Home > Theory > Hydrodynamics > Hydrodynamics of offshore floating structures

Hydrodynamics of offshore floating structures

This part describes the theoretical background to study the motion of floating structures in waves, referred as hydrodynamic or sea-keeping theories. In this section, hydrodynamic background for a cylinder is presented. Indeed, a major part of hydrodynamics for offshore structures deals with circular cylinders and can be extended to more complex offshore systems, with some assumptions and modifications, since these structures mostly consist of cylinders assembly. This page is based on Clement's PhD thesis [49].

Wave theory

Most of this part focusing on waves is based on the book of Molin [27].

Regular wave characteristics

Only regular waves are considered in this report. Irregular waves are commonly defined as a sum of regular waves but are not detailed here. A regular wave is progressive if it propagates in one direction. Regular wave consists in harmonic wave characterised by several parameters depicted below:

Regular wave main characteristics.

H: Wave height (crest to trough) [m]
A: Wave amplitude [m]
h: Mean water depth [m]
T: Wave period [s]
g: Gravitational acceleration [\unit{m.s^{-2}}]
$\lambda$: Wavelength [m]

Potential theory

In the orthonormal basis $(Oxyz)$, the velocity potential $\Phi$ is defined as:

\[ \vec{V}(x,y,z,t)=\bigtriangleup \Phi (x,y,z,t) \]

Considering $u$, $v$ and $w$ the velocity components of the flow along $(Ox)$, $(Oy)$ and $(Oz)$ axes, the potential difference between two points A and B in the fluid is introduced as:

\[ \bigtriangleup \Phi_{A \longrightarrow B}=\int_{A}^{B}\vec{V}\cdot d\vec{s}=\int_{A}^{B} (u\cdot dx+v\cdot dy +w\cdot dz) \\ =\int_{A}^{B} (\frac{\partial \Phi}{\partial x}dx+\frac{\partial \Phi}{\partial y}dy +\frac{\partial \Phi}{\partial z}dz)=\int_{A}^{B} d\Phi =\Phi(B)-\Phi(A) \]

The potential function $\Phi$ can thus be defined by the three following equalities:

\[ u=\frac{\partial \Phi}{\partial x} \qquad v=\frac{\partial \Phi}{\partial y} \qquad w=\frac{\partial \Phi}{\partial z} \]

Considering an infinite ocean of water filled by an incompressible, non-viscous, irrotationnal (i.e. null velocity gradient) flow, the four conditions below have to be verified:

Laplace equation (mass conservation):

\[ \tag{1} \bigtriangleup \Phi =0 \]

No-leak boundary condition: The vertical velocity of a water particle at the sea bed is zero.

\[ \tag{2} \frac{\partial \Phi}{\partial z} =\Phi_{z}=0 \qquad z=-h \]

Free surface dynamic boundary condition:

The pressure $p$ of the fluid at the free surface ($z=\eta (x,y,t)$) is equal to the atmospheric pressure $p_{0}$. This leads to:

\[ \tag{3} \Phi_{t}+\frac{1}{2}(\nabla \Phi)^{2} +gz=0 \qquad \textrm{at} \quad z=\eta(x,y,t) \]

Free surface kinematic boundary condition:

The vertical velocity of a particle at the free surface is equal to the vertical velocity of the free surface itself. This means that the free surface is considered as a material surface.

\[ \tag{4} \eta_{t}+\Phi_{x}\eta_{x}+\Phi_{y}\eta_{y}=\Phi_{z} \qquad \textrm{at} \quad z=\eta(x,y,t) \]

$\Phi$ can also be developed in a Taylor series for z along water depth as: $$ \Phi(x,y,z,t)=\Phi(x,y,0,t)+z\Phi_{z}(x,y,0,t)+\cdots \quad 0\leqslant z\leqslant \eta(x,y,t) $$

The resolution of the system formed by (1), (2) ,(3) and (4) for the potential, with Taylor development leads to Stokes wave theory. Stokes order depends on the order of the Taylor development realised. Le Méhauté diagram (below) details the wave theories and the order of Taylor development that should be selected to correctly model a wave depending on the wave height, wave period (noted $\tau$ in the graph) and water depth.

Airy waves

First the wave is assumed periodic in time and in space. The flow is restrained to the $(Oxz)$ vertical plane. The resolution of the system above using first order Taylor development for the velocity potential leads to the following expression of the free surface elevation:

\[ \eta^{(1)}=A\cos(kx-\omega t) \]

The first order velocity potential is expressed as:

\[ \tag{5} \Phi^{(1)}=\frac{Ag}{w}\frac{\cosh k(z+h)}{\cosh kh} \sin (kx -\omega t) \]

The free surface kinematic boundary condition leads to the dispersion relation valid only for regular waves:

\[ \omega ^{2}=gk\tanh kh \]

The wavelength $\lambda=\frac{2\pi}{k}$ can then be expressed as:

\[ \lambda=\frac{gT^{2}}{2\pi}\tanh \frac{2\pi h}{\lambda} \]

This relation can be solved iteratively. For infinite depth (i.e. $\tanh \frac{2\pi h}{\lambda} \simeq 1$), it can be simplified as:

\[ \lambda=\frac{gT^{2}}{2\pi}=1.56T^{2} \]

From the velocity potential (5), and the dispersion relation, the kinematics of water particle can be expressed by the orbital velocities expressed below:

\[ u=\frac{\partial \Phi}{\partial x}=\frac{\partial x}{\partial t}=\frac{Agk}{\omega}\frac{\cosh k(z+h)}{\sinh kh}\sin(kx-\omega t) \]

\[ w = \frac{\partial \Phi}{\partial z}=\frac{\partial z}{\partial t}=\frac{Agk}{\omega}\frac{\sinh k(z+h)}{\sinh kh}\sin(kx-\omega t) \]

Depending on water depth, velocity fields will be different as shown below:

Velocity field (left side) and velocity orbitals (right side) in deep, intermediate and shallow water.

Also, with the dispersion relation, the phase velocity (velocity of peaks) $c=\omega /k$ can be determined as:

\[ c=\sqrt{\frac{g}{k}\tanh kh} \]

In deep water, $\tanh kh=1$ leads to $c=g/\omega$.

In shallow water, $\tan kh=kh$ leads to $c=\sqrt{gh}$

\medbreak

Pressure field for Airy waves

The pressure and velocity are key variables in hydrodynamics. For Airy waves, the pressure field can be obtained by solving Bernoulli equation with the expression of the velocity potential given in (5).

For an unsteady and irrotational flow, the Bernouilli equation leads to:

\[ \frac{\partial \Phi}{\partial t}+\frac{1}{2}(u^{2}+v^{2}+w^{2})+\frac{p}{\rho} +gz=Constant \]

In two dimensions ($v=0$) and assuming the waves have a small steepness ($u$ and $w$ are small), this equation becomes:

\[ p=-\rho gz -\rho \frac{\partial \Phi}{\partial t} \]

From the wave velocity potential (5), the following expression of the linearised pressure at first order is obtained:

\[ p=-\rho gz + \rho g A \frac{\cosh k(h+z)}{\cosh kh}\cos(kx-\omega t) \]

The first term of this expression is the hydrostatic pressure. The second one is the dynamic pressure, associated to waves.

In deep water ($kh\geq3$), the linearised pressure is:

\[ p=-\rho gz + \rho g A e^{kz}\cos(kx-\omega t) \]

In shallow water ($kh\ll1$), the pressure becomes:

\[ p=-\rho gz + \rho g A cos(kx-\omega t) \]

Stokes 2^nd order waves

Stokes second order wave potential is the sum of the velocity potential $\Phi^{(1)}$, derived in the previous paragraph, and the second order potential $\Phi^{(2) }$. The three boundary conditions ((2), (3) and (4)) combined with Laplace aquation (1) leads to the following expression:

\[ \Phi^{(2)}(x,z,t)=\frac{3}{8} \frac{A^{2}w}{\sinh^{2}kh}\cosh 2k(z+h) \sin2(kx-\omega t) \]

The free surface elevation is obtained as:

\[ \eta(x,t) = \eta^{(1)}(x,t) + \eta^{(2)}(x,t) \]

in which $\eta^{(2)}(x,t)$ is the second order component of the free surface elevation defined as as:

\[ \eta^{(2)}(x,t)=\frac{A^{2}k}{4} (3\coth^{3}kh-\coth kh)\cos2(kx-wt) \]

If one deals with deep water waves (i.e $kh\geq3$), $\Phi^{(2)}$ is null and the second order free surface elevation component becomes:

\[ \eta^{2}(x,t)=\frac{1}{2} A^{2}k\cos2(kx-wt) \]

Stokes 3^rd order waves

In this section, the waves are assumed infinite. The velocity potential developed to the third order is defined in Eq. (6) as:

\[ \tag{6} \Phi(x,z,t)=\frac{Ag}{\omega} e^{k'z} \sin (k'x-\omega t) \]

where $k'=(1-A^2k^2)k$.

The free surface elevation is defined in Eq. (7) as:

\[ \tag{1} \eta(x,t)= (1-\frac{3}{8}A^2 k^2)A\cos{\theta} + \frac{1}{2} A^2 k \cos(2 \theta ) + \frac{3}{8} A^3 k^2 \cos 3 \theta \]

where $\theta = (k'x - \omega t)$.

The development of the velocity potential to the third order does not impact the prediction of the peak-to-peak amplitude but can lead to a difference on the predicted value of the wavelength compared to Airy and Stokes 2^nd order theories.

Cylinder in an oscillating flow

This section aims at describing the hydrodynamic loads on a cylinder in an oscillating flow representing wave effects. It focuses on the dynamic loads obtained due to the oscillating flow. In this case, the hydrostatic load is the buoyancy. First, the potential theory is applied to describe dynamic loads on the cylinder referring to the book of Molin [27]. Then, Morison experimental formula is introduced.

As for wave theory described previously, the following fluid hypotheses have to be respected in order to be in potential flow condition: incompressible, non viscous, continuous and homogeneous fluid.

The inline flow on a given cylinder, represented in the figure below (($R$,$\theta$) the cylinder frame), is considered.

The flow is supposed oriented along $(Ox)$ and $u(t)$ is the velocity along this axis. This leads to the incident potential $\Phi_{I}$ respecting the following equation :

\[ \Phi_{I}=u(t)x =u(t)R\cos\theta \]

The cylinder is considered as infinite and loads on the cylinder are given per cylinder length in the following subsections.

Fixed cylinder in an oscillating flow

The cylinder is considered fixed. The resulting forces on the cylinder in $(Ox)$ direction is given by:

\[ F_{x}(t)=\int_{S}p(R,\theta,t)\vec{dr}\cdot\vec{n}=\int_{0}^{2\pi}p(R,\theta,t)R\cos\theta d\theta=2\rho \pi R^{2} \dot u(t) \]

where $p$ is the dynamic pressure.

$F_{x}$ is also known as Froude-Krilov force: the force resulting of the integration of pressure over the surface of the cylinder in an undisturbed flow.

Moving cylinder in a flow at rest

Considering a moving cylinder in a flow at rest, $\ddot X(t)$ its acceleration in $(Ox)$ direction, the hydrodynamic force is given by:

\[ F_{x}=- \rho \pi R^{2} \ddot{X}(t) \]

Generalisation to a moving cylinder

Finally, hydrodynamic loads on a moving cylinder in an oscillating flow applying potential theory are expressed by:

\[ \tag{7} F_{x}=(1+C_{axx})\rho S \dot{u}(t)-C_{axx} \rho S \ddot{X}(t) \]

\[ \tag{8} F_{y}=(1+C_{ayy})\rho S \dot{v}(t)-C_{ayy} \rho S \ddot{Y}(t) \]

where $S$ is the cylinder section in $(Oxy)$ plan, $u(t)$ and $v(t)$ the flow velocity component in $(Ox)$ and $(Oy)$ directions, $X(t)$ and $Y(t)$ the position of the cylinder in $(Oxy)$ coordinate system. $C_{axx}$ and $C_{ayy}$ are the added mass coefficient for radiation in each direction (equal to 1 for a circular section in potential flow theory).

In equations (7) and (8), due to the assumption of a perfect flow, no viscosity is taken into account. To include viscous loads and detached flow on the cylinder, a specific empirical formula was derived for oscillating viscous flows on a cylinder: Morison formula (see Morison forces).