Study on Wind Waves Similarity and Wind Waves Spectrum Characteristics in Limited Waters

Abstract

Wind waves is an important factor affecting navigation safety and water environment in limited waters such as lakes and bays. Wind wave spectrum represents the frequency domain features of wind waves and has always been the focus of research. Based on the field observation and flume experimental method, the system analysis of similarity of two kinds of situations, discussed nonlinear response of the relationship of the spectral shape parameter of balance field α, β and wind waves basic frequency between factors like wind speed, wind blowing fetch and water depth. By means of wind tunnel flume and prototype observation data of nonlinear regression analysis, The relation formulas of wind wave frequency prediction considering the comprehensive influence of wind speed, wind blowing fetch and water depth is established. Relevant research is of great significance for revealing the evolution characteristics of wind waves in limited waters and guiding navigation safety and water environment management.

1 Introduction

Wind waves are the product of the interaction between water and air and exist widely in natural waters. Because wind waves carry a certain amount of energy. Strong wind and wave may lead to shipwreck accident. Wave force is also one of the driving forces affecting the energy and material conversion of muddy-water interface. Therefore, wind waves research has been a hot spot in the field of water transportation and water environment. Among them, wind waves in lakes, harbors and other limited waters are not only restricted by wind speed, but also affected by wind speed, wind blowing fetch, and their morphological characteristics become more complex (Karimpour et al. 2017).

Field observation is the most direct method to study the characteristics of wind wave spectrum, but the field conditions are greatly affected by external factors, the observation is not controllable, and systematization of data is poor. The wind tunnel flume generalizes the field environment to a certain extent, and has the advantages of being controllable and easy to measure, so that the evolution characteristics of wind wave spectrum can be systematically and carefully studied (Zakharov et al. 2015). Both methods play an important role in wind waves research, but there is no discussion on the similarity of wind waves under the two scenarios.

Because natural wind waves are affected by uneven and unstable wind field, the wind waves process at a certain point is the superposition of wind waves at different scales, and its total energy is the sum of the energy carried by wind waves at different scales. Wind wave spectrum is often obtained through energy spectrum analysis for further study of wind waves characteristics (Zakharov et al. 2015). The high frequency part of wind wave spectrum exists balance field. In this region, energy intake by wind and consumption by breakage of wind waves tends to balance (Grare et al. 2013), and the wind tunnel experimental research shows that balance and frequency domain energy relationship can use S(ω) = αg²ω^β to express (ω = 2πf) (Hidy and Plate 1966). The α decreases with the increase of wind path and β characterization of balance between the different scale wave energy transfer efficiency (The smaller the absolute value, the faster the energy travels). For wind waves in shallow water, the effect of the bottom wall on wind and wave enhances its nonlinearity, inhibits the propagation of small-scale wind and wave to large-scale wind and wave, and results in significant changes in the spectral shape. For example, the spectrum width of wind waves increases, and the absolute value of β decreases (the energy transfer of wind and wave at different scales accelerates) (Young and Verhagan 1996; Nair and Kumar 2017; Fedele et al. 2019).

Wind waves dominant frequency (f_d) represents the size of the dominant wind wave scale. At present, the relationship between the dominant frequency and wind speed (U₁₀) and wind blowing fetch (F) is mostly established in the form of formula (1) (Hwang 2005). Statistical studies have found that when a is between 1.72~3.50, and b is between −0.24~−0.33 the two coefficients are highly discrete in different studies. Karimpour et al. (2017) found that for the wind waves in the fully developed shallow water area, the influence of relative wind blowing fetch F/d(D is water depth) on f_du₁₀/g and gF/u₁₀² should be considered, and the dominant frequency prediction formula (2) considering the influence of water depth is given. However, this formula is only applicable to fully developed wind waves, and it does not consider the wind tunnel experimental data.

$$ f_{\text{d}} U_{10} /{\text{g}} = a\left( {{\text{g}}F/U_{10}^2 } \right)^b $$

(1)

$$ f_{\text{d}} U_{10}/{\text{g}} = 3.5\left( {{\text{g}}F/U_{10}^2 } \right)^{ - 0.75\left( {F/d} \right)^{ - 0.1} } $$

(2)

In this study, the similarity of wind wave spectrum shape and spectrum parameters in the two scenarios was analyzed by combining field observation and wind tunnel flume experiment, and the response relationship between the spectrum parameters α and β in the equilibrium domain and wind speed, wind blowing fetch and water depth was further summarized. This paper analyzes the reasons for the large deviation of parameters a and b in the existing dominant frequency prediction relation, and puts forward a dominant frequency prediction method suitable for limited waters. Relevant research is of great significance for revealing the evolution characteristics of wind waves in limited waters and guiding navigation safety and water environment governance.

2 Research Methods

2.1 Prototype Observation

Lake Tai, a typical shallow water lake, is selected for prototype observation. The data of airflow, wind wave and water depth were collected from the monitoring platform at the baymouth of Zhushan Bay (Fig. 1). The wind speed and wind direction sensor was located 2.5 m above the water surface, and the water surface fluctuation data were measured by the pressure sensor located at the bottom of the lake (Gao et al. 2022).

Fig. 1.

Air flow and wind wave station arrangement

2.2 Flume Experiment

The experiment was carried out in the wind tunnel flume of Nanjing Hydraulic Research Institute (Fig. 2). Six measuring cross section (F4.5~F19.5) were arranged along the wind tunnel flume, and the cross section spacing was 3 m. A wind speed measuring line is arranged on the middle line of each measuring section, and 16 hot wire anemometers are arranged along each measuring line vertically. Accordingly, a pair of capacitive wave altimeters is arranged along the wind direction 10 cm from each measuring line, with a total of 12 (Gao et al. 2022).

Considering the influence of shallow lake water depth on wind waves, the experimental water depth d was 0.10 m, 0.15 m and 0.30 m respectively. A total of 7 levels of reference wind speed U_r were set (the average wind speed of wind tunnel cross section, which is carefully controlled by transducer), including the prevailing wind speed range of Lake Tai, as shown in Table 1. Two groups of tests were carried out for each wind speed, and the relative errors of the test results were all within 5%, and the average of the two results was the final result.

Fig. 2.

Wind tunnel flume and instrument layout

Table 1. Test group design

3 Result Analysis

3.1 Characteristics of Wind Wave Spectrum

3.1.1 Prototype Observation

Figure 3 shows the relationship between wind wave spectrum and wind blowing fetch of Lake Tai at the same wind speed (U_2.5 = 2.9 m/s) (wind wave spectrum is obtained by fast Fourier transform method). The solid line in the figure is the smoothing fitted line of scattered data. It can be seen that with the increase of wind blowing fetch, the spectrum peak gradually rises and the dominant frequency shows a slight decreasing trend (the dominant frequency wind wave cycle increases). It reflects the characteristics of wind wave developing gradually with the increase of wind blowing fetch under the condition of limited wind blowing fetch.

Figure 4 shows the relationship between wind wave spectrum and wind speed in Lake Tai at the same wind blowing fetch (F = 33 km). It can be seen that with the increase of wind speed, the spectrum peak gradually rises and the dominant frequency gradually decreases, indicating that the wind wave scale of the dominant frequency increases. In addition, when the wind speed is high, the amplitude of the dominant frequency and spectrum peak energy decreases with the increase of wind speed, indicating that the wind wave tends to be saturated with the increase of wind speed. When the wind speed is 15.3 m/s, the dominant frequency of wind waves is about 0.31 Hz.

Fig. 3.

Wind wave spectrum changes with wind blowing fetch in the field (u_2.5 = 2.9 m/s)

Fig. 4.

Wind wave spectrum changes with wind speed in the field (F = 33 km)

3.1.2 Wind-Tunnel Test

Figure 5 shows the change of wind wave spectrum with wind blowing fetch when U_r = 7.7 m/s. The solid line in the figure is a smooth fitted line for scatter data. It can be seen that with the increase of wind blowing fetch, the dominant frequency gradually decreases and the peak value rises gradually. The spectral shape near the dominant frequency gradually changes from wide and low to sharp and narrow, indicating that with the increase of wind blowing fetch, there is a dominant wave scale which is relatively consistent with the dynamic and boundary conditions at this time, and the wave energy of other frequencies gradually shifts to the dominant frequency wave, and the spectral shape changes rapidly when the wind blowing fetch is small, but slowly when the wind blowing fetch is large, which is consistent with the law of measured wind wave spectrum.

Fig. 5.

Wind wave spectrum changes with wind blowing fetch in the flume

Figure 6 shows the change of spectral shape with the increase of wind speed when d = 0.15 and F = 13.5 m. It can be seen that the overall energy increases with the increase of wind speed, the spectral peak energy increases with the increase of wind speed, the spectral peak frequency decreases with the increase of wind speed, and the spectral shape near the main frequency gradually changes from wide and short to sharp and narrow with the increase of wind speed. It shows that the wind wave energy of other frequencies is gradually transferred to the dominant frequency wave, and the spectral shape changes rapidly with the increase of wind speed when the wind speed is small, while the spectral shape changes slowly when the wind speed is large, which is consistent with the response relationship between measured wind wave spectral shape and wind speed (Fig. 4).

Fig. 6.

Wind wave spectrum changes with wind speed in the flume

The influence of water depth on spectral shape is mainly reflected in the influence of bottom wall on wave shape and dynamic process when wave scale is large enough. Figure 7 shows the wind wave spectra at three water depths when F = 19.5 m U_r = 12.3 m/s. It can be seen that with the increase of water depth, the overall energy spectrum line increases, especially the increase of the main peak, and its spectral shape becomes more sharp and thin, indicating that under the same wind speed and wind blowing fetch, the greater the water depth, the more easily the energy is transferred to the main frequency wave, and the main frequency energy is larger.

Fig. 7.

Wind wave spectrum changes with water depth in the flume

3.1.3 Comparison Between Prototype Observation and Wind Tunnel Experiment

Figure 8 shows the comparison between the flume experiment and the measured wind wave spectrum in Lake Tai. For the convenience of comparison, the wind wave spectrum is normalized, in which frequency is normalized by dominant frequency (f_d) and energy is normalized by energy of dominant frequency (S_d). It can be seen that the spectral shapes obtained in field and flume are generally similar, but the dispersion degree of field measured data is significantly greater than that of flume data, which is believed to be caused by the variable external factors such as field wind speed and direction.

Fig. 8.

Wind wave spectrum comparison between Lake Tai and flume experiment

3.2 Variation Rule of Parameters in Spectrum Equilibrium Domain

3.2.1 Prototype Observation

Figure 9 shows the relationship between the spectrum parameters α and β in the wind wave equilibrium domain observed at U_2.5 = 2.9 m/s and the wind blowing fetch. It can be seen that α ranges from 1.6 $\times$ 10^–3 to 7.3 $\times$ 10^–3 and shows a decreasing power function trend with the wind blowing fetch.

Figure 10 shows the relationship between α and β with wind speed when F = 33 km. It can be seen that α is between 4.0 $\times$ 10^–5 and 1.6 $\times$ 10^–3, and decreases with wind speed as a power function. β was between −8.76 and −3.20, and β showed a logarithmic increase trend with the increase of wind speed, indicating that the wave energy transfer efficiency in the equilibrium region tended to be saturated with the increase of wind speed, indicating that wind waves were broken under strong wind force and dissipated part of energy rapidly. In addition, the interaction between wind waves and bottom wall in limited water depth also dissipated part of energy.

Fig. 9.

Relationship between α and β with wind blowing fetch

Fig. 10.

Relationship between α and β with wind speed

3.2.2 Wind Tunnel Experiment

Figure 11 shows the relationship between the spectral parameters α and β and the wind blowing fetch when d = 0.15 m U_r = 7.7 m/s. It can be seen that α is between 2.5 10⁻⁴ and 2.1 10⁻³ and the wind blowing fetch shows a decreasing power function trend. β was in the range of −4.53~−3.89, and β showed a linear increase trend with the increase of blowing distance, indicating that the wave energy transfer rate and blowing distance had a linear relationship with the measured wind wave law in Lake Tai.

Figure 12 shows the relationship between α and β and wind speed when d = 0.15 m and F = 13.5 m, indicating that α was in the range of 2.8 $\times$ 10⁻⁴~1.0 $\times$ 10⁻², decreases as a power function with respect to wind speed; β was between −5.53 and −3.67, and β showed a logarithmic increase trend with the increase of wind speed, indicating that the wave energy transfer efficiency in the equilibrium region tended to saturation with the increase of wind speed, which was consistent with the measured wind wave law in Lake Tai.

Figure 13 (a) shows the relationship between α and β and water depth when F = 7.5 m and U_r = 4.6 m/s. It can be seen that α is positively correlated with water depth, while β is negatively correlated with water depth under test conditions. Figure 13 (b) shows the relationship between α and β when F = 19.5 m and Ur = 12.3 m/s. It can be seen that α is positively correlated with water depth under experimental conditions, while β is not significantly correlated with water depth. The relationship between β and water depth was different under the two test conditions. The analysis found that for the F = 7.5 m U_r = 4.6 m/s group, the corresponding F_c values (F_c = g^1.25H^0.5T^2.5/d^1.75) (Zhu 2015) were 9.8, 4.4 and 2.0 respectively under the three water depths (0.10 m, 0.15 m and 0.30 m), both less than 10, and wind wave nonlinear weak; The corresponding F_c values of F = 19.5 m U_r = 12.3 m/s at three water depths were 81.9, 45.4 and 27.4 respectively, both greater than 10, indicating weak wind wave nonlinearity. The comparison of the two cases shows that water depth only has an effect on the energy transfer efficiency in the balance domain when the nonlinear effect of wind and wave is weak, and when the nonlinear effect is strong, water depth has a weak effect on the energy transfer efficiency in the balance domain.

Fig. 11.

Relationship between α and β with blowing distance

Fig. 12.

Relationship between α and β with wind speed

Fig. 13.

Relationship between α and β with water depth

3.3 Variation Rule of Wind Wave Dominant Frequency

3.3.1 Prototype Observation

F_d reflects the scale of dominant wave groups in wave sequence and is an important parameter to quantify the shape of wind wave spectrum. The following formula is used to calculate f_d (Mo et al. 2018):

$$ f_d = \frac{m_1 }{{m_0 }} $$

(3)

where m_k (k = 0, 1) represents the k-order central moment of the spectrum.

Figure 14 shows the influence of measured wind speed (a) and wind blowing fetch (b) on the dominant frequency of wind waves in Lake Tai. It can be seen that with the increase of wind speed, the dominant frequency shows a non-linear decreasing trend, and the damping decreases with the increase of wind speed. The dominant frequency decreases linearly with the increase of wind blowing fetch.

Fig. 14.

Influence of wind speed (a) and wind blowing fetch (b) on dominant frequency of wind waves in Lake Tai

3.3.2 Wind Tunnel Experiment

Figure 15 for three water depth under the experiment and the relationship between frequency and wind speed and fetch, obviously with the increase of wind speed and fetch the decrease trend of dominant frequency is nonlinear. When fetch and wind speed is small the damping of dominant frequency is larger, with the increase of fetch and wind speed, the damping gradually become smaller, trends related to the intensity of the nonlinear wave, and measured in Lake Tai is consistent. The change law of nonlinear frequency show that at the beginning of the interaction between water and gas, water surface changes in response to flow boundary and dynamic conditions and produce scale small waves and diversity is strong. With the increase of fetch both interaction, a concentrate towards the dominant wave after wave absorb wind energy scale increase gradually and the main frequency gradually decreases with the increase of wind blowing fetch. However, when large scale wind waves are generated under high wind speed and blowing range, the nonlinear effect caused by limited water depth inhibits the transfer of non-dominant frequency wave energy to dominant frequency wave to some extent, and the amplitude of dominant frequency decreases with the increase of blowing range and wind speed.

Fig. 15.

Relationship between f_d with wind blowing fetch and wind speed under different water depths

3.4 Wind Wave Dominant Frequency Prediction

The relationship between experimental and prototype observation of wind wave dominant frequency and wind blowing fetch in Lake Tai was compared with previous studies (Formula (1)). The method of converting measured U_2.5 in Lake Tai to U₁₀ was referred to (Gao et al. 2022), and the profile gradient method was used to extrapolate experimental and prototype observation for U₁₀. Figure 16 shows the relationship of f_dU₁₀/g and gF/U₁₀² of the experimental and prototype observations and three fitting relationship lines of existing studies are also shown in the figure (CERC 1977; Donelan et al. 1985; Donelan et al. 1992). It can be seen that under experimental and prototype observation conditions, both of them are negatively correlated with power functions. Experimental and measured data of Lake Tai are basically located between the three fitting lines, indicating that the data are generally consistent with the trend of existing research. But there is a certain deviation between three fitting line. It is considered that one of the reasons for the difference is that the water depth of each research area is different and the water depth is not considered when establishing the dominant frequency relationship.

Based on the above results can be inferred formula (1) without considering the influence of the depth of the water, It has the problem of poor suitability in limited waters and we consider in establishing f_dU₁₀/g and gF/U₁₀² depth was developed for the relation between the factors. The formula (4) is constructed by referring to the form of formula (2). When a and e are 0, the relation degenerates into the form of formula (1). According to the experimental and measured data of Lake Tai, relevant literature data (Ding 2005; Johnson et al. 1998; Donelan et al. 2006), 134 groups of wind tunnel test and 96 groups of field observation, the water depth (d) ranged from 0.1 to 4.0 m, the wind blowing fetch (F) ranged from 4.5 to 33000 m, the wind speed (U₁₀) ranged from 2.36 to 24.02 m/s, and the dominant frequency (f_P) ranged from 0.28 to 5.87 Hz. After nonlinear regression analysis of the relation, the relation Eq. (5) was obtained, with correlation coefficient 0.97 and determination coefficient 0.94. This equation shows that the dominant frequency is negatively correlated with the wind speed, wind blowing fetch and water depth, which is consistent with the previous results and the conclusion in reference (2017).

Figure 17 shows the comparison between the measured f_dU₁₀/g and the calculated values by formula (5). It can be seen that the experimental and field observation data points are evenly and compact distributed on both sides of the 45° line. Figure 17(b) shows the measured f_dU₁₀/g compared with the data calculated by formula (2). It can be seen that the prototype observation data points are evenly and compactly distributed on both sides of the 45° line, but the experimental data are at around f_dU₁₀/g > 2, data points and the 45° line deviation increases gradually, and show that type formula (5) does not apply to wind waves of the wind tunnel sink. However, both wind tunnel flume experiments and field observations, the interaction mechanism of water and wind is consistent with each other, thus suggests that there are some defects on the similarity between the wind tunnel experiment and field observation data in formula (2).

Formula (5) takes into account the wind wave similarity relationship between wind tunnel flume and shallow lake. By introducing water depth as a modulation factor, the prediction relationship of wind wave dominant frequency considering the comprehensive influence of wind speed and water depth is established, which makes up for the deficiency of Formula (2) without considering the wind tunnel experimental data.

$$ f_{\text{d}} U_{10} /{\text{g}} = a + b\left( {{\text{g}}F/U_{10}^2 } \right)^{c*\left( {F/d} \right)^e } $$

(4)

$$ f_{\text{d}} U_{10} /{\text{g}} = - 0.847 + 3.443\left( {{\text{g}}F/U_{10}^2 } \right)^{ - 0.233*\left( {F/d} \right)^{ - 0.069} } $$

(5)

Fig. 16.

Relationship between $f_{\text{d}} U_{10} /{\text{g }}\;{\text{and}}\;{\text{ g}}F/U_{10}^2$

Fig. 17.

Comparison of calculated and measured values

4 Conclusion

• The normalized spectrum of measured wind waves in Lake Tai is similar to that in flume experiment, but the dispersion degree of measured data in Lake Tai is greatly affected by unstable wind speed and direction. The measured parameters of wind wave spectrum in balance domain and the variation of dominant frequency, wind speed and wind blowing fetch in Lake Tai are consistent with the wind tunnel experiment.

• The spectrum parameter α decreases as a power function with the increase of wind speed and wind blowing fetch, and increases with the increase of water depth. The parameter β increases linearly with the increase of wind blowing fetch and logarithmically with the increase of wind speed. When the nonlinearity of wind wave is weak, the parameter β decreases with the increase of water depth. When the nonlinearity is strong, the parameter β does not change with the increase of water depth.

• The main frequency of wind wave is negatively correlated with wind speed and blowing range. The amplitude reduction is large at low wind speed and wind blowing fetch, and becomes smaller with the increase of wind speed and blowing range. With the increase of wind speed and wind blowing fetch, the dominant frequency energy increases first and then hold steady.

• By considering the similarity relationship between wind tunnel experiment and shallow lake, and by introducing water depth as the modulation factor, the dominant frequency prediction relationship of wind wave considering the comprehensive influence of wind speed, wind blowing fetch and water depth is established, which makes up for the deficiency that wind tunnel experiment data is not taken into account in existing studies.