A survey of the generation of ocean waves in a test basin

A Survey of the Generation of Ocean Waves in a Test Basin

Ben T. Nohara

Electronics Research & Development Department, Technical Headquarters, Mitsubishi Heavy Industries, 1-1Iwatsuka Nakamura-ku, Nagoya Japan

At present stage the analytical design of wave tolerance for floating structures and vessels is still imperfect due to the mutually complex and nonlinear phenomena between structures and waves. Wave tolerance design is usually carried out through iterative evaluations of results from model tests in a wave basin, and this is done in order to reach a final structural design. The wave generation has then become an important technology in the field of the coastal and ocean engineering.

This paper summarizes the facilities of a test basin and a wave maker in Japan and also surveys the methodology of the generation of ocean waves in a test basin.

Keywords: Complex Waves, Generation of Ocean Waves, Real-time Algorithm, Wave Basin

Introduction

The analytical design of wave tolerance for floating structures and vessels is still imperfect due to the mutually complex and nonlinear phenomena between structures and waves. Especially, we have no design experience of a huge floating structure planned for an airport (Research and Development of Mega-float, 1996). How to maintain a high level of wave tolerance performance is an important research item in the construction of floating or coastal structures. Wave tolerance design is usually carried out through iterative evaluations of results from model tests in a wave basin, and this is done in order to reach a final structural design. The wave generation using a wave maker in a test basin has then become an important technology in the field of the coastal and ocean engineering.

We have been able to clarify wave characteristics using numerical formulas of ocean waves developed during 19^th century. Major wave theories from the 19^th century are as follows (Lamb, 1932):

- 1802 Gerstner finite amplitude wave theory(trochoidal wave) - 1844 Scott Russell discovery of solitary wave - 1845 Sir G. B. Airy Airy linear wave theory - 1847 Stokes Stokes wave theory - 1871 Boussinesq solitary wave theory - 1876 Rayleigh solitary wave theory - 1895 D.J.Korteweg and G.de Vries cnoidal wave

The fundamental theories of ocean waves were established in the 19^th century, however, it took another half century for marine and civil engineers to put these theories to practical use. Sverdrup and Munk developed the theory of relations for wave forecasting (Sverdrup and Munk, 1947) based on the concept of the significant wave ^*. The significant wave, with a wave elevation and a wave period, represents the diversity of the irregular waves. By applying this theory to the landing operations on Normandy during World War II, the Allied Powers won a decisive victory. Pierson analyzed the creation, development, and propagation of waves using the spectrum concept which is based on the inconsistency of the irregular waves (Pierson, Neumann, and James, 1955).

The irregularity of ocean waves is currently considered as the fundamental characteristic upon which wave tolerance designs of floating structures and vessels are based. It has been difficult to generate multi-directional irregular waves, so-called short crested waves, in the test basin in order to evaluate the wave tolerance performance. How to generate multidirectional irregular waves iteratively, efficiently, and with good accuracy is a recently established technology.

This paper summarizes the facility of a test basin and a wave maker and also surveys the methodology of the generation of ocean waves using a wave maker in a test basin (Nohara, 1997C). The contents include

(1)the outline of a test basin and some wave maker types,

(2)the generation of ocean waves in a test basin,

(3)the real-time algorithm for generation of ocean waves,

(4)future problems to be solved,

(5)concluding remarks.

Nomenclature

a- Amplitude of the water surface elevation

b - Wave paddle width

g - Acceleration due to gravity

h - Uniform water depth in the test basin

j - Time step

k - Wave number

t - Time

F - Transfer function for wave generation

G - Directional distribution function

M - Number of component waves in direction

N - Number of component waves in frequency

S - Frequency spectrum

S_h- Directional spectrum

e - Random phase lag

g - Index number of the directional concentration

h - Water surface elevation

q - Propagation direction

s - Angular frequency

G - Gamma function

Suffix

i - denotes the i-th segmented wave paddle

n - denotes the n-th frequency

m - denotes the m-th propagation direction

^ - denotes the long-crested irregular wave

~ - denotes the short-crested irregular wave

Test Basin and Wave Maker

Most test basins used for ocean engineering applications as well as coastal engineering applications are equipped with segmented wave makers on one side only. Passive wave absorbers are then generally placed on the basin termination opposite to the wave maker, whereas the two lateral sides are either left reflective. In order to generate multidirectional irregular waves with a given directional wave spectrum uniformly throughout the basin, wave reflection on the lateral side walls of the basin is utilized positively.

Figure 1 shows the test basin of Mitsubishi Heavy Industries, which is not only for seakeeping and manoeuvring tests of vessels but also for wave tolerance tests of coastal structures. This basin is 160 m long, 30 m wide and 3.3 m deep, one of the largest scale facilities in the world. The carriage suspending a test body can be moved along X and Y directions simultaneously to evaluate the directional performance against the wave. The multi-segmented wave maker which consists of 75 wave paddles of 0.4 m width is installed in the shorter side.

Figure 2 shows the driving mechanism which is driven by the brushless electric servo motors and by the combined mechanism of the ball screws and the linear guides. Figure 3 shows the front view of wave paddles. This is the "flap type" ^** wave maker. The motion of the flap shows an arc around the hinge under the water. Major dimensions of this wave maker are shown in Table 1. The converted wave height at the real sea state is equivalent to 6 m to 30 m based on the model scale.

Generation of Ocean Waves

A fundamental wave of ocean waves is formulated as a sinusoidal function, which is based on the potential wave theory (Crapper, 1984). Therefore various kinds of ocean waves can be written as a linear superposition of a large number of sinusoidal waves. Above all, the so-called short crested waves are multi-directional irregular waves, all traveling independently of one another in different directions with different frequencies.

Model tests using multi-directional irregular waves is in the process of becoming a current practice. However, other several waves must be used to obtain the basic data.

The following linear waves such as regular waves, long-crested irregular waves, multi-directional irregular waves, and complex waves are obtained using small amplitude wave theory (Crapper, 1984) under the linearization of the boundary condition.

Regular Waves

The regular wave is the artificial but fundamental wave for obtaining the basic data acting on the test body. The regular wave with its propagation in q direction is generated by the wave paddle motion described in the following. The wave paddle motion h_i(t) of the i-th segmented wave paddle at time t can be formulated as

(1)

where a, s, b and t denote amplitude, angular frequency, wave paddle width and time, respectively, and F indicates transfer function for wave generation. F is represented by

(2)

for flap type wave paddle (Biesel and Suquet, 1951), where h means the uniform water depth in the basin. Moreover, k is wave number and a solution of the dispersion relation as follows.

s² = kg tanh (kh) (3)

Here g indicates the acceleration due to gravity.

Figure 4 shows oblique regular waves, with a wave height of 0.2 m and a wave period of 2.0 s generated by using the real-time algorithm described the next section.

Long-Crested Irregular Waves

The long-crested irregular wave has its original spectrum such as ISSC (Huang, Tung and Long, 1990), JONSWAP (Hasselmann, et al., 1973), and Bretschneider-Mitsuyasu (Mitsuyasu, 1970) etc., therefore can be considered as a linear superposition of a large number of sinusoidal waves with different frequencies. However this wave also progresses toward a direction which is the same of the regular wave.

In the same manner as the regular wave, the flap motion h_i(t) can be formulated as

(4)

where N, s _n, k_n ,e_n and F_n are the number of component waves in frequency, the angular frequency of n-th component wave, the wave number of n-th component wave, the random phase lag defined from 0 to 2p of n-th component wave and the transfer function of n-th component wave, respectively. F_n is represented by the following equation for flap type wave paddle.

(5)

Here k_n is obtained by

s_n² = k_ng tanh (k_nh)^. (6)

Moreover, a_n is the amplitude of n-th component wave and represented by

(7)

where S denotes the frequency spectrum and Ds_n represents the minute spectrum bandwidth.

To calculate Eq. (7) comes into question in terms of the stationary characteristics of generated random waves. Nohara indicated that the energy equivalence frequency division method resulted in astronomical figures for the iteration period (Nohara, 1997A: Nohara, 1998).

Multi-Directional Irregular Waves

This wave is the so-called short crested wave which has a number of directional components added to the frequency components of the irregular wave. To generate this wave, the wave paddle motion h_i(t) can be formulated by the double summation model (Takayama and Hiraishi, 1989)as

(8)

where M, a_nm, q_m, and e_nm are the number of component waves in a given direction, the amplitude of nm-th component wave, the m-th propagation direction, and the random phase lag defined from 0 to 2p of nm-th component wave, respectively. The amplitude of nm-th component wave, a_nm, is represented by

(9)

where Dq_m represents the directional resolution and Sh denotes directional spectrum which is given b

S_h (s, q) = S(s)G(q)^. (10)

G(q) denotes directional distribution function which is represented by

(11)

where G, g and q₀ mean Gamma function, the index number of the directional concentration and main wave stream line, respectively (Maeda, et al., 1995).

Figure 5 shows multi-directional irregular waves, with a significant wave height of 0.2 m, and a significant wave period of 2.0 s for a Bretschneider-Mitsuyasu spectrum, generated by using the real-time algorithm described the next section.

Complex Waves (Nohara, 1997B)

This wave is defined by the summation of long-crested irregular waves and short-crested irregular waves. Complex waves represent the waves, which propagate a long distance, superposed by the waves, which propagate a relatively short distance. The former waves normally become long-crested irregular waves while progressing with vanishing short wavelength components and the latter waves are short-crested irregular waves.

The wave paddle motion h_i(t) for the complex wave can be formulated as the summation of some regular waves and irregular waves; it then follows from Eq. (1) and Eq. (4) that it can be mathematically written as:

(12)

In this equation, the circumflex ^ and ~ indicate parameters of the long-crested irregular wave and the short-crested irregular wave, respectively. The number of is relatively small compared with the number of .

Real-Time Algorithm For Generation Of Ocean Waves

Numerical generation of wave signals, especially multi-directional irregular waves and /or complex waves, is a troublesome task because it needs the evaluation of many trigonometric functions. To complete the computation within a time-step of the wave generation is too hard even by a high performance computer. Therefore wave generation signals for all wave maker units usually have to be calculated beforehand. That is, an operator of a wave maker has to wait a long stand-by time to compute generation signals defined by a next wave condition, because the preparation of calculation of irregular wave needs a much longer computation time than the duration of actual wave generation (Matsuura, Yamaguchi, Yamamoto, and Nohara, 1997).To eliminate the problem described above, the author has formulated and employed the iterative computation method for the generation of various kinds of waves in real-time (Nohara, 1995: Nohara, Yamamoto, and Matsuura, 1995) and established its associated control hardware (Nohara, Yamamoto, and Matsuura, 1996A). Moreover, the unified algorithm for the generation of various kinds of waves is obtained (Nohara, 1997B).

Regular Waves

Let Eq. (1) be changed to the discrete time equation as follows:

(13)

where j denotes the j-th time step.

Here

(14)

(15)

then

h_ij = x_i cos(jsDt) + z_i sin(jsDt)^. (16)

Now let j_ij and F_ij be

(17)

then the following iterative equation is obtained. That is,

(18)

where

a = cos(sDt) (19)

b = sin(sDt)^. (20)

As a result of that, and by calculating Eq. (14), Eq. (15), Eq. (19), and Eq. (20) in advance, , i.e., is obtained by Eq. (18) without the calculation of a large number of the trigonometric function.

Long-Crested Irregular Waves

Similarly, in the case of the irregular wave, the following way be shown.

Let Eq. (4) be changed to the discrete time equation as follows:

(21)

Here

(22)

(23)

then

(24)

Now let j_nij and F_nij be

(25)

then the following iterative equation is obtained. That is,

(26)

where

a_n = cos(s _nDt) (27)

b_n= sin(s _nDt). (28)

By calculating Eq. (22), Eq. (23), Eq. (27), and Eq. (28) in advance, is obtained by Eq. (26) without the calculation of a large number of the trigonometric function.

Finally, is obtained by Eq. (24) as follows:

(29)

Multi-Directional Irregular Waves

The case of the multi-directional irregular wave is shown similarly.

Let Eq. (8) be changed to the discrete time equation as follows:

(30)

Here

(31)

(32)

then Eq. (24) is obtained.

Now let j_nij and F_nij be given by Eq. (25). Then Eq. (26) and Eq. (29) are obtained.

Complex Waves

The calculation procedure is similar and, therefore, only resulting expressions are shown.

The discrete time equation is:

(33)

the parameters involved are:

(34)

(35)

The equation which is rewritten by using Eq. (34) and Eq. (35) in Eq. (33) is :

(36)

where j_nij, F_nij are given by:

(37)

(38)

(39)

(40)

leading to the following iterative equation:

(41)

where

(42)

(43)

(44)

(45)

(46)

(47)

Structure of the Real-Time Algorithm and its Performance

The structure of the iterative Eq. (18), Eq. (26) and Eq. (41) is now considered. The matrices of coefficients of Eq. (18), Eq. (26) and Eq. (41) are the orthogonal matrices which represent rotation. So, j_* and F_* are guaranteed to be stable numerically depending on this calculation process.

Figure 6 shows the structure of Eq. (18), Eq. (26) and Eq. (41) by expressing them in the form of digital filters. This structure is a lattice filter in the field of signal processing (Zelniker and Taylor, 1994) and has the calculation error-less property (Nohara, 1997B).

The performance of the developed real time algorithm can be evaluated as follows. For example, the generation of multi-directional irregular waves needs M(N+1)-times calculation of trigonometric functions per a time step and a wave paddle (Eq.(30)). In contrast, the real time algorithm needs some sum of product calculations only (Eq.(26) and Eq.(29)). Under the condition of M=70 and N=200, the direct calculation method takes 143 ms, but the real time algorithm takes only 85 ms (Nohara, 1996C) on the operation of the computational hardware of TMS320C30 (Texas Instruments, 1993). This means that the real time algorithm has the performance of about 1600 times faster in the execution speed compared with the direct method.

Figure 7 shows the theoretical (expected) directional spectrum, which has the parameters: a significant wave height of 0.1 m, a significant wave period of 1.0 s for a ISSC frequency spectrum. Figure 8 shows the measured directional spectrum with the same parameters of Figure 7. Figure 8 obtained from the generated waves using the developed algorithm in the wave basin is a good agreement with a theoretical one.

Future Problems

This paper does not present the non-stationary waves, but describes the stationary waves. The research on the generation of the non-stationary waves, such as a transiently concentrated wave, a tsunami, etc., has just begun. The research of the mechanism of the non-stationary waves is also in an initial stage.

Moreover, the wave which has the very low frequency spectrum of the envelope of the water surface elevation can do severe damage to coastal structures. The author’s group is trying the generation of the irregular wave with a given envelope (Nohara and Matsuura, 1998). The author considers that a seismic wave has similar characteristics to this.

We can get the valuable and precise data from the model test in a test basin. However, this process consumes time and costs a great deal. The author has proposed the numerical basin (Nohara, Yamamoto, Matsuura, 1996B) in which the software program can calculate the mutual phenomena between the test body and the wave. However, the hydrodynamic calculation of waves and a test body takes a lot of computer resources such as CPU power and memory amount, etc. The high speed and cost effective calculation method of the hydrodynamics which includes non linear phenomena is an important problem.

Concluding Remarks

This paper describes the facilities of a test basin and a wave maker. The author surveys the mathematical formulation of the generation of ocean waves using a wave maker in a test basin. The mathematical formulation for linear waves such as regular waves, long-crested irregular waves, multi-directional irregular waves, and complex waves is given by small amplitude wave theory under the linearization of the boundary condition.

The author shows that the real time algorithm for the generation of linear waves is formulated by the iterative computational method. The structure of the real time algorithm is a lattice filter in the signal processing field. The calculation speed performance of the developed algorithm is about 1600 times faster than the direct calculation method.

Some future problems to be solved such as the generation of the non-stationary waves and the very low frequency envelope spectrum, etc., are also presented.

71 % of the earth’s surface is water. We live on an Ocean Planet. The seas play an important role in the resources as well as the environment of the earth. For example, there is the cobalt-rich crust, which consists of 0.5 % cobalt or more, in the sea bed. The North Pacific Ocean bed has several times the amount of cobalt deposits than land deposits. The manganese nodule which contains high purity (15 % to 30 %) of manganese could possibly be found in the sea bed at depths of 3,000 m or more.

The sea is the origin of life from an evolutionary point of view. The sea fosters thousands of kinds of bacteria per gram of submarine sediment. Moreover, the sea is a rich repository of natural food and we have been using the sea as a means of transportation since the Age of Great Voyages.

The sea is precious to our lives. Ocean development will play a key role for our future evolution. Ocean space management as well as land management should be a most significant subject of the 21^st century.

The author will find great pleasure if this survey paper should contribute to the reference of a broad range of scientists and engineers.

Invited at COBEM97 – 14th Brazilian Congress of Mechanical Engineering, 8 – 12 December 1997, Bauru – SP/Brasil

Manuscript received: August 1999, Technical Editor: Haus Iugo Weber.

** The another common type of a wave maker is the "piston type" which works in a back-and-forth motion like a piston of a reciprocal engine.

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*

significant wave originally defined as the mean height or period of the highest 1/3 of the irregular waves.

Publication Dates

• Publication in this collection
  18 Dec 2000
• Date of issue
  2000

History

• Received
  Aug 1999