Hydrodynamic Force of Resistance of Tourist Underwater Vehicle’s Bare Hull with Diff erent Heads using OpenFOAM Hidrodinamička sila otpora trupa turističke podmornice s različitim

Power reduction is the central goal to maximize cruising duration of tourist underwater vehicles (UV) that can be achieved by shaping the hull. So, in this paper, hydrodynamic force of resistance of the tourist UV’s bare hull is analysed. A numerical model based on Computational Fluid Dynamics in OpenFOAM is developed to simulate the longitudinal movement of an UV in a viscous and incompressible fl uid for the infi nite water depth. Three head geometries, including both spherical heads (S-S), spherical bow and elliptical stern head (S-E), and UV with both elliptical heads (E-E) are compared. At the fi rst step, the eff ects of the length-to-diameter ratio and forward speed is studied for the S-S UV. The mesh size is calibrated using Grid Convergence Index, provided by ASME, while the model validation is based on the results for cube and sphere as well as by comparison with resistance coeffi cient of a SUBOFF bare hull. S-E and E-E UVs are then analysed for typical length-to-diameter ratio, comparing their force of resistance to the S-S type. The elongated elliptical heads are in many cases found favourable compared to the spherical heads. The results of this study may be useful for the conceptual design of tourist UV and for verifi cation of the complex numerical models that are necessary to account for the infl uence of appendages on the force of resistance of such innovative UV


INTRODUCTION / Uvod*
In design of the naval submarines, the primary concern is usually their acoustic signature [1][2][3]. For a tourist underwater vehicle (UV), however, the central objective becomes power requirement to maximize cruising duration. Power reduction can be achieved by adopting energy-saving propulsion systems, controlling the boundary layer on the surface of the UV, and shaping the hull. Hydrodynamic shaping is therefore an important part of submarines and other UVs design.
Optimization of the bare hull form of submarine was presented in [4], while DREA (Defense Research Establishment speeds. DARPA developed the SUBOFF project to evaluate diff erent fl ow fi eld predictions for an axisymmetric hull, with and without appendages [9]. Accordingly, hull shapes with bullet noses and sharp tails with length-to-diameter ratio equal to 7.14 had the best performance. Equations were defi ned for the optimal design of submarines. For instance, Myring's hull profi le equations were developed to reduce underwater force of resistance when designing the hull shape of hybrid underwater gliders [10], while these equations are then used in practical design [11]. Autonomous and unmanned UVs are often designed using equations presented in Groves et al. [12]. Joung et al. [13] developed a CFD model to optimize the AUV's profi le based on minimum force of resistance, using Myring's equations, while De Barros et al. [14] experimentally studied the submerged bodies designed by the same relationships. Saghi et al. [15] suggested equations based on Artifi cial Neural Network (ANN) to estimate the resistance coeffi cient of a submarine with spherical heads, variable length-to diameter ratio, and for diff erent forward and transverse speeds.
Other studies emphasized the optimization of the submarine's appendages, such as the rudder and tail, to minimize its force of resistance. An example is the tail cone angle, which is one of the major parameters of a submarine's geometrical design. The tail cone angle of the DARPA Suboff AFF8 UV was studied by Ozden et al. [16]. A modifi cation of the depth rudder system by Kiciński and Jurczak [17] was used to investigate the force of resistance characteristics of submarines. This was done by modelling two kinds of rudder geometry: parallel and X-type.
There has been much discussion about the eff ect of surface roughness caused by several reasons such as welding seams, coatings, or biofouling on the force of resistance characteristics of submarines. In a study conducted by Uzun et al. [18], the impact of barnacle type biofouling roughness on submarine performance is examined. The eff ects of antifouling coating system, with a range of roughness and fouling conditions, on the force of resistance and powering characteristics were evaluated by Schultz [19].
A submarine operation in diff erent environments such as submerged and free surface conditions (including calm and wave conditions) has been investigated. The added force of resistance of submarines due to deck wetness at surface conditions was studied by Moonesun et al. [20]. They determined deck fl ooding of submarines caused by the wave making pattern at the bow. Similarly, Dong et al. [21] simulated a submarine sailing near the surface with long-crested waves. They found that irregular waves cause fl uctuations in the hydrodynamic force exerted on the submarine, even below the surface. The eff ect of the water depth, including free surface conditions were studied by Gatin et al. [22], where the eff ect of the external structures was also discussed.
As mentioned earlier, submarines and other UVs designs should incorporate hydrodynamic shaping, and some research has been conducted on optimizing submarine bare hull forms [4][5][6][7][8][9]. In this regard, , the purpose of this paper is to present Computational Fluid Dynamics (CFD) analysis of UV's bare hull geometry with spherical heads (S-S), spherical bow and elliptical stern (S-E), and both elliptical heads (E-E) (see Figure 1), with respect to the forward speed in the infi nitely deep water.
The open-source CFD software OpenFOAM is used for that purpose. Other options of head's geometries are also possible, i.e., partially conical head with spherical dome or tory-spherical head, but not considered herein. The present study is inspired by a tourist UV with acrylic cylindrical hull, designed to provide visitors a view of the surrounding ocean in a comfortable manner [23]. The overall length of the studied UV is about 25 m, while the external diameter of the acrylic hull reads 2.64 m. The designed diving depth is 50 meters, and the maximum forward speed is 2.5 knots. The hull of the UV has spherical heads on both ends. The studied UV can accommodate 48 passengers and 2 crew members. The geometry variation in the present study is limited to the variation of geometry of the UV's steel heads since the central part consists of the acrylic cylinders and its shape is therefore fi xed. Length-to-diameter ratio of the bare hull is also varied to observe its impact on the force of resistance while keeping the internal volume constant. The present study deals with the bare hull of the UV while the complete design consists also of external structures [22], which increases the hydrodynamic force of resistance. Such considerations are beyond the scope of the study while the present results may be used in the conceptual design and for verifi cation of the complex numerical calculations of the hull with appendages.

METHODOLOGY / Metodologija
In the present study, a submerged bare hull of a UV was modelled in a viscous and incompressible fl uid so that the fl uid fl ow was considered turbulent. Accordingly, Reynolds Averaged Navier Stokes (RANS) equations are used as the governing equations [24][25][26][27]: (1) The nomenclature is presented at the end of the paper.
To solve the RANS equations, the Pressure Implicit Method with Pressure Linked Equations (PIMPLE) algorithm was developed, and the pimpleFOAM solver was used. Diff erent terms of the discretized equations, such as derivative terms, gradient parameters, Laplace derivative terms, and divergence terms, were discretized using 1 st order implicit Euler, 2 nd order centre Gauss linear, skewness corrected centre Gauss linear correction, and Upwind schemes, respectively [28]. By using block mesh and refi nement techniques, a cylindrical domain and a Cartesian structured grid were generated. Diff erent boundary conditions summarized in Table 1 were used for the velocity, pressure, kinetic energy, and dissipate rate on the boundaries shown in Figure 2.
Parametric equations for the spherical and elliptical bow and stern read: for spherical bow and stern (3) for elliptical bow (4) for elliptical stern (5) To satisfy the symmetry boundary condition for around the domain, the domain length (L t ) and diameter (D t ) were considered as L t =1.4L oa and D t =4D s , respectively [29].
In addition, the K-Epsilon two-equation model was used to account for turbulence [30][31].
where To estimate pressure force of resistance and friction force of resistances along x axis, dynamic pressure and shear stress are integrated over the surface of the body as follows: Total force of resistance is calculated as follows: (10) The resistance coeffi cient reads: (11) where S is the surface area of the UV's bare hull.

MESH SIZE CALIBRATION / Kalibracija veličine mreže
To determine the appropriate mesh size for the numerical model, a cube with edge length of L c = 0.45 m was placed at a cylinder of 6 m length and 2 m diameter (see Figure 3(a)). The total force of resistance of the cube against the fl uid fl ow is estimated by considering the input boundary condition as a constant velocity. The mesh size dependency was examined for a cube against a constant fl uid fl ow with a velocity of 10 m/s. According to Figure 3(b), the total force of resistance was estimated at diff erent mesh sizes (ms), and the numerical results showed that the uniform mesh sizes with ms<0.03m (ms/L C < 0.067) did not signifi cantly aff ect the total force of resistance.
F igure 2 Schematic sketch of the UV geometry, domain, and boundary conditions. (x axis is oriented from the bow to the stern The Grid Convergence Index (GCI) method, provided by ASME was used to evaluate the mesh size convergence. To do this, and were estimated as follows [32]: where is estimated by solving equations (14) to (16), and by using fi xed-point iteration as: and The parameters , and are the relative error, which are also informative parameters. So, the parameters , , and were estimated for four groups of meshes (fi ne, middle, and coarse) and outlined in Table 2. Table 2 shows that in row 4, the and for ms=0.03m are much lower than the maximum acceptable value of 1%, which indicates that the mesh size is small enough for future computations. Accordingly, that mesh size is used to estimate the resistance coeffi cient of the cube against fl uid fl ows with diff erent Reynolds numbers . The results are compared with   Haider and Levenspiel [33] and Hassan Khan et al. [34] and shown in Figure 4-a. The results indicate a good agreement with the resistance coeffi cient calculated by other researchers.
In another test, the resistance coeffi cient of a sphere with the diameter of 2.64 m was estimated against fl uid fl ows with diff erent Reynolds numbers , as shown in Figure 4-b. The average drag coeffi cient of 0.45 was calculated for diff erent Reynolds numbers. In the reference of Mikhailov et al. [35], the resistance coeffi cient of a sphere is estimated between 0.4 and 0.5 for Reynolds numbers greater than 10 4 . Consequently, the average of 0.45 was found to be close to the amount suggested in Mikhailov et al. [35].

RESULTS / Rezultati
The main objective of the present study is the modelling of the bare hull of a UV against diff erent forward speeds, to determine infl uence of the geometry on the hydrodynamic force of resistance. For that purpose, credible range of tourist UV geometries, represented by the ratio L/D S is defi ned and shown in Table 3. In Table 3, L represents the length of the cylindrical hull, L oa represents the overall length including the heads, and D S represents the diameter of the cylindrical hull of the UV. The geometries are defi ned by keeping the total internal volume of the UV as constant.
Each of 9 geometry variations given in Table 3 was generated by using blockMesh, which is a utility in OpenFOAM for generating multiblock hexahedral meshes [28].
As the geometry of the UV is diff erent compared to the cube, the mesh convergence parameters GCI and ε are calculated for the forward speed 1 m/s of case 5 of the UV and shown in Table 4.
From the last row in Table 4, it may be seen that and of ms=0.03 m are less than the limiting value of 1%, indicating that the mesh size 0.03 m is appropriate.

Hydrodynamic analysis of the bare hull of the UV with both spherical heads (S-S) / Hidrodinamička analiza trupa podmornice s dva sferna kraja (S-S)
A total of 117 diff erent cases, as described in the previous section, are modelled, and analysed in OpenFOAM. In these cases, the radius of the bow and stern heads of the UV were considered as (see Fig. 2). The analysis is performed using supercomputer Isabella, based in SRCE -University of Zagreb Computing Centre. Isabella consists of 135 worker nodes, 3100 processor cores, 12 GPUs, and 756 TiB of data storage [36]. OpenFOAM computation for each of 117 cases lasted for about 18000 s, so the total running time was about 35100 min. Results of the analysis are presented in Figure 5, where hydrodynamic characteristics are presented against L/D S and forward speed UV.   According to the results shown in Figure 5(a) and Figure  5(b), the pressure and total force of resistance of the UV has a direct and reverse relationship with V f and L/D S , respectively. Moreover, by increasing the forward speed, the L/D S parameter's eff ectiveness on the pressure and total force of resistance is increased. As an example, the pressure force of resistance for V f =0.25 m/s and for L/D S =10.2 and 5.9 is 111 N and 173 N, respectively. Consequently, by decreasing the parameter L/D S from 10.2 to 5.9, the pressure force of resistance is increased by 55%. In contrast, the increment of pressure force of resistance for V f =1.5 m/s for the same amounts of L/D S is around 76% (from 2599 N to 4571 N). In terms of total force of resistance, the eff ectiveness of the parameters L/D S and V f is less than those in comparison with the pressure force of resistance, so the total force of resistance is increased by around 24% and 54% by decreasing L/D S from 10.2 to 5.9 and for V f =0.35 m/s and 1.5m/s, respectively. Furthermore, the results show that the relationship between total force of resistance (R) and L/D S is linear for low forward speeds. However, as the forward speed increases, the relationship becomes nonlinear. For all L/D S , expectedly, the relationship between force of resistance and forward speed is nonlinear.
In Figure 5(c), it is evident that the pressure force of resistance is in most cases dominant over the friction force of resistance. It is always the case that R P is larger than R f for diff erent L/D S values and forward speeds greater than 0.9 m/s. For UVs with L/D S < 6.9 and forward speeds less than 0.9 m/s, the pressure force of resistance is also larger than the friction force of resistance. The only cases when friction force of resistance exceeds pressure force of resistance, occur for the slender UVs (L/D S > 8) and for the low forward speeds (V f =0.6 m/s). As an example, the ratio of pressure force of resistance to friction force of resistance is 0.82 for V f =0.25 m/s and L/D S =10.2. In contrast, R p /R f is 3.77 for V f =1.5 m/s and L/D S =5.9.
The UV's resistance coeffi cient is shown in Figure 5(d). As forward speed increases, the resistance coeffi cients decrease so that maximum coeffi cients follow minimum forward speed (V f =0.25 m/s). However, these coeffi cients are minimal for V f =1.1 m/s, and they increase slightly for higher speeds.
It is interesting to compare results presented in Figure 5(d) with the results for the resistance coeffi cient of a SUBOFF bare hull, presented by Divsalar [8], where the resistance coeffi cient of the bare hull with bullet nose, sharp tail, for L oa /D S =7 and V f =1.5 m/s reads around 0.02. On the other hand, the present results obtained for the resistance coeffi cient of the UV with spherical heads and L oa /D S = 7 reads around 0.023 ( Figure 5(d)). So, we can claim that the results are consistent, even though the geometry of the SUBOFF bare hull is diff erent from the geometry of the present UV. It should be noted that Divsalar [8] obtained excellent agreement of his CFD analysis with the experimental results.

Hydrodynamic analysis of the bare hull of the UV with spherical bow and elliptical stern head (S-E) / Hidrodinamička analiza trupa podmornice sa sfernim pramcem i eliptičnom krmom (S-E)
The bare hull of the UV S-S with L/D S = 8 is used further as the representative case for the purpose of comparison with diff erent head geometries. Thus, a UV S-E with L/D S = 8, spherical bow with a radius of 1.32 and elliptical stern with changing longitudinal semi-axis R s (see Figure 2) are modeled, and diff erent scenarios are summarized in Table 5.
Based on scenarios summarized in Table 5, the total force of resistance of the UV is calculated, and the results are shown in Figure 6. F igure 6 The variation of the total force of resistance of diff erent UV cases S-E (see Table 5) and forward speeds ( V f ) (L/D S = 8). Figure 6 shows that the bare hull of the UV S-E with R S =2.07m (case 10 in Table 5) has the minimum force of resistance, so that the force of resistance of the UV in that case is reduced by around 38% compared to the UV S-S for V f =1.5 m/s.

Hydrodynamic analysis of the bare hull of the UV with both elliptical heads (E-E) / Hidrodinamička analiza trupa podmornice s dva eliptična kraja (E-E)
In this step, UVs of the type E-E with elliptical bow with a longitudinal semi-axis R b , and elliptical stern with a longitudinal semi-axis of 2.07 were modelled (see Figure 2 and Table 6), and the total force of resistance is shown in Figure 7 for V f =0.5 m/s, 1 m/s, and 1.5 m/s. F igure 7 The variation of the total force of resistance of diff erent UV cases E-E (see Table 4) and forward speeds (V f ) (L/D S = 8). Figure 7 illustrates that the force of resistance of the UV E-E is decreased as the longitudinal semi-axis of the ellipse R b is increased. For instance, the UV's force of resistance for R b =2.07 (case 10 in Table 6) and for V f =1.5 m/s is 2417 N. For the same forward speed, the force of resistance of the spherical UV reads 4195 N. This means that the force of resistance of the UV E-E with R b =2.07 is around 57% of the force of resistance of the spherical UV.

DISCUSSION / Rasprava
H ere, we compare the total force of resistance of the selected bare hulls (case 5 in Table 3, case 10 in Table 5, and case 10 in Table 6) for diff erent UV forward speeds. Figure 8 shows comparison of the force of resistance of S-S, S-E, and E-E geometries for diff erent Reynolds numbers . The geometrical parameters of the selected S-S, S-E, and E-E UVs are provided in tables 3,4, and 5 respectively.
According to Figure 8, increasing the Reynolds number (forward speed) decreases the ratio of the force of resistance of the UV E-E to the UV S-S. For example, the force of resistance of the UV E-E (case 10 in Table 4) for R e = 1.26×10 7 (V f =0.5 m/s) is 527 N, which is about 87% of the force of resistance of the suggested UV S-S at the same forward speed (601 N). In contrast, the force of resistance of the UV E-E for R e = 3.78 × 10 7 (V f =1.5 m/s) (2417 N) is about 57% of that of the UV S-S (4195 N). Due to this, the effi ciency of the geometry of the UV E-E increases as the UV forward speed increases as compared to the geometry of the UV S-S.  Fi gure 8 Comparison of the force of resistance of the UVs S-S (case 5), S-E (case 10), and E-E (case 10) for diff erent Reynolds numbers.

Slika 8. Usporedba sile otpora podmornice tipa S-S (slučaj 5), S-E (slučaj 10) i E-E (slučaj 10) za različite Reynoldsove brojeve
The pressure to friction force of resistance ratio for these UVs and for diff erent Reynolds numbers are shown in Figure 9. The results show that the pressure to friction force of resistance ratio of the concept E-E is lower than that of the concept S-S so it reads approximately 1.0 for the UV E-E for R e = 3.78 × 10 7 . However, the ratio is around 2.6 for the same Reynolds number for the UV S-S.
To clarify the eff ect of the geometry on the pressure distribution around the UVs' bare hull, Figure 10 shows the pressure distribution around the three types of UV heads (S-S, S-E, E-E), for the same forward speed (V f =1.5 m/s). The results shown in Figure 10 indicate that the low-pressure area around the elliptical stern is reduced compared to the spherical stern, thereby reducing the force of resistance. Similarly, a reduction in the high-pressure area of the elliptical bow leads to a lower force of resistance.

CONCLUSION / Zaključak
A Computational Fluid Dynamic numerical model in OpenFOAM is developed for the analysis of tourist underwater vehicle's (UV's) bare hull with diff erent length-to-diameter aspect ratios and head geometries. To evaluate shape of the UV, hydrodynamic force of resistance is calculated. The mesh size is calibrated using Grid Convergence Index, provided by ASME, and employed on the case of the cube and UV. Model is validated based on the results of resistance coeffi cient for cube and sphere as well as by comparison with resistance coeffi cient of a SUBOFF bare hull with the same ratio of length to diameter (L oa /D S ). Based on the results of total of 177 cases analysed, the following conclusions may be drawn: -total force of resistance of the UV has a direct and reverse relationship with V f and L/D S , respectively, -the pressure force of resistance is in most cases dominant over the friction force of resistance, -the results showed signifi cant improvement of the hydrodynamic characteristics of the UV if the elongated elliptical heads are used instead of spherical heads, because of the reduction of the high-and low-pressure areas in the bow and stern regions respectively for the UV with elliptical heads. Thus, -the total hydrodynamic force of resistance of the UV with spherical bow and elliptical stern, and with L/D=8 is about 60-90 % of the corresponding case with both spherical heads, -the total force of resistance of the UV with both elliptical heads is about 55-85 % of the case with spherical heads. The results are intended for the conceptual design of tourist UVs and for verifi cation of complex numerical computations for UV's hull with external structure that signifi cantly increases their hydrodynamic force of resistance.

Funding:
The project is co-fi nanced by the European Union from the European Regional Development Fund within the Operational Program "Competitiveness and Cohesion 2014-2020", project KK.01.2.1.02.0339 -Development of the multipurpose luxury touristic and research submarine.
Confl ict of interest: None.