Slamming theory

Slam loads, usually of short duration and potentially of large magnitude, occur when an object penetrates the water surface with significant velocity. In the real, physical world, these loads may well be highly sensitive to precise local conditions at the time the object enters (or leaves) the water, such as the exact angle between the object and the orientation of the sea surface. This makes the time history and peak value of slam force hard to model accurately. However, the total impulse due to the slam force, which corresponds to the transfer of momentum between the water and the object, will in general be less sensitive and so the overall effect on the motion of the object can be modelled more accurately.

Slam forces in OrcaFlex can be calculated for 6D buoys and lines. Slam force can be applied to both water entry and water exit, and slam data are defined independently for water entry and exit in order to model these two different situations. In each case, the data may be given in the form of either a constant slam coefficient or as varying with submergence below the surface.

The OrcaFlex slamming models are based on circular cylinders. These correspond directly to the cylinders which make up a spar buoy or towed fish, to a notional surface-normal cylinder representing a lumped buoy, or to the half-segments which make up a line. Clearly the slam force on a cylinder will vary as it passes through the water surface, and the source of this variation differentiates the constant slam coefficient and variable data models: in the former, the variation is due to the changing waterplane area as the cylinder penetrates, and in the latter, the coefficient data themselves encapsulate the variation with submergence.

Note:

The slam models used by OrcaFlex (and described below) make no allowance for any variation in surface slope and water velocity due to waves along the length or across the diameter of each cylinder. Instead, the surface outward normal $\vec{n}$ is the instantaneous value at the surface point vertically above or below the centre of volume of the cylinder, and the relative normal velocity $\vn$ is the instantaneous value at a specific point on the cylinder. If the horizontal extent of the cylinder is large enough to be a significant fraction of wavelength for a sufficiently large portion of the wave spectral energy, then this may lead to the slam models giving poor results in those circumstances. We therefore recommend that short waves should be suppressed when modelling slamming on a large object in a random sea. This can be done by setting the maximum relative frequency on the waves data form.

Constant slam data

For constant slam coefficients, the slam $(\fs)$ or water exit $(\fe)$ force on a cylinder is given by \begin{align} \fs &= +\tfrac12 \rho\ C_\mathrm{s} A_\mathrm{w} \vn^2\ \vec{n} \\ \fe &= -\tfrac12 \rho\ C_\mathrm{e} A_\mathrm{w} \vn^2\ \vec{n} \end{align} where

$\rho=$ water density.

$C_\mathrm{s}$ and $C_\mathrm{e}$ are slam coefficients for entry (subscript s for slam) and exit (subscript e) respectively. These are the user-specified constant slam data values.

$\vn$ is the component in the surface normal direction of the cylinder velocity relative to the fluid velocity. Note that this velocity is measured at different points for buoys and lines.

$\vec{n}$ is a unit vector in the water surface outward normal direction. This (together with the minus sign in the exit slam force formula) ensures that the slam force opposes the penetration of the water surface in both directions.

$A_\mathrm{w}=$ slam waterplane area. In general, this is the instantaneous waterplane area, the area of the intersection of the water surface and the cylinder, which is calculated by OrcaFlex. Hollow buoys and lumped buoys are, separately, treated as special cases.

When a constant slam or water exit coefficient is used, the slam or water exit force is only applied while the cylinder is surface-piercing; no force is applied when the cylinder is fully-submerged, since the waterplane area is by definition zero at that point.

This is a very simple model of slamming which does not take into consideration the details of the physics behind the process. DNV RP-H103 (sections 3.2.9 and 3.2.11) or Faltinsen (chapter 9) provide more background on the theoretical arguments for this simplified model, and give information on suitable values for the constant coefficients.

Variable slam data

Alternatively you can apply slam data that are variable with submergence of the centre of volume of the cylinder below the surface. The slam data are specified by giving the rate of change of the added mass with submergence in the non-dimensional form \begin{equation} \frac{\ud \Ca \big(\frac{h}{s}\big)}{\ud \big(\frac{h}{s}\big)} \end{equation} where

$\Ca=$ added mass coefficient, relative to the total fully-submerged displaced mass. This is the added mass for motion normal to the surface, divided by the fully-submerged displaced mass, $\rho \VT$, where $\rho$ is water density and $\VT$ total cylinder volume.

$h=$ submergence of the centre of volume of the cylinder. This is the distance from the surface to the centre of volume, measured positive in the inwards surface normal direction.

$s=$ half-span of the cylinder in the surface normal direction. The half-span is half the instantaneous span of the cylinder in the surface normal direction; this is given by $s = r\sin\theta + \frac12 L \cos\theta$, where $r$ is cylinder outer radius, $L$ is cylinder length and $\theta$ is the angle between the cylinder axis and the outward surface normal direction $(0 \leq \theta \leq \pi/2)$.

$\hovers=$ normalised (non-dimensional) submergence. This varies from -1 when the cylinder is just about to enter the water, to 0 when it is half submerged, to +1 when it is just fully-submerged, and continues increasing with depth of the fully-submerged cylinder below the surface.

The user-specified variable slam data therefore represent the gradient of the added mass coefficient when viewed as a function of the non-dimensionalised submergence, $\hovers$. For a horizontal cylinder this is the same as the non-dimensional quantity shown by the dotted line in DNV recommended practice RP-H103 figure 3.5; for a cylinder inclined to the surface it is the natural generalisation of that quantity.

Notes:	Added mass is frequency dependent, and the rate of change of high frequency added mass coefficient should be used for variable slam data that are used for water entry, whereas the rate of change of low frequency added mass coefficient should be used for water exit. See DNV RP-H103 sections 3.2.9 and 3.2.11.
	These rates of change of high- and low frequency added mass used to represent variable slam data might well not be consistent with any variable added mass data that you specify for the added mass effects on the object. The variable added mass data which you supply should be for the expected dominant frequency of oscillation of the object.

Force formulae with variable slam data

When using variable slam data, the slam $(\fs)$ or water exit $(\fe)$ force is given by \begin{align} \fs &= \rho\ V_\mathrm{T} \frac{\ud C_a}{\ud \big(\frac{h}{s}\big)} \frac1s\ \vn^2\ \vec{n} \\ \fe &= \frac12 \rho\ V_\mathrm{T} \frac{\ud C_a}{\ud \big(\frac{h}{s}\big)} \frac1s\ \vn^2\ \vec{n} \\ \end{align} where

$\rho=$ water density

$V_\mathrm{T}=$ total cylinder volume

$\frac{\ud C_a}{\ud (h/s)}=$ user-specified non-dimensional variable slam data

$\vn=$ component in the surface normal direction of the cylinder velocity relative to the fluid velocity. Note that this velocity is measured at different points for buoys and lines

$n=$ unit vector in water surface outward normal direction.

Notes:

For a cylinder whose axis is parallel to the surface plane, the half-span $s$ is equal to the radius $r$ of the cylinder. The above formula for slam force is then exactly equivalent to that given for $f_s$ (which is slam force per unit length) in DNV recommended practice RP-H103 section 3.2.13.7. And the above formula for water exit force is exactly equivalent to that given for $F_e$ (water exit force per unit length) in DNV RP-H103 section 3.2.13.9.

The formula for entry slam force is based on conservation of added momentum, whereas the formula for water exit force is based on conservation of energy. This is the source of the extra factor $\frac12$ that appears in the formula for water exit force.

Variable slam data requirements

For water entry, the variable slam data are the rates of change of the high frequency added mass coefficient with normalised submergence; these normally take positive values, and so the entry slam force acts in the upward direction. For water exit, on the other hand, the variable slam data are the rates of change of the low frequency added mass coefficient with normalised submergence: these are usually negative-valued, and so the exit slam force is downwards.

OrcaFlex variable slam data must satisfy the following requirements:

they must contain at least two values of normalised submergence, $\hovers$.
The smallest value of normalised submergence for which data are provided must be -1, and the specified rate of change of $\Ca$ for this value of normalised submergence must be zero. The rate of change of $\Ca$ may well rise very quickly as the object enters the water, over a short range of normalised submergence $\hovers$, as in our example. But it must ramp up from zero at $\hovers=-1$ in order to avoid a discontinuous slam or water exit force as the object enters or leaves the water.
The rate of change of $\Ca$ for the largest value of normalised submergence given must also be zero.

The data are truncated at both ends of the range, so the rate of change of added mass will be taken to be zero outside the range of values of normalised submergence that you give. This ensures that the slam or water exit force is zero when the object is out of the water, and that the force also reduces to zero once the object gets far enough below the water surface.

There is no upper limit on the value of normalised submergence. So if you want slam or water exit force to be applied even when the object is far below the surface, then you can enter as large a value as you like for normalised submergence.

Differences from DNV RP-H103

This slam force model used in OrcaFlex closely matches that given in DNV-RP-H103. However in order to cater for the more general situation covered by OrcaFlex, the OrcaFlex model extends that of DNV RP-H103 in a number of ways described above. These are, in summary,

The OrcaFlex model includes surface movement due to waves. Therefore the velocity used, $\vn$, is the normal component of cylinder velocity relative to the water surface allowing for the velocity of the surface due to waves, and the slam force is applied in the surface normal direction allowing for waves. In DNV RP-H103, the slam force is always in the vertical direction. For variable coefficients a still, horizontal sea surface is assumed (3.2.13.7), and for constant coefficients, only the vertical component of wave velocity is considered (3.2.9.3).
For constant slam data, the instantaneous waterplane area is used for the slam area $A_\mathrm{w}$. In DNV RP-H103 (section 3.2.9) the projected area is used.
For variable slam data the normalised submergence, which is $\small{h/r}$ in DNV RP-H103, is generalised to $\hovers$, where $s$ is the half-span of the object in the surface normal direction. This then permits non-horizontal cylinder orientation.