Heidari et al 2019a - Revision history

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Rimni: /* 2.4 Mesh Convergence */

2021-06-02T13:47:13Z

‎2.4 Mesh Convergence

Rimni at 13:28, 2 June 2021

2021-06-02T13:28:36Z

Rimni at 13:25, 2 June 2021

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Rimni at 13:07, 28 May 2021

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Rimni at 11:02, 23 April 2020

2020-04-23T11:02:10Z

← Older revision		Revision as of 11:22, 8 June 2021
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	In this study, the angular configuration of trimaran side hulls was considered. With regards to the coordinate axes, the side hull was rotated around the <math>y</math>-axis at determined angles. Then, this motion was investigated at different speeds, including its effect on vessel resistance. The transverse distance of the side hull from the main hull was 0.209 m, and the transoms of all three hulls were in the same direction. The effect of trim angle of the side hull was investigated relative to the main hull at four Froude numbers and four trim angles of 0º, 0.4º, 0.8º and 1.2º ([[#tab-5\|Table 5]] and [[#img-14\|Figure 14]]). The results showed that resistance of the side hull reduced as the trim angle of the side hull increased. Also, at all speeds, the total resistance of the trimaran vessel reduced with growing relative trim angle. Furthermore, with the increase of Froude number, the percentage of reduction in hull resistance increased due to relative trim angle. The wetted surfaces of the vessel were investigated for different trim angles and Froude numbers. At Froude number 0.4 and speed of 2.98 m/s, the trim angle of the main hull was 0.2°; and at Froude number 0.45 and speed of 3.356 m/s, the trim angle of the main hull increased to 0.44°. The increasing trend retained until it peaked at 0.77° for the speed of 4.47 m/s and Froude number 0.6.		In this study, the angular configuration of trimaran side hulls was considered. With regards to the coordinate axes, the side hull was rotated around the <math>y</math>-axis at determined angles. Then, this motion was investigated at different speeds, including its effect on vessel resistance. The transverse distance of the side hull from the main hull was 0.209 m, and the transoms of all three hulls were in the same direction. The effect of trim angle of the side hull was investigated relative to the main hull at four Froude numbers and four trim angles of 0º, 0.4º, 0.8º and 1.2º ([[#tab-5\|Table 5]] and [[#img-14\|Figure 14]]). The results showed that resistance of the side hull reduced as the trim angle of the side hull increased. Also, at all speeds, the total resistance of the trimaran vessel reduced with growing relative trim angle. Furthermore, with the increase of Froude number, the percentage of reduction in hull resistance increased due to relative trim angle. The wetted surfaces of the vessel were investigated for different trim angles and Froude numbers. At Froude number 0.4 and speed of 2.98 m/s, the trim angle of the main hull was 0.2°; and at Froude number 0.45 and speed of 3.356 m/s, the trim angle of the main hull increased to 0.44°. The increasing trend retained until it peaked at 0.77° for the speed of 4.47 m/s and Froude number 0.6.
−

	<div class="center" style="font-size: 75%;">'''Table 5'''. The effect of trim angle on hydrodynamic behavior of trimaran vessel</div>		<div class="center" style="font-size: 75%;">'''Table 5'''. The effect of trim angle on hydrodynamic behavior of trimaran vessel</div>

← Older revision		Revision as of 09:05, 3 June 2021
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	[[#tab-9\|Table 9]] shows the critical yaw angles side hull in different Froude number. Generally, the resistance of trimaran was significantly increased by creating the yaw angle of +6º in all Froude numbers. The highest total resistance was seen at Froude number 0.55. At the Froude number in 0.4 and 0.45, creating yaw angle of side hull did not decrease resistance. However, at higher speeds, the yaw angles of -4ºand -2º can reduce vessel resistance.		[[#tab-9\|Table 9]] shows the critical yaw angles side hull in different Froude number. Generally, the resistance of trimaran was significantly increased by creating the yaw angle of +6º in all Froude numbers. The highest total resistance was seen at Froude number 0.55. At the Froude number in 0.4 and 0.45, creating yaw angle of side hull did not decrease resistance. However, at higher speeds, the yaw angles of -4ºand -2º can reduce vessel resistance.
−

	<div class="center" style="font-size: 75%;">'''Table 9 '''. Critical yaw angles in different Froude Number</div>		<div class="center" style="font-size: 75%;">'''Table 9 '''. Critical yaw angles in different Froude Number</div>

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 ===2.3 Computational domain and boundary conditions===
-In this study, various dimensions of the computational domain were considered. Therefore, all models were imported to Star-CCM [https://en.wikipedia.org/wiki/CD-adapco (CD-adapco, USA)] software for analysis. Then, the results for each computational domain were compared with an experimental study [20]. [[#img-6|Figure 6]] shows the comparison between the numerical and experimental results in terms of computational domain volume. The simulations were performed at the highest speed (Froude number 0.6). Based on Figure 5, at the volume of fluid (VOF) of 235  <math display="inline">{m}^{3}</math>, the numerical results were reasonably close to the experimental results and there was a very small error. However, with further increase in domain size, the results did not change significantly, whereby this only led to an increase in unnecessary computations. From here, a domain with a volume of 235 <math display="inline">{m}^{3}</math> was chosen for the simulations in this study. [[#img-7|Figure 7]] shows the dimensions of this domain. The width of the computational domain was 3 m as well.
+In this study, various dimensions of the computational domain were considered. Therefore, all models were imported to Star-CCM [https://en.wikipedia.org/wiki/CD-adapco (CD-adapco, USA)] software for analysis. Then, the results for each computational domain were compared with an experimental study [20]. [[#img-6|Figure 6]] shows the comparison between the numerical and experimental results in terms of computational domain volume. The simulations were performed at the highest speed (Froude number 0.6). Based on [[#img-5|Figure 5]], at the volume of fluid (VOF) of 235  <math display="inline">{m}^{3}</math>, the numerical results were reasonably close to the experimental results and there was a very small error. However, with further increase in domain size, the results did not change significantly, whereby this only led to an increase in unnecessary computations. From here, a domain with a volume of 235 <math display="inline">{m}^{3}</math> was chosen for the simulations in this study. [[#img-7|Figure 7]] shows the dimensions of this domain. The width of the computational domain was 3 m as well.
 <div id='img-6'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
@@ Line 378: / Line 378: @@
 </div>
 <div id='tab-3'></div>
-{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
+{| style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Viscosity (Pa-s)'''

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 </div>
 <div id='tab-3'></div>
-{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:65%,"
+{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Viscosity (Pa-s)'''

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 ===2.4 Mesh Convergence===
-Mesh convergence was carried out using the Star-CCM software. Three types of mesh, including surface, prism layer, and trimmer were used in this study. The surface mesh was used for the surface of the hull as well as the outer boundaries of the computational domain. The finest surface mesh was defined on the side hulls by considering the dimensions of the side hull in comparison with the main hull. At the free surface of the contact area of the two fluids (air and water), the cells contain two phases of fluid that are combined in the presence of a spray flow of these two fluids. Otherwise, water is placed, as the heavier fluid on the cell floor and the cell contains both fluids. To determine the exact location of each fluid within the cells in the free surface, smaller cells must be created in the area. For this purpose, a free-timer network was used. As shown in Figure 9(a), at the free surface, the height of the cells decreases along the ''z''-axis in a controlled manner. Two cubes were defined around each side hull and main hull. In these volumes, using the surface network of the cells, the model was minified as desired, thus reducing calculation error. Figure 9(b) shows the grid on the surface of the main and side hulls. Due to the smaller dimensions of the side hull, a smaller mesh was chosen for its surface. With regards to the base size, each cell in the Star-CCM software was 0.3 m with a growth rate of 1.3, the maximum cell size was estimated to be about 5000 times the size of the base. By moving away from the hull and approaching the outer boundaries of the computational domain, the mesh size was increased by the growth rate of 1.3, hence, generating the largest cell size at the outer boundaries. Also, to accurately resolve the interface between air and water, a trimmer mesh was used for this region. Six parallel layers of prism layer mesh with a thickness of 0.007 ''m'' were generated around the hull. This type of mesh helps to improve simulation results.
+Mesh convergence was carried out using the Star-CCM software. Three types of mesh, including surface, prism layer, and trimmer were used in this study. The surface mesh was used for the surface of the hull as well as the outer boundaries of the computational domain. The finest surface mesh was defined on the side hulls by considering the dimensions of the side hull in comparison with the main hull. At the free surface of the contact area of the two fluids (air and water), the cells contain two phases of fluid that are combined in the presence of a spray flow of these two fluids. Otherwise, water is placed, as the heavier fluid on the cell floor and the cell contains both fluids. To determine the exact location of each fluid within the cells in the free surface, smaller cells must be created in the area. For this purpose, a free-timer network was used. As shown in [[#img-9|Figure 9]](a), at the free surface, the height of the cells decreases along the ''z''-axis in a controlled manner. Two cubes were defined around each side hull and main hull. In these volumes, using the surface network of the cells, the model was minified as desired, thus reducing calculation error. [[#img-9|Figure 9]](b) shows the grid on the surface of the main and side hulls. Due to the smaller dimensions of the side hull, a smaller mesh was chosen for its surface. With regards to the base size, each cell in the Star-CCM software was 0.3 m with a growth rate of 1.3, the maximum cell size was estimated to be about 5000 times the size of the base. By moving away from the hull and approaching the outer boundaries of the computational domain, the mesh size was increased by the growth rate of 1.3, hence, generating the largest cell size at the outer boundaries. Also, to accurately resolve the interface between air and water, a trimmer mesh was used for this region. Six parallel layers of prism layer mesh with a thickness of 0.007 ''m'' were generated around the hull. This type of mesh helps to improve simulation results.
+<div id='img-9'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
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-A mesh convergence was performed at Froude number 0.6. Fundamentally, the resistance coefficient was obtained using different mesh sizes and the results were compared with an experimental study under the same conditions (Figure 10). The computational domain size was the same for all simulations. As the number of cells approached 741,868, the numerical results approached closer to that of experimental with a decrease in value of error. Then, with a further increase in the number of cells, the simulation time increased without any significant change in accuracy. Consequently, the mesh with 741,868 cells was chosen as the optimal mesh to perform the simulations at different Froude numbers.
+A mesh convergence was performed at Froude number 0.6. Fundamentally, the resistance coefficient was obtained using different mesh sizes and the results were compared with an experimental study under the same conditions ([[#img-10|Figure 10]]). The computational domain size was the same for all simulations. As the number of cells approached 741,868, the numerical results approached closer to that of experimental with a decrease in value of error. Then, with a further increase in the number of cells, the simulation time increased without any significant change in accuracy. Consequently, the mesh with 741,868 cells was chosen as the optimal mesh to perform the simulations at different Froude numbers.
+<div id='img-10'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-

@@ Line 345: / Line 345: @@
 ===2.3 Computational domain and boundary conditions===
-In this study, various dimensions of the computational domain were considered. Therefore, all models were imported to Star-CCM [https://en.wikipedia.org/wiki/CD-adapco (CD-adapco, USA)] software for analysis. Then, the results for each computational domain were compared with an experimental study [20]. Figure 6 shows the comparison between the numerical and experimental results in terms of computational domain volume. The simulations were performed at the highest speed (Froude number 0.6). Based on Figure 5, at the volume of fluid (VOF) of 235  <math display="inline">{m}^{3}</math>, the numerical results were reasonably close to the experimental results and there was a very small error. However, with further increase in domain size, the results did not change significantly, whereby this only led to an increase in unnecessary computations. From here, a domain with a volume of 235 <math display="inline">{m}^{3}</math> was chosen for the simulations in this study. Figure 7 shows the dimensions of this domain. The width of the computational domain was 3 m as well.
+In this study, various dimensions of the computational domain were considered. Therefore, all models were imported to Star-CCM [https://en.wikipedia.org/wiki/CD-adapco (CD-adapco, USA)] software for analysis. Then, the results for each computational domain were compared with an experimental study [20]. [[#img-6|Figure 6]] shows the comparison between the numerical and experimental results in terms of computational domain volume. The simulations were performed at the highest speed (Froude number 0.6). Based on Figure 5, at the volume of fluid (VOF) of 235  <math display="inline">{m}^{3}</math>, the numerical results were reasonably close to the experimental results and there was a very small error. However, with further increase in domain size, the results did not change significantly, whereby this only led to an increase in unnecessary computations. From here, a domain with a volume of 235 <math display="inline">{m}^{3}</math> was chosen for the simulations in this study. [[#img-7|Figure 7]] shows the dimensions of this domain. The width of the computational domain was 3 m as well.
+<div id='img-6'></div>
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 |}
+<div id='img-7'></div>
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 These computational dimensions for vessel simulations were used to ensure that vessel waveforms do not match the magnitude of the waveform. Thus, for further investigations, smaller dimensions may be selected such that the necessary conditions can be created with unnecessary calculations reduced, ultimately minimizing time spent for analysis. Therefore, by changing the dimensions of the computational range of the star Star-CCM software and comparing the numerical results in each domain with the experimental results, the optimal dimensions were chosen without any significant effect on the final results.
-After determination of the computational domain, it was necessary to define the boundary conditions of this domain with respect to the physics of the problem (Figure 8). At the input, the boundary condition was velocity, acting perpendicular to the boundary. The tangential velocity was at the upper and lower boundaries. At the output, the conditions were defined such that output pressure is equal to hydrostatic pressure. The boundaries at the right and left sides of the vessel were considered symmetric. Table 3 shows the physical properties of water and air [20]. The fluids in the computational domain were defined as the Eulerian multiphase, which includes air, water, and the combination of the two phases. Furthermore, due to the interaction between the water and the moving cylinder under the water surface, the flow regime was considered turbulent. The <math>k-\varepsilon</math> model was also used to solve the equations governing the turbulent flow regime.
+After determination of the computational domain, it was necessary to define the boundary conditions of this domain with respect to the physics of the problem ([[#img-8|Figure 8]]). At the input, the boundary condition was velocity, acting perpendicular to the boundary. The tangential velocity was at the upper and lower boundaries. At the output, the conditions were defined such that output pressure is equal to hydrostatic pressure. The boundaries at the right and left sides of the vessel were considered symmetric. [[#tab-3|Table 3]] shows the physical properties of water and air [20]. The fluids in the computational domain were defined as the Eulerian multiphase, which includes air, water, and the combination of the two phases. Furthermore, due to the interaction between the water and the moving cylinder under the water surface, the flow regime was considered turbulent. The <math>k-\varepsilon</math> model was also used to solve the equations governing the turbulent flow regime.
+<div id='img-8'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
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 <div class="center" style="font-size: 75%;">'''Table 3'''. Physical properties of fluid
 </div>
+<div id='tab-3'></div>
-{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%,"
+{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:65%,"
 |-
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Viscosity (Pa-s)'''

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 ===2.2 Resistance components===
-The study of resistance in trimaran vessel is slightly different due to the positioning of the three hulls alongside each other, therefore, the interaction between the hulls must be considered. The resistance of a vessel consists of several components. Figure 5 shows the resistance components of a vessel, separately [21]. Hence, the main components of the resistance are shown, whereas other resistances, such as wind resistance, dormancy strength, and resistance to motion in shallow water are excluded. This pattern has always been used for a variety of vessels, and with small changes, it can be generalized to a trimaran vessel. Of course, in this model, the main components of the resistance have been cited, and other resistances, such as wind resistance have been ignored. With regards to the dependence of each component of the resistance on a dimensionless number, the model's resistance is generalized using the dimensionless number scale corresponding to the total resistance of the vessel. Based on the type of vessel, some components of resistance shown in Figure 5 are more important than others. The vessel resistance in water is divided into two parts of residual and frictional resistance. To calculate total resistance in this study, the residual resistance, (it has two components; pressure resistance and shell resistance), shell frictional resistance, wave-making resistance, wave resistance, frictional resistance, viscous resistance, and viscous pressure were considered.
+The study of resistance in trimaran vessel is slightly different due to the positioning of the three hulls alongside each other, therefore, the interaction between the hulls must be considered. The resistance of a vessel consists of several components. [[#img-5|Figure 5]] shows the resistance components of a vessel, separately [21]. Hence, the main components of the resistance are shown, whereas other resistances, such as wind resistance, dormancy strength, and resistance to motion in shallow water are excluded. This pattern has always been used for a variety of vessels, and with small changes, it can be generalized to a trimaran vessel. Of course, in this model, the main components of the resistance have been cited, and other resistances, such as wind resistance have been ignored. With regards to the dependence of each component of the resistance on a dimensionless number, the model's resistance is generalized using the dimensionless number scale corresponding to the total resistance of the vessel. Based on the type of vessel, some components of resistance shown in [[#img-5|Figure 5]] are more important than others. The vessel resistance in water is divided into two parts of residual and frictional resistance. To calculate total resistance in this study, the residual resistance, (it has two components; pressure resistance and shell resistance), shell frictional resistance, wave-making resistance, wave resistance, frictional resistance, viscous resistance, and viscous pressure were considered.
+<div id='img-5'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
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 where <math>\rho</math> is the water density, <math display="inline">{A}_{S}</math> the wetted surface of the vessel, and <math display="inline">{R}_{w}</math> the pressure section of the resistance applied to the hull when the vessel moves in calm water. This resistance includes the wave resistance and the viscous compressive resistance of the hull. The wave resistance includes wave breaking resistance and wave-making resistance. Using Eqs.(4) and (12), the residual coefficient in the trimaran can be calculated.
-Speed of vessel was calculated in each Froude number using Eqs.(1) and (2) based on Figure 5, mentioned equations, and the initial conditions used in the simulation of trimaran in calm water. In Eq.(2), <math display="inline">{F}_{z}</math> is a vertical force and  <math display="inline">{m}_{tri}</math>  the total weight of the trimaran which is obtained from Eq.(3). To obtain the resistance, it was necessary to calculate the wetted surface and the resistance coefficient. The level of the wetted surface of the vessel can be calculated for the main hull and the side with using Eq.(5). In the trimaran, the residual coefficient of resistance is obtained from Eq.(4) which consists of two components; the coefficient of wave-making resistance and the resistance coefficient of the form of the hull in the shell, (it is, in fact, the friction resistance). Eq.(10) is used to calculate the coefficient of wave-making resistance. The coefficient of wave-making resistance for main and side hulls is obtained by Eqs.(11) and (12). <math display="inline">{R}_{w}</math> is obtained from the results of the simulation.
+Speed of vessel was calculated in each Froude number using Eqs.(1) and (2) based on [[#img-5|Figure 5]], mentioned equations, and the initial conditions used in the simulation of trimaran in calm water. In Eq.(2), <math display="inline">{F}_{z}</math> is a vertical force and  <math display="inline">{m}_{tri}</math>  the total weight of the trimaran which is obtained from Eq.(3). To obtain the resistance, it was necessary to calculate the wetted surface and the resistance coefficient. The level of the wetted surface of the vessel can be calculated for the main hull and the side with using Eq.(5). In the trimaran, the residual coefficient of resistance is obtained from Eq.(4) which consists of two components; the coefficient of wave-making resistance and the resistance coefficient of the form of the hull in the shell, (it is, in fact, the friction resistance). Eq.(10) is used to calculate the coefficient of wave-making resistance. The coefficient of wave-making resistance for main and side hulls is obtained by Eqs.(11) and (12). <math display="inline">{R}_{w}</math> is obtained from the results of the simulation.
 ===2.3 Computational domain and boundary conditions===

@@ Line 59: / Line 59: @@
 ===2.1 Model geometry===
-The hull model and hull lines were selected from the model tested by Brizzolara et al., which is a trimaran with a main hull and two side hulls [20]. The dotted lines of the hull lines were made using the plot digitizer software; then the coordinates of the obtained results were entered into the AutoCAD ([https://en.wikipedia.org/wiki/Autodesk Autodesk] , USA) software, and the same hull lines were drawn and modeled (Figure 1). Afterward, the lines of the body were plotted, and the shape of the float was obtained.
+The hull model and hull lines were selected from the model tested by Brizzolara et al., which is a trimaran with a main hull and two side hulls [20]. The dotted lines of the hull lines were made using the plot digitizer software; then the coordinates of the obtained results were entered into the AutoCAD ([https://en.wikipedia.org/wiki/Autodesk Autodesk] , USA) software, and the same hull lines were drawn and modeled ([[#img-1|Figure 1]]). Afterward, the lines of the body were plotted, and the shape of the float was obtained.
+<div id='img-1'></div>
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-Figure 2 shows a 3-dimensional (3D) model of the trimaran, which was simulated using AutoCAD software. The full-scale vessel has a length of 105.6 m. The main and side hulls were both selected from the 64 series and had round bilge. This standard series was introduced in 1974 and includes several hull lines to design planing hulls. Table 1 presents the principal dimensions of the main and side hulls [20].
+[[#img-2|Figure 2]] shows a 3-dimensional (3D) model of the trimaran, which was simulated using AutoCAD software. The full-scale vessel has a length of 105.6 m. The main and side hulls were both selected from the 64 series and had round bilge. This standard series was introduced in 1974 and includes several hull lines to design planing hulls. [[#tab-1|Table 1]] presents the principal dimensions of the main and side hulls [20].
+<div id='img-2'></div>
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 <div class="center" style="font-size: 75%;">'''Table 1'''. Dimensions of the trimaran model
 </div>
+<div id='tab-1'></div>
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-Due to the large dimensions of the full-scale vessel, it was necessary to provide a scaled model according to the laboratory conditions to test the towing tank. The choice of the appropriate scale for the model always depends on the dimensions of the towing tank. The initial model of the vessel was scaled and modeled in AutoCAD software. Table 2 shows the different modes of hull and side hulls to indicate the two-heel orientation [20]. Generally, the position of the side hull is defined in the same direction as the main hull. This position is expressed in terms of longitudinal, transverse and vertical distances of the hulls from each other. The longitudinal distance is the distance between the stern of the main hull and the stern of the side hull, which is measured along the x-axis. The transverse distance is the distance between the longitudinal centerline of the main hull and the longitudinal centerline of the side hull, which is measured along the y-axis. The third position determines the position of the hulls relative to each other, which is the vertical position of the side hull relative to the main hull. In this study, the side hull was 0.077 m above the main hull. Also, the side hull was at a longitudinal distance of 0 m and a transverse distance of 0.209 m from the main hull (Figure 3).
+Due to the large dimensions of the full-scale vessel, it was necessary to provide a scaled model according to the laboratory conditions to test the towing tank. The choice of the appropriate scale for the model always depends on the dimensions of the towing tank. The initial model of the vessel was scaled and modeled in AutoCAD software. [[#tab-2|Table 2]] shows the different modes of hull and side hulls to indicate the two-heel orientation [20]. Generally, the position of the side hull is defined in the same direction as the main hull. This position is expressed in terms of longitudinal, transverse and vertical distances of the hulls from each other. The longitudinal distance is the distance between the stern of the main hull and the stern of the side hull, which is measured along the x-axis. The transverse distance is the distance between the longitudinal centerline of the main hull and the longitudinal centerline of the side hull, which is measured along the y-axis. The third position determines the position of the hulls relative to each other, which is the vertical position of the side hull relative to the main hull. In this study, the side hull was 0.077 m above the main hull. Also, the side hull was at a longitudinal distance of 0 m and a transverse distance of 0.209 m from the main hull ([[#img-3|Figure 3]]).
+<div id='img-3'></div>
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 <div class="center" style="font-size: 75%;">'''Table 2'''.  Characteristics of the trimaran model
 </div>
+<div id='tab-2'></div>
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-Based on Figure 4, a right-handed coordinate system was considered at the center line of the main hull. The origin of this coordinate system coincides at the intersection point of keel and stern. The x-axis is along the vessel and its positive direction is towards the bow of the vessel. The z-axis is defined in the vertical direction, and the positive direction of the y-axis is towards the starboard of the vessel. The position of the side hull relative to the main hull can be calculated relative to the coordinate axes.
+Based on [[#img-4|Figure 4]], a right-handed coordinate system was considered at the center line of the main hull. The origin of this coordinate system coincides at the intersection point of keel and stern. The x-axis is along the vessel and its positive direction is towards the bow of the vessel. The z-axis is defined in the vertical direction, and the positive direction of the y-axis is towards the starboard of the vessel. The position of the side hull relative to the main hull can be calculated relative to the coordinate axes.
+<div id='img-4'></div>
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-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
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 |style="padding:10px;"|  [[File:35.jpg|600x600px]]