Resistance Coefficient Estimation for a Submarine’s Bare Hull Moving in Forward and Transverse Directions

Abstract

Resistance of the bare hull of the tourist submarine with spherical heads, moving in forward and transverse directions is analyzed in OpenFOAM using Computational Fluid Dynamics. The resistance coefficients of the submarine are estimated for different length-to-diameter ratios and Reynolds numbers. The Artificial Neural Network with the optimum number of neurons is then trained to predict the resistance coefficients. Two simplified Artificial Neural Network models and Nonlinear Least Squares Marquardt-Levenberg algorithm are employed to fit the results in the form of equations that may be used in the initial design of this type of submarines. The comparative analysis of different prediction models is performed and guidelines for their practical application are given.

1. Introduction

The present study is inspired by the tourist submarine with the transparent acrylic hull, designed to provide visitors a view of the surrounding ocean in a comfortable manner [1,2]. The overall length of the studied submarine is about 25 m, while the external diameter of the acrylic hull is 2.64 m. The hull of the submarine has spherical heads on the ends, where one head is made of steel, while the other one has large acrylic dome to provide visibility to the pilot of the submarine. The specific feature of tourist submarines is their ability to move in different directions with low speed, while the power requirements are a central consideration in their design to maximize the cruising duration. Although power reduction can be achieved in different ways, e.g., by adopting energy-saving propulsion systems or controlling the boundary layer on the submarine’s surface, the shaping of the submarine’s hull is considered as the most efficient approach. So, the purpose of the present paper is twofold:

• To develop accurate and efficient numerical hydrodynamic model for the resistance coefficient of the submarine moving in forward and transverse directions.
• To develop prediction tool for the resistance coefficients of the submarine in the initial design phase, without need to perform complex hydrodynamic analysis.

Hydrodynamic calculations are performed by the numerical model in OpenFOAM, using Computational Fluid Dynamic (CFD). Practical engineering tool replacing hydrodynamic computations is developed using the Artificial Neural Network (ANN) with optimal number of neurons. As designers of marine structures may be disinclined for using ANN as a “black box”, relatively simple design equations are proposed using two ANN models with small number of neurons, and by Nonlinear Least Squares Marquardt-Levenberg algorithm. Although the actual resistance coefficient of the tourist submarine is higher because of the appendices and additional structures, there is a great need to have preliminary estimate of the bare hull resistance, which is the main benefit of the present study [2].

The paper is organized as follows. In the second part of Introduction, literature review regarding hydrodynamic analysis of submarines and combinations of physics with neural networks is provided. After the explanation of the governing equations and boundary conditions in the second section, the developed model was validated in the third section. Section 4 presents the results and comparative analysis of different prediction models for the resistance coefficients. Finally, the conclusions are presented in Section 5.

Literature Review

Among the key aspects of naval architecture is predicting the resistance of ships and submerged bodies. In this regard, Reynolds Averaged Navier Stokes Equations (RANS) based solutions were studied broadly in the literature. Larsson et al. [3] compared the results for benchmark vessels investigated by various institutions and universities with codes implementing RANS. Wang et al. [4], investigated resistance and wave patterns of a submarine model at different depths, and the influence of free surface on the resistance was discussed. Moonesun et al. [5] studied flow behavior on a model of an underwater vehicle with a tango-shaped nose and presented some formulas for the resistance of submarine bare hulls in deep water. Lastly, they compared the equation to determine the optimum resistance coefficient for the submarine.

Sukas et al. [6] applied RANS-based CFD to the numerical simulation of the flow field around a surface piercing and a fully submerged body to estimate the total resistance of the submarine. A CFD analysis was presented by Moonesun et al. [7] on the bare hull form of submarines or torpedoes to minimize resistance. They studied the bare hull form without free surface effect since the required power will always be estimated for submerged navigation. CFD was used by Ahmed [8] to determine the viscous resistance of a tourist submarine suitable for work in the Red Sea region. In that study, the resistance and hydrodynamic characteristics of the flow surrounding the tourist submarine under different speeds were investigated using the finite volume RANS code CFX. Shen et al. [9] used model-scale submarine resistance tests to predict full-scale resistance. It was assumed that the residual resistance coefficient is independent of the Reynolds number and is measured at the model-scale.

Utina et al. [10] evaluated experimentally and numerically the pressure and frictional force in the opposite direction of the mini submarine movement. Anh et al. [11] calculated the resistance coefficient of an exploratory submarine with a displacement of 6.8 tons moving forward, backward, diving, and rising in different directions. A full-scale SUBOFF model was investigated by Liu et al. [12] for different forward speeds, submerged positions, and fluid densities. They showed that the submarine hydrodynamic performance is significantly affected by the forward speed and submerged depth. Based on Star CCM+ fluid simulation software, the resistance coefficients under different submarine depths and speeds were calculated based on the specific resistance characteristics of submarines sailing near the surface by Chen et al. [13].

Combining physics models with neural networks has been used in fluid dynamics during the last few years [14,15,16,17,18]. As an example, convolutional neural networks were used by Takaaki et al. [14] to model the flow around a circular cylinder at 100 Reynolds numbers. Brenner et al. [17] used machine learning for advancing fluid mechanics. The authors demonstrated that the suggested model is likely to have a positive impact provided outcomes are held to the long-held critical standards that should guide flow physics research. In another research, Brunton et al. [18] reviewed the history and development of machine learning in fluid mechanics, as well as opportunities for the future. A discussion of fundamental machine learning methodologies and their use in understanding, modeling, optimizing, and controlling fluid flow was presented in that paper. The models in these studies were used as a “black box” without explicit equations to estimate unknown parameters.

2. Governing Equations and Boundary Conditions

In the present study, a submerged bare hull of a submarine was modeled in a viscous and incompressible fluid so that the fluid flow was considered turbulent. Accordingly, RANS are used as the governing equations [19]:

$$\nabla .\left(\rho \underset{\_}{V}\right)=0$$

(1)

$$\frac{\partial }{\partial t}\left[\rho \underset{\_}{V}\right]+\nabla .\left[\rho \underset{\_}{VV}\right]=-\nabla \underset{\_}{p}+\nabla .\left[\underset{\_}{\tau }-\rho \underset{\_}{V^{\prime }{V}^{\prime }}\right]+\underset{\_}{f_b}$$

(2)

The nomenclature is presented at the end of the paper. To solve the RANS, the Pressure Implicit Method with Pressure Linked Equations (PIMPLE) algorithm was applied. Different terms of the discretized equations, such as derivative terms, gradient parameters, Laplace derivative terms, and divergence terms, were discretized using 1st order implicit Euler, 2nd order centered Gauss linear, skewness corrected centered Gauss linear correction, and Upwind schemes, respectively [20]. Using block mesh and refinement techniques, a cylindrical domain was generated by using a Cartesian structured grid. Different boundary conditions summarized in

Table 1. Different boundary conditions used for different parameters.

Boundary	Velocity	Pressure	Kinetic Energy (k)	Dissipation Rate (ε)
Inlet	Fixed Value	Zero Gradient	Fixed Value	Fixed value
Outlet	Inlet Outlet	Fixed Value	Inlet Outlet	Inlet Outlet
Body	Moving wall velocity	Zero Gradient	Wall Function	Wall function
around	Symmetry	Symmetry	Symmetry	Symmetry

were used for the velocity, pressure, kinetic energy, and dissipate rate on the boundaries shown in Figure 1. According to Allmendinger [21], if the submarine moves at a depth greater than five times the diameter of its hull, the effect of surface and wave interactions can be ignored.

In addition, the K-Epsilon two-equation model was used to account for turbulence [22].

$$\frac{\partial }{\partial t}\left[\rho k\right]+\nabla .\left[\rho Vk\right]=\nabla .\left[{\mu }_{eff,k}\nabla k\right]+P_k-\beta \rho k\epsilon$$

(3)

$$\frac{\partial }{\partial t}\left[\rho \epsilon \right]+\nabla .\left[\rho V\epsilon \right]=\nabla .\left[{\mu }_{eff,\epsilon }\nabla \epsilon \right]+C{\epsilon }_1\frac{\epsilon }{k}{P}_k-C_{ϵ2}\rho \frac{{\epsilon }^2}{k}$$

(4)

where

$$C_{\epsilon 1}=1.44C_{\epsilon 2}=1.92\beta =0.09$$

$${\mu }_{eff,k}=\mu +\frac{{\mu }_t}{{\sigma }_k} {\mu }_{eff,\epsilon }=\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}{\sigma }_k=1.0{\sigma }_{\omega }=1.3$$

The resistance force and coefficient were calculated by integrating pressure and shear stress over the surface of the submarine as follows [23]:

$$R={\int }_{S}p{n}_{x}dA+{\int }_S{\mathsf{\tau }}_{xy}{n}_{y}dA+{\int }_S{\mathsf{\tau }}_{xz}{n}_{z}dA$$

(5)

$$C_R=\frac{R}{\frac{1}{2}\rho S{V}_f^2}$$

(6)

where S is the surface area of the submarine’s bare hull.

2.1. Artificial Neural Network (ANN)

The back propagation (BP) ANN is a multilayer feedforward ANN used for benchmarking prediction performance [24,25,26]. There are three layers in BP ANN: input, hidden, and output. Each hidden layer receives the input signal through the input layer, and finally, the output layer receives it. Error signals are sent back to the hidden layer and the input layer. After that, the gradient descent algorithm is applied to adjust the weight and threshold of each neuron so that the BP analogy output is close to the expected value (See Figure 2). The number of neurons in the hidden layer is selected by repeated experiments to optimize the performance of the neural network.

As ANN with many neurons can be used only as the “black box”, the approach may be inconvenient for submarine designers, who are usually not familiar with ANN. Therefore, it is of great practical interest to develop design equations for practical engineering usage. Since the equation derived from a neural network with many neurons would be very long and unusable, two ANN structures with one and two neurons in the hidden layer are considered (See Figure 3 and Figure 4).

Convolution produces the following output for the ANN shown in Figure 3 and Figure 4, respectively:

$$C_R=f_2\left(b_2+w_3{f}_1\left(w_1\frac{L}{D_s}+w_{2}log\left(R_e\right)+b_1\right)\right)$$

(7)

$$C_R=f_2\left(b_3+w_5{f}_1\left(w_1\frac{L}{D_s}+w_{3}log\left(R_e\right)+b_1\right)+w_6{f}_1\left(w_2\frac{L}{D_s}+w_{4}log\left(R_e\right)+b_2\right)\right)$$

(8)

Activation functions introduce non-linearity into neural networks and help to capture the non-linear characteristics of input data [28,29]. There are different activation functions including Sigmoid, hyperbolic tangent, and Identity used in the neurons of the hidden and output layers of ANN [27]. According to the preliminary results of this study, ANNs with hyperbolic tangents $\left(f_1\left(x\right)=tanh\left(x\right)\right)$ and Identity $\left(f_2\left(x\right)=x\right)$ activation functions in the hidden and output layers provides credible results.

2.2. Nonlinear Least Squares Levenberg-Marquardt Algorithm (NLLS)

Due to its high convergence efficiency to obtain the global optimal solution, the NLLS algorithm has been widely used [30,31,32,33]. By considering the equation $C_R=f\left(\frac{L}{D_s},log\left(R_e\right), p\right)$, the problem-solving nonlinear equations are expressed as:

$$F\left(x,p\right)=C_R-f\left(\frac{L}{D_s},log\left(R_e\right), p\right)$$

(9)

The nonlinear equation $f\left(x,p\right)$ needs to be solved, where x is the time series, y is the observations, and p is the nonlinear equation parameters. A nonlinear model is fitted by minimizing the sum of the square of errors, which can be expressed as:

$$min{\displaystyle \sum }_{i=k}^{n}R-f\left(\frac{L}{D_s},log\left(R_e\right), p_k\right)=min{\displaystyle \sum }_{i=k}^n{\epsilon }_k$$

(10)

where $p_k$ is the parameter of the kth iteration, and ${\epsilon }_k$ is the residual of the kth iteration. Taylor expansion was used to approximate the nonlinear function of $f\left(x,p_{k+1}\right)$ for solving Equation (10) as follows:

$$f\left(\frac{L}{D_s},log\left(R_e\right), p_{k+1}\right)=f\left(\frac{L}{D_s},log\left(R_e\right), p_k+\Delta \right)=f\left(x,p_k\right)+J\left(p_k\right)\Delta$$

(11)

where J is the Jacobian matrix and ∆ are the steps of the (k + 1)th iteration. By solving the following equation, we can get the residual of the (k + 1)th iteration as follows:

$$C_R-f\left(\frac{L}{D_s},log\left(R_e\right), p_{k+1}\right)=C_R-f\left(\frac{L}{D_s},log\left(R_e\right), p_k\right)-J\left(p_k\right)\Delta ={\epsilon }_k-J\left(p_k\right)\Delta =0$$

(12)

It is known as Newton’s method. This method, however, cannot be used to solve equations with overdetermined matrices. This problem was solved by multiplying a transposed matrix to reduce the overdetermined matrix’s dimension by Gauss-Newton (Equation (13)).

$$J{\left(p_k\right)}^{T}J\left(p_k\right)\Delta =J{\left(p_k\right)}^{T}J\left(p_k\right){\epsilon }_k=g$$

(13)

By developing the Levenberg-Marquardt (LM) algorithm, the Gauss-Newton method was improved, since it does not work when the Hessian matrix is singular. To use the LM method, a constant, the trust-region radius, must be included in the equation as follows.

$${\Delta }_k^{LM}=-{\left(J{\left(p_k\right)}^{T}J\left(p_k\right)+{\mu }_{k}I\right)}^{-1}g$$

(14)

The parameter ${\mu }_k$ is an iteration parameter introduced to overcome constraints caused by singularities or near singularities of $J\left(p_k\right).$ This parameter is also used to preventing ${\Delta }_k^{LM}$ from being too large when the Hessian matrix ($J{\left(p_k\right)}^{T}J\left(p_k\right)$) is nearly singular. ${\Delta }_k^{LM}$ is well defined in this case if ${\mu }_k$ is positive. In comparison to the steepest gradient method, Newton’s method, and the Gauss-Newton method, the LM method is the most widely used nonlinear fitting method. Near to the solution, ${\mu }_k$ may be very small. In contrast, when the solution is far away, the value of ${\mu }_k$ may be very large; therefore, controlling ${\mu }_k$ can lead to an optimal solution.

3. Mesh Size Calibration

Figure 5a shows how a cube with L_C = 0.45 m edge length was placed inside a cylinder of 6 m length and 2 m diameter to determine mesh size. By considering the input boundary condition as a constant velocity, the total resistance of the cube against fluid flow is estimated. A cube was analysed against a constant 10 m/s fluid flow to examine the mesh size dependency. The numerical results showed that uniform mesh sizes (ms) with ms/L_C less than 0.09 did not significantly affect the total resistance as shown in Figure 5b.

Accordingly, the developed model was used to estimate the resistance coefficient (C_R) of the cube by using Equation (15) against fluid flows with different velocities (V_f), and the results are shown in Figure 6a. Based on the results shown in Figure 6a, the average resistance coefficient for different speeds is calculated to be about 1.13. According to the literature [34,35,36], the resistance coefficient of a cube is estimated to be between 1.05 and 1.20. Therefore, the average of 1.13 was found to be in very good agreement with the amount suggested by other researchers.

$$C_R=\frac{2R}{\rho V_f^2{L}^2}$$

(15)

The cube exhibits significantly more vortex shedding than the submarine’s bare hull, so another mesh size verification is performed using the sphere. Thus, a sphere with a diameter of 2.64 m is also computed with the same mesh size. Figure 6b shows the resistance force and coefficient for fluid flows with different Reynolds numbers, where an average resistance coefficient of 0.45 was calculated. The resistance coefficient of a sphere is estimated in the range of 0.4 and 0.5 for Reynolds numbers greater than 10⁴ [36]. Consequently, the average of 0.45 was found to be very close to the amount suggested in [37].

4. Results

The present study aims to model the bare hull of a submarine against speeds to derive equations for resistance coefficients in forward and transverse directions. To that end, a range of credible tourist submarine geometries, represented by the ratio L/D_S, is defined, and shown in

Table 2. The dimensions of a submarine for different L/D_S ratios.

Case	L (m)	DS (m)	L/DS
1	25	2.44	10.2
2	24	2.49	9.6
3	23	2.54	9.1
4	22	2.59	8.5
5	21.06	2.64	8.0
6	20	2.7	7.4
7	19	2.76	6.9
8	18	2.83	6.4
9	17	2.9	5.9

, where L represents the length of the cylindrical hull, L_oa is the overall length including the heads, and D_S represents the diameter of the cylindrical hull. The geometries are defined by keeping the total internal volume of the submarine nearly constant. Isabella, a supercomputer housed at SRCE—University Computing Centre of the University of Zagreb, is used for the analysis. Isabella consists of 135 worker nodes, 3100 processor cores, 12 GPUs, and 756 TiB of data storage [38].

4.1. Forward Motion

A total of 117 different cases, as described in the previous section, were modeled in OpenFOAM for 13 different speeds (0.25, 0.35, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4 and 1.5 m/s). Therefore, all 117 cases listed in Appendix A were analyzed. OpenFOAM computation for each of the 117 cases lasted for about 18,000 s, so the total running time was about 35,100 min. Results of the analysis were presented in Figure 6. The resistance coefficient for the forward motion in Figure 7 were presented against L/D_S and Reynolds number, given by the following expression:

$$R_{ef}=\frac{V_f{D}_S}{\upsilon }$$

(16)

In the present case, $\upsilon$ is the dynamic viscosity of ${10}^{-6}\frac{{\mathrm{m}}^2}{\mathrm{s}}$ [39].

The results shown in Figure 7 indicate that the relationship between the resistance coefficient in the forward direction (C_D–f) and L/D_S is almost linear. However, the change of this parameter is somewhat nonlinear at the low (e.g., 0.5 m/s) and high (e.g., 1.5 m/s) speeds. On the other hand, for all L/D_S, the relationship between C_D–f and forward speed (V_f) is nonlinear.

Next, NLLS and ANN methods were applied to fit the best curve to the results shown in Figure 7 and find a nonlinear equation to estimate the resistance coefficient of the bare hull based on two dimensionless parameters, L/D_S and Reynolds number (R_ef). The following equations were derived for the resistance coefficient of the submarine with the forward speed using the developed code in MATLAB. For the ANN, 70%, 15%, and 15% of data (corresponding to 83, 17, and 17 data) were used in training, testing, and evaluation steps, respectively:

NLLS:

$$\begin{array}{cc}{C}_{R-f}=5.372 -\hfill & 0.34\frac{L}{D_S}+1.625log(R_{ef})+0.0031{\left(\frac{L}{D_S}\right)}^2+0.096\frac{L}{D_S}log(R_{ef})+0.125{\left(log(R_{ef})\right)}^2\hfill \\ & -0.0000534{\left(\frac{L}{D_S}\right)}^3-0.0002102{\left(\frac{L}{D_S}\right)}^2\left(log(R_{ef})\right)+0.007171\frac{L}{D_S}{\left(log(R_{ef})\right)}^2\hfill \end{array}$$

(17)

ANN (Figure 3):

$$C_{R-f}=0.0971+0.0827tanh\left(-0.146\frac{L}{D_S}-1.584log(R_{ef}\right)+9.578)$$

(18)

ANN (Figure 4):

$$\begin{array}{cc}{C}_{R-f}=0.112+\hfill & 0.0547\mathit{tanh}\left(-0.161\frac{L}{D_S}+1.971log(R_{ef})-13.066\right)+0.047tanh(-0.119\frac{L}{D_S}\hfill \\ \hfill & -2.004log(R_{ef})+12.325)\hfill \end{array}$$

(19)

To assess that accuracy of Equations (17)–(19), ANNs with larger number of neurons (NON) in the hidden layer (see Figure 8) are also trained.

In ANN, the optimal NONs in the hidden layers is determined by trial and error and not straightforwardly. However, it was advised to use 2–4 times the input layer’s number of nodes [40,41]. So, by considering the number of nodes in the input layer = 2, ANNs with 3–10 neurons in the hidden layer (as a “black box”) were used to estimate the resistance coefficient of the submarine for the speeds of 0.1, 0.45, 1.15, and 1.7 m/s, and Sum Square Error ($SSE={\left({\left(C_{R-f}\right)}_{CFD}-{\left(C_{R-f}\right)}_{Pre}\right)}^2$) of the models were estimated and shown in Figure 9. ${\left(C_{R-f}\right)}_{CFD}$ and ${\left(C_{R-f}\right)}_{Pre}$ are the predicted resistance coefficient by using CFD and ANN, respectively.

Figure 9 shows that ANN with six neurons in the hidden layer (NON = 6) has the best performance for predicting resistance coefficients.

Equations (17) to (19) and the trained ANN with NON = 6 were used to predict the resistance coefficient of the submarine for the speeds 0.1, 0.45, 1.15, and 1.7 m/s, and the results were shown in Figure 10. The error estimated as $Error=100\ast \frac{{\left(C_{R-f}\right)}_{CFD}-{\left(C_{R-f}\right)}_{Pre}}{{\left(C_{R-f}\right)}_{CFD}}$ was also calculated for different methods and shown in Figure 11.

According to Figure 11a, Equations (18) and (19) are unable to estimate accurately the resistance coefficient for speeds lower than those used in training the artificial neural network. The model error varies from 15 to 30 percent depending on the submarine’s length and diameter. NLLS (Equation (17)) estimates the resistance coefficient with a maximum error of 15%. ANN with NON = 6 estimates the resistance coefficient with a maximum error of 3% and 10% for the submarines with L/D_S > 7.5 and L/D_S < 7.5, respectively.

As shown in Figure 11b,c, Equations (17) and (19) and ANN with NON = 6 provide high accuracy for estimating resistance coefficients at speeds used for training ANNs and NLLS.

The results for the forward speed 1.7 m/s (Figure 11d) which was higher than the range of speeds used in the extraction of relationships indicates that Equations (17) and (19) estimate the resistance coefficient with a maximum 10% and 7% error, respectively. ANN with NON = 6 shows a better performance than other methods, with maximum error of 3%.

4.2. Transverse Motion

Tourist submarines are equipped with transverse thrusters to enable transverse motion and maneuvering. Additionally, they are used to prevent the transverse motion by acting in the opposite direction to the sea current. Therefore, there is an interest in studying the transverse resistance of the submarine as well as the transverse motion of the submarine, and to suggest equations to estimate the resistance coefficient in the transverse direction. To accomplish this, 90 different cases are modeled in OpenFOAM using the developed code for 10 different transverse speeds (0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.4, 0.45 and 0.5 m/s). Therefore, 90 cases listed in Appendix B were analyzed and are shown in Figure 12. OpenFOAM computation for each of the 90 cases lasted for about 18,000 s, so the total running time was about 27,000 min.

Figure 12 shows that the resistance coefficient in the transverse direction (C_D-t) is linearly related to the L/D_S. Nevertheless, for all L/D_S, the relationship between C_D-t and transverse speed (V_t) is nonlinear. NLLS and ANNs were applied to fit the best curve to the obtained results shown in Figure 13 and to find a nonlinear equation to estimate the resistance coefficient of the bare hull based on two dimensionless parameters, L/D_S and Reynolds number (R_et). The following equations were derived for the resistance coefficient of the submarine with transverse speed using the code in MATLAB. In ANN, 70%, 15%, and 15% of data (corresponding to 64, 13, and 13 data) were also used in training, testing, and evaluation steps, respectively:

NLLS:

$$\begin{array}{cc}{C}_{R-t}=232.71-\hfill & 5.211889\frac{L}{D_S}-105.35826\left(log(R_{et})\right)+0.083091{\left(\frac{L}{D_S}\right)}^2+1.46633\frac{L}{D_S}\left(log(R_{et})\right)\hfill \\ & +16.000585{\left(log(R_{et})\right)}^2-0.0133027{\left(\frac{L}{D_S}\right)}^{2}log(R_{et})-0.101747\frac{L}{D_S}{\left(log(R_{et})\right)}^2\hfill \\ & -0.815011{\left(log(R_{et})\right)}^3\hfill \end{array}$$

(20)

ANN (Figure 3):

$$C_{R-t}=1.772+1.513tanh(-0.093\frac{L}{D_S}-1.92log(R_{et})+10.361)$$

(21)

ANN (Figure 4):

$$\begin{array}{cc}{C}_{R-t}=1.516+\hfill & 1.22\mathit{tanh}\left(-0.1093\frac{L}{D_S}-2.278log(R_{et})+12.466\right)+0.0374tanh(-0.534\frac{L}{D_S}\hfill \\ & -3.695log(R_{et})+26.177)\hfill \end{array}$$

(22)

The data in Appendix B has also been used to train ANNs with more than two neurons in the hidden layer (see Figure 8). The resistance coefficient was estimated at 0.02 m/s as a test of the ability of the proposed models to estimate it at different speeds. This speed is lower and outside the range used for the extraction of relations. In addition, the resistance coefficient of the submarine at transverse speeds of 0.125 and 0.375 was estimated, which was included in the range of speeds used in the extraction of relationships, but not in the training. In a similar manner, the resistance coefficient was estimated for a speed of 0.6, which was higher than the range of speeds used to extract the relationships. Training ANNs with different NONs were used to predict the coefficient of resistance of the submarine at speeds of 0.02 m/s, 0.125, 0.375, and 0.6, and SSE is shown in Figure 13.

It is shown in Figure 13 that ANN with seven neurons in the hidden layer (NON = 7) performs the best at predicting resistance coefficients. So, Equations (20) to (22) and the trained ANN with NON = 7 were used to estimate the resistance coefficient at speeds of 0.02 m/s, 0.125, 0.375, and 0.6, and the results were shown in Figure 14. The estimation error for the resistance coefficient is shown in Figure 15.

Figure 15a shows that the derived equations (Equations (21) and (22)) by using ANN are not able to estimate accurately resistance coefficients at lower speeds than those used in the ANN training, while the NLLS method (Equation (20)) can estimate the resistance coefficient for the transverse direction with a maximum error of 15%. Even ANN with NON = 7 is not performing well, having a maximum error of 20%.

As shown in Figure 15b,c, the proposed equations and ANN with NON = 7 are effective in estimating resistance coefficients at speeds inside the training range.

According to the results for 0.6 m/s transverse speed (Figure 15d), which is higher than the range of speeds used to extract relationships, Equation (21) can estimate resistance coefficients with a maximum of 10% for the submarine with L/D_S > 7.5, while Equation (20) shows maximum error of 12% in all cases. ANN with NON = 7 is slightly better, with 10% prediction error.

5. Discussion

The range of the estimation error is summarized in

Table 3. The range of the estimation error (%) for the velocities below, within, and above those used for the training.

Parameter	Equation	Below the Range		Within the Range		Above the Range
		L/DS > 7.5	L/DS < 7.5	L/DS > 7.5	L/DS < 7.5	L/DS > 7.5	L/DS < 7.5
CR-f	Equation (17) (NLLS)	10–15	11–13	7–9	6–7	6–10	1–11
	Equation (18) (ANN)	15–30	16–17	1–13	6–18	5–23	24–26
	Equation (19) (ANN)	15–21	24–30	0.2–5	1–4	0.2–7	1–6
	ANN (NON = 6)	1–3	7–10	2–4	1–4	0.1–2	0.8–3
CR-t	Equation (20) (NLLS)	11–12	13–16	4–6	2–4	3–7	3–12
	Equation (21) (ANN)	28–30	31–30	1–9	2–8	1–10	7–9
	Equation (22) (ANN)	25–30	21–27	2–3	0.4–3	5–11	12–15
	ANN (NON = 7)	13–15	16–20	0.6–1	0.1–0.2	2–4	3–10

.

The following conclusions can be made on the accuracy of the procedures for estimating the resistance coefficient:

➢

For the velocities lower than the range used for training, ANN with NON = 6 shows much better ability to estimate the resistance coefficient for the forward direction compared to the equations. Surprisingly, for transverse movement the NLSS performs better than the ANN.

➢

For forward velocities higher than those used for the training, ANN with optimal NON has the lowest error. However, for transverse directions, differences between equations and ANN are small.

➢

For transverse velocities within training range, ANN with NON = 7 is more accurate than the equations. For forward motion, however error of Equation (19) is close to the error of ANN with NON = 6.

6. Conclusions

A Computational Fluid Dynamics numerical model in OpenFOAM is developed to compute the resistance coefficient of a tourist submarine’s bare hull with a variable length-to-diameter ratio at different speeds, moving in forward and transverse directions.

To propose a practical design estimation method for hydrodynamic resistance of a submarine, the Artificial Neural Network is trained with optimal number of neurons, which is found to be 6 and 7 for forward and transverse direction, respectively. Furthermore, two simple Artificial Neural Network models with small number of neurons and Nonlinear Least Squares Marquardt-Levenberg algorithm are employed to develop design equations for the resistance of the bare hull in two directions. Although the proposed equations perform relatively well compared to the numerical results, the Artificial Neural Network with optimal number of neurons leads to the more reliable results in most of the cases. The care should be taken when employing Artificial Neural Network outside the training range, as large errors could arise, as shown in Table 3.

The actual resistance coefficient of the tourist submarine will be higher because of the appendages and external structures [2]. Nevertheless, equations and results developed herein may be used for preliminary estimate of the bare hull resistance, which is necessary step in evaluation of the total submarine’s resistance.