad\qquad\qquad}\) : external cross-sectional area
\(\mathrm {m_a^x,m_a^y,m_a^z\quad \;\;\,\,}\) : 2D added mass in local x-, y-and z-directions
\(\mathrm {(r_x,r_y,r_z)\qquad\:\:}\) : translational structural displacement components in the local coordinate system
\(\mathrm {C_D^x,C_D^y,C_D^z\quad \;\;\:}\) : dimensional quadratic drag coefficients in local x-, y- and z-directions
\(\mathrm {C_{DL}^x,C_{DL}^y,C_{DL}^z}\) : dimensional linear drag coefficients in local x-, y- and z-directions
\(\mathrm {(U_x^c,U_y^c,U_z^c)\quad \;\;}\) : current velocity components in the local coordinate system
\(\mathrm {(u_x^I,u_y^I,u_z^I)\qquad\;}\) : velocity component of undisturbed wave field in the local coordinate system

See also the User Manual for the relation between these force coefficients and commonly used non-dimensional coefficients.

It should be noted that oscillatory forces due to vortex shedding are neglected. The drag coefficients and added mass coefficients will in general depend on structural form, Reynolds number, roughness ratio, Keulegan-Carpenter number, current velocity relative to wave induced velocity, etc. The nature of the orbital motion of the undisturbed local wave field will have an influence. For circular cylinders there is some information available in the literature.

1.2. Rotation symmetric cross section

The linear part of the drag force and the inertia forces, which are also linear, are calculated as for the bi-symmetric cross section. The longitudinal quadratic drag force is also calculated as for the bisymmetric cross section.

The transverse quadratic drag force is calculated differently, using the total relative velocity normal to the line axis:

\[F_y^D=F\cos\phi\]

\[F_z^D=F\sin\phi\]

where

\[F=dxC_D^y[(U_y^c+u_y^I-\dot r_y)^2+(U_z^c+u_z^I-\dot r_z)^2]\]

\[\phi =\textrm{arctan}[\frac{U_z^c+u_z^I-\dot r_z}{U_y^c+u_y^I-\dot r_y}],\quad (0\leq\phi \leq2\pi )\]

The hydrodynamic roll-moment (i.e. moment about the x-axis) on an element of length \(\mathrm {dx}\) is given by:

\[M_x=dx(\rho [\iint\limits_A\textrm{d}{y}\textrm{d}{z}(u^2-z^2)+A_{44}^{(2D)}]\frac{\partial }{\partial y}\dot u_z^I-A_{44}^{(2D)}\ddot \eta _4)\]

Here \(\mathrm {\ddot \eta _4}\) is the roll angle. Further \(\mathrm {A_{44}^{(2D)}}\) is the two-dimensional added moment in roll with respect to the local x-axis. For steady current we can write the Munk moment as

\[M_x=(A_{33}^{(2D)}-A_{22}^{(2D)})U_z^cU_y^c\]

\(\mathrm {M_x}\) is not implemented in the present version.

At every time step, forces and moments are calculated along the whole instantaneously submerged length of the riser. It is earlier discussed how the free surface position \(\mathrm {\eta }\) is calculated, which is important information to calculate the submerged length of the riser.

In the case of additional buoys, the hydrodynamic forces is handled in a similar way as described for the pipe cross sections.

2. MacCamy-Fuchs with Quadratic Drag

For structures with relatively large diameter compared to the wave length, near-field diffraction effects become more important. An analytical solution for the potential around a vertical, surface-piercing circular cylinder was developed by MacCamy and Fuchs.

For elements with MacCamy-Fuchs type loading, the formulation for the inertia and diffraction forces follows (MacCamy and Fuchs, 1954 and Dean and Dalrymple, 1991), with modifications for the irregular wave history. The force per unit length (\(\mathrm {dF}\)) at \(\mathrm {x=0}\) for a given elevation \(\mathrm {z}\) is given by:

\[dF(z)=\sum_{n=1}^{N}\frac{4\rho ga_n}{k_n}\frac{\cosh{k_n(z+h)}}{\cosh{k_nh}}G\cos{(\omega _nt-\epsilon_n-\alpha _n)}\]

where

\[\tan{\alpha _n}=\frac{J_1^\prime(k_na)}{Y_1^\prime(k_na)}\]

and

\[G=\frac{1}{\sqrt{(J_1^\prime(k_na))^2+(Y_1^\prime(k_na))^2}}\]

and the wave components are defined as for the wave potential.

The resulting \(\mathrm {C_m}\) and phase angle for a section with diameter 10 m are shown in Figure 3.

Figure 3. Effective mass coefficient and force phase, D=10m

The model is only applicable for circular cross-sections and approximately vertical sections. The forces are computed based on the z-coordinate in the center of the element and are applied in a lumped manner.

In addition to the inertia force, a quadratic drag force based on linear wave kinematics may also be included when using MacCamy-Fuchs. The drag force is according to Morison’s Equation, with the same implementation as in section Section 1.

3. Netloads

The method for calculation of current forces on fish farms is described in detail in Løland, G. (1991). The main equations are shown in this section. For details see Løland (1991) and also Aaarsnes et al (1998), Ormberg (1991).

The method is based on that the mean drag and lift force on a net panel can be written as

\[F_D=\frac{1}{2}\rho C_D(\alpha)AU^2\]

\[F_L=\frac{1}{2}\rho C_L(\alpha)AU^2\]

where

\(\mathrm {C_D}\) = drag coefficient as a function of the angle between net normal and flow direction
\(\mathrm {C_L}\) = lift coefficient as a function of the angle between net normal and flow direction
\(\mathrm {A}\) = area of net panel
\(\mathrm {U}\) = relative velocity
\(\mathrm {\rho}\) = density of water
\(\mathrm {\alpha}\) = angle between the flow direction and the net normal vector in the direction of the flow

The relative velocity is the difference between the wave and current velocity and the structural velocity. The current velocity is corrected using the reduced current velocity factor (the ratio between reduced current speed and ambient current speed due to upstream net shadowing effect).

The drag force is defined as the force in the direction of the flow. The lift force is defined as the force normal to the flow direction. In the global axis system, the force components can be written as

\[\boldsymbol{F}=F_D \boldsymbol{n}(x,y) + F_L\boldsymbol{l}(x,y)\]

\[\boldsymbol{n}(x,y)=\frac{\boldsymbol{U}}{|U|}=\boldsymbol{e}_U\]

\[\boldsymbol{l}(x,y)=(\boldsymbol{e}_T\times\boldsymbol{e}_U)\times\boldsymbol{e}_U\cdot(\boldsymbol{e}_U\cdot\boldsymbol{e}_T)\]

\(\mathrm {\boldsymbol{e}_T}\) is the unit tangential vector defining the direction from the upper end to the lower end point of the net panel.

The drag and lift coefficient are function of the solidity ratio. The solidity ratio is degined as the ratio between the area covered by the threads in the screen and the total area of the screen.

\[S_n=\frac{2D}{\lambda}+\frac{1}{2}(\frac{D}{\lambda})^2\]

\(\mathrm {\lambda}\) is the mesh size and \(\mathrm {D}\) the twine diameter. The relationship between the drag coefficient and the solidity ratio is given as

\[C_D=0.04+(-0.04+S_n-1.24S_n^2+13.7S_n^3)\cos(\alpha)\]

\[C_L=(0.57 S_n-3.54S_n^2+10.1S_n^3)\sin(2\alpha)\]

The factor 0.04 is introduced to take into account the drag on a net panel parallel to the flow. The factor is assumed to be independent of the solidity ratio. This is an approximation. The model test indicated that the factor is almost independent of the solidity ratio, \(\mathrm {S_n}\) , for the solidity ratio range used in the model test. This range was 0.13 to 0.317 as well as a angle \(\mathrm {\alpha}\) from 0-90 degrees.

Figure 4. Drag and lift coefficient as a function of the solidity ratio

3.1. References

Milne, P.H. (1970): Fish farming: A Guide to the Design and Construction of NetEnclosures, Department of agriculture and Fisheries for Scotland, Marine ResearchRep. No. 1, 1970

Aarsnes, J.V., Lien E. and Oltedal, G. (1988): SIFICA - Theory manual, MARINTEK report: 513004.05.02, Trondheim

Løland, G. (1991): Current Forces on and Flow Through Fish Farms, Doctoral thesis, NTH, Trondheim, Norway

Ormberg, H. (1991): Non-linear Response Analysis of Floating Fish Farm Systems. Dr.ing dissertation,The Norwegian Institute of Technology. ISBN 82-7119-330-9

Løland (1993): Current forces on, and water flow through and around, floating fish farms ,Aquaculture International 1, Pages 72-89

Kristiansen, T. et al (2012): Modelling of current loads on aqua culture net cages Journal of Fluids and Structures Volume 34, October 2012, Pages 218-235

4. Loads due to Vortex-Induced Vibrations (VIV)

4.1. Load Formulation

The time domain VIV load per unit length is given by:

\[\begin{aligned} F &= (C_A + 1)\rho\frac{\pi D^2}{4}\dot{u}_n - C_A\rho\frac{\pi D^2}{4}\ddot{x}_n + \frac{1}{2}\rho C_D D v_n|v_n| \\ &+ \frac{1}{2}\rho D C_{v,CF}|v_n|(j_3 \times v_n)\cos\Phi_{v,CF} \\ &+ \frac{1}{2}\rho D C_{v,IL}|v_n|v_n\cos\Phi_{v,IL} \\ &+ \frac{1}{2}\rho D C_{hh}|v_n|(j_3 \times v_n)\cos(\Phi_{v,CF} + \Phi_{v,IL}) \end{aligned}\]

Where:

The first line is equivalent to the ordinary Morison load
The second line is the cross-flow vortex shedding force term
The third line is the in-line vortex shedding force term
The fourth line is the higher harmonic vortex shedding force term

4.2. Synchronization Model

The cross-flow vortex shedding force is given by:

\[F_{v,CF} = \frac{1}{2}\rho D C_{v,CF}|v_n|(j_3 \times v_n)\cos\Phi_{v,CF}\]

where \(\Phi_{v,CF}\) is the time varying instantaneous phase of the cross-flow vortex shedding force.

For a fixed force frequency this phase would be:

\[\Phi_{v,CF} = 2\pi f_{v,CF}\, t\]

Instead, the frequency is adjusted to synchronize force and response.

\(\Phi_{v_n}\): Instantaneous phase of the relative velocity
\(\Phi_{v,CF}\): Instantaneous phase of the CF force
\(\theta\): Instantaneous phase difference between relative velocity and force, \(\theta = \Phi_{v_n} - \Phi_{v,CF}\)

Adjust the phase of the cross-flow load \(\Phi_{v,CF}\) to reduce the phase difference.

The IL synchronization model is similar to the CF one, as illustrated in Figure 5.

Figure 5. Synchronization curve illustrating the lock-in (frequency) bandwidth and the phase difference between the external forcing and the structural response under synchronization (phase-locking).

The non-dimensional frequency is defined as:

\[\hat{f} = \frac{f_{osc} \cdot D}{U}\]

The phase is updated at each time step as:

\[\Phi_{v,CF,upd} = \Phi_{v,CF} + \frac{2\pi|v_n|}{D}\,\hat{f}\,\Delta t\]

Figure 6. Illustration of the synchronization concept. The phase difference between vortex shedding force and relative structure velocity is reduced by increasing or decreasing instantaneous frequency of the force at every time step.

4.3. References

Thorsen, M. J. (2016): Time Domain Analysis of Vortex-Induced Vibrations. PhD Thesis, Norwegian University of Science and Technology, Trondheim, Norway.
Ulveseter, J. V. (2018): Advances in semi-empirical time domain modelling of vortex-induced vibrations. PhD Thesis, Norwegian University of Science and Technology.
Kim, S. (2021): Non-Linear Time Domain Analysis of Deepwater Riser Vortex-Induced Vibrations. PhD Thesis, Norwegian University of Science and Technology, Trondheim, Norway.