1 Introduction

Real-world optimization engineering problems are usually very challenging and hard to solve [1]. Many topics such as multi-objective, possibility, multi-conditioned, and uncertainties involved in solving problems, including being multi-objective, refer to the multi objectivity of a problem [2]. Generally, there is no individual optimal solution to these problems. One possible solution may be better for one of the objectives but inappropriate to other objectives. So, there is a set of solutions known as the Pareto-optimal set to answer multiple objective problems. The Pareto-optimal set includes quiet optimized solutions to show the best possible balance between the objectives is of a particular form [3].

Due to several conflicting objectives, numerous involving parameters, and intricate environmental conditions, marine propeller design is one of the most complicated engineering problems [4].

Previously, the selection or design of efficient propellers was only possible based on the experimental measurements [5]. Nevertheless, this method is not able to describe problems such as propeller cavitation and vibration [6, 7]. Therefore, analytical methods, such as that of Rankine [8] and Anderson [9] theories, were used. Then, 3D theories like levels and boundary elements were used to fix the error caused in the final results of 2D calculations [10].

Based on different developed and used theories, various parameters affecting the propeller design case [11, 12]. As a result, the optimization problems were possible, focusing on the influence of these parameters [13]. The marine propeller’s efficient design’s main objectives include maximizing the produced axial force, minimizing the torque, optimal performance, designing cost, the construction possibility, and the lowest probability of cavitation occurrence [14].

In this process, optimum values must be calculated, given that increases the desirable efficiency is in some conflicting factors. For example, a large step increases the desirable efficiency, leading to an undesirable increase in cavitation. Accordingly, if any such factors are not in the optimal range, the initial assumption reselects should achieve the optimal result. Many researchers have tried to use meta-heuristic algorithms to optimize different parameters, given the limitations and complexity of the propeller’s design problem on the one hand and the ability of meta-heuristic algorithms on the other hand [15, 16].

Some optimized algorithms were developed based on using the Wageningen series to design marine propellers [6]. The genetic-based algorithm with a high ability to optimize complex problems was another excellent revealed process. This algorithm spends much lower time in driving system design [17]. Furthermore, optimized genetic algorithms were utilized in many studies [18, 19]. A comprehensive study on applying different meta-heuristic algorithms in propeller design up to 2016 has been provided in reference [20]. This study pointed out that stochastic nature algorithms and indefinite problem-solving methods afford more reliable results than classical algorithms. The ant-lion optimizer (ALO) is another novel meta-heuristic algorithm used in propeller design [21]. According to the no free lunch (NFL) Theorem, there is no meta-heuristic algorithm best suited for all optimization problems [22]. Therefore, there may be a meta-heuristic algorithm that performs well in some cases.

Generally speaking, in the literature on propeller design, two approaches, namely deterministic and stochastic methods, have been proposed [23, 24]. The propeller design based on the first method is very complicated and time-consuming, which makes them inapplicable. On the other hand, several deterministic models have been proposed to solve various optimization problems [25]. However, deterministic models need to know the optimization problem’s property and some info about the gradient or subgradient info. Metaheuristic optimization algorithms (MOA) have recently become increasingly popular in optimization tasks as one of the options. In this field, some well-known MOAs are genetic algorithms (GA) [26, 27], ant colony optimization (ACO) [28,29,30,31], particle swarm optimization (PSO) [32,33,34], differential evolution (DE) [28], evolutionary strategy (ES) [35], and evolutionary programming (EP) [36, 37].

Despite the merits of MOAs, according to the no-free-lunch (NFL) theorem [38], there is no MOA to solve all optimization problems as the best method successfully. Therefore, researchers have tried to develop novel MOAs for solving various optimization problems with a bit misunderstood. However, most of these methods resemble a modified variant of the previous methods such as DE, GA, and PSO attached with a new metaphor, ignoring the fact that the metaphor has no application in computer science and only hinders understanding the core mechanisms of an algorithm. Such a trend made metaheuristics biased towards the useless focus on the “novel” metaphors instead of focusing on performance. One of the motivations of this study is to show how the base methods’ core mechanisms. Examples of the methods that are a kind of metaphor-based or less evolutionary methods are Biogeography-based optimization [39,40,41,42], gray wolf optimizer [43, 44], salp swarm algorithm (SSA) [45,46,47], whale optimization algorithm (WOA) [48], moth-flame optimization algorithm (MFO) [49], sine cosine algorithm (SCA) [50], dragonfly algorithm (DA) [51, 52], ant lion optimizer (ALO) [53], and stochastic fractal search (SFS) [54, 55]. However, researchers later discovered that most so-called novel methods are metaphor-based with no novelty [29]. Therefore, in this paper, we try to exploit the simplicity and applicability of PSO instead of novel metaphor-based algorithms.

In recent decades, the pervasive problem of all MOAs, i.e., local minima stagnation and premature convergence, is vastly addressed by stochastic algorithms so that the optimum result cannot be reliably achieved. It is associated with the various parameters and objectives, given the type of problem, lots of design parameters, and the problem’s high dimensions [56,57,58,59,60,61]. Therefore, we propose the chaotic self-governing groups of particles concept to address the shortcomings mentioned above.

Given the chaotic self-governing (with the different ability to search problem space), the algorithm can avoid local optima and uncertainty in marine problem design. Thus, reach its global optimum (the best parameter design). Hence, a cost function and the search agents must be defined. The design parameters should be checked and, and a few critical parameters should be selected for the topic that will be explained in the following sections.

The paper is organized as follows: Sect. 2 describes the algorithm. Section 3 addresses the definition of the MGPSO propeller design optimization problem, and Sect. 4 provides simulation results and discussions. Sequentially, the final section is the conclusion.

2 Optimization methodology

In this section, first, the two main PSO equations will be mentioned, and then the concept of the MGPSO algorithm will be introduced.

2.1 PSO algorithm

The PSO algorithm is motivated by birds’ social behavior. The method uses the number of particles (candidate solutions) to obtain the best solution in the search space. Meanwhile, all the answers move toward the best solution on their way. In other words, particles considered their best solution as the best overall solution. In the particle swarm optimizer, each particle should consider parameters such as current position, current speed, distance from pbest, and distance from gbest to change its position. The mathematical modeling of PSO is as Eq. 1 and Eq. 2.

$$\nu_{i}^{t + 1} = w\nu_{i}^{t} + c_{1} \times rand \times \left( {p{\text{best}}_{i} - \chi_{i}^{t} } \right) + c_{2} \times rand \times \left( {g{\text{best}} - \chi_{i}^{t} } \right),$$
(1)
$$\chi_{i}^{t + 1} = \chi_{i}^{t} + \nu_{i}^{t + 1} .$$
(2)

2.2 MGPSO

As a fact, the community’s people are not quite the same in terms of intelligence and abilities, but all do their duties as group members [62]. In such a community, each person’s ability can be useful in a particular situation. For example, termites are named soldiers, workers, nannies, and queens in a termite colony. They have diverse capabilities, but the difference is needed for colony survival. Each kind of termite can be recognized as an independent group with a common objective, and it is the colony’s survival [63].

In standard PSO, all particles act similarly in the local and global search. In this circumstance, the particles can be counted as a group with a search strategy. However, theoretically, any population-based optimization algorithm can realize stochastic and direct search results simultaneously. This can be reached using different independent groups with a common objective. The paper used independent groups proposed by reference [64] in updating C1 and C2.

We establish four groups based on termite colonies, each of which uses its unique pattern to look for global and local problems. We attempt to examine the influence of a wide variety of functions on the performance of PSO. As a result, these functions were chosen to demonstrate the efficacy of various slopes, curvatures, and interception sites in enhancing standard PSO performance. These independent groups use their own patterns to explore the problem space locally and generally. The algorithm’s dynamic coefficients are presented in Table 1 and Fig. 1. In this table, T represents the maximum number of iterations, and t is the current iteration.

Table 1 Updating strategies
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Fig. 1
figure 1

MGPSO mathematical models, where blue and black curves represent C1 and C2, respectively. As can be seen, C1 decreases as the iteration progresses, whereas C2 increases. Obviously, when C1 is bigger than C2, particles have a better capacity for local search. By contrast, when C2 is larger than C1, particles scan the search space more globally

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In MGPSO, first, all particles are placed randomly in the problem search space. After that, the particles were randomly divided into some predefined independent groups. In each iteration gbest, pbest, and the proportion of particles are defined. Each particle has its own group strategy implementation to update C1 and C2. C1 and C2 coefficients are calculated, and then the speed and position of a particle will be updated using Eq. 1 and Eq. 2.

3 Discussion and experimental results

In this section, we first examine the ability of MGPSO in finding the global minimum and convergence rate. Proven MGPSO’s ability to work with high-dimensional data, the algorithm will be used in practical marine propeller design.

3.1 Benchmark test functions

As shown in Table 2, six benchmark high-dimensional functions are applied to test the MGPSO performance to find the global minimum [64,65,66,67]. Dim represents the Vh functions dimension in the tables, d indicates the scope of search space, and fmin represents the minimum value of the benchmark function. Note that several benchmark functions with 30 dimensions were chosen to study the proposed method’s performance in the face of high-dimensional problems. MGPSO is compared with ALO and BBO to investigate its performance. Table 3 shows ALO and BBO Initial parameters. Inertia weight w for the MGPSO algorithm reduces linearly from 0.9 to 0.4. It must be noted that the number of particles is 100 with a maximum of 500 iterations for MGPSO and all benchmark algorithms.

Table 2 Six multi-dimensional benchmark functions
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Table 3 Initial parameters of the benchmark algorithms
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Results for benchmark functions are outlined in Table 4 and Fig. 2. These standard functions have many local minima increasing exponentially with the increasing size of the problem. Therefore, they are suitable to investigate the efficiency of the algorithm to avoid local minima. As the results show, the MGPSO acts more genuine than other algorithms in many several benchmark functions. The results (Table 4 and Fig. 2) show that MGPSO with autonomous chaotic groups decreases local minima stagnation probability. Figure 2 shows the better rate of convergence of the MGPSO algorithm compared to other algorithms. This MGPSO ability improvement is the outcome of independent chaotic groups. This has been made with a more random search compares to ALO and BBO algorithms. In this case, the particles cannot easily fall into the trap of local minima.

Table 4 Parameters obtained with a five-blade propeller
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Fig. 2
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The comparison between the convergence curves of various algorithms

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Figure 2 represents the comparison between the convergences curve of the MGPSO and other algorithms in the aforementioned benchmark functions.

3.2 Propeller design

The relatively high density of water makes marine propellers more important than their air classes. Propeller’s efficiency returns to engine power and the conversion of power to the driving force. In addition to efficiency, power conversion should be performed with the least amount of vibration and noise. The third feature of a good propeller is low surface corrosion induced by its cavitation. Find the balance among these three modes is a challenging task that should be counted during the propeller’s design process. The blade is a major portion of the propeller. The geometry of the blade must satisfy the requirements mentioned above.

Propeller adds ΔV speed to the V speed surrounding it. The acceleration and speed are being created in two parts: the first in the front half and the second on the rear half of the propeller. A propeller turns fluid surrounding it by its rotation. The amount of rotation depends on the speed of rotation of the motor and energy losses. Propeller’s effective waste for rotation equals 1 to 5 percent of their power. Propellers driving force is calculated as follows:

$$T = \frac{\pi }{4}D^{2} \left( {V + \frac{\Delta V}{2}} \right)\rho \Delta V,$$
(3)

where T is the driving force, D denotes propellers’ diameter, V present input fluid speed, and ΔV denotes the added speed. ρ is the fluid density. Power is defined as power multiplied by displacement on time-unit. The power required to drive the vehicle by amount V is calculated by Eq. 4.

$$P_{a} = TV.$$
(4)

One of the goals of optimization in propellers is the production of the most thrust with the least amount of power possible. Propellers’ efficiency is expressed by Eq. 5.

$$h = \frac{{P_{a} }}{{P_{{{\text{engine}}}} }} = \frac{T\upsilon }{{P_{{{\text{engine}}}} }}.$$
(5)

Also, propeller efficiency is expressed by Eq. 6:

$$\eta \left( x \right) = \frac{{JK_{T(X)} }}{{2\pi K_{Q(X)} }} = \frac{{V_{a} K_{T(X)} }}{{2\pi nDK_{Q(X)} }},$$
(6)

where J is a progress number, KT denotes propellers driving factor, and KQ is the propellers Torque coefficient. Also, J is defined as Eq. 7 [7]:

$$J = \frac{{V_{a} }}{nD},$$
(7)

where Va is the axial speed, n represents the rotation speed and D indicates the propellers’ diameter. Equation 7 can be rewritten by replacing efficiency J as Eq. 8.

$$\eta \left( x \right) = \frac{{V_{a} K_{T(X)} }}{{2\pi nDK_{Q(X)} }},$$
(8)

where KT driving factor and KQ propeller torque coefficient are calculated as follows:

$$K_{T} = \sum\limits_{n = 1}^{39} {C_{{T_{n} }} } \left( J \right)^{{s_{n} }} \left( \frac{P}{D} \right)^{{t_{n} }} \left( {\frac{{A_{e} }}{{A_{o} }}} \right)^{{u_{n} }} \left( Z \right)^{{\upsilon_{n} }} ,$$
(9)
$$K_{Q} = \sum\limits_{n = 1}^{47} {C_{{Q_{n} }} } \left( J \right)^{{s_{n} }} \left( \frac{P}{D} \right)^{{t_{n} }} \left( {\frac{{A_{e} }}{{A_{o} }}} \right)^{{u_{n} }} \left( Z \right)^{{\upsilon_{n} }} ,$$
(10)

where P/D is the step rate, \({{A_{e} } \mathord{\left/ {\vphantom {{A_{e} } {A_{o} }}} \right. \kern-\nulldelimiterspace} {A_{o} }}\) represents the propellers disk rate, Z is the number of the blades and \(C_{{T_{n} }}\), \(C_{{Q_{n} }} \upsilon_{n}\), \(S_{n}\), \(t_{n}\), \(u_{n}\) and \(\upsilon_{n}\) are corresponding reducing coefficients. Cavitation is another important problem in the design of the propellers.

When the blade of a propeller moves in the water at high-speed, low-pressure areas form the shape of water accelerator particles passing the blade. This causes the formation of bubbles that burst and can induce strong local wild waves. The result is corrosion on blades. The propeller’s sensitivity to cavitation is calculated as Eq. 11.

$$\sigma_{n.0.8} = \frac{{\left( {p_{a} + \rho gh_{0.8} - p_{v} } \right)}}{{0.5\rho \left( {\pi nD} \right)^{2} }},$$
(11)

where Pa is atmospheric pressure, Pv represents water vapor pressure, g is gravity acceleration and h0.8 indicates immersion of 0.8 radial blade h when the blade is located at 12.00 o’clock location. The final aim of this paper is to design propellers with the highest efficiency and lowest sensitivity to cavitation. In other words, for finding the final geometry of blades (Fig. 3), the National Advisory Committee for Aeronautics (NACA) patterns were used. According to Fig. 3, two parameters define an airfoil’s shape: (a) maximum thickness and (b) the chord length. The current paper examines 10 airfoils for the blade. Therefore, there are 20 parameters to be optimized. The final vector parameter will be in the form of Eq. 12.

$$\overrightarrow {X}_{i} = \left( {T_{1} ,C_{1} ,T_{2} ,C_{2} ,.....T_{10} ,C_{10} } \right),$$
(12)

where Ti and Ci are the ith airfoil thickness and the length of the arc along the blade, finally, the marine propeller design optimization problem may be expressed as Eq. 13. In fact, this equation will be the cost function for the MGPSO algorithm.

$$\begin{gathered} \overrightarrow {{\mathbf{X}}}_{{\mathbf{i}}} = \left( {T_{i} ,C_{i} } \right), \, i = 1,2,....10 \hfill \\ {\text{maximise}}:\eta \left( X \right) \hfill \\ {\text{minimise}}:V_{C} \left( X \right) \hfill \\ \ge 10000\,{\text{propulsion}}{.} \hfill \\ \end{gathered}$$
(13)
Fig. 3
figure 3

An airfoil along the blade defines the shape of the propeller

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3.3 Construction processes

In the previous section, the cost function required by the MGPSO algorithm was proposed. Given the algorithm parameters and taking 60 search agents into account, we tried to optimize this cost function with the results shown in Fig. 5 and Tables 4 and 5. In this figure, f1 corresponds to efficiency and f2 corresponds to cavitation. To construct the propeller, the chord and optimum thickness coordinates were obtained as the best MGPSO parameters and recorded. Then, given the obtained parameters, the propeller map was plotted in openprop software and entered Catia software to change the format. Then, the file was recalled by PowerMill software (CNC machine operational software). The three-axis lathes reading the map and turning it into code began implementing cutting of the material (phosphorus bronze) with the help of Simco software, and then the propeller was obtained finally. Figure 4 shows a summary of the design process (from the beginning until the final product).

Fig. 4
figure 4

Summary of the construction process

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Table 5 Parameters were obtained with a six-blade propeller
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Fig. 5
figure 5

Comparison between selected algorithms with three model propellers for simulation results to achieve the most optimal efficiency and lowest cavitation

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4 Simulation by the experimental setup and discussion

4.1 Setup properties

It should be noted that in the above example design, all the parameters were not used in the mathematical relationship and all restrictions have not been imposed because of a large number of design parameters and mathematical complexity. To obtain a reliable cavitation level, which is defined by sound pressure level (SPL), a closed-circuit cavitation water tunnel model NA-10 England is employed. In the experimental tests, three model propellers are utilized to perform the relevant tests with small differences in their dimensions. Characteristics and design specifications of the propellers are tabulated in Table 6.

Table 6 Characteristics and design specifications of the propellers
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In this table, \(AE/AO\) signifies the area ratio for the blade, while \(P/D\) signifies the pitch-diameter ratio, \({T}_{0}\) indicates thrust in open water, and \(J\) signifies the advanced coefficient. In the experiments, the revolution rate of the propeller is set at \(1800\mathrm{ RMP}\), while the flow speed is set at \(4\mathrm{ m}/\mathrm{s}\). Moreover, two hydrophones (model B&K 8103) are used to record the radiated sound measurements. In addition, a data acquisition board (\(\mathrm{UDAQ}\_\mathrm{Lite}\)) is connected to the hydrophones. Finally, Fig. 6 presents the test scenario and the configuration of the hydrophones.

Fig. 6
figure 6

The test scenario and the configuration of the propeller and the hydrophones

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4.2 Phases of noise and reflection removal

In this section, it is assumed that \(x(t)\) signifies the reverberant-noise signal for the first hydrophone, while \(y(t)\) signifies the reverberant-noise signal for the second hydrophone. Moreover, the undistorted source of the propeller is denoted by \(s(t)\). The mathematical models for the two reverberant-noise signals are expressed as follows:

$$x(t) = \int\limits_{ - \infty }^{t} {h(t - \tau } )s(\tau )\,\,{\text{dt,}}$$
(14)
$$y(t) = \int\limits_{ - \infty }^{t} {g(t - \tau )s(\tau )\,\,{\text{dt}}} .$$
(15)

In these equations, \(g(t)\) and \(h(t)\) signify the square testing section-response functions. It is assumed that these response functions are not known. However, the tails of \(g\) and \(h\) are not correlated. It is clear that the first few echoes do not reach the two hydrophones at the same time.

Based on the underwater acoustics standard (ITTC, 2017), the usual unit for SPL (i.e., the sound pressure level) is \(dB re 1 \mathrm{\mu Pa}\). However, the hydrophone’s output is stated in terms of \(\mathrm{\mu v}/\mathrm{Pa}\). Hence, the output from the hydrophone must be multiplied by \({10}^{6}\). To achieve the frequency domain SPL, the hamming windows and fast Fourier transforms are applied. It should be noted that generally, the values of SPL measured in each of the \(1/3\) octave bands are decreased using Eq. 16 below to set them equally at the \(1 \mathrm{Hz}\) bandwidth (ITTC, 2017).

$${\text{SPL}}_{1} = {\text{SPL}}_{m} - 10\log \Delta f.$$
(16)

In this equation, \({\mathrm{SPL}}_{m}\) signifies the SPL measured at each center frequency in terms of \(\mathrm{dB re }1\mathrm{ \mu Pa}\), while \(\Delta f\) signifies the bandwidth for each of the \(1/3\) octave band filters. Moreover, \({\mathrm{SPL}}_{1}\) signifies the SPL decreased to \(1 \mathrm{HZ}\) bandwidth, which is stated in terms of \(\mathrm{dB re} 1 \mathrm{\mu Pa}\).

Furthermore, the correction for the background noise measurements will be applied to the measured total levels of noise as a means to remove the noise of the propeller. To do so, instead of the designed propellers, a dummy hub is utilized for recording the background noise by Bert Schneider. The correction is applied according to the level of the difference based on the procedure outlined in the reference [68]. Based on this procedure, if the difference is less than \(3 \mathrm{dB}\), the result must be neglected. However, when the difference ranges from \(3\) to \(10 \mathrm{dB}\), Eq. 17 is utilized to modify the results. In addition, when the difference exceeds \(10 \mathrm{dB}\), there is no need for correction. In a nutshell, in cases where the data have a low signal-to-noise ratio, i.e., lower than \(3\mathrm{ dB}\), the signal is omitted.

$${\text{SPL}}_{P} = 10\,\,\log \left[ {10^{{^{{({\text{SPL}}_{T} /10)}} }} - 10^{{^{{({\text{SPL}}_{B} /10)}} }} } \right].$$
(17)

In this equation, \(T\) signifies the total noise, while \(P\) signifies the propeller noise, and \(B\) signifies the background noise. Afterward, using the following equation, SPLs are modified to a standard measuring distance of \(1 \mathrm{m}\).

$${\text{SPL}}_{1} = {\text{SPL}}_{1} + 20\log \left( r \right).$$
(18)

It should be noted that in this equation, \({\mathrm{SPL}}_{1}\) signifies the \(1\mathrm{ Hz}\) bandwidth SPL stated in terms of \(\mathrm{dB re }1\mathrm{ \mu Pa}\), while \(\mathrm{SPL}\) is the equivalent \(1\mathrm{ Hz}\) at \(1\mathrm{ m}\) distance SPL stated in terms of \(\mathrm{dB re }1\mathrm{ \mu Pa}\). Moreover, \(r\) indicates the distance between the center of the propeller disk, which is the noise source, and the hydrophone [68]. The procedural phase for removing the background noise and performing the preprocessing is presented in Fig. 7.

Fig. 7
figure 7

Preprocessing and removal of background noise

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Once the background noise is removed, the wall reflections must be removed next. As noted earlier, the preprocessing steps are performed based on individual frequency bands. As the first step, the delays present between coherent parts or early echoes in each of the frequency bands are eliminated through a phase shift. This phase of the preprocessing is known as the ‘co-phase and add in bands’ by Bertschneider [68]. Next, a normalized cross-correlation function is used for each individual frequency band to regulate the gain. Afterward, to obtain the estimate for the signal, denoted by \(\widehat{S}\), the resulting signals in individual frequency bands are combined. In addition, Fig. 8 indicates the stage for removing the wall reflections.

Fig. 8
figure 8

Removing the wall reflections

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4.3 The comparison of the cavitation generated by the designed propeller with standard propeller

It is worth mentioning that these conditions apply to each individual frequency band. In other words, in cases where \(x\) and \(y\) are not correlated at one frequency, but they are correlated at another frequency, then the uncorrelated part is removed while the correlated pat is kept. Moreover, since these operations are defined based on a moving average, everything can change as a function of time. The ITTC discusses the different components of the previous process in detail.

Figure 9 only depicts three conventional presentations of the recorded propeller noise along with their Fourier transform, as well as the power spectral density (PSD) function. For instance, the specific frequency characteristics of the physical phenomena are only presented in Fig. 9c. On the other hand, to better understand the noise results, these interpretations can be generalized for other propellers and their operational conditions.

Fig. 9
figure 9

Conventional presentations of recorded noise of the propellers, their Fourier transform, and their PSD function.‬

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It should be noted that around some frequencies, an anomalous peak can be observed, as shown in Fig. 9c. This noise component can be because of the vibration in the foils or other internal components. However, other tonal components are higher than \(90\mathrm{ Hz}\). It is worth mentioning that the characteristic frequencies of these components are multiples of the shaft rate as well as the fundamental frequency. The following equation can be utilized for computing the fundamental frequency in Hertz.

$$f_{{_{{{\text{Fundamental}}}} }} = \frac{{{\text{RPM}}\, \times \,{\text{Number}}\;{\text{of}}\;{\text{blades}}}}{60}\user2{.}$$
(19)

For instance, for propeller C, which has three blades and a revolution rate of \(1800\mathrm{ RPM}\), the fundamental frequency will be \(90\mathrm{ Hz}\), while the other tonal components are multiples of the fundamental frequency. Nonetheless, the elevated levels of the continuous portion of the spectra indicate the effects of the cavitation noise. Therefore, identifying different spectral components in the propeller noise can be very difficult. In general, any cavitation phenomenon results in tonal noise as well as broadband noise.

Therefore, it can be concluded that the cavitation of a propeller is generally represented by a specific random component that affects the continuous portion of the spectrum. Nevertheless, as the frequency goes up, there is a slight decrease in the SPL. In a nutshell, the broadband noise created by the suction-side cavitation and.

The above figure shows the cavitation diagrams, and Fourier transforms when the water flows in the tunnel, has a speed of 4 m per second, and the propeller rotates in 1800 RPM rotations. The noise caused by the bursting of bubbles in the cavity is a unipolar type noise with spherical distribution and uniform dispersion in all directions. Therefore, noise in cavitation has more intensity and power due to the bursting of bubbles evident in propellers’ noise at 1800 RPM rotation diagrams. According to the 1800 RPM rotation noise diagram, the MGPSO designed propeller noise equals 106.79 dB, while the benchmark propeller designed in reference [69] shows noise equal to 107.49 dB, which indicates the improved performance of the propeller in case of the presence of cavitation. According to the experiments, the cavitation noise level is 20 dB, which is higher than that of in the absence of cavitation in the propeller that indicates the cavitation far greater role in detecting and identifying the vessel. The simulations and experiments show that the MGPSO designed a propeller, which is more efficient in terms of efficiency and producing cavitation noise than that of the standard propeller designed.

5 Conclusions

This paper uses the MGPSO algorithm for the sake of marine propeller design. The algorithm can search for high dimension complex problems. Findings show that this algorithm uses four completely Modified Group searches in the problem space, and thus the local optimization complex problems with many local optimizations do not get stuck. Using this algorithm, we were able to achieve a propeller with maximum efficiency and minimal cavitation. The designed propeller has three or five blades. The obtained maximum efficacy is 0.7211 that is desirable. The results show that a five or six blades propeller with a rotational speed of 1800 to 1900 RPM has a lower noise level. On average, the designed propeller produces 106.79 dB noises in case of cavitation and 86.64 dB in case of absence of cavitation so that it produces 1.24 dB and 0.8 dB fewer noises compared to the benchmarks. These designs were proposed in controlled situations and without considering the complexity of the underwater environment. Therefore, the generalizability of these algorithms to provide an algorithm able to consider other propeller’s design parameters, designed to perform more efficiently, can be considered as further work.