Environment: Setting up a random sea

This section gives information on how to set up a random sea using OrcaFlex's modelling facilities.

The most common requirement is to produce a realistic wave train which includes a design wave of specified height $H_\textrm{max}$ and associated period $T_\textrm{ass}$. Other requirements are also encountered however, and it is sometimes useful to impose additional conditions for convenience in results presentation, etc.

The height and period of the maximum design wave may be specified by the client, but on occasion we have to derive appropriate values ourselves, either from other wave statistics (for example a wave scatter table, giving significant wave heights $\Hs$ and average periods $\Tz$) or from a more general description of weather (such as wind speed).

Having decided on the values of $H_\textrm{max}$ and $T_\textrm{ass}$, we select an appropriate wave train as follows, using the facilities available in OrcaFlex.

• Set the significant wave height ($\Hs$) and average period ($\Tz$) for the design storm, and the wave spectrum.
• Set the number of wave components (typically 100).
• Search the time history of wave height for a particular wave rise (trough to crest) or fall (crest to trough) which has the required total height and period. If no wave with the required characteristics can be found, then adjust $\Hs$ and $\Tz$ slightly and repeat.
• When the required design wave has been located, set the wave time origin and duration so that the design wave occurs within the simulation time.

When choosing the simulation time origin and duration, you should allow sufficient time before and after to avoid starting transients and to collect all important responses of the system to the design wave. A typical random sea simulation may represent 5 or 6 average wave periods (say 60-70 seconds for a design storm in the North Sea) plus a build-up period of 10 seconds. If the system is widely dispersed in the wave direction, then the simulation may need to be longer, to allow time for the principal wave group to pass through the whole system. Since short waves travel more slowly than long ones, this affects simulations of mild sea states more than severe seas.

Setting the sea state data

The ISSC spectrum (also known as Bretschneider or modified Pierson-Moskowitz) is appropriate for fully-developed seas in the open ocean. The JONSWAP spectrum is a variant of the ISSC spectrum in which a peak enhancement factor, $\gamma$, is applied to give a greater concentration of energy in the mid-band of frequencies. JONSWAP is commonly specified for the North Sea. The Ochi-Hubble and Torsethaugen spectra enable you to represent sea states that include both a remotely generated swell and a local wind generated sea.

Two parameters are sufficient to define an ISSC spectrum – we use $\Hs$ and $\Tz$ for convenience. For the JONSWAP spectrum, five parameters are required: $\Hs$, $\Tz$, $\gamma$, and two additional parameters $\sigma_a$ and $\sigma_b$ (labelled $\sigma_1$ and $\sigma_2$ in OrcaFlex) defining the bandwidth over which the peak enhancement is applied. As a minimum, you must supply values for $\Hs$ and $\Tz$; you may then choose to specify some or all of the remaining parameters. For the North Sea it is common to set $\gamma{=}3.3$. If you have to do a systematic series of analyses in a range of wave heights, there are advantages in keeping $\gamma$ constant. Note that a JONSWAP spectrum with $\gamma{=}1.0$ is identical to the ISSC spectrum with the same $\Hs$ and $\Tz$.

Choice of wave spectrum can cause unnecessary pain and suffering to the beginner. For present purposes, the important point is to get the design wave we want embedded in a realistic random train of smaller waves. The spectrum is a means to this end, and in practice it matters little what formulation is used. The one exception to this sweeping statement may be double peaked spectra (e.g. Ochi-Hubble or Torsethaugen).

Setting the number of components

OrcaFlex generates a time history of wave height by dividing the spectrum into a number of component sine waves of constant amplitude and phase. The phases associated with each wave component are pseudo-random. OrcaFlex uses a random number generator and user-defined seed to assign phases. The sequence is repeatable, so the same seed will always give the same phases and consequently the same train of waves. The wave components are added assuming linear superposition to create the wave train. Ship responses and wave kinematics are also generated for each wave component and added assuming linear superposition. You may specify the number of wave components to use; more components give greater realism but a greater computing overhead.

The time history generated is just one of an infinite number of possible wave trains which correspond to the chosen spectrum – in fact there are an infinite number of wave trains which could be generated from 100 components, a further infinite set from 101 components and so on.

Ideally, we would use a full Fourier series representation of the wave system which would typically have several thousand components (the number depends on the required duration of the simulation and the integration time step). This would be prohibitively expensive in computing time however, so we use a much reduced number of components, as noted above. This necessarily involves some loss of randomness in the time history generated. For a discussion of the consequences of this approach, see Tucker et al (1984).

Finding a suitable design wave

A frequent requirement is to find a section of random sea which includes a wave corresponding in height and period to a specified design wave. OrcaFlex provides preview facilities for this purpose. If you are looking for a large wave in a random sea, say $H_\textrm{max}=1.9\Hs$, then use the list events command (on the waves preview page of the environment data form) to ask for a listing of waves with height greater than $1.7\Hs$, say. It is worth looking over a reasonably long period of time at first – say $t = 0\textrm{s}$ to $50{,}000\textrm{s}$ or even $100{,}000\textrm{s}$. OrcaFlex will then search that time period and list wave rises and falls which meet your criterion.

Suppose that the list shows a wave fall at $t = 647\textrm{s}$ which is close to your requirement. Then you can use the view profile command to inspect this part of the wave train, by asking OrcaFlex to draw the sea surface elevation for the period from $t = 600\textrm{s}$ to $t = 700\textrm{s}$, say. You will then see the large wave with the smaller waves which precede and follow it.

Note that, when you use the preview facility, you must specify both the time and the location (X,Y coordinates) since a random wave train varies in both time and space. For waves going in the positive $X$-direction (wave direction = 0°), the wave train at $X{=}0$ differs from that at $X{=}300\textrm{m}$.

You can use the preview facility to examine the wave at different critical points for your system. For example, you may be analysing a system in which lines are connected between ship A at $X{=}0$ and ship B at $X{=}300\textrm{m}.$ it is worth checking that a wave train which gives a design wave at ship A does not simultaneously include an even higher wave at ship B. If you want to investigate system response to a specified design wave at both ship A and ship B, then you will usually have to do the analysis twice, once with the design wave at ship A and once at ship B.

If no wave of the required characteristics can be found, then adjust $\Hs$ and $\Tz$ slightly and repeat. As we noted above, the important point is to get the design wave we want embedded in a realistic random train of smaller waves. This is often conveniently done by small adjustments to $\Hs$ and $\Tz$. We need make no apology for this. In the real world, even in a stationary sea state, the instantaneous wave spectrum varies considerably and $\Hs$ and $\Tz$ with it. For further discussion see Tucker et al (1984).

If you are using an ISSC spectrum, or a JONSWAP spectrum with constant $\gamma$, then you can make use of some useful scaling rules at this point. In these two cases, provided the number of wave components and the seed are held constant, then

• For constant $\Tz$, wave elevation at any time and any location is directly proportional to $\Hs$. For example, if you have found a wave at time $t$ which has the period you require but is 5% too low in height, then increasing $\Hs$ by 5% will give you the wave you want at the same time $t$.
• For constant $\Hs$, the time between successive wave crests at the spatial origin $(X{=}0, Y{=}0)$ is proportional to $\Tz$. For example, if you have found a wave at the origin at time $t$ which has the height you require but the period between crests is 5% less than you want, increasing $\Tz$ by 5% will give you the wave you want, but at time $1.05t$.

These scaling rules can be helpful when conducting a study of system behaviour in a range of wave heights. We can select a suitable wave train for one wave height and scale to each of the other wave heights. This gives a systematic variation in wave excitation for which we may expect a systematic variation in response. If the wave trains were independently derived, then there would be additional scatter.