Journal of Marine Science and Engineering

Article Hydrodynamic Model of Diver–DPV Coupled Multi-Body and Its Underwater Cruising Numerical Simulation

Hansheng Li, Fenglei Han *, Haitao Zhu, Jiawei Zhang, Weipeng Zhang and Yuliang Wu

Shipbuilding Engineering College, Harbin Engineering University, Harbin 150001, China; [email protected] (H.L.); [email protected] (H.Z.); [email protected] (J.Z.); [email protected] (W.Z.); [email protected] (Y.W.) * Correspondence: [email protected]; Tel.: +86-18345152097

Abstract: Diver propulsion vehicles (hereinafter referred to as DPV) are a kind of small vehicle with underwater high-speed used by divers, who are able to grasp or ride on, and operate the volume switch to change the speed. Different from unmanned underwater vehicles (UUVs), the interference caused by diver’s posture changing is a unique problem. In this paper, a Diver–DPV multi-body coupling hydrodynamic model considering rigid body dynamics and fluid disturbance is established by analyzing the existing DPV related equipment. The numerical simulation of multi-body articulated motion is realized by using Star-CCM+ overlapping grid and DFBI 6-DOF body motion method. Five cases of DPVs underwater cruising in a straight-line when restraining diver movement is simulated, and five cases with free diver movement are simulated too. Finally, the influence of the diver’s posture changing on the cruising speed resistance is analyzed, and the motion equation including the disturbance is solved. The final conclusion is that, the disturbance is favorable at high speed,


which can reduce the cruising resistance, and unfavorable at low speed, which increases the cruising
 resistance. The friction resistance Ff always accounts for the main part in all speed cases. Citation: Li, H.; Han, F.; Zhu, H.; Zhang, J.; Zhang, W.; Wu, Y. Keywords: diver propulsion vehicle; numerical simulation; multi-body coupling; hydrodynamic Hydrodynamic Model of Diver–DPV interference; overlapping grid; CFD Coupled Multi-Body and Its Underwater Cruising Numerical Simulation. J. Mar. Sci. Eng. 2021, 9, 140. https://doi.org/10.3390/ 1. Introduction jmse9020140 Diver propulsion vehicle (hereinafter referred to as DPV) is a kind of small vehicle with underwater high-speed used for divers [1], who are able to grasp or ride on, and Academic Editor: Alessandro Ridolfi operate the volume switch to change the speed. The numerical simulation of DPV cruising Received: 28 December 2020 in water is different from that of HOVs (human-occupied vehicles) and UUVs (unmanned Accepted: 26 January 2021 Published: 29 January 2021 underwater vehicles). That is, when DPV cruising, the diver’s body will swing with the flow field, and its posture change will have a nonlinear disturbance to the surrounding

Publisher’s Note: MDPI stays neutral flow field, which has a huge impact on the speed and stability of the DPV. Meanwhile, with regard to jurisdictional claims in the DPV’s hydrodynamic shape and propeller arrangement will also have an impact on published maps and institutional affil- the diver body’s posture. There is also the problem of the propeller wake impacting the iations. diver’s body. In the traditional numerical simulation method, diver and DPV are regarded as a relatively fixed rigid body. While in the multi-body coupling model, the joints of diver body and DPV are regarded as a complex hinge connecting rod structure, which has active and driven motion, and the two have the influence of fluid disturbance and rigid hinge force. Using CFD (compulation fluid dynamics) method to simulate the underwater Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. cruising of a multi-body coupling model, and finally predicting the performance and This article is an open access article analyzing the posture of the diver are the keys to solving the design optimization problem distributed under the terms and of high-performance DPVs. conditions of the Creative Commons Since the last century, scholars from all over the world have begun to study the design Attribution (CC BY) license (https:// theory of DPVs and other diving equipment. In 1987, Presswood, Clark G, Godshall, David creativecommons.org/licenses/by/ and others [2] first proposed a set of propeller matching theory applicable to DPV, and 4.0/). applied to the optimization of Tekna DV-3X and MIL-UNIT MOD S-5100 DPV.

J. Mar. Sci. Eng. 2021, 9, 140. https://doi.org/10.3390/jmse9020140 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 2 of 26

Godshall, David and others [2] first proposed a set of propeller matching theory applica- ble to DPV, and applied to the optimization of Tekna DV-3X and MIL-UNIT MOD S-5100 DPV. CFD method is widely used in the numerical simulation of DPV linear cruising. In 2017, M.R. Sadeghizadeh and B.Daranjam [3] studied the numerical simulation of the diver–DPV coupling model by CFD in detail, and put forward the DPV optimization the- J. Mar. Sci. Eng. 2021, 9, 140 2 of 24 ory of improving the rapidity and reducing the impact on the human face is proposed. The problem that part of diver body is exposed to the external flow field is considered in the modeling. Finally, the model experiment of towing tank is carried out and the accu- racyCFD of the method numerical is widely simulation used is in verified, the numerical and its simulation CFD velocity of DPVcloud linear chart cruising.is presented In 2017,in Figure M.R. 1. Sadeghizadeh In particular, the and k- B.Daranjamω SST Turbulence [3] studied model the is numericalused to simulate simulation the complex of the diver–DPVvortex region coupling at the connection model by CFDbetween in detail, the diver’s and putbody forward and DPV, the and DPV the optimization mesh of the theorylocal prism of improving layer is encrypted the rapidity on and the reducinghuman face. the impact on the human face is proposed. The problemIn the CFD that method part of diverof DPV, body the is movable exposed hu toman the externalbody modeling flow field is isalso considered the research in thefocus. modeling. At the end Finally, of the the 20th model century, experiment the body of towing hydrodynamic tank is carried modeling out andand theCFD accuracy numer- ofical the simulation numerical of simulation swimmers is and verified, divers and were its began CFDvelocity to take. cloudIn 1996, chart Bixler, is presented B. S. and inM. FigureSchloder1. Inet particular,al. [4] from theUniversityk-ω SST of TurbulenceCalgary firstly model put isforward used to CFD simulate method the to complex simulate vortexthe dynamic region and at the posture connection model between of swimmers, the diver’s so as bodyto predict, and DPV, analyze, and and the meshimprove of the the localperformance prism layer of swimmers. is encrypted on the human face.

FigureFigure 1. 1.CFD CFD velocityvelocity cloud cloud of of DPV–diver DPV–diver coupled coupled model model in in M.R.Dadeghizadeh’s M.R.Dadeghizadeh’s research research [3 [3].].

InIn the terms CFD of method 3D action of DPV,model the of movable swimmers, human Motomu body modelingNakashima is， alsoKen the Datou, research and focus.Yasufumi At theMiura end [5] of proposed the 20th century, a method the for body the establishment hydrodynamic of modeling a numerical and simulation CFD nu- mericalmodel of simulation swimming of human swimmers body and considering divers were rigid began body to dynamics take. In 1996, and nonlinear Bixler, B. S.hydro- and M.dynamic Schloder interference et al. [4] from in 2007. University In this ofmodel, Calgary the firstly human put limb forward model CFD rotates method in the to simu-water, lateand the its dynamicrotation angle and posture and root model bending of swimmers, moment are so measured, as to predict, and analyze, the splitting and improvesteps are theshown performance in Figure of 2. swimmers. Although the error of the model is 10%, it can better reflect the un- steadyIn termsfluctuation of 3D of action experimental model of data. swimmers, In addition, Motomu it also Nakashima, simulates the Ken position Datou, of and the Yasufumihuman body Miura that [5 can] proposed generate a methodthe gliding for lift the to establishment determine the of tangent a numerical resistance simulation coeffi- modelcient. Finally, of swimming a complete human six-step body considering freestyle simulation rigid body example dynamics is given. and nonlinear The simulation hydro- dynamicspeed is 7.5% interference lower than in 2007. the actual In this swimming model, the speed. human limb model rotates in the water, and itsCFD rotation method angle of swimming and root bendinghuman is moment divided areinto measured,grid method and and the grid splitting free method. steps areCOHEN, shown R.C.Z. in Figure et al.2. [6] Although proposed the a errorcomputational of the model method is 10%, for human it can better swimming reflect using the unsteadysmoothed fluctuation particle hydrodynamics of experimental (SPH data. method) In addition, in the it alsoyear simulatesof 2009. It thepoints position out that of the the humanSPH method body that can can better generate solve the the gliding influence lift to of determine turbulence, the complex tangent resistance free surface coefficient. motion Finally,(including a complete gas splash six-step and entrainment) freestyle simulation and the example rapid deformation is given. The of simulationswimmers’ speed geomet- is 7.5%ric structure. lower than The the preliminary actual swimming simulation speed. of SPH traction for male and female swimmers at differentCFD method speeds of with swimming fixed gliding human posture is divided is carried into grid out. method In the experiment, and grid free laser method. scan- COHEN,ning is used R.C.Z. to establish et al. [6] proposeda more accurate a computational three-dimensional method for model human of human swimming body, using and smoothed particle hydrodynamics (SPH method) in the year of 2009. It points out that the SPH method can better solve the influence of turbulence, complex free surface motion (including gas splash and entrainment) and the rapid deformation of swimmers’ geometric structure. The preliminary simulation of SPH traction for male and female swimmers at different speeds with fixed gliding posture is carried out. In the experiment, laser scanning is used to establish a more accurate three-dimensional model of human body, and the dynamic model of articulation is generated based on the surface mesh deformation of human bones. Finally, the underwater video shot of swimmers is captured to verify the model. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 3 of 26

the dynamic model of articulation is generated based on the surface mesh deformation of J. Mar. Sci. Eng. 2021, 9, 140 3 of 24 human bones. Finally, the underwater video shot of swimmers is captured to verify the model.

Figure 2. Swimming dynamics modeling and dynamic process simulation proposed by Nakashima et al. [5]. Figure 2. Swimming dynamics modeling and dynamic process simulation proposed by Nakashima et al.In [5]. the aspect of grid method, M.R. Sadeghizadeh and B. Daranjam et al. [7] conducted numerical simulation of three-dimensional two-phase turbulent flow on the basis of fluid In the aspectvolume of method grid method, (VOF) of M.R. FLUENT Sadeghi softwarezadeh and and tetrahedral B. Daranjam mesh et in al. 2012, [7] so conducted as to evaluate numerical simulationthe effective of power three- requireddimensional by high-speed two-phase underwater turbulent flow swimmers. on the basis The speed of fluid of the volume methodswimmer (VOF) model of FLUENT is increased software to 8 m/s and by using tetrahedral the towing mesh tank in experiment, 2012, so as and to evalu- the results ate the effectiveare consistent power required with CFD. by high-speed underwater swimmers. The speed of the swimmer modelIn is 2010,increased Zaidi to H. 8 etm/s al. by [8 ]using analyzed the towing different tank types experiment, of turbulence and models the results in CFD of swimmers in order to better predict resistance in fluid environment. The test of two are consistent with CFD. turbulence models shows that the standard k-ω model can accurately predict the resistance In 2010,and Zaidi capture H. theet al. vortex [8] analyzed structure formed different by the types back of and turbulence buttocks of models underwater in CFD swimmers, of swimmers inwhile order the to standard better predictK-ω model resistance underestimates in fluid theenvironment. value of resistance. The test Finally, of two thetur- flow bulence modelsvisualization shows that experiment the standard in insep k- (Nationalω modelInstitute can accurately of Sports) predict verifies the the resistance rationality of and capturevortex the vortex structure structure under K– formedω model. by the back and buttocks of underwater swim- mers, while the Instandard 2012, Ramos K-ω etmodel al. [9 ]underestimates used CFD method the to value analyze of theresistance. influence Finally, of water the depth flow visualizationon the resistance experiment ofswimmers in insep (National during underwater Institute of gliding, Sports) and verifies calculated the rationality the resistance of vortex structurecoefficient under (increment K–ω model. is 0.5 m/s) in the speed range of 1.5–2.5 m/s. It is found that the resistance decreases with the increase of depth when the center line of swimmer’s modeling In 2012, Ramos et al. [9] used CFD method to analyze the influence of water depth is in different depths of 0–0.75 m (increment is 0.25 m), while when the depth is greater on the resistancethan 0.75 of m,swimmers the resistance during value under is almostwater the gliding, same stay and the calculated same. the resistance coefficient (incrementCompared is 0.5 with m/ swimmers,s) in the speed the underwater range of motion1.5–2.5 ofm/s. diver It is wearing found scubathat the diving resistance decreasesequipment with is more the increase complex. of In dep 2016,th when Darko the Valenko center et line al. of [10 swimmer’s] firstly established model- the ing is in differentdynamic depths model of and 0–0.75 simulation m (increment method ofis scuba0.25 m), diver while and when buoyancy the controldepth is device greater (BCD), than 0.75 m,simulated the resistance the complete value heaveis almost motion the process, same stay and the compared same. the simulation or results with Comparedexperimental with swimmers, results. the underwater motion of diver wearing scuba diving equipment is moreThe complex. problem of In divers 2016, driving Darko DPV Valenko in the et water al. [10] is essentially firstly established the mutual disturbancethe dy- of multiple underwater objects, such as the problem of the submarine running near the namic model and simulation method of scuba diver and buoyancy control device (BCD), wall, the problem of two vehicles meeting or exceeding in narrow waters, the mutual simulated theinterference complete between heave the motion legs of process, the jacket and platform, compared and so on.the Previoussimulation studies or onresults the two- with experimentaldimensional results. analytical solutions of multi-body perturbation problems under the condition The problemof potential of divers flow havedriving been DPV carried in the out wa in detail.ter is essentially the mutual disturbance of multiple underwater objects, such as the problem of the submarine running near the wall, the problem of two vehicles meeting or exceeding in narrow waters, the mutual in- terference between the legs of the jacket platform, and so on. Previous studies on the two-

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J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 4 of 26 dimensional analytical solutions of multi-body perturbation problems under the condi- J. Mar. Sci. Eng. 2021, 9, 140 tion of potential flow have been carried out in detail. 4 of 24 In 2010, A.A. Tchieu, D. Crowdy, et al. [11] studied the interaction between two arbi- dimensional analytical solutions of multi-body perturbation problems under the condi- trary bodies in two-dimensional non viscous fluid, and gave the linear velocity and angu- tion of potential flow have been carried out in detail. lar velocityInIn 2010,2010, ofA.A. A.A. the Tchieu, Tchieu,object. D. D.For Crow Crowdy, thedy, more et et al. comple al. [11] [11 studied]x studied problem the theinteraction with interaction divers, between betweenit can two be twoarbi-regarded as arbitrarythetrary problem bodies bodies in of two-dimensional inunderwater two-dimensional multi-body non nonviscous viscous articu fluid, fluid,lated and gave andmotion, gavethe linear which the linear velocity has velocity a wider and angu- and application angularinlar thevelocity field velocity of of the ofnumerical object. the object. For simulation Forthe themore more comple and complex xexperimental problem problem with with divers,research divers, it can itof can bebionic beregarded regarded fish asrobots. In as2018,the the problem problemLijun Li,of of underwater Gen underwater Li, et al. multi-body multi-body [12] carried articu articulated oulatedt numerical motion, motion, whichsimulation has aa widerwiderand motion applicationapplication control sim- inulationin the the field field on of ofthe numerical numerical multi simulationfin simulation propulsion and and experimental of experimental puffer fish. research researchIda ofLouise bionic of bionic G. fish Borlaug robots. fish robots. In and 2018, InKristin Y. LijunPettersen2018, Li,Lijun Gen proposedLi, Li, Gen et al. Li, [ 12aet ]generalizedal. carried [12] carried out numerical over out numericaltwist simulation algo simulationrithm and in motion 2019and motion control[13,14], control simulation which sim- is used to on the multi fin propulsion of puffer fish. Ida Louise G. Borlaug and Kristin Y. Pettersen trackulation the on angle, the multi position, fin propulsion and direction of puffer of thefish. base Ida Louisejoint of G. the Borlaug 6-DOF and of KristinAIAUV Y. (a biotic proposed a generalized over twist algorithm in 2019 [13,14], which is used to track the snakePettersen AUV, proposed shown a ingeneralized Figure 3). over twist algorithm in 2019 [13,14], which is used to angle,track the position, angle, and position, direction and of direction the base of joint the of base the 6-DOFjoint of of the AIAUV 6-DOF (a of biotic AIAUV snake (a AUV,biotic shownsnake AUV, in Figure shown3). in Figure 3).

Figure 3. The Biotic AIAUV Eelume™ is repairing the underwater pipeline [13]. FigureFigure 3.3.The The BioticBiotic AIAUVAIAUV Eelume™Eelume™ isis repairingrepairing thethe underwaterunderwater pipeline pipeline [ 13[13].].

InInIn the thethe CFDCFD CFD methodmethod method ofof multiofmulti multi jointjoint joint bioticbiotic biotic fishfish robots,robots,fish robots, the the main main the difficulty difficultymain difficulty lies lies in in using using lies in using moremore accurate accurateaccurate moving moving grid grid grid technology technology technology and and and gr gridid grdeformation deformationid deformation technology technology technology to todiscretize discretize to discretize the the theregion, region, and and and using using using efficient efficient efficient solver solver solver to make to to make make the convergencethe the convergence convergence speed speedspeed catch catchup catch with up up the with with move- the the move- movementment andand deformation anddeformation deformation speed. speed. speed. Qing Qing Qing Xiao, Xiao, Xiao, Ke suKe Ken, sun,suHaon, HaoHaoLiu, Liu, etLiu, al. et [15]et al. al. [ 15have ][15] have studied have studied studiedthe the thehydrodynamic hydrodynamic performance performance performance of theof of thewave wave NACA NACA0012NACA0012 0012plate plate platenear near wake near wake cylinder,wake cylinder, cylinder, which which can which can cansimulate simulate the the themulti-body multi-body multi-body disturbance disturbance disturbance of the of of rigidthe the rigid rigid AUV AUV shell shelltoshell the to toBCF the the tail BCF BCF fin. tail tail The fin. fin. neph- The The neph- nephogramogram ofof vorticity vorticity of vorticity is isshown shown is shown in Figurein in Figure Figure 4 4.

Figure 4. Multibody disturbance of cylinder and wing [16]. Figure 4. Multibody disturbance of cylinder and wing [16]. FigureRuoxin 4. Multibody Li, Qing disturbance Xiao, Yuan of cylinder Liu, Jianxin, and wing et al. [16 [16]]. proposed a multi-body dynamics algorithm in 2016, which combines the incompressible fluid flow with the bionic multi- Ruoxin Li, Qing Xiao, Yuan Liu, Jianxin, et al. [16] proposed a multi-body dynamics body Ruoxindynamics Li, system. Qing Xiao, The tool Yuan has Liu, shown Jianxin, its powerful et al. [16] ability proposed to solve a a multi-body variety of bio- dynamics algorithm in 2016, which combines the incompressible fluid flow with the bionic multi- algorithmmechanical infish 2016, swimming which problems,combines including the incompressible self-propelled fluid multi flow DOF with with the rigid bionic un- multi- body dynamics system. The tool has shown its powerful ability to solve a variety of bodydulator. dynamics system. The tool has shown its powerful ability to solve a variety of bio- biomechanical fish swimming problems, including self-propelled multi DOF with rigid mechanical fish swimming problems, including self-propelled multi DOF with rigid un- undulator.Yang Luo, Qing Xiao, Guangyu Shi, et al. proposed the multi body dynamic algo- dulator.rithmYang advanced Luo, Qing code Xiao, in Guangyu2019 [17] Shi,to carry et al. out proposed numerical the multi simulation body dynamic on the algorithmrobot fish advancedequippedYang codewith Luo, insoft Qing 2019 bionic [ 17Xiao, ]fins. to carryGuangyu It is a out fluid numerical Shi, structure et al. simulation interaction proposed on (FSI) the the robotsolver,multi fish bodywhich equipped dynamic carries algo- withrithmout finite soft advanced bionic element fins. calculationcode It is ain fluid 2019 based structure [17] on toCalculiX interactioncarry software.out (FSI)numerical solver, whichsimulation carries on out the finite robot fish elementequipped calculation with soft based bionic on fins. CalculiX It is software.a fluid structure interaction (FSI) solver, which carries out finite element calculation based on CalculiX software.

J. Mar. Sci. Eng. 2021, 9, 140 5 of 24

In this paper, the flexible deformation of divers’ fins is ignored in numerical simulation, so the function of Star-CCM+ software can be used to realize the numerical simulation of underwater multi-body articulated motion. There has been accurate CFD methods for the underwater motion of human body, which splits the swimming posture, simulates and integrates each step separately, and proposes the grid model and turbulence model with the most accurate calculation results. However, for the DPV underwater cruising, predecessors have fixed the diver body with DPV rigidly in the model, without active and passive motion of human joints. That is to say, the influence of nonlinear disturbance of multi rigid body is ignored. On this basis, the main research contents are as follows: (1) A diver–DPV coupled model with changeable posture is established. (2) The cruising resistance of rigid connection immobility model is solved by two CFD methods of overlapping grids and DFBI multibody motion. (3) The influence of DPV resistance and movable human driving posture disturbance on the underwater cruising is studied.

2. Materials and Methods 2.1. Numerical Methodology In this paper, Star-CCM+ software is used to simulate the nonlinear motion of un- derwater multibody. The following laws are satisfied for the Solver: conservation of mass, conservation of momentum, and conservation of energy. If the flow state is turbu- lent, the system also needs to follow the turbulent transport equation. It is defined as: Re = ρUL/µ = UL/γ, where U is the velocity of simulated free flow, and γ is the kinematic viscosity of fluid. According to the minimum length of the smallest part in this study, the minimum Reynolds number Remin occurs on the smallest appendage, and Remin is about 44005, which is shows that the flow field is turbulent, and the Reynolds number of the whole model’s cruising speed Re is 6816272, which belongs to high Reynolds number. In this paper, K–ω SST turbulence model is adopted, which is a two-way eddy viscosity model. The formula used in the interior of the boundary layer makes the model can be directly used to the wall through the viscous sublayer. Therefore, K–ω SST model can be used as turbulence model with low Re number and without any additional damping function. The SST formula is also converted to K–ω method in free flow, thus avoiding the common problem that the model is too sensitive to the turbulent characteristics of the inlet free flow. The K–ω SST model has good performance in adverse pressure gradient and separation flow [18]. In this paper, K–ω SST turbulence model is used. It meets the following requirements: Continuity equation: → → ∇ · u= 0 (1) → → where ∇ is Hamilton operator, u is velocity vector. Momentum equation:

t ∂ ∂P ∂tij ∂τij ∂x (ρuiuj) = − ∂x + ∂x + ∂x j  i i i t = µ ∂ui + ∂ui (2) ij ∂xj ∂xi t 0 0 τij = −ρu iu j

t where P is the preesure, tij is the stress tensor, τij is Reynolds stress term. ui, uj is the fluctuating velocity component in the i, j direction fluid density, µ is the fluid dynamic viscosity, and ρ is the fluid density. Because in turbulence, the actual velocity of each particle is the sum of instantaneous 0 0 and fluctuating components. The last term ρu iu j, the Reynolds stress, should be obtained using a suitable turbulence model. Predecessors have proved which model is more effective than other models with the same conditions. For example, Zaidi et al. (2010) proposed that k-ω SST model is more J. Mar. Sci. Eng. 2021, 9, 140 6 of 24

accurate for predicting human underwater swimming resistance. According to their model, k and ω are ∂ (ρk) + ∂ (ρku ) = ∂ (Γ ∂w ) + G − Y + S ∂t ∂xi i ∂xj k ∂xj k k k ∂ ∂ ∂ ∂ω (3) (ρω) + (ρωu ) = (Γw ) + Gω − Yω + Sω ∂t ∂xi i ∂xj ∂xj where the parameter “ω” is the energy dissipation rate (or specific dissipation rate) per unit volume, and “k” is the turbulent kinetic energy per unit mass. Γk and Γw are the effective diffusivities of k and ω. Gk and Gw are the turbulence results of k and ω. Yk and Yω are the turbulent dissipations of k and ω. Sk and Sω are source terms of k and ω. Therefore, the resistance of DPV can be obtained by this method, and the propulsion power Pp related to the required cruising speed V can be calculated as

Pp = Ft · V (4)

where Ft is real time resistance of hull. On the other hand, if the propulsion power is electric, other parameters will come into − question, such as propulsion efficiency ηp (composed of propeller and motor efficiency) and the total weight of diver and DPV. Under this condition, the input power Pi is defined as the formula Pp Pi = − (5) ηp

2.2. 6-DOF Body Splitting and Model Building Firstly, the hydrodynamic shape of DPV is modeled. Whether DPV or AUV, its hydrodynamic shape directly affects the resistance and posture of underwater cruising. In this paper, a streamlined DPV with pump-jet propulsion is designed as the research object, which is referring to the Rotinor®Blackshadow 730 DPV [19]. Its parameters are in Table1.

Table 1. Parameters of four typical DPV

Main Indexes Notations Value

Length LDPV 950 mm Width BDPV 400 mm Height HDPV 300 mm Displacement m 12 kg (neutral buoyancy) Maximum cruise speed V5 2.5 m/s Maximum diving depth Dmax 50 m

The model abandons the exposed propeller layout of the normal DPV and adopts pump jet propulsion [20]. Which is, the inflow enters from the bottom-front openings, accelerates through the propeller in the duct, and ejects at a high velocity. In this model, when calculating the resistance of DPV, the model of propeller only retains the hub part. In this model, the posture of diver driving DPV is considered as drag-and-drop. Which is, the diver navigates by holding the operating lever of the DPV with both hands, and uses torque to change the head direction of the DPV to carry out oblique navigation and steering movement like typical SUEX DPV [21]. This model assumes that the X-direction motion of DPV is locked, and a fixed velocity X-direction flow is used to simulate cruising. The local coordinate system was established with the centroid point P0 of the hip part of diver. At this time, the typical cruising posture and force situation of diver and DPV are shown in Figure5. The equation of motion is

max = Ft − Ff (6)

may = G − Fb − Fl (7)

where ax and ay is the acceleration in X, Y directions. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 7 of 26

accelerates through the propeller in the duct, and ejects at a high velocity. In this model, when calculating the resistance of DPV, the model of propeller only retains the hub part. In this model, the posture of diver driving DPV is considered as drag-and-drop. Which is, the diver navigates by holding the operating lever of the DPV with both hands, and uses torque to change the head direction of the DPV to carry out oblique navigation and steering movement like typical SUEX DPV [21]. This model assumes that the X-direction motion of DPV is locked, and a fixed velocity X-direction flow is used to simulate cruising. The local coordinate system was established with the centroid point P0 of the hip part of diver. At this time, the typical cruising posture and force situation of diver and DPV are shown in Figure 5. The equation of motion is

max=Ft-Ff (6)

J. Mar. Sci. Eng. 2021, 9, 140 may=G-Fb-Fl (7) 7 of 24

where ax and ay is the acceleration in X, Y directions.

Figure 5. Forces on diver–DPV coupled model. Figure 5. Forces on diver–DPV coupled model. For the convenience of calculation, the diver body posture at design speed is given For the convenienceas T0 posture. of calculation, According the to thedive Professionalr body posture Association at design of Divingspeed Instructorsis given (PADI) as T0 posture. AccordingDPV driving to the course, Professional the typical Association driving posture of Diving should Instructors be: DPV, (PADI) diver’s DPV trunk and fins driving course, theare typical parallel driving to the cruising posture direction, should be: DPV DPV, is parallel diver’s to trunkx-axis; and thighs fins are are slightly par- upward J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 8 of 26 allel to the cruisingbending; direction, knees DPV and is elbows parallel are to bending x-axis; at thighs a certain are angle; slightly forearms upward are bend- parallel to DPV. ing; knees and elbowsAfter quantification,are bending at the a modelcertain in angle; Figure 6forearms is obtained, are in parallel which the to anglesDPV. After are assumed to be the positive direction of acute angle with x-axis. quantification, the model in Figure 6 is obtained, in which the angles are assumed to be the positive direction of acute angle with x-axis. That is, θp1 is the acute angle between the body’s central axis and the x-axis; θp1 is the acute angle between the body’s central axis and the x-axis; θp2 and θp3 is the acute angle between the upper arm’s central axis and the x-axis, and θp3=θp2; θp4 is the acute angle between the thigh’s central axis and the x-axis; θp5 is the acute angle between the crus’s central axis and the x-axis;

Figure 6. Notations of model’s dimension and posture. Figure 6. Notations of model’s dimension and posture. That is, θp1 is the acute angle between the body’s central axis and the x-axis; θp1 is the acute angle between the body’s central axis and the x-axis; θp2 and θp3 is the acute angle betweenIn order the to upper make arm’s the hydrodynamic central axis and themodelingx-axis, andof thisθp3 =paperθp2; θ p4quantitativeis the acute angleand more universal,between the the thigh’sdimensions central and axis anddriving the x -axis;postureθp5 isof the four acute typical angle betweenDPV as the SEADOO crus’s 、 U.FLIGHT,central axis SUEX, and the ROTINORx-axis; are analyzed through image data Figure 7, and the follow- ing TableIn 2 order is obtained. to make the hydrodynamic modeling of this paper quantitative and more universal, the dimensions and driving posture of four typical DPV as SEADOO, U.FLIGHT, SUEX, ROTINOR are analyzed through image data Figure7, and the following Table2 is obtained. In this paper, a typical diver’s body is selected, equipped with back-flying BCD (Buoyancy Control Device) and open breathing apparatus [23]. According to Table1, the posture angles of DPV with different models and speeds are slightly different. Since the shape and max cruising speed of DPV in this paper are similar to ROTINOR, the T0 posture angle is selected as Table3.

Figure 7. Driving posture of four typical DPV [19,21,22].

Table 2. Parameters of four typical DPV

Length of DPV Cruising Speed Body Elbow Thigh Knee DPV Type LDPV (m) V (m/s) θP1 θP3 θP4 θP5 SEADOO [22] 0.55m 1 −26.6° 37.3° 11° 38.6° SUEX [21] 0.814m 1.5 −26° 41° −4.5° 52° U·FLIGHT 0.95m 2 −25.9° 43.5° −4.5° 47.8° ROTINOR [19] 1.766m 2.5 −7.3° 42.4° 5.4° 60.9°

In this paper, a typical diver’s body is selected, equipped with back-flying BCD (Buoyancy Control Device) and open breathing apparatus [23]. According to Table 1, the posture angles of DPV with different models and speeds are slightly different. Since the shape and max cruising speed of DPV in this paper are similar to ROTINOR, the T0 pos- ture angle is selected as Table 3.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 8 of 26

Figure 6. Notations of model’s dimension and posture.

In order to make the hydrodynamic modeling of this paper quantitative and more universal, the dimensions and driving posture of four typical DPV as SEADOO 、 J. Mar. Sci. Eng. 2021, 9, 140 8 of 24 U.FLIGHT, SUEX, ROTINOR are analyzed through image data Figure 7, and the follow- ing Table 2 is obtained.

FigureFigure 7. 7.DrivingDriving posture posture of of four typicaltypical DPV DPV [19 [19,21,22].,21,22].

TableTable 2. 2. ParametersParameters of fourfour typicaltypical DPV DPV

Length of DPV Cruising Speed Body Elbow Thigh Knee DPV Type Length of DPV Cruising Speed Body Elbow Thigh Knee DPV Type LDPV (m) V (m/s) θP1 θP3 θP4 θP5 LDPV (m) V (m/s) θP1 θP3 θP4 θP5 SEADOO [22] 0.55m 1 −26.6◦ 37.3◦ 11◦ 38.6◦ SEADOOSUEX [22] [21 ] 0.814m 0.55m 1.5 1 −26−26.6°◦ 4137.3°◦ −4.5◦ 11° 52◦ 38.6° SUEXU· FLIGHT[21] 0.814m 0.95m 2 1.5 −25.9−26°◦ 43.541°◦ −4.5−◦4.5° 47.8◦ 52° ◦ ◦ ◦ ◦ U·FLIGHTROTINOR [19] 1.766m0.95m 2.5 2 −7.3−25.9° 42.443.5° 5.4−4.5° 60.9 47.8° ROTINOR [19] 1.766m 2.5 −7.3° 42.4° 5.4° 60.9° Table 3. Posture angle at T0 moment In this paper, a typical diver’s body is selected, equipped with back-flying BCD Definitions Notations Value (Buoyancy Control Device) and open breathing apparatus [23]. According to Table 1, the Length of DPV LDPV 960 mm posture angles of DPV with different models and speeds are slightly different.◦ Since the Body posture θp1 7.3 ◦ shape andShoulder max cruising posture speed of DPV in thisθp2 paper are similar to ROTINOR,42.4 the T0 pos- ◦ ture angle Elbowis selected posture as Table 3. θp3 42.4 ◦ Thigh posture θp4 5.4 ◦ Knee posture θp5 60.9

In addition, the elbow joint angle θp3 of all the four DPV is within 12%, and the knee joint angle θp5–θp4 of the other three DPV is within 7.4% except SEADOO. Therefore, in order to further facilitate the calculation, the model is simplified as follows:

(1) The angle of knee joint and elbow joint remained unchanged at T0; (2) All joints only rotate around y-axis; (3) The DPV is rigidly fixed to the forearm and parallel to the x-axis when cruising; (4) All solids have the same density as water According to the above conditions, the diver–DPV coupling model are divided into three articulated 6-DOF rigid bodies. As shown in Figure8, they are named DPV and arms, body and hip, and legs respectively. The hinge points of them are P2 and P4, and the quality attributes are shown in Table4. Three reference coordinate systems of them named O1-X1Y1Z1,O2-X2Y2Z2,O3-X3Y3Z3 are established. The symbols for the force components are defined in Figure8. J. Mar.J. Mar. Sci. Sci. Eng. Eng. 20212021, ,99,, x 140 FOR PEER REVIEW 9 of 24 10 of 26

FigureFigure 8. 8.Simplified Simplified three three 6-DOF 6-DOF rigid rigid bodies. bodies. Table 4. Mass properties of three 6-DOF bodies. Table 4. Mass properties of three 6-DOF bodies 2 2 2 Model G (kg) G (x,z) Ix (kg·m )Iy (kg·m )Iz (kg·m ) Model G (kg) G (x,z) Ix (kg·m2) Iy (kg·m2) Iz (kg·m2) 840.232 DPV and arms 42.13840.232 0.643297 2.539773 2.733827 DPV and arms 42.13 −171.604 0.643297 2.539773 2.733827 227.264 Body and hip 53.48–171.604 0.822940 3.110566 3.404638 227.264 114.077 −0.515 Body and hip 53.48Legs 13.8 0.8229400.471254 1.0332883.110566 1.4092763.404638 114.077 0.155 –0.515 Legs 13.8 0.471254 1.033288 1.409276 2.3. Force Components and Disturbance0.155 Analysis The causes of the external force components are explained as follows: According to the2.3. driving Force Components principle of DPV,and Disturbance when the propeller Analysis is started, the DPV generates axial thrust Ft, whichThe iscauses transmitted of the toexternal the diver’s force arms. componen At thets shoulder are explained hinge point as follows: P2, FP2Y ,According and to the driving principle of DPV, when the propeller is started, the DPV generates axial thrust Ft, which is transmitted to the diver’s arms. At the shoulder hinge point P2, FP2Y, and FP2X are applied to the body and hips. At the same time, the DPV and arms are subjected to hydrodynamic forces, which are decomposed into Fl,da and Ff,da. Body and hips, and legs

J. Mar. Sci. Eng. 2021, 9, 140 10 of 24

FP2X are applied to the body and hips. At the same time, the DPV and arms are subjected to hydrodynamic forces, which are decomposed into Fl,da and Ff,da. Body and hips, and legs are also subject to the tension at the hinge point and hydrodynamic forces, and the fluid-solid coupling and solid–solid coupling between multi rigid bodies exist in it [24]. Next, the coupling disturbance components are analyzed: for DPV and arms, in addition to the theoretical value calculated from hydrodynamic coefficient, the disturbance generated in the flow field due to the motion and posture of body and hip, and legs need to add a disturbance term Fdis1. According to the external force analysis of the 6-DOF body in the Figure, the disturbance terms of DPV and arms in the x-axis and z-axis directions of geodetic coordinate system 0-XYZ at the initial T0 time are obtained as

mdaaZ,da(t) = Gda − Fb,da − FP2y − Fdis1,y (8)

2 where Gda = mdag, Fb,da = ρwgVda, Ff,da = 1/2CL,daρwv(t) Sda

mdaaX,da(t) = Ft − FP2x − Ff,da − Fdis1,x (9)

2 where Ff,da = 1/2Ct,daρwv(t) Sda CL,da is the lift coefficient, ρw is the density of water, FP is the hinge point force between the 6-DOF bodies, Fb is the buoyancy of the 6-DOF bodies, G is the gravity of the 6-DOF bodies, Ft is the propeller trust, Fdis is the disturbing force of the other two 6-DOF bodies. Similarly, for body and hip

mbhaZ,bh(t) = Gbh + FP2y − Fb,bh − FP4y − Fdis2,y (10)

mbhaX,bh(t) = FP2x − Ff,bh − Fdis2,x − FP4x (11) for legs, we get mlegdaZ,legs(t) = Glegs + FP4y − Fb,legs − Fdis3,y (12)

mlegdaX,legs(t) = FP4x − Ff,legs − Fdis3,x (13) To sum up, when the DPV curise at a uniform speed, the acceleration a is zero. Combined with the above formula, along the x-axis of the geodetic coordinate system, there are Ft = Ff = Ff,da + Ff,bh + Ff,legs + Fdis1,x + Fdis2,x + Fdis3,x (14)

where Ff,da + Ff,bh + Ff,legs = Ff0. That is, the resistance in the direction of x-axis is restrained. Fdis1,x + Fdis2,x + Fdis3,x = Fdis,x. That is, the posture disturbance term in the x-axis direction. In the fourth section, by solving and discussing the size and direction of Fdis, the disturbance factors caused by diver motion are determined.

2.4. Computational Region and Meshing Firstly, the overlapping mesh is generated. In the common CFD calculation of under- water vehicle, the background region is usually set as the water tank involved in. While in order to save computational resources, the small area around the model and the area of its wake flow always set finer grid, which is called the encryption region, and the other flow area uses thicker grid. In order to fully develop the incoming flow and avoid the interference from the boundary of the external field, the size of the background region (water tank) is adjusted to be 4-times the total length of the diver–DPV coupled model, 2.5-times of the full width, and 5-times of the full height. Furthermore, the encryption grid area is set to increase the calculation accuracy. After integer, the size is as follows in Table5. J. Mar. Sci. Eng. 2021, 9, 140 11 of 24

Table 5. Dimensions of three regions.

Region Length, L (m) Width, B (m) Heigth, H (m) J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 12 of 26 Model 2.429 0.843 0.764 Encryption 4.000 1.300 2.400 Water tank 9.000 2.000 4.000

TheThe side side length length of the of grid the in thegrid water in the tank water region istank 10cm, region and the is side 10cm, length and of thethe side length of gridthe ingrid the encryptionin the encryption area is 1cm area (0.4% is of 1cm the model (0.4% length). of the The model grid numberlength). of waterThe grid number of tankwater is 6704321,tank is and6704321, that of and the encryptionthat of the is encryption 1052876. Finally, is 1052876. the calculated Finally, areas the are calculated areas shownare shown in Figure in9 Figure. 9.

FigureFigure 9. 9.Calculation Calculation regions regions of water of tank. water tank.

InIn order order to to simulate simulate the disturbancethe disturbance of diver of jointdiver motion joint onmotion the hydrodynamic on the hydrodynamic per- performance simulation calculation, the following provisions are made on the basis of regionformance division: simulation calculation, the following provisions are made on the basis of region (1)division: The rigid body of DPV and arms is bound on the water tank region, that is, the region (1) moves The rigid together body with of the DPV motion and of arms DPV, is which bound can ensureon the that water the DPVtank will region, not run that is, the region outmoves of the together water tank with during the cruising; motion of DPV, which can ensure that the DPV will not run (2) The dynamic region is set based on Body and Hip and Legs. In order to improve out of the water tank during cruising; the calculation accuracy, its contour envelops the whole diver–DPV model, which is (2) shown The dynamic in Figure 10 region. is set based on Body and Hip and Legs. In order to improve the (3) Incalculation particular, the accuracy, overlapping its gridcontour relationship envelo is establishedps the whole between diver–DPV the surfaces model, which is ofshown the three in overlappingFigure 10. mesh continuum to ensure that the fluid converges when (3) passing In particular, through thethe interface. overlapping grid relationship is established between the surfaces of (4) The encrypted region is set large enough to ensure that the overlapped grid is always includedthe three in when overlapping in motion, mesh so as not continuum to diverge due to toensure different that scales the after fluid contacting converges when pass- withing thethrough external the flow interface. field. (4) Finally, The encrypted the side length region of the is set overlapped large enough grid region to ensure is also that 1cm (thethe overlapped same as the grid is always encryptionincluded region), in and when the overlappedin motion, grid so numberas not to of di bodyverge and due hip, andto different legs are 492833 scales after contact- and 462466.ing with The numberthe external of grids flow in all field regions is with a total of 7659620. Finally, the side length of the overlapped grid region is also 1cm (the same as the encryption region), and the overlapped grid number of body and hip, and legs are 492833 and 462466. The number of grids in all regions is with a total of 7659620.

J. Mar. Sci. Eng. 2021, 9, 140 12 of 24 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 13 of 26

Figure 10.FigureOverlapping 10. Overlapping grid and grid encryption and encryption region. region.

AfterAfter allall thethe regionsregions areare established,established, the DFBIDFBI (Dynamic Fluid Body Interaction) mo- motion functiontion function of Star-CCM+ of Star-CCM+ software software is is used used to to makemake the the 6-DOF 6-DOF body body produce produce constrained constrained motion.motion. The DFBI body body model model can can simulate simulate the the passive passive motion motion of ofrigid rigid body body under under the the actionaction ofof fluid.fluid. That That is, is, the the DOF DOF of of the the ri rigidgid body body can can be be determined determined according according to the to the needs,needs, andand thethe specificspecific DOF can can be be constrain constrained/released.ed/released. The The mass, mass, moment moment of inertia, of inertia, andand centroidcentroid coordinates of of the the three three rigid rigid bodies bodies of ofdiver–DPV diver–DPV model model are aredescribed described in in TableTable2 ,2, while while thethe specificspecific degrees of of freedom freedom constraints constraints of of each each DFBI DFBI body body are are described described inin SectionSection 3.23.2.. The joint joint and and shoulder shoulder coordinate coordinate are are needed needed to tobe besupplement: supplement: P2 (–0.455196 m,0.0 m,0.234889 m) P4 (–0.3711 m,0.0 m, –0.12 m) The boundary conditions of regions are set as follows: The upper, lower and the in- P2 (−0.455196 m, 0.0 m,0.234889 m) P4 (−0.3711 m, 0.0 m, −0.12 m) flow surface of the water tank region are set as the velocity-inlet, the back surface is the pressure-outlet,The boundary and conditionsthe left- and of right-side regions are surface set as are follows: set as symmetrical-planes. The upper, lower and An the inflowoverlapping surface grid of boundary the water is tank established region betw are seteen as the the outer velocity-inlet, surface of the the leg back region, surface the is thebody pressure-outlet, region, and the and water the tank left- region, and right-side while the surfacemodel surfaces are set asin symmetrical-planes.the three overlapping An overlappingregions are set grid as the boundary wall-surface. is established The k–ω SST between turbulence the outer model surface is used of to the close leg the region, N-S the bodyequation. region, The and full they+ wall water function tank region, method while suitable the modelfor high surfaces Reynolds in number the three isoverlapping used for regionswall treatment. are set asFirst the boundary wall-surface. layer mesh The k–is ωobtainedSST turbulence by y+=30, and model the iscase used y+ distribu- to close the N-Stion equation.is shown in The Figure full 11.y+ Finally,wall function 10 layersmethod of prismatic suitable mesh for are highadopted, Reynolds and the number thick- is usedness of for the wall first treatment. prismatic Firstmesh boundary is 0.0073 mm layer according mesh is to obtained y+. by y+=30, and the case y+ distribution is shown in Figure 11. Finally, 10 layers of prismatic mesh are adopted, and the thickness of the first prismatic mesh is 0.0073 mm according to y+. In the aspect of solver, the time step is selected first. When the time step is too small, the calculation time is too long, and when the time step is too large, the calculation accuracy is not enough. The resistance time step is generally 0.005 L/V~0.01 L/V, where V is the velocity, and the corresponding Courant number of this model is 0.2–0.4. Therefore, 0.005 s is taken as the time step, which satisfies the CFL condition. Set the maximum number of internal iterations to 5. The SIMPLE (Semi Implicit Method for Pressure Linked Equation) algorithm is used to solve the Navier–Stokes equations. The viscous term in the discrete equation adopts the second-order central difference scheme, and the convective term adopts the second-order upwind scheme. In terms of initial conditions, assuming that the experimental condition is infinite water depth, the fresh water temperature is 24 °C, g = 9.81 m/s2, and the kinematic viscosity coefficient v = 0.9167 × 106 m2/s. The necessary DOF of the rigid bodies are constrained, and the x-axis cruising is simulated by uniform inflow, that is, the absolute motion is replaced by relative motion to save calculation time.

J. Mar. Sci. Eng. 2021, 9, 140 13 of 24 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 14 of 26

Figure 11. The case y+ distribution nephogram. Figure 11. The case y+ distribution nephogram.

3. ResultsIn the aspect of solver, the time step is selected first. When the time step is too small, the calculation time is too long, and when the time step is too large, the calculation accu- racy Thisis not sectionenough. showsThe resistance the CFD time results step is thegenerally straight-line 0.005L/V cruising~0.01L/V, where motion V is of the the diver– DPVvelocity, coupling and the model corresponding in the water. Courant That number is, the of motionthis model in is the 0.2–0.4.x-axis Therefore, direction 0.005s is simulated throughis taken as the the uniform time step, incoming which satisfies flow, andthe CFL the translationcondition. Set movement the maximum in the numberz-axis of direction isinternal constrained. iterations Firstly, to 5. The the SIMPLE diver and (Semi DPV Implicit are regardedMethod for as Pressure a rigid Linked body by Equation) restraining the humanalgorithm motion; is used thento solve the the human Navier–Stokes motion equations. is released The as viscous a multi-rigid-articulated term in the discrete model; finally,equation the adopts diver–DPV the second-order disturbance central components difference are scheme, analyzed and by the comparison. convective term adopts the second-order upwind scheme. 3.1. CaseIn terms of Restricting of initial Humanconditions, Motion assuming that the experimental condition is infinite ℃ 2 waterThis depth, section the fresh only water simulates temperature constrained is 24 , g straight-line = 9.81 m/s , and cruising. the kinematic All DOF viscosity of the three coefficient v = 0.9167*106 m2/s. The necessary DOF of the rigid bodies are constrained, and 6-DOF models are fixed and restrained. the x-axis cruising is simulated by uniform inflow, that is, the absolute motion is replaced by relativeFirstly, motion mesh to independence save calculation and time. convergence are analyzed by taking the incoming velocity with the DPV’s maximum speed V5 = 2.5 m/s as an example. The sub relaxation J. Mar. Sci. Eng. 2021, 9, x FOR PEER factorREVIEW of velocity and pressure is 0.5. After 10000 time steps of simulation, the convergence15 of 26 3. Results resultThis of resistancesection showsFf is the shown CFD in results Figure the 12 stra. Theight-line velocity cruising nephogram motion is of shown the diver– in Figure 13. DPV coupling model in the water. That is, the motion in the x-axis direction is simulated through600 the uniform incoming flow, and the translation movement in the z-axis direction is constrained. Firstly, the diver and DPV are regarded as a rigid body by restraining the human motion; then the human motion is released as a multi-rigid-articulated model; fi- nally,500 the diver–DPV disturbance components are analyzed by comparison.

3.1. Case of Restricting Human Motion This section only simulates constrained straight-line cruising. All DOF of the three 6- 400 DOF models are fixed and restrained. (N)

f Firstly, mesh independence and convergence are analyzed by taking the incoming F velocity with the DPV’s maximum speed V5=2.5 m/s as an example. The sub relaxation factor300 of velocity and pressure is 0.5. After 10000 time steps of simulation, the convergence result of resistance Ff is shown in Figure 12. The velocity nephogram is shown in Figure 13. 200

100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Time (s) FigureFigure 12.12. ResistanceResistance convergenceconvergence resultsresults ofof constrainedconstrained straight-linestraight-line cruisingcruising simulation.simulation.

Figure 13. Velocity nephogram at 3s moment of constrained straight-line cruising simulation.

Then the straight-line thrust Ft required by V5=2.5 m/s is obtained. Because of the resistance tends to converge after 1s, the resistance value in the period of 1–3 s is taken as the arithmetic average. That is, Ff=Ft=300.4N，z-axis lift Fl =-14.2N. Next, the resistance prediction of constrained straight-line cruising under V5=2.5 m/s is selected for grid independent. According to ITTC (International Towing Tank Confer- ence) [25], the grid side length is set as 1/2000–5/2000 of the model length L, which is 1 cm in this paper. There set three different length as 1 cm,√2 cm, and 2 cm to generate coarse mesh. The total number of meshes is 7659620, 2708494, and 957453. When the other pa- rameters and settings are the same, the change trend of resistance value with the founda- tion size is observed as Figure 14.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 15 of 26

600

500

400 (N) f F

300

200

100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

J. Mar. Sci. Eng. 2021, 9, 140 Time (s) 14 of 24 Figure 12. Resistance convergence results of constrained straight-line cruising simulation.

FigureFigure 13. 13. VelocityVelocity nephogram nephogram at at 3s 3s moment moment of cons constrainedtrained straight-linestraight-line cruising cruising simulation. simulation.

ThenThen the the straight-line straight-line thrustthrust FFt trequired required by byV 5V=5=2.5 2.5 m/sm/s isis obtained.obtained. Because Because of of the the resistanceresistance tends tends to to converge converge after after 1s, thethe resistanceresistance value value in in the the period period of of 1–3 1–3 s iss takenis taken as as thethe arithmetic arithmetic average. average. That That is, is, FFff==FtF=300.4Nt = 300.4， N,z-axisz-axis lift lift FlF =-14.2N.l = −14.2 N. Next,Next, the the resistance resistance predictionprediction of of constrained constrained straight-line straight-line cruising cruising under underV5 = V 2.55=2.5 m/s m/s is isselected selected for for grid grid independent. independent. According According to ITTCITTC (International(International Towing Towing Tank Tank Confer- Confer- ence)ence) [25], [25 ],the the grid grid side side length length is isset set as as 1/2000–5/2000 1/2000–5/2000 of the of the√ model model length length L, whichL, which is 1 is cm 1 cm in this paper. There set three different length as 1 cm, 2 cm, and 2 cm to generate in this paper. There set three different length as 1 cm,√2 cm, and 2 cm to generate coarse coarse mesh. The total number of meshes is 7659620, 2708494, and 957453. When the mesh. The total number of meshes is 7659620, 2708494, and 957453. When the other pa- J. Mar. Sci. Eng. 2021, 9, x FOR PEER otherREVIEW parameters and settings are the same, the change trend of resistance value with16 of the 26 rametersfoundation and size settings is observed are the as same, Figure the 14 change. trend of resistance value with the founda- tion size is observed as Figure 14.

600 Ff,mesh1

Ff,mesh2

Ff,mesh3 500

400 (N) f,mesh F 300

200

100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Times (s) Figure 14. Convergence values of resistanceresistance underunder threethree kindskinds ofof grids.grids.

Finally, the mean resistance Ff,mesh1=300.4N, Ff,mesh2=321.2N, Ff,mesh3=378.1N. So there are:

FFf ,3mesh f ,2 mesh =>=1.18 1.07 (15) FF f ,2mesh f ,1 mesh Therefore, with the gradual refinement of the grid, the numerical simulation value of resistance gradually decreases, and the calculation accuracy gradually converges as 6.9%, and the number of grids increases sharply as 180%. Considering the calculation accuracy and cost, it is reasonable to take the grid side length of 1 cm. Then, the numerical simulation of constrained straight-line cruising under the other four speed conditions is carried out. The four inflow velocities are set as V1=0.5 m/s, V2=1.0 m/s, V3=1.5 m/s, and V4=2.0 m/s. So the linear relationship between Vn2 and Ff is obtained. The resistance coefficient Ct is introduced here. Because the rigid body under water straight-line cruising has Ff=1/2Ctρwv2S, the ideal value Ct0 can be obtained from the slope of the fitting curve. After numerical simulation, the relationship between resistance Ff and V2 is shown in Figure 15.

J. Mar. Sci. Eng. 2021, 9, 140 15 of 24

Finally, the mean resistance Ff,mesh1 = 300.4 N, Ff,mesh2 = 321.2 N, Ff,mesh3 = 378.1 N. So there are: Ff ,mesh3 Ff ,mesh2 = 1.18 > 1.07 = (15) Ff ,mesh2 Ff ,mesh1 Therefore, with the gradual refinement of the grid, the numerical simulation value of resistance gradually decreases, and the calculation accuracy gradually converges as 6.9%, and the number of grids increases sharply as 180%. Considering the calculation accuracy and cost, it is reasonable to take the grid side length of 1 cm. Then, the numerical simulation of constrained straight-line cruising under the other four speed conditions is carried out. The four inflow velocities are set as V1= 0.5 m/s, 2 V2 = 1.0 m/s, V3 = 1.5 m/s, and V4 = 2.0 m/s. So the linear relationship between Vn and Ff is obtained. The resistance coefficient Ct is introduced here. Because the rigid body 2 under water straight-line cruising has Ff = 1/2Ctρwv S, the ideal value Ct0 can be obtained J. Mar. Sci. Eng. 2021, 9, x FOR PEER fromREVIEW the slope of the fitting curve. After numerical simulation, the relationship between17 of 26 2 resistance Ff and V is shown in Figure 15.

350 300.4 300

250

200 190.95 (N)

f F 150 105.68 100

49.23 50 11.96 0 0 0123456 V 2 (m/s) n

2 2 FigureFigure 15. 15.Relation Relation curve curve between betweenVn VnandandF fF. f.

−3 C −3 Finally,Finally, the the resistance resistance coefficient coefficient of diver–DPV of diver–DPV model model is obtained is obtained as t0 =as 27.2984/10 Ct0= 27.2984/10. . 3.2. Case of Releasing Human Motion 3.2. Case of Releasing Human Motion This section only carries out the simulation of straight-line cruising with released humanThis motion. section That only is, carries releasing out the thex -simulation and z-axis translationof straight-line DOF cruising and the rotationwith released DOF hu- aroundman motion. the y-axis That of twois, releasing 6-DOF bodies the ofx- bodyand andz-axis hips, translation and legs to DOF simulate and thethe articulated rotation DOF motionaround of the diver y-axis body of intwo water. 6-DOF bodies of body and hips, and legs to simulate the articu- lated motion of diver body in water. 3.2.1. Maximum Cruising Speed

3.2.1.Firstly, Maximum the numerical Cruising simulationSpeed of the maximum cruising speed V5 = 2.5 m/s is carriedFirstly, out. Thethe numerica velocity cloudsl simulation at the of moment the maximum of 2 s is cruising shown inspeed Figure V5=2.5 16. m/s And is the carried variationout. The curvevelocity of resistance clouds atF ftheand moment lift force Fofl are2 s shownis shown in Figures in Figure 17 and 16. 18 And. The the posture variation angle changes of 6-DOF model legs’ θ1, and body and hip’s θ4 are obtained as shown curve of resistance Ff and lift force Fl are shown in Figures 17 and 18. The posture angle in Figures 19 and 20. changes of 6-DOF model legs’ θ1, and body and hip’s θ4 are obtained as shown in Figures 19 and 20.

Figure 16. Velocity nephogram at 2s moment of straight-line cruising with releasing motion at V5 speed.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 17 of 26

350 300.4 300

250

200 190.95 (N)

f F 150 105.68 100

49.23 50 11.96 0 0 0123456 V 2 (m/s) n

Figure 15. Relation curve between Vn2 and Ff.

Finally, the resistance coefficient of diver–DPV model is obtained as Ct0= 27.2984/10−3.

3.2. Case of Releasing Human Motion This section only carries out the simulation of straight-line cruising with released hu- man motion. That is, releasing the x- and z-axis translation DOF and the rotation DOF around the y-axis of two 6-DOF bodies of body and hips, and legs to simulate the articu- lated motion of diver body in water.

3.2.1. Maximum Cruising Speed

Firstly, the numerical simulation of the maximum cruising speed V5=2.5 m/s is carried out. The velocity clouds at the moment of 2 s is shown in Figure 16. And the variation J. Mar. Sci. Eng. 2021, 9, 140 curve of resistance Ff and lift force Fl are shown in Figures 17 and 18. The posture angle16 of 24 changes of 6-DOF model legs’ θ1, and body and hip’s θ4 are obtained as shown in Figures 19 and 20.

J. Mar. Sci. Eng. 2021Figure, 9, x 16. FOR Velocity PEER REVIEW nephogram at 2s moment of straight-line cruising with releasing motion at V5 speed. 18 of 26 J. Mar. Sci. Eng. 2021, 9, x FOR PEER FigureREVIEW 16. Velocity nephogram at 2 s moment of straight-line cruising with releasing motion18 of at26

V5 speed.

-5 -5 ） ° ） ° （ 1 （ θ 1

θ -10 -10

-15 -15 Body and hipmotion Body and hipmotion

(2, -18.26774) (2, -18.26774) -20 -20 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Time (s) Time (s) Figure 17. Change process of 6-DOF model body and hip posture angle. Figure 17. Change process of 6-DOF model body and hip posture angle.angle.

10 10 (2, 9.5859) (2, 9.5859) ） ° ） ° （ 4 （ θ 4 θ Leg motion Leg Leg motion Leg

5 5

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Time (s) Time (s) Figure 18. Change process of 6-DOF model legs posture angle. Figure 18. Change process of 6-DOF model legs posture angle.angle. 500 500

400 400

300 300 (N) f (N) F f F 200 200

100 100

0 0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Time (s) Time (s) Figure 19. Convergence process of cruising resistance. Figure 19. Convergence process of cruising resistance.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 18 of 26

-5 ） ° （ 1 θ -10

-15 Body and hipmotion

(2, -18.26774) -20 0.0 0.5 1.0 1.5 2.0 2.5 Time (s) Figure 17. Change process of 6-DOF model body and hip posture angle.

10 (2, 9.5859) ） ° （ 4 θ Leg motion Leg

5

0.0 0.5 1.0 1.5 2.0 2.5 J. Mar. Sci. Eng. 2021, 9, 140 17 of 24 Time (s) Figure 18. Change process of 6-DOF model legs posture angle.

500

400

300 (N) f F 200

100

0 0.0 0.5 1.0 1.5 2.0 2.5 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW Time (s) 19 of 26

FigureFigure 19. 19.Convergence Convergence process process of of cruising cruising resistance. resistance.

200

100 (N)

l 0 F

-100

-200 0.0 0.5 1.0 1.5 2.0 2.5 Time (s) Figure 20.20.Convergence Convergence process process of of cruising cruising lift. lift.

TheThe analysisanalysis shows shows that that the the angle angleθP1 θofP1 of Body Body and and Hip Hip gradually gradually converges converges to the to the stable range, and θ of legs first fluctuates in a certain range, and finally converges to the stable range, and P4θP4 of legs first fluctuates in a certain range, and finally converges to the stable range.range. That That is, is, the the diver diver body body will will swing swing up up and and down down during during cruising. cruising. At the Atsame the same time, the resistance also fluctuates in a certain range. Because the resistance converge after time, the resistance also fluctuates in a certain range. Because the resistance converge after 1 s and fluctuates in a small range,and the human body posture reached the inflection point 1s and fluctuates in a small range,and the human body posture reached the inflection angle at 2 s, the resistance value in the period of 1–2 s is taken as the arithmetic average. point angle at 2 s, the resistance value in the period of 1–2 s is taken as the arithmetic That is, Ff,V5 = 272.72 N, and the lift Fl,V5 finally converges near 0. average. That is, Ff,V5 = 272.72N, and the lift Fl,V5 finally converges near 0. 3.2.2. Other Cruising Speeds 3.2.2. Other Cruising Speeds Next, the numerical simulation of other speeds is carried out. The x-axis and z-axis translationNext, the and numerical rotation around simulationy-axis of of other body speeds and hip, is andcarried legs out. are released,The x-axis and and the z-axis ztranslation-axis translation and rotation of DPV andaround arms y is-axis restrained. of body Set and five hip, inflow and velocities legs are asreleased,V1 = 0.5 and m/s, the z- Vaxis2 = 1.0translation m/s, V3 =of 1.5 DPV m/s, andV4 =arms 2.0 m/s, is restrained. and V5 = 2.5 Set m/s, five then inflow the posturevelocities angle as ofV1 diver=0.5 m/s, bodyV2=1.0 and m/s, its V influence3=1.5 m/s, on V the4=2.0 speed m/s, of and straight-line V5=2.5 m/s, cruising then arethe obtained.posture angle of diver body and itsIn fluidinfluence mechanics, on the a speed dimensionless of straight-line variable cruising characterizing are obtained. the relative magnitude of inertialIn forcefluid andmechanics, gravity ofa dimensionless fluid is called Froude variable number characterizingFr, which representsthe relative the magnitude ratio of of 2 theinertial inertial force force and to thegravity magnitude of fluid of is gravity. called ThatFroude is, Fr number = v /gL, Fr where, whichv is represents the straight-line the ratio cruisingof the inertial speed offorce rigid to body, the mag isgnitude the acceleration of gravity. of gravity,That is,L Fr=vis the2/gL characteristic, where v is length the straight- of the object. line cruising speed of rigid body, g is the acceleration of gravity, L is the characteristic length of the object. In this paper, the resistance of the other four cases of different speeds are calculated according to Fr number from low to high. After CFD convergence, the mean resistance, mean lift and posture are shown in Table 6, and the velocity scalar nephogram is shown in Figure 21. The instantaneous value of posture angle is obtained at 2s moment, and the resistance is the average value of 1~2s.

Table 6. DPV resistance and posture angle at different speeds.

Vn (m/s) Body and Hip θP1 Legs Ff Fl Fr (n = 1~5) (°) θP4 (°) (N) (N) 0.00867 0.5 −30.24 13.80 18.68 −0.54 0.03469 1.0 −28.42 13.30 52.64 −0.66 0.07944 1.5 −27.07 13.35 100.30 1.55 0.13875 2.0 −26.32 14.00 185.13 1.03 0.21680 2.5 −18.27 9.59 272.72 5.49

J. Mar. Sci. Eng. 2021, 9, 140 18 of 24

In this paper, the resistance of the other four cases of different speeds are calculated according to Fr number from low to high. After CFD convergence, the mean resistance, mean lift and posture are shown in Table6, and the velocity scalar nephogram is shown in Figure 21. The instantaneous value of posture angle is obtained at 2 s moment, and the resistance is the average value of 1~2 s.

Table 6. DPV resistance and posture angle at different speeds.

Vn (m/s) Body and Hip θP1 Legs Ff Fl Fr ◦ ◦ (n = 1~5) ( ) θP4 ( ) (N) (N) 0.00867 0.5 −30.24 13.80 18.68 −0.54 0.03469 1.0 −28.42 13.30 52.64 −0.66 0.07944 1.5 −27.07 13.35 100.30 1.55 − J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW0.13875 2.0 26.32 14.00 185.13 1.0320 of 26 0.21680 2.5 −18.27 9.59 272.72 5.49

Figure 21.21. Velocity scalarscalar nephogramnephogram atat 22s s mo momentment at other four cruising speeds.

The analysis shows that with the increase of speed V, the Body and Hip posture angle θP1 decreasesdecreases with the increase of speed, the Legs posture angle θP4P4 is also the same,same, andand the resistance Ff gradually decreases. The fitting fitting curvecurve of the relationship betweenbetween FFff and V2 isis shown shown in in Figure Figure 22,22, and and the the fitting fitting equation equation is is in in Table Table7 7.. It can be seen that the resistance velocity relationship of traditional submersible is not 300 2 consistent with Ff = 1/2Ctρwv S, This is due to the change of human body posture angle under the action of flow field, which leads to the real-time272.72 change of resistance coefficient Ct, and finally tends to the minimum value. The specific process is as follows: According to the Figure 20, the absolute value of lift |Fl| under various working conditions has experienced a state from large to small, and finally tends to 0N. Therefore, 200 in straight-line navigation, the lift force brought by the real-time posture angle θp (t) will 175.13 drive） the human body to approach the posture with the minimum resistance Fl (t). When theN final lift tends to 0, the resistance is the minimum, and the posture angle reaches the （ targetf value. F When the resistance F converges, there is still a difference between the resistance of 100 f 100.3 the constrained diver–DPV model, which is defined as the posture disturbance term Fdis. When Fdis > 0, it is called favorable disturbance, and when Fdis < 0, it is called adverse 52.64

18.68 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 V (m/s) n

Figure 22. Fitting curve of the relationship between Ff and Vn.

Table 7. Fitting curve equation of the relationship between Ff and V

Equation Ff =a*v^b a 48.51346 ± 2.39272 b 1.87543 ± 0.06026 Reduced chi-square 18.02041 R2 (COD) 0.99867 Adjusted R2 0.99834

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 20 of 26

J. Mar. Sci. Eng. 2021, 9, 140 Figure 21. Velocity scalar nephogram at 2s moment at other four cruising speeds. 19 of 24

The analysis shows that with the increase of speed V, the Body and Hip posture angle θP1 decreases with the increase of speed, the Legs posture angle θP4 is also the same, and disturbance.the resistance Its Ff relationshipgradually decreases. with other The resistance fitting cu componentsrve of the relationship and speed between is discussed Ff and in theV2 is next shown section. in Figure 22, and the fitting equation is in Table 7.

300 272.72

200 175.13 ） N （ f F 100 100.3

52.64

18.68 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 V (m/s) n

FigureFigure 22. 22.Fitting Fitting curve curve of theof the relationship relationship between betweenFf andFf andVn. Vn.

Table 7. Fitting curveTable equation7. Fitting ofcurve the relationshipequation of betweenthe relationshipFf and V between. Ff and V

EquationEquation Fff =a*v^ba*vˆb a 48.5134648.51346 ± ± 2.39272 b 1.87543 ± ± 0.06026 ReducedReduced chi-square 18.02041 R2 (COD) 0.99867 2 AdjustedR (COD) R2 0.998670.99834 Adjusted R2 0.99834 4. Discussion This section analyzes the relationship between the X-axis resistance disturbance term Fdis, speed and posture. Firstly, the total resistance components of the diver–DPV cruising in the water are analyzed. The results show that the total resistance of the diver–DPV coupled model in the water is Ff = Ff0 + Fdis, where Ff0 is the resistance value of restricting the direct navigation movement. Ignoring the influence of near surface and bottom effect, the component of the resistance of Ff0 is usually divided into the following points: friction resistance R f , residual resistance Rrs, Therefore, the total resistance can be expressed as

Ff = Ff0 + Fdis = Rf + Rrs + Fdis (16)

According to Hughes, the ratio of viscous resistance coefficient to frictional resistance coefficient is constant. By introducing the shape factor to deal with the three-dimensional flow of the diver–DPV model, the resistance conversion between the model and the real vehicle is realized. The resistance coefficient relationship is   Ct = Ctm − (1 + k) Cf m − Cf s + ∆Cf (17)

where Ctm is the resistance coefficient of the model, (1 + k) is the shape coefficient, Cfm is the friction resistance coefficient, Cfs is the actual friction resistance coefficient, 4Cf is the roughness conversion subsidy coefficient. J. Mar. Sci. Eng. 2021, 9, 140 20 of 24

According to the formula of ITTC (International Towing Tank Conference) [25], the friction and resistance coefficient of a real vehicle can be calculated by Reynolds number (Re)

0.075 Cf = (18) (lgRe − 2)2

UL Re = (19) v where v is the water viscosity coefficient affected by temperature. The corresponding subsidy coefficient of roughness conversion can be expressed as " #  K 1/3 ∆C = 105 s − 0.64 × 10−3 (20) f L

Ks is the height of rough surface. This CFD model is a full-scale model. The fresh water temperature is 25 ◦C and the kinematic viscosity coefficient of water is v = 0.9167 × 6 2 −6 10 m /s. Without considering the cruising wear, this paper takes Ks = 150 × 10 . In conclusion, according to the main scale of diver–DPV model, the roughness conversion subsidy coefficient can be obtained, and then the resistance coefficient conversion between the CFD model and the actual DPV can be realized. In order to effectively analyze the change of resistance components, the model resis- tance is dimensionless. Assuming that the wet surface area of diver–DPV model is constant, 2 there is still friction resistance f = 1/2Cfρwv S. The empirical formula of friction resistance coefficient is 0.445 Cf = (21) (lgRe)2.58 2 The total resistance coefficient is still calculated as Ff = 1/2Ctρwv S, while 2 Ff 0 = 1/2Ct0ρwv S. That is: Ct = Ct0 + Cdis = Cf + Crs + 4Cf + Cdis (22) 2 −3 According to that the wet surface area Sw = 3.521381 m , we get Ct0 = 27.2984/10 . The results are shown in Table8.

Table 8. Dimensionless results of diver–DPV resistance.

Total Friction Residual Rough Reynolds Disturbance Froude Resistance Resistance Resistance Subsidy Number Coefficient NumberFr Coefficient Coefficient Coefficient −3 Coefficient Re −3 −3 −3 Cdis/10 −3 Ct/10 Cf/10 Crs/10 4Cf/10 1324.86 0.00867 42.4381 23.5864 18.1827 6.6641 2649.72 0.03469 29.8993 18.6005 11.2988 4.5793 3974.58 0.07944 25.3188 16.3426 8.97622 −7.8424 0.669 5299.44 0.13875 24.8670 14.9653 9.90173 −17.1235 6624.30 0.21680 24.7833 14.0056 10.7777 −27.6766

According to Table8, the resistance coefficient Ct of the diver–DPV coupled model is plotted as Figure 23 according to the variation curve. It can be seen from Figure 23 that the total resistance coefficient of model tends to decrease with the increase of Froude number Fr. The friction resistance Rfs in those speed cases is the main part of the total resistance, and the correlation between the value and the speed change trend gradually decreases. When Fr > 0.06, Cdis < 0, which means it is favorable disturbance; when Fr < 0.06, Cdis > 0, which means it is adverse disturbance. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 22 of 26

Without considering the cruising wear, this paper takes Ks = 150*10-6. In conclusion, ac- cording to the main scale of diver–DPV model, the roughness conversion subsidy coeffi- cient can be obtained, and then the resistance coefficient conversion between the CFD model and the actual DPV can be realized. In order to effectively analyze the change of resistance components, the model re- sistance is dimensionless. Assuming that the wet surface area of diver–DPV model is con- stant, there is still friction resistance f=1/2Cfρwv2S. The empirical formula of friction re- sistance coefficient is 0.445 C = f 2.58 (21) (lg Re)

The total resistance coefficient is still calculated as Ff=1/2Ctρwv2S, while Ff0 =1/2Ct0ρwv2S. That is:

Ct =Ct0 +Cdis =Cf+Crs+△Cf+ Cdis (22)

According to that the wet surface area Sw=3.521381m2, we get Ct0= 27.2984/10-3. The results are shown in Table 8.

Table 8. Dimensionless results of diver–DPV resistance

Friction Residual Reynolds Total Resistance Disturbance Rough Subsidy Froude Number Resistance Resistance Number Coefficient Coefficient Coefficient Fr Coefficient Coefficient Re Ct/10−3 Cdis/10−3 △Cf/10−3 Cf/10−3 Crs/10−3 1324.86 0.00867 42.4381 23.5864 18.1827 6.6641 2649.72 0.03469 29.8993 18.6005 11.2988 4.5793 3974.58 0.07944 25.3188 16.3426 8.97622 −7.8424 0.669 5299.44 0.13875 24.8670 14.9653 9.90173 −17.1235 J. Mar. Sci.6624.30 Eng. 2021 , 9, 140 0.21680 24.7833 14.0056 10.7777 −27.6766 21 of 24 According to Table 8, the resistance coefficient Ct of the diver–DPV coupled model is plotted as Figure 23 according to the variation curve.

Ct

Cf C 40 rs Cdis

20 C

0 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 23 of 26

-20It can be seen from Figure 23 that the total resistance coefficient of model tends to 0.00 0.05 0.10 0.15 0.20 0.25 decrease with the increase of Froude number Fr. The friction resistance Rfs in those speed cases is the main part of the totalFr resistance, and the correlation between the value and the Figurespeed 23. 23.change VariationVariation trend of of resistancegradually resistance coefficient coefficientdecreases. of ofDPV When DPV underwater underwater Fr > 0.06, cruising. C cruising.dis < 0, which means it is favora- ble disturbance; when Fr < 0.06, Cdis > 0, which means it is adverse disturbance. According to various various resistance resistance coefficients, coefficients, the the actual actual resistance resistance Ff, Fresistancef, resistance of re- of restraintstraint movement movement Ff0F, f0disturbance, disturbance term term FdisF disandand speed speed V areV are shown shown in Table in Table 9. 9.

Table 9.9. ResistanceResistance andand speed.speed.

n f dis f0 VVn Ff F Fdis F Ff0F ((nn= = 1~5)1~5) (N)(N) (N)(N) (N)(N) 0.50.5 18.6818.68 6.666.66 12.0212.02 1.01.0 52.6452.64 4.584.58 48.0648.06 1.51.5 100.30100.30 −7.84−7.84 108.14 108.14 2.0 175.13 −17.12 192.26 2.0 175.13 −17.12 192.26 2.5 272.723 −27.68 300.40 2.5 272.723 −27.68 300.40

The fitting curves of F and Vn is as follows in Figure 24: The fitting curves of Fdisdis and Vn is as follows in Figure 24：

10 6.66 4.58

0 0

-7.84 (N) -10 dis F

-17.12 -20

-27.68 -30 0123 V (m/s) n

Figure 24. Fitting curve of the relationship between Fdisdis andand VVn.n.

Finally, the relationship between the total resistance Ff and Vn is obtained as

=+=1 ρ 223+ + + + FVDVfnnnFFfdistwn00 CVSABVC * * * (23) 2 The diver–DPV disturbance in the x-axis is

 =+++1 ρ 223+ mVCVDVVCVSABntwnn()tt0 ()()()() * ttt * n * n (24) 2 The values of the constants are shown in the Table 10.

Table 10. Fitting curve equation of the relationship between Fdis and Vn

Equation Fdis = A + B*Vn + C*Vn2+ D*Vn3 A −0.0299 ± 1.48536 B 27.55986 ± 5.77036

J. Mar. Sci. Eng. 2021, 9, 140 22 of 24

Finally, the relationship between the total resistance Ff and Vn is obtained as

1 2 2 3 Ff = Ff 0 + Fdis = 2 Ct0ρwVn S + A + B ∗ Vn + C ∗ Vn + D ∗ Vn (23)

The diver–DPV disturbance in the x-axis is . 1 2 2 3 mVn(t) = 2 Ct0ρwVn(t) S + A + B ∗ Vn(t) + C ∗ Vn(t) + D ∗ Vn(t) (24) The values of the constants are shown in the Table 10.

Table 10. Fitting curve equation of the relationship between Fdis and Vn.

2 3 Equation Fdis = A + B*Vn + C*Vn + D*Vn A −0.0299 ± 1.48536 B 27.55986 ± 5.77036 C −29.82083 ± 5.73532 D 5.76419 ± 1.50635 Reduced Chi-Sqr 2.2975 R2(COD) 0.9949 Adjustment R2 0.9872

From the conclusion, it can be concluded that in the design of DPV, if the maximum cruising speed index is too low, the diver’s attitude will always be in the range of adverse disturbance, and the navigation efficiency will be reduced. However, if the maximum cruising speed is too high, the disturbance coefficient will not change, and the cruising efficiency will not be further improved. It can be understood that the high-speed flow field has already impacted the diver’s body to the horizontal attitude, and will not change further. Through this CFD model, we can get the most suitable speed range index in the DPV design, which is of great significance to the design of various shape of DPV in the future.

5. Conclusions In summary, by analyzing the existing DPV (diver propulsion vehicle) equipment, a set of diver–DPV multi-body coupling model considering rigid body dynamics and fluid disturbance is established in this paper. The split and motion of human joints is realized by overlapping grid of Star-CCM+ software and DFBI 6-DOF model motion method. The nonlinear disturbance caused by diver posture transformation under 10 different cruising cases and its influence on the rapidity and diver posture characteristics of DPV are concluded as follows: When a diver driving DPV and cruising straight line, the angle of diver body’s posture is related to the speed of the vehicle. The movement of this posture change produces a disturbing potential to the surrounding flow field, and produces a resistance disturbance term to the diver–DPV coupled model, which makes the overall cruising resistance different from the resistance at the same speed under constraint human body conditions. The relationship between the sign and volume of the disturbance term Fdis and the speed is 2 3 as follows: Fdis = A + B*Vn + C*Vn + D*Vn , when Fr > 0.06, Fdis < 0, which means it is favorable disturbance and can reduce cruising resistance. When Fr < 0.06, Fdis > 0, which means it is adverse disturbance. In addition, within the design maximum speed of DPV, the friction resistance Ff is always the main part. Through this multi-body hydrodynamic model, we can get the most suitable speed range index in the DPV design. It is of great significance for the future design of DPV with various shapes to improve the underwater cruising efficiency. J. Mar. Sci. Eng. 2021, 9, 140 23 of 24

Author Contributions: Conceptualization, H.L. and H.Z.; methodology, F.H.; software, W.Z.; valida- tion, H.L and Y.W.; formal analysis, J.Z.; investigation, H.L and H.Z.; data curation, J.Z.; writing— original draft preparation, H.L; writing—review and editing, W.Z. and F.H.; visualization, W.Z.; supervision, H.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key R&D Program of China, grant number 2018YFC0309402. Data Availability Statement: Not applicable. Acknowledgments: First of all, I would like to appreciate for Harbin DepTech marine technology Co., Ltd. and other corporations for the DPV product data. Second, thanks to every scuba diving instructors from PADI for their detailed introduction on the professional DPV driving posture. Finally, I would like to thank all the editors and reviewers for their careful guidance and help. This is the first time that I have written an international top-level journal paper as the first author. Thanks a lot for your tolerance of minor errors in the process of paper revision. Wish there will be more cooperation in the future. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

Remin Reynolds number of minimum rigid body in multi joint model Fr Froude number, Fr = v2/gL V Dynamic viscosity coefficient of fresh water LDPV Length of DPV BDPV Width of DPV HDPV Height of DPV Dmax Maximum diving depth m Weight of DPV L Length of the diver–DPV coupled model B Width of the diver–DPV coupled model H Height of the diver–DPV coupled model M Weight of the diver–DPV coupled model ax, ay Acceleration of x-axis and y-axis cruising Pn Hinge points of multi rigid body joint model, n = 1, 2, 3, 4, 5 θPn Angle between diver joint and x-axis at hinge point n Ct Total resistance coefficient Cdis Disturbance resistance coefficient Cf Friction resistance coefficient Crs Residual resistance coefficient 4Cf Rough subsidy coefficient Cdis Disturbance coefficient Vn Underwater cruising speed of DPV, n = 1, 2, 3, 4, 5 Sw Wet surface area of the model Ix,Iy,Iz Moment of inertia for X, Y, Z axis Ft Propeller thrust Ff Cruising resistance of the diver–DPV model Ff,Vn Cruising resistance at different speeds Ff,meshn Resistance at maximum speed V5 with different computational grid, n = 1, 2, 3 Fb Buoyancy Fl Lift force Fl,da Z-axis lift force acting on DPV and arms Ff,da Resistance in x-axis direction acting on DPV and arms Fb,da Buoyancy of DPV and arms Gda Gravity of DPV and arms FP2X,FP2Y Tensile force at hinge point P2 Fl,bh Lift force acting on body and hip in z-axis direction Ff,bh Resistance in x-axis direction acting on body and hip Fb,bh Buoyancy of body and hip J. Mar. Sci. Eng. 2021, 9, 140 24 of 24

Gbh Gravity of body and hip FP4Y,FP4X Tensile force at hinge point P4 Fl,leg Z-axis lift acting on legs Ff,legs Resistance in x-axis direction acting on legs Fb,legs Buoyancy of legs Glegs Gravity of legs Fdis Disturbance force Fdisn,x Disturbance force on different 6-DOF body in x-axis, n = 1, 2, 3

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