The Propagation of the Waves in the CTO S.A. Towing Tank

Abstract The paper presents the results of research focused on the wave propagation in the CTO S.A. deepwater towing tank. In the scope of paper, the wavemaker transfer function was determined for regular waves, based on the Biésel Transfer Function and further for irregular waves, based on Hasselman model of nonlinear energy transfer. The phenomena: wave damping, wave breakdown and wave reflection, were measured, analyzed and mathematically modeled. Developed mathematical models allow to calculate the impact of mentioned phenomena on the wave propagation and furthermore to calculate the wave characteristics along the whole measurement area in the CTO S.A. deepwater towing tank, based on wavemaker flap motion control.


INTRODUCTION
During the hydromechanical tests with a model of the object in a sea wave, such as the tests carried out in the CTO S.A. deepwater towing tank (Fig. 1), it is crucial to obtain the spectrum of the generated wave consistent with the modeled sea wave in a reduced scale. The wave should be properly modeled along whole measurement area. The transfer function of the wavemaker and moreover the damping, breakdown and reflection phenomena, have impact on modeling the marine environment conditions.
The knowledge of the above factors is important for realization of the model tests. In the scope of this paper, mentioned factors have been analyzed. The analysis was carried out on the basis of measurements made in CTO S.A.

TOWING TANK
The CTO S.A. deepwater towing tank is a research object with 270 m length, 12 m width and 6 m depth, as shown in Fig. 2. Investigations carried out in the towing tank consist in testing the properties of objects floating freely (vessels), anchored (oil rigs) or permanently fixed to the bottom (offshore wind turbines). The tests are conducted for models of objects in a reduced scale. The models are tested in the uniform flow (simulated by motion of the carriage at constant speed) or subjected to the impact of the waves for modeling the impact of the marine environment.

WAVEMAKER
The waves corresponding to ocean waves in a reduced scale are generated by the rigid flap wavemaker with single articulation above channel bed (and below water level). The flap is an actuator with 4.15 m height and 12.00 m width, articulated 2.30 m above channel bed, as it shown in Fig. 2.

SURVEY OF RELATED WORKS
In 1951, F. Biésel and F. Suquet published formulas known as Biésel Transfer Functions BTF. The BTF allow to calculate the wave height in a towing tank for several types of wavemakers, also for wavemaker with rigid flap and single articulation above channel bed, like in CTO S.A. [1].
The research carried out in 1972, in the Laboratório Nacional de Engenharia Civil (Lisbon, Portugal) has shown that BTF is proper for regular waves. However, for irregular waves, the measured transfer function is inconsistent with theoretical BTF. [2] One possible reason indicated by the authors was the lack of linearity and energy losses.
In 1985, S. Hasselmann and K. Hasselmann described a mechanism of the formation of the JONSWAP spectrum and proposed a model of the nonlinear energy transfer in wind waves [3]. The model has two parameters proposed by the authors. Proper parameterization of the model allows to calculate an energy stream transferred from higher frequency bands to lower frequency bands of the wind wave spectrum.
Van Vledder in 2012 discussed the ambiguity of selection of values of the C and λ parameters [4].
In 2014 group of people from Kyushu University (Japan), carried out simulations and study for the method mentioned and for a few others methods of wave model implementation [5].
The control system of the wavemaker in the CTO S.A. deepwater towing tank was modernized in 2015 -an analog control system was replaced by a new digital control system [6]. In 2017 research was conducted on optimization of the digital control system [7].
Research described in this paper is intended to develop mentioned works to enable the prediction of wave characteristics in CTO S.A. deepwater towing tank.

OBJECTIVES AND SCOPE OF THE WORK
The objectives of research were to determine the way of wave propagation and to develop the method of prediction of the wave characteristics in the CTO S.A. deepwater towing tank. For this purpose, transfer function for the wavemaker and further the impact of the damping, breakdown and reflection were investigated. The scope of research included the measurements of the waves along whole measurement area and analysis of registered signals.

MEASUREMENTS
The measurements were made along whole measurement area of the towing tank, between 1 st end on the beach side and 2 nd end on the wavemaker side. The 1 st end is in 205 m distance from the wavemaker. The 2 nd end is in 60 m distance from the wavemaker. The measurements were made during generation of the waves: regular and irregular. The regular wave is a harmonic wave. The irregular wave is a synthesis of harmonic waves corresponding to the determined spectrum. Parameters of the generated and measured waves are shown in Tab. 1.
The waves were measured by the resistance probe of the water level ( Fig. 3), manufactured by CTO S.A. The resistance probe was mounted to the towing carriage in the towing tank axis, i.e. on the actual track of the models tested in waves. During wave generation, the displacement of the wavemaker flap was measured by linear displacement sensor WDS-1000-P60-SR-U manufactured by MicroEpsilon.

TRANSFER FUNCTION
For rigid f lap wavemaker with single articulation above channel bed, the wave height HW is dependent on flap displacement a, wave frequency f, water depth h and articulation height above channel bed h0 [1]. The formulas representing this dependence are given as (1) and (2). The Biésel Transfer Function BTF is understood as the relation (3). The BTF for mentioned type wavemaker was experimentally checked for two types of waves -regular and irregular, in the scope consistent with Tab. 1. For this purpose, two signals were measured: the wave HWmeasured at the 2nd end of the measurement area using resistance probe and the flap displacement a -measured using the displacement sensor. (1) (3)

Regular waves
The BTF for regular waves in the CTO S.A. deepwater towing tank with the rigid flap wavemaker was checked. The theoretical BTF was corrected as it shown in (4). Implementation of the correction factor Cr=0.8 was necessary to obtain the agreement of the calculations with the results of measurements. As shown in Fig. 4, the results of the measurements are consistent with the calculated BTFcorr with satisfactory accuracy.

Irregular waves
The transfer function for irregular waves TF was obtained, based on the FFT analysis of the signal HW and signal a. The signals were measured for irregular waves no. 26-28 (Tab. 1). As it shown in Fig. 5, the measured TF is consistent with the calculated BTFcorr in scope from 0.3 Hz to 0.6 Hz. For frequencies higher than 0.6-0.7 Hz, TF is significantly increasing and further rapidly falling below BTFcorr characteristic.
The rapidly falling characteristics below the theoretical transfer function has been also observed in other hydrodynamics laboratory [2]. It is mentioned there that this phenomenon may be due to lack of linearity and probably caused by energy losses as effect of the variations of velocity and acceleration in irregular waves.
Other works devoted to gravity waves, like [3] and further [4] and [5], describes and proves that in the wind waves there are nonlinear interactions that cause the nonlinear energy transfer. It occurs in this way that the energy stream Snl is transferred from higher frequency bands to lower frequency bands of the wind wave spectrum. A similar phenomenon can occur in irregular waves generated in the towing tank. It can be seen in Fig. 5.
These papers ( [3], [4], [5]) describe the physics of the phenomenon and the methods of calculation Snl for wind waves spectra.
The mathematical model of nonlinear energy transfer proposed in [3] is described by formula (5), where: δSnl relates to the frequency f, δSnl + relates to the frequency (6) and δSnlrelates to the frequency (7). Furthermore, C is the strength of the nonlinear interaction parameter [3] and λ is the Hasselmann's shape parameter [8]. Values of mentioned parameters were proposed as λ=0.25 and C=3×106. The values have been chosen empirically as optimal. Another proposition was given as λ=0.15 and C=3.75×105 [4].
The ambiguity of selecting the parameters may indicate necessity of individual optimization for given towing tank or even for given spectrum.
Following this path, the λ and C were chosen individually for each spectrum of the waves no. 26-28 (Tab.1), using binary search algorithm to obtain the best compatibility of the nonlinear transfer function NTF (8) with the measured TF. Chosen parameters are listed in Tab. 2. The results are shown in Fig. 6-8. The tnl (9) is the time of nonlinear transfer of Snl at the distance xnl=60 m between wavemaker flap and resistance probe.
As can be seen in Fig. 6-8 the results of the calculated NTF are consistent with the measured TF with satisfactory accuracy in the frequency scope up to 1.0 Hz.

WAVE DAMPING
The wave damping was investigated on the axis of the towing tank along the whole measurement area. For this purpose, the HW for waves no. 6-15 (Tab. 1) was measured at the 2 nd end (HW2 nd in Tab. 3) and then at the 1st end (HW1 st in Tab. 3). The measurements were made while the towing carriage was stationary.
Relative damping factor DF, understood as (10) allows to calculate the DF as function DF(x), where x=145 m is the distance traveled by generated wave from 2 nd end to 1 st end. Dependence of DF on f, shown in Fig. 9, was linearly approximated (11). Finally, the resulting height of the wave under damping conditions as function HW DF (HW2 nd , f, x) can be calculated as (12).

WAVE BREAKDOWN
The wave breakdown was investigated on the axis of the towing tank between 1 st end and 2 nd end of the measurement area. The goal of investigation was to determine the content of the power spectral density of harmonic wave at main frequency in the generated waves along whole measurement area. By the main frequency, we mean the frequency generated by the wavemaker. The HW was measured on five sections between 1 st end and 2 nd end during generation of the regular waves listed in Tab. 1. Each section had length dx=29 m. Based on the spectrum in frequency domain, obtained from measured HW using FFT, the percentage content of the power spectral density of harmonic wave at main frequency -P % in the spectrum was calculated according to (13). The HW M (f) means power spectral density of the main frequency band. The HW 0.0-1.5Hz (f) means power spectral density of the frequency bands up to 1.5 Hz. Characteristics of P % as function of x are shown in Fig. 10. As shown, there are a three groups of regular waves due to the dynamics of P % changes: • group A: the waves with frequencies from 0.3 Hz to 0.9 Hz -approximated CP1A as (14), • group B: the waves with frequencies from 1.0 Hz to 1.1 Hz -approximated CP1B as (15), • group C: the waves with frequencies from 1.1 Hz to 1.2 Hz -approximated CP1C as (16). As shown in Fig. 10, the P % for waves from the A group is relatively constant. For waves from B group, the P % decreases according to S-characteristic and in midway of the measuring area falls below 50%. For waves from C group, the P% decreases according to J-characteristic and in midway of the measuring area falls to about 20%.

WAVE REFLECTION
The wave reflection was investigated on the axis of the towing tank between 1 st end and 2 nd end of the measurement area. The goal of investigation was to determine the reflections arising from the wave reflections from the artificial beach ( Fig. 1.) For this purpose, two signals were measured: HW and velocity of the driving towing carriage V. The towing carriage was driven from the beach to the wavemaker.
The Doppler effect for measured waves, when towing carriage was driven, allows to extract the mean height of the generated waves HWg and the mean height of the reflected waves HWr from the wave spectrum in frequency domain.
The frequency bands were appropriately shifted depending on measured V.
The mean reflection coefficient R, understood as (17), calculated for measured HWg and HWr, is listed in Tab. 4. The numbers in the first column of Tab. 4 are consistent with Tab. 1.
Characteristic of R as function of f is shown in Fig. 11. As shown, R ranges from below 10% up to 25%, depending on the f. This dependency was approximated as (18).

CONCLUSIONS
In the range of waves listed in Tab. 1, it is valid to use BTF for prediction of HW of regular waves in CTO S.A. towing tank. Moreover, it is possible to use BTF and properly parameterized Snl model for prediction HW of irregular waves.
The wave damping phenomenon is dependent on the f. As f increases, DF will increases. It is possible to predict HW on given x and for given f basing on the HW2 nd .
The wave breakdown phenomenon is dependent on the f. P % is constant along the whole measurement area for f up to 0.9 Hz. From f equal to 1.0 Hz the phenomenon is noticeable and for f above 1.1 Hz the phenomenon is significant and waves are rapidly collapsing.
The wave reflection phenomenon arising from reflections from the artificial beach is dependent on f. As f increases from 0.5 Hz to 1.2 Hz, R increases from about 10% to 25%.
Using the developed models of the phenomena: wave damping, wave breakdown, wave reflection, nonlinear energy transfer and the transfer function, it will possible to predict how the wave will propagate along the measurement area in the CTO S.A. towing tank, basing on wavemaker flap motion control.
Finally, hitherto the wavemaker control system will be developed in CTO S.A. to the adaptive wave control system.