|
Line theory: Hydrodynamic and aerodynamic loads |
|
|
Line theory: Hydrodynamic and aerodynamic loads |
Hydrodynamic and aerodynamic drag forces, represented by the drag term in Morison's equation, are applied to the line. The same drag formulation is used for hydrodynamic and aerodynamic drag forces.
| Note: | Aerodynamic drag is only included if the include wind loads on lines option is enabled in the environment data. |
The drag forces applied to a line are calculated using the cross-flow principle. That is, the fluid velocity relative to the line $\vr$ is split into its components $\vn$ and $\vz$ normal and parallel to the line axis. The drag force normal to the line axis is then determined by $\vn$ and its $x$- and $y$-components $\vx, \vy$; the drag force parallel to the line axis is determined by $\vz$.
The drag force formulae use drag coefficients, $\CDx$, $\CDy$ and $\CDz$, and the drag areas appropriate to each direction. The drag coefficients are given on the line type data form, but can also be modified to model wake interference from other lines upstream.
For the directions normal to the line axis ($x$ and $y$) the drag area is taken to be the projected area $\dn l$, where $\dn$ is the normal drag / lift diameter and $l$ the length of line represented by the node. For the axial direction the drag area is taken to be the skin surface area $\pi \da l$ where $\da$ is the axial drag / lift diameter.
OrcaFlex offers a choice of formulations for the drag force $\fD$. These differ in the way in which the drag force components vary with the incidence angle $\phi$ between the flow and the line axial direction. Some methods are more suited to particular configurations than others: they are reviewed in Casarella and Parsons.
Notation:
$\rho=$ fluid density
$p=$ proportion wet (hydrodynamics) or proportion dry (aerodynamics), as appropriate
$\fD = (\fDx,\fDy,\fDz)$ in local coordinates.
\begin{align} \fDx &= \tfrac12\ p\ \rho\ \dn l\ \CDx \vx \lvert\vn\rvert \label{fDxStd} \\ \fDy &= \tfrac12\ p\ \rho\ \dn l\ \CDy \vy \lvert\vn\rvert \label{fDyStd} \\ \fDz &= \tfrac12\ p\ \rho\ \pi\da l\ \CDz \vz \lvert\vz\rvert \end{align} This formulation is the most commonly-used, and was originally the only one offered by OrcaFlex. It has been promoted by various authors, including Richtmyer, Reber and Wilson, and is appropriate for general flow conditions.
\begin{align} \fDx &= \tfrac12\ p\ \rho\ \dn l\ \CDx \vx \lvert\vn\rvert \label{fDxPode} \\ \fDy &= \tfrac12\ p\ \rho\ \dn l\ \CDy \vy \lvert\vn\rvert \label{fDyPode} \\ \fDz &= \pm\tfrac12\ p\ \rho\ \pi\da l\ \CDz \lvert \vec{v}\rvert^2 \end{align} The expressions for $\fDx$ (\ref{fDxPode}) and $\fDy$ (\ref{fDyPode}) are the same as the standard formulation, (\ref{fDxStd}) and (\ref{fDyStd}) respectively. The sign of $\fDz$, i.e. whether it is towards end A of the line or towards end B, is the same as the axial component of the relative flow vector.
This formulation is preferred by some analysts for systems with near-tangential flow.
| Warning: | The Pode formula for $\fDz$ is discontinuous at $\phi{=}90\degree$, since the direction of $\fDz$ is undefined (the axial component of the flow vector is zero). In this case OrcaFlex sets $\fDz$ to zero. |
\begin{align} \fDx &= \tfrac12\ p\ \rho\ \dn l\ \CDx \vx \lvert\vn\rvert + \tfrac12\ p\ \rho\ \pi\da l\ \CDz \vx(\lvert\vec{v}\rvert-\lvert \vn\rvert) \\ \fDy &= \tfrac12\ p\ \rho\ \dn l\ \CDy \vy \lvert\vn\rvert + \tfrac12\ p\ \rho\ \pi\da l\ \CDz \vy(\lvert\vec{v}\rvert-\lvert \vn\rvert) \\ \fDz &= \tfrac12\ p\ \rho\ \pi\da l\ \CDz \vz \lvert\vec{v}\rvert \end{align}
These drag force formulae can be presented instead in a form which shows more clearly the way in which drag force varies with incidence angle.
Consider the case where the line is axially symmetric, so $\CDx=\CDy=\CDn$, say. Let $\phi$ be the incidence angle between the flow vector $\vec{v}$ and the line axis, then $\vz = \lvert\vec{v}\rvert\cos\phi$ and $\lvert\vn\rvert = \lvert\vec{v}\rvert\lvert\sin\phi\lvert$. Define
$R = \frac12\ p\ \rho\ d\ l\ \CDn \lvert\vec{v}\rvert^2\ $ with $\ d = \dn = \pi\da$
$\mu = \pi\ \CDz / \CDn = \dfrac{\text{(drag force in axial flow)}}{ \text{(drag force in normal flow of same velocity)}}$
$\fDn =$ normal component of drag force $\fD$
Then the various formulations can be written
| Standard: | $\lvert\fDn\rvert = R\ {\sin^2}\phi$ | $\lvert\fDz\rvert = R\ \mu\ {\cos^2}\phi$ |
| Pode: | $\lvert\fDn\rvert = R\ {\sin^2}\phi$ | $\lvert\fDz\rvert = R\ \mu$ |
| Eames: | $\lvert\fDn\rvert = R[(1-\mu)\ {\sin^2}\phi + \mu\sin\phi]$ | $\lvert\fDz\rvert = R\ \mu \cos\phi$ |
You may choose to define normal drag coefficients $\CDx$ and $\CDy$ which vary with Reynolds number. The variable data table specifies the drag coefficient as a function of Reynolds number, $\CD(\Reyn)$.
Reynolds number can be calculated in a number of different ways, as specified by the Reynolds number calculation data item. You should set the Reynolds number calculation data item to match the data source used for your variable drag coefficient data.
Alternatively the normal drag coefficients can be specified to vary with height above seabed, $\zb$. More precisely, we define $\zb$ to be the vertical height above the seabed of the underside of the node, allowing for outer contact diameter. The actual drag coefficient used by the OrcaFlex calculation is given by \begin{equation} \CD = \lambda \C{D_2} + (1-\lambda) \C{D_1} \end{equation} where
$\C{D_1}$ is the drag coefficient on the seabed
$\C{D_2}$ is the drag coefficient away from the seabed
$\lambda$ is the drag variation factor.
These data are all defined on the variable data form.
The drag variation factor $\lambda$ is a function of the height above the seabed normalised by drag diameter, $\zb/\dn$. For nodes lying on the seabed $\CD$ should equal $\C{D_1}$, since $\zb/\dn = 0$, so $\lambda(0)$ should have the value 0. Similarly, for nodes well away from the seabed we expect $\CD$ to be equal to $\C{D_2}$ and so $\lambda$ should take the value 1 for large values of $\zb/\dn$.
The drag coefficients $\C{D_1}$ and $\C{D_2}$ may themselves be specified as varying with Reynolds number, thus allowing specification of a drag force which varies with both Reynolds number and height above seabed.
The hydrodynamic lift force is defined as \begin{equation} \label{fL} \vec{f}_\mathrm{L} = p\ \lvert\un{\times}\uz\rvert\ \frac12\rho\ \dn l\ \C{L}\ \lvert\vt\rvert^2 \ul \end{equation} where
$\un$ is the unit vector in the seabed outward normal direction
$\uz$ is the unit vector in the node z-direction
$\ul$ is the unit vector in the lift force direction $= (\uz{\times}\ut)/\lvert\uz{\times}\ut\rvert$. This is the direction normal to the line axis in the plane of that axis and the seabed normal
$\ut$ is the flow direction for lift purposes $= (\un{\times}\uz)/\lvert\un{\times}\uz\rvert$. This is the transverse direction that is normal to the line axis and in the seabed plane
$\C{L}$ is the lift coefficient
$\vt$ is the component of $\vr$ in the transverse direction $\ut$.
In equation (\ref{fL}), the lift force magnitude is scaled by $\lvert\un{\times}\uz\rvert$, which is equal to $\cos\theta$ where $\theta$ is the angle of the line axis to the seabed plane. The purpose is to scale down the lift force as the line axis becomes more inclined to the seabed plane. If the line axis is parallel to the seabed plane, this factor is 1 and the lift force is not reduced. But as the line axis inclines further the factor reduces until, when $\theta=90\degree$, the factor is zero and no lift force at all is applied.
Another way of thinking about this factor is that it generalizes the standard "parallel" lift force formula $\vec{f}_\mathrm{L} = \frac12\rho\ \dn l\ \C{L}\ \lvert\vt\rvert^2 \ul$ to cases where the line axis is inclined to the seabed. The effect of the factor is equivalent to assuming that the standard formula gives the lift force per unit projected length, instead of per unit arc length. By projected length we mean the length of the projection of the node onto the seabed plane.
The lift coefficient can be specified to vary with Reynolds number in an identical manner to that for drag coefficients.
Reynolds number can be calculated in a number of different ways, as specified by the Reynolds number calculation data. You should set the Reynolds number calculation data to match the data source used for your variable lift coefficient data.
Alternatively the lift coefficient can be specified to vary with height above seabed. In this case the actual lift coefficient used by the OrcaFlex calculation is given by \begin{equation} \C{L} = \alpha\ \C{L_1} \end{equation} where
$\C{L_1}$ is the seabed lift coefficient
$\alpha$ is the lift coefficient decay factor.
These data are all defined on the variable data form.
The lift coefficient decay factor $\alpha$ is a function of the normalised height above the seabed, $\zb/\dn$. For nodes lying on the seabed $\C{L}$ should equal $\C{L_1}$, so $\alpha(0)$ should have the value 1. For nodes well away from the seabed the lift effect dies away and we expect $\C{L}$ to be equal to 0, so $\alpha$ should be 0 for large values of $\zb/\dn$.
The lift coefficient $\C{L_1}$ can itself be specified as varying with Reynolds number which allows specification of a lift force which varies with both Reynolds number and height above seabed.
Separate added mass $(\Ca)$ and added inertia $(\Cm)$ coefficients are given for flow normal to ($x$- and $y$-directions) and axially along ($z$-direction) the line. The added mass coefficients for the $x$ and $y$ directions may be variable; the $z$-coefficient must be constant. Added inertia coefficients in the $x$- and $y$-directions may take the special value '~', which means that they depend on the value of $\Ca$ (which in turn may be itself variable). The different formulations for constant and variable added mass coefficients, and the dependency of $\Cm$ on $\Ca$, follow below.
In its simplest form, the added mass load on each line segment follows the inertia term of Morison's equation. Writing $\al$ for the line segment acceleration relative to earth, we have \begin{equation} \fA = \Cm\Delta\ \af - \Ca\Delta\ \al \end{equation} where
$\fA$ is the added mass load on the line segment
$\Delta$ is the reference mass: the mass of fluid displaced instantaneously by the segment, determined by the proportion wet
$\Cm$ is the added inertia coefficient
$\Ca$ is the constant added mass coefficient
$\af$ is the fluid acceleration relative to earth
$\al=\af-\ar$, where $\ar$ is the fluid acceleration relative to the line segment.
The first term in this expression is often written as $(1{+}\Ca)\Delta\ \af$, and this is the form which is applied when the added inertia coefficient $\Cm$ is assigned the value '~'. The $1$ in $(1{+}\Ca)$ represents what is known as the Froude-Krylov force; the '$\Ca$' term gives rise to the so-called added mass force.
The same expression applies in each of the $x$,$y$ and $z$ component directions for the coefficients $\Cax,\Cay,\Caz$ and $\Cmx,\Cmy,\Cmz$, so for the constant coefficient case the added mass load is \begin{align} \fAx &= \Cmx\Delta\ \afx - \Cax\Delta\ \alx \\ \fAy &= \Cmy\Delta\ \afy - \Cay\Delta\ \aly \\ \fAz &= \Cmz\Delta\ \afz - \Caz\Delta\ \alz \end{align}
You may, if you wish, specify either or both of the normal-direction added mass coefficients $\Cax$ and $\Cay$ in a more precise form, using tabular variable data. Using a constant value, as above, means that the variation with depth of the added mass load is calculated by OrcaFlex through the displaced mass (for force) and proportion wet (for moment): consequently, the variation occurs only while the line segment is surface-piercing. Choosing instead a variable form for $\Ca$ means that you may also define the nature of any variation when the segment is fully submerged and at any depth.
The variable added mass model is based on cylinders which correspond to line half-segments. Each segment is notionally divided centrally across its diameter into a pair of equal-sized cylinders, each of which is therefore associated with a single node positioned at one end of the cylinder. Conversely, each node (other than end nodes) has two such cylinders associated with it, one to either side.
OrcaFlex allows you to define two differing forms of added mass variation: close to the sea surface, and close to the seabed.
Specifically, depth in this case is represented as it is for slamming, by normalised submergence, $\small{h/s}$, for a circular cylinder in a generalisation of DNV RP-H103 3.2.13.
As all the variation is in the coefficient values, the reference mass is constant in this case; we take it to be the mass of fluid displaced by the whole of the segment were it to be fully submerged, regardless of its actual instantaneous submergence. We denote this fully-submerged reference mass by $M$.
The interpretation of the symbol ~ for $\Cm$ is rather more subtle in this case: the reference mass for the $1$ in $(1{+}\Ca)$ is the displaced mass, $\Delta$, but for the (now variable) $\Ca$ part it is the fully-submerged mass $M$. Hence, writing $\Ca\!(\frac{h}{s})$ to signify the variable data form for $\Ca$ in each $(x,y)$ case, we have for $\Cmx{\neq}\sim$ \begin{equation} \fAx = \Cmx\Delta\ \afx - \Cax\!(\tfrac{h}{s})\ M\ \alx \end{equation} and for $\Cmx{=}\sim$ \begin{equation} \fAx = \left(\Delta + \Cax\!(\tfrac{h}{s})\ M\right)\ \afx - \Cax\!(\tfrac{h}{s})\ M\ \alx \end{equation} Likewise for the $y$ component direction we have for $\Cmy{\neq}\sim$ \begin{equation} \fAy = \Cmy\Delta\ \afy - \Cay\!(\tfrac{h}{s})\ M\ \aly \end{equation} and for $\Cmy{=}\sim$ \begin{equation} \fAy = \left(\Delta + \Cay\!(\tfrac{h}{s})\ M\right)\ \afy - \Cay\!(\tfrac{h}{s})\ M\ \aly \end{equation} Since the axial added mass coefficient $\Caz$ is not allowed by OrcaFlex to be variable, only the $\fAx$ and $\fAy$ components differ from the constant coefficient case.
Some example data for $\Ca$ against $\small{h/s}$ are available in DNV, RP-H103, 3.2.13, for instance. You should be aware that the data there represent the high frequency limit so may well be inappropriate for your own model.
| Note: | If you use variable added mass close to surface, you must provide data for at least two values of normalised submergence $\small{h/s}$. The smallest value of $\small{h/s}$ must be -1 (corresponding to the cylinder just about to enter the water), and on physical grounds the corresponding Ca value must be zero. There is no upper limit on the range of $\small{h/s}$: the data will be truncated at both ends of the range. |
To specify variation in added mass close to the seabed, it is more convenient to use height above seabed rather than depth below surface. As with drag and lift, we use vertical height above seabed, but this time normalised by outer diameter $\OD$, so normal added mass coefficients may be defined in terms of $\zb/\OD$. Again as with drag and lift, the height is the vertical height above the seabed of the underside of the node, allowing for outer contact diameter, so $\zb{=}0$ corresponds to the line cylinder just touching the seabed.
In this case, we choose the reference mass to be the instantaneous displaced mass $\Delta$. This may seem counter-intuitive, but consider: (i) close to the seabed, where the added mass coefficient is expected to vary most rapidly, the cylinder is likely to be fully submerged so $\Delta$ is effectively constant in that region, and (ii) we must ensure that added mass falls to zero as the cylinder exits the sea surface. In the close-to-surface case using normalised submergence this happens naturally by forcing the coefficient to take the value zero at $\small{h/s}{=}-1$, but when measuring height above seabed we need to use displaced mass as the reference to achieve this. Hence, for $\Cmx{\neq}\sim$ \begin{equation} \fAx = \Cmx\Delta\ \afx - \Cax\!(\tfrac{\zb}{\OD})\ \Delta\ \alx \end{equation} and for $\Cmx{=}\sim$ \begin{equation} \fAx = \left(1 + \Cax\!(\tfrac{\zb}{\OD})\right)\ \Delta\ \afx - \Cax\!(\tfrac{\zb}{\OD})\ \Delta\ \alx \end{equation} Likewise for the y component direction we have for $\Cmy{\neq}\sim$ \begin{equation} \fAy = \Cmy\Delta\ \afy - \Cay\!(\tfrac{\zb}{\OD})\ \Delta\ \aly \end{equation} and for $\Cmy{=}\sim$ \begin{equation} \fAy = \left(1 + \Cay\!(\tfrac{\zb}{\OD})\right)\ \Delta\ \afy - \Cay\!(\tfrac{\zb}{\OD})\ \Delta\ \aly \end{equation} Some example data for $\Ca$ against normalised height are available in DNV, RP-C205, figure 6-9.
Again, since the axial added mass coefficient $\Caz$ is not allowed by OrcaFlex to be variable, only the $\fAx$ and $\fAy$ components differ from the constant coefficient case.
| Note: | If you use variable added mass close to seabed, you must provide data for at least two values of normalised height above seabed, $\zb/\OD$. There are no restrictions on the values of $\zb/\OD$ other than that they may not be negative. The data will be truncated at both ends of the range. |
If the mass flow rate is non-zero then the centrifugal and Coriolis effects due to this flow are included.
Note that pressure effects of flow rate are not included.
Slam loads on lines are calculated following the OrcaFlex model for slamming on circular cylinders. The slam force is calculated for both water entry and water exit, and each may (independently) use either constant slam coefficients or variable slam data.
The cylinders which this theory requires are the same line half-segments as those used for variable added mass coefficients. The slam load on each of these half-segments is calculated according to the general theory, and this calculated slam load is then added to the load on the associated node for each half-segment.
For both constant and variable slam coefficients, OrcaFlex is required to determine $v\urm{n}$, the component in the surface normal direction of the cylinder velocity relative to the fluid velocity, for the slam force calculation. Both of these velocities are measured, for lines, at the node associated with the cylinder.
The slam force on each half-segment is applied at the associated line node. This is the case for both constant and variable slam data.