Numerical Analysis of the Take-Off Performance of a Seaplane in Calm Water

applied sciences

Article Numerical Analysis of the Take-Off Performance of a Seaplane in Calm Water

Yang Guo , Dongli Ma, Muqing Yang * and Xing’an Liu

School of Aeronautical Science and Engineering, Beihang University, Beijing 100191, China; [email protected] (Y.G.); [email protected] (D.M.); [email protected] (X.L.) * Correspondence: [email protected]

Featured Application: This paper provided a numerical approach to predict the resistance and attitude of a seaplane taking off in calm water. The investigations on take-off performance of the seaplane would provide some guidance in the integrated design of aerodynamics and hydrodynamics of the seaplanes.

Abstract: Nowadays, with the escalating tensions in maritime dispute and the development of marine economy, there has been renewed interest in seaplanes for their special capacity of taking off and landing on water. Prediction of take-off performance, involving aerodynamic analysis and hydrodynamic analysis, is a main challenge in seaplane design, while the prediction methods have been little improved since the 1960s. This paper aims to investigate the attitude and resistance characteristics of a seaplane at different speeds during the take-off by numerically modeling the air-water ﬂow ﬁeld using RANS equations with VOF method. The trim and heave motion of seaplane in response to aerodynamic forces, hydrodynamic forces, hydrostatic forces, and propeller thrust was

realized by solving rigid body dynamics equations and adopting dynamic overset mesh technique. The variations in heave, trim angle, and resistance characteristics during the takeoff were investigated,

Citation: Guo, Y.; Ma, D.; Yang, M.; and their inherent relationships with the aerodynamic, hydrodynamic, and hydrostatic performance Liu, X. Numerical Analysis of the were revealed. Particular investigation on the hydrodynamic resistance indicates that the stagnation Take-Off Performance of a Seaplane in line located at the convex bow would contribute a considerable increase of pressure resistance at the Calm Water. Appl. Sci. 2021, 11, 6442. first hump, and the trim angel of a seaplane should be operated in an optimum trim range, typical https://doi.org/10.3390/app11146442 between 4–6 deg, to minimize the hydrodynamic resistance at the second hump. Additionally, the dynamic motion convergence study proves that the utilization of damping terms was an effective Academic Editor: Emmanuel way to accelerate the convergence of the dynamic motion ending with a quasi-static state. P. Bénard Keywords: CFD; seaplane; hydrodynamics; aerodynamics; resistance; overset mesh; VOF Received: 18 June 2021 Accepted: 9 July 2021 Published: 13 July 2021 1. Introduction Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in Seaplanes are powered fixed-wing aircraft capable of taking off and landing on water. published maps and institutional affil- They once played an important role in the early stage of modern aviation. But the rapid iations. development of land-based aircraft and the growing investments in airports after WWII drastically reduced the demand for seaplanes, which resulted in the stagnation of seaplane technique for nearly half a century [1]. Recently, the development of the marine economy and the escalating tensions in maritime disputes has drawn people’s attention back to water- based aircrafts for their wide applications in military and civil fields, such as maritime Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. reconnaissance, anti-submarine, logistics service, sea rescue, etc. [2]. This article is an open access article The design of a seaplane is an art of tradeoff between aerodynamics and hydrodynam- distributed under the terms and ics with the ultimate goal to achieve the desired hydrodynamic performance with the least conditions of the Creative Commons loss in aerodynamic efficiency. The main challenge faced by designers for evaluating the Attribution (CC BY) license (https:// take-off performance is the prediction on the resistance characteristics, since during take-off creativecommons.org/licenses/by/ the seaplane is free to trim and heave in subject to the forces of hydrodynamics, hydrostat- 4.0/). ics, aerodynamics, and propulsion system. Evaluation of hydrodynamic performance is

Appl. Sci. 2021, 11, 6442. https://doi.org/10.3390/app11146442 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 6442 2 of 19

the most challenging job because the seaplane disturbs the water surface into complicated wave flow that is difficult to predict. It is known that besides the fuselage geometry, the hydrodynamic resistance mainly depends on speed, water load, and trim angle which is defined as the angle between the keel of forebody and the water level. The water load and trim angle would be influenced by the aerodynamic lift and pitching moment respectively, which basically results in the coupling of hydrodynamic performance and aerodynamic performance that makes the prediction more complicated [3]. There are three available methods for investigating the resistance characteristics of seaplanes: experimental method, empirical method and numerical method. Towing tank tests are the common experimental implements to investigate the resistance characteristics of seaplanes in take-off. Since the towing tanks always have a limitation on the model scale and planing speed, a dilemma that researchers have to face in the scaling model tests is the inability to satisfy the similarity of theFroude number and Reynold number [4]. Moreover, the experimental method is too expensive and time-consuming for the preliminary design and optimization of a seaplane. Empirical methods are more efficient for the preliminary design. Several attempts [5,6] have been made to estimate the resistance characteristics of seaplanes by adding aerodynamic models into Savitsky equations. Whereas, since the Savitsky methods were devel- oped for the sample non-stepped planing hulls [7], the empirical methods lack accuracy for evaluating the hydrodynamics of unconventional hulls, especially for seaplane hulls characterized by a distinctive step at the bottom. The step introduces a sharp corner on the bottom which divides the fuselage into forebody and afterbody. The water flow is forced to separate at the step giving room to an air-filled cavity, which could reduce the wetted surface on the afterbody and prevent unfavorable suction that keeps the afterbody from leaving the water [8]. A recent study by Savitsky expanded the empirical methods to estimate the hydrodynamics of stepped hulls through predicting the wake profile of forebody [9]. But the application of the methods is still limited to the range of parameters which are not applicable for seaplanes. Nowadays, numerical methods, especially Computational Fluid Dynamics (CFD) simulations, have become fundamental support for the analysis of aerodynamics and hydrodynamics. Reynolds-averaged Navier-Stokes (RANS) equations with turbulence models have been widely applied for the simulations of turbulence flow. Besides, the Volume of Fluid (VOF) method, which was proposed by Hirt and Nichols [10] in 1981, has been broadly used in solving the free surface problem. Recently, there have been some CFD studies focused on the hydrodynamic performance of seaplane hulls. Xiao et al. [11] investigated the hydrodynamic performance of a single seaplane hull numerically and experimentally. They adopted the model with fixed trim and heave to examine the effects of velocity, trim, and draft on hydrodynamic performance. Khoo and Koe [12] investigated the hydrodynamics of a Wing-In-Gound (WIG) vehicle with towing tank experiment and CFD simulation. The fuselage together with floats were modeled resulting in the average error in trim angle and water resistance in comparison to experimental data, 4.3% and 8.4% respectively. Zheng et al. [13] verified the feasibility of URANS equations with VOF method on hydrodynamic analysis of a seaplane hull. They reported that the average error in trim angle and water resistance in comparison to experimental data are 3.26% and 3.12% respectively. However, the numerical investigations on the take-off performance of seaplanes com- bining the analysis of aerodynamics and hydrodynamics have been rarely published. Qiu and Song [14] simulated the take-off process of a fixed-trim amphibious aircraft with an efficient decoupling method that calculated aerodynamic and hydrodynamic performance separately in CFX and FLUENT. Ma et al. [15] supplemented the decoupling method with an approximate equilibrium hypothesis which could efficiently estimate the acceleration in heave. It is evident that the decoupling method could save computing resources by simplifying the two-phase flow field around the seaplane, but a certain model error is expected owing to the interactions between the aerodynamics and hydrodynamics are ignored. Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 19

acceleration in heave. It is evident that the decoupling method could save computing re- Appl. Sci. 2021, 11, 6442 sources by simplifying the two-phase flow field around the seaplane, but a certain3 model of 19 error is expected owing to the interactions between the aerodynamics and hydrodynamics are ignored. Moreover, both studies adopted a fixed trim angle for the seaplane model, which is impractical for real operations. As the largest deviation in the resistance evalua- Moreover, both studies adopted a fixed trim angle for the seaplane model, which is imprac- tions are related to the errors in the evaluation of trim angle, the rigid body dynamics tical for real operations. As the largest deviation in the resistance evaluations are related to should be introduced to determine the equilibrium trim angle of the seaplane at different the errors in the evaluation of trim angle, the rigid body dynamics should be introduced speeds in response to aerodynamic forces, hydrodynamic forces, hydrostatic forces, and to determine the equilibrium trim angle of the seaplane at different speeds in response to propeller thrust. aerodynamic forces, hydrodynamic forces, hydrostatic forces, and propeller thrust. This paper aims to investigate the take-off performance of a seaplane in calm water This paper aims to investigate the take-off performance of a seaplane in calm water by numerically modeling the air-water two-phase flow around a seaplane that is free to by numerically modeling the air-water two-phase flow around a seaplane that is free to trimtrim and and heave heave at at different different speeds speeds ranging ranging from from standstill standstill to to leave leave the the water water based based on on commercialcommercial software software STAR-CCM+ STAR-CCM+ 12.02.12.02. The rema remainderinder of of this this paper paper isis organized organized as asfol- follows:lows: First, First, the the numerical numerical model model is is introduced, introduced, and and the the numerical numerical setups setups are are reported reported in indetail. detail. The The computational computational domain domain size, size, boundary boundary conditions, conditions, and and grid grid refinements refinements are aredescribed. described. Second, Second, the thenumerical numerical model model is va islidated validated with with benchmark benchmark data data of F6 of wing F6 wingbody body and andFridsma Fridsma hull hullrespectively. respectively. Then, Then, the theseaplane seaplane planing planing on speeds on speeds ranging ranging from fromthe standstill the standstill to the to takeoff the takeoff are simulated are simulated to investigate to investigate the variations the variations in heave, in heave, trim trimangle, angle,and resistance and resistance characteristics characteristics against against speed. speed.In particular, In particular, a detailed a detailedanalysis analysisof the hydro- of thedynamic hydrodynamic resistance resistance characteristics characteristics is presented. is presented. Additionally, Additionally, a convergence a convergence study is per- studyformed is performed to examine to the examine influence the influence of initial of attitude initial attitudeand damping and damping factors factorson the onconver- the convergencegence time timeof 2DOF of 2DOF motion. motion.

2.2. Seaplane Seaplane Model Model TheThe unmanned unmanned seaplane seaplane model model investigated investigated in in this this study study is is shown shown in in Figure Figure1. The1. The seaplaneseaplane is is a typicala typical flying flying boat boat configuration configuration with with high-wing high-wing and and V-tails. V-tails. The The fuselage fuselage is is similarsimilar to to a steppeda stepped hull hull where where the the bottom bottom of of the the fuselage fuselage is is divided divided into into forebody forebody and and afterbodyafterbody in in order order to to reduce reduce resistance resistance at at high high speeds. speeds. Motor-driven Motor-driven propeller propeller is is high high mountedmounted above above the the fuselage fuselage to to prevent prevent the the blades blades from from the the impact impact of of water water spray. spray. The The thrustthrust line line of of the the propeller propeller is parallel is parallel to the to the keel. keel. The The floats floats on wingtip on wingtip are used are used to keep to thekeep lateralthe lateral attitude attitude at low at planing low planing speed. Detailspeed. parameters Detail parameters of the seaplane of the modelseaplane are model given inare Tablegiven1. in Table 1.

FigureFigure 1. 1.The The seaplane seaplane proﬁle profile (longitudinal (longitudinal section section every every 0.03 0.03 m) m) and and body-plan body-plan (transversal (transversal section section every 0.2 m). every 0.2 m).

Appl. Sci. 2021, 11, 6442 4 of 19

Table 1. Details of the seaplane model.

Parameter Unit Value Wingspan m 3.6 Wing area m2 1.296 Length of fuselage m 2.9 Beam width m 0.26 Longitudinal position of step(measured from the nose) m 1.35 Step height m 0.02 Deadrise angle of forebody deg 20 Deadrise angle of afterbody deg 22 Afterbody keel angle deg 8 Weight kg 20 Pitching moment of inertia kg·m2 7.26 Longitudinal CG(measured from the nose) m 1.24 Vertical CG(measured from the forebody keel) m 0.28 Distance from thrust line to CG m 0.4 Thrust-weight ratio at rest 0.5

3. Numerical Model 3.1. Governing Equations The incompressible viscous two-phase ﬂow ﬁeld surrounding the seaplane was described using the Reynold-Averaged Navier-Stokes (RANS) equations with turbulence model. The Reynolds averaged forms of mass conservation equation and the momentum conservation equation can be written as follows:

∂ρ ∂u + i = 0 (1) ∂t ∂xi ! ! ∂ ∂ ∂p ∂ ∂ui ∂uj 0 0 (ρui) + ρuiuj = − + µ + − ρuiuj + gi (2) ∂t ∂xj ∂xj ∂xj ∂xj ∂xi

where, xi and xj are Cartesian coordinates, ui and uj represent the averaged components of velocity vector in Cartesian coordinates. p is the averaged pressure, ρ is the density, and gi the component of gravity acceleration along the coordinate axes of inertia. In 0 0 addition, µ is the effective viscosity coefﬁcient of dynamic viscosity, and ρuiuj represents the Reynolds stresses. The k-ω shear stress transport (SST) turbulence model was introduced to close the equations. Concretely, the transport equations for k and ω are written as:

∂ ∂ ∂ ∂k (ρk) + (ρkui) = Γk + Gk − Υk (3) ∂t ∂xi ∂xi ∂xi

∂ ∂ ∂ ∂ω (ρω) + (ρωui) = Γω + Gω − Υω (4) ∂t ∂xi ∂xi ∂xi

where Gk and Gω represent the turbulence kinetic energy generation items, and Υk and Υω represent the turbulence dissipation rate of k and ω, respectively. A segregated flow solver with Semi Implicit Method for Pressure Linked Equations (SIMPLE) was used to conjugate pressure field and velocity field, and an Algebraic Multi- Grid solver was applied to accelerate the convergence of the solution. The convection terms were discretized by 2nd order upwind schemes. Appl. Sci. 2021, 11, 6442 5 of 19

The VOF method was used to capture the water-air interface by introducing a volume Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 19 fraction α for each cell, defined as the ratio between the volume occupied by the phase and the volume of the control volume. When a cell is full of water, α = 1; while when the cell is full of air, α = 0; and when there is a fluid interface in the cell, 0 < α < 1. The transport equationthe cell is for fullα isof as air, follows: α = 0; and when there is a fluid interface in the cell, 0 < α < 1. The ∂α transport equation for α is as follows: + ∇ · (αu) = 0 (5) ∂α∂t +∇⋅()0αu = (5) In each control volume, the two∂t phases share the same velocity, pressure and tempera- ture. Hence,In each the control continuity volume, and the momentum two phases equations share the were same solved velocity, for a mixedpressure fluid and whose tem- densityperature.ρ and Hence, viscosity the continuiµ werety calculated and momentum by following equations equations: were solved for a mixed fluid whose density ρ and viscosity μ were calculated by following equations: ρ ρ=+−=αραρw + (α1 −ρ α)ρa (6) wa(1 ) (6) μ =+−αμ α μ µ = αµwaw(1+ (1 )− α )µa (7)(7) TheThe VOFVOF modelmodel isis very very sensitive sensitive to to the the discretization discretization of of this this convection convection term. term.Low Low orderorder schemes schemes often often lead lead to to smearing smearing of of the the interface interface due due to numerical to numerical diffusion, diffusion, while while the usethe ofuse high of high order order schemes schemes results results in numerical in nume oscillationsrical oscillations due to due their to unstable their unstable behavior. be- Inhavior. this paper, In this the paper, High-Resolution the High-Resolution Interface CapturingInterface Capturing (HRIC) scheme (HRIC) is usedscheme to discretizeis used to thediscretize convection the convection term of volume term of fraction volume function. fraction The function. HRIC The scheme HRIC is currentlyscheme is thecurrently most successfulthe most successful advection schemeadvection and scheme is extensively and is extensively used in many used CFD in codes.many CFD For improving codes. For theimproving stability the and stability robustness and of robustness the scheme, of thethe softwarescheme, the blends software the HRIC blends with the the HRIC upwind with differencingthe upwind (UD)differencing scheme depending(UD) scheme on depe the Courant-Friedrichs-Lewynding on the Courant-Friedrichs-Lewy (CFL) number. (CFL) And annumber. upper And limit an CFL upperU and limit a lower CFLU limit and CFLa lowerL are limit introduced CFLL are to introduced adjust the to blend. adjust For the simulationsblend. For simulations ending with ending a steady with state, a steady the value state, of CFLthe valueU and of CFL CFLL mustU and be CFL largerL must than be thelarger local than CFL the to local ensure CFL the to HRICensure scheme the HRIC is applied. scheme is In applied. this way, In a this large way, time a large step sizetime couldstep size be used could whilst be used a sharp whilst interface a sharp is interface retained. is retained.

3.2.3.2. RigidRigid BodyBody DynamicsDynamics

TheThe forcesforces andand momentsmoments actingacting onon aa seaplane seaplane inin take-off take-off are are shown shown in in Figure Figure2 .2.L La,a, RRaa,, MMaa areare respectivelyrespectively thethe aerodynamicaerodynamic lift,lift,aerodynamic aerodynamic resistance,resistance, andand aerodynamic aerodynamic pitchingpitching momentmoment acting on on CG; CG; △4, R, wR, wM, wM arew are the thewater water load, load, water water resistance, resistance, and pitch- and pitchinging moment moment acting acting on CG on produced CG produced by water, by water, respectively; respectively; W isW theis theweight weight of the of thesea- seaplane,plane, T isT theis the propeller propeller thrust, thrust, τ isτ isthe the trim trim angle, angle, d isd isthe the draft draft of of keel keel atat transom, transom, and and dT disT isthe the distance distance between between thrust thrust line line and and CG. CG.

FigureFigure 2. 2.The The forcesforces andand moments moments acting acting on on a a seaplane seaplane in in take-off. take-off.

AA globalglobal fixedfixed right-handright-hand Cartesian Cartesian coordinate coordinate system system (GS) (GS) was was used used to to describe describe the the flowflow field, field, defined defined with with the the XZ-plane XZ-plane coinciding coinciding with with the the symmetrysymmetry planeplane ofof thethe seaplane,seaplane, andand thethe XY-planeXY-plane paralleled paralleled to to the the calm calm water water surface. surface. TheThepositive positivex-axis x-axis is is consistent consistent withwith thethe velocityvelocity directiondirection andand positivepositive z-axisz-axis pointspoints inin thethe oppositeopposite directiondirection ofof gravity.gravity. Besides,Besides, a a body-fixed body-fixed right-hand right-hand Cartesian Cartesian coordinate coordinate system system (BS) (BS) locating locating its origin its origin at the at CGthe wasCG utilizedwas utilized to define to define the position the position and attitudeand attitude of seaplane. of seaplane. TheXZ-plane The XZ-plane coincides coin- withcides the with symmetry the symmetry plane of plane the seaplane. of the seaplane. The x-axis The is x-axis pointing is pointing forward inforward the longitudinal in the longitudinal axis of seaplane which is parallel to the keel and the positive y-axis points to the port side. The trim angle τ is defined as the angle between the x-axis of GS and the x-axis of BS. In order to determine the equilibrium trim angle and heave of the seaplane at different speeds, the Dynamic Fluid Body Interaction (DFBI) model was activated to simulate the 2 degree of freedom (2DOF) motion of the seaplane in response to the forces the fluid Appl. Sci. 2021, 11, 6442 6 of 19

axis of seaplane which is parallel to the keel and the positive y-axis points to the port side. Appl. Sci. 2021, 11, x FOR PEER REVIEWThe trim angle τ is defined as the angle between the x-axis of GS and the x-axis of BS.6 of 19 In order to determine the equilibrium trim angle and heave of the seaplane at different speeds, the Dynamic Fluid Body Interaction (DFBI) model was activated to simulate the 2 degree of freedom (2DOF) motion of the seaplane in response to the forces the fluid exerts andexerts the and external the external forces. Particularly,forces. Particularly, as we only as we concerned only concerned the forces the at forces final equilibriumat final equi- state,librium damping state, damping terms were terms introduced were introduced to reduce to thereduce convergence the convergence time. The time. governing The gov- equationserning equations for the 2DOFfor the motion 2DOF aremotion written are aswritten follows: as follows: 𝜕𝑤 𝑚∙∂w =𝐹 +𝐹 −𝜉 ∙𝑤 (8) m· 𝜕𝑡= F ,+ F ,− ξ ·w (8) ∂t f luid,z ∑ ext,z w

𝜕𝜔 ∂ωy 𝐼 ∙ =𝑀, +𝑀, −𝜉 ∙𝜔 (9) Iy· 𝜕𝑡 = M f luid,y + Mext,y − ξωy ·ωy (9) ∂t ∑ where 𝐹 and 𝑀 are the z component of resultant force and the resultant pitch- where Ff luid,,z and M f luid,,y are the z component of resultant force and the resultant pitching 𝐹 momenting moment acting acting on the on seaplane the seaplane calculated calculated by the two-phaseby the two-phase ﬂow solver. flowF solver.ext,z are external, are forcesexternal in z-axisforces includingin z-axis including gravity W gravityand the W zand component the z component of propeller of propeller thrust T zthrust. Mext T,yz. is𝑀 the, external is the external pitching pitching moment, moment, that is, thethat pitching is, the pitching moment moment caused bycaused propeller by propeller thrust t 𝜉 𝑤 𝜉 Mthrustt. ξw isM the. damping is the damping factor to thefactor vertical to the velocity verticalw velocityand ξωy is theand damping is the factor damping to the pitchingfactor to angular the pitching velocity angularωy. velocity 𝜔. InIn summary,summary, aa brief workflow workﬂow of of the the DFBI DFBI module module is isillustrated illustrated in inFigure Figure 3. 3The. The sea- seaplaneplane was was released released at atinitial initial trim trim angle angle and and heave, heave, then then was was free to trimtrim andand heave.heave. ForFor eacheach timetime step,step, thethe aerodynamicaerodynamic andand hydrodynamichydrodynamic forcesforces actingacting onon the the seaplane seaplane were were calculatedcalculated byby thethe URANS&VOFURANS&VOF solver.solver. TheThe propellerpropeller thrustthrust andand moment moment were were yielded yielded basedbased on on data data base base and and introduced introduced into into the the DFBI DFBI model model as as external external inputs. inputs. Then, Then, the the forces forces andand moments moments were were brought brought into into Equations Equation (8)(8) andand (9)(9) toto determinedetermine thethe verticalvertical accelerationacceleration andand pitchingpitching angularangular acceleration.acceleration. Hence,Hence, thethe variationsvariations inin heaveheave andand trimtrim angleangle werewere yieldedyielded and and the the overset overset grids grids were were relocated relocated for for the the next next time time step. step. TheseThese proceduresprocedures were were repeatedrepeated until until the the seaplane seaplane reached reached the the equilibrium equilibrium state. state.

Figure 3. Flow chart of the DFBI module. Figure 3. Flow chart of the DFBI module.

3.3. Computational Domain and Boundary Conditions Overset mesh, also known as “Chimera” or overlapping mesh, was applied to move the grids with the dynamic seaplane for its accuracy, efficiency, and high-adaptability to the wide variations of trim and heave [16]. As for only the longitudinal performance was Appl. Sci. 2021, 11, 6442 7 of 19

3.3. Computational Domain and Boundary Conditions Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 19 Overset mesh, also known as “Chimera” or overlapping mesh, was applied to move the grids with the dynamic seaplane for its accuracy, efﬁciency, and high-adaptability to the wide variations of trim and heave [16]. As for only the longitudinal performance was concerned,concerned, a a half half model model with with symmetry symmetry boundary boundary was was employed. employed. The The computational computational do- do- mainmain and and boundary boundary conditions conditions are are shown shown in Figurein Figure4 where 4 where the dimensionsthe dimensions are expressed are expressed inin terms terms of of the the length length of of fuselage, fuselage, L. L.

FigureFigure 4. 4.Computational Computational domain domain dimensions dimensions and and boundary boundary conditions. conditions.

TheThe computational computational domain domain combined combined a static a static background background region region and a movingand a moving overset over- region.set region. The backgroundThe background enclosed enclosed the entire the computational entire computational domain and domain the overset and the region overset moved with seaplane as a rigid body that ensured the grid quality near the seaplane. In region moved with seaplane as a rigid body that ensured the grid quality near the sea- the computational domain, the cells are grouped into active, inactive, or acceptor cells. At plane. In the computational domain, the cells are grouped into active, inactive, or acceptor the overset boundary, the acceptor cells are used to couple the solutions on the two regions, andcells. divide At the active overset and boundary, inactive cells the in acceptor the background cells are region. used Theto couple discretized the solutions governing on the equationstwo regions, are and solved divide within active active and cells inactive which cells update in the with background the moving region. overset The region. discretized A lineargoverning interpolation equations scheme are solved was usedwithin for active linking cells the which solution update between with the the background moving overset regionregion. and A linear the overset interpolation region for scheme its efficiency was used and for accuracy linking [17 the]. solution between the background region and the overset region for its efficiency and accuracy [17]. 3.4. Grid Generation and Refinement 3.4. GridThe gridGeneration design and and Refinement topology and appropriate refinements were found crucial in obtainingThe grid reasonable design results. and topology Trimmed and cell appropriate mesher was refinements applied to providewere found high-quality crucial in ob- hexahedraltaining reasonable cells in bothresults. the Trimmed regions due cell tomesher the hexahedral was applied cells to alignedprovide with high-quality the flow hex- directionahedral cells and in interface both the making regions less due numerical to the hexahedral diffusion. cells For aligned accurately with describing the flow direction the boundary layer flow, prism mesh with 10 layers and a stretching ratio of 1.2 was generated and interface making less numerical diffusion. For accurately describing the boundary near the wall. The first layer was set to keep wall y+ in the range from 30–100 and All layer flow, prism mesh with 10 layers and a stretching ratio of 1.2 was generated near the y+ Wall Treatment was utilized in all simulations. wall.Grid The refinements first layer werewas utilizedset to keep for obtainingwall y+ in an the accurate range description from 30–100 of the and flow All field, y+ Wall particularlyTreatment inwas the utilized place where in all large simulations. flow gradient and complex flow occurred. Anisotropic hexahedralGrid refinements cells were used were in gridutilized refinements for obtaining which an allow accurate specifying description the cell of size the in flow each field, coordinateparticularly direction, in the place which where could large achieve flow desired gradient accuracy and complex substantially flow reducingoccurred. the Aniso- numbertropic hexahedral of grids in comparisoncells were used with in the grid isometric refinements refinement which approach. allow specifying As demonstrated the cell size inin Figure each coordinate5, besides the direction, refinement which in thecould air flowachieve field, desired there wereaccuracy four substantially refinement zones reducing thatthe arenumber important of grids for thein comparison description of with water the flow: isometric refinement approach. As demon- •stratedThe in water Figure surface 5, besides zone. Athe sharp refinement free surface in the is theair preconditionflow field, there of accurate were four estimation refinement zonesof hydrodynamicthat are important performance. for the description The grids located of water at waterflow: level should be well-refined • The water surface zone. A sharp free surface is the precondition of accurate estimation of hydrodynamic performance. The grids located at water level should be well- refined to improve the resolution to capture the water surface. Note that a uniform cell height should be adopted at the free surface to avoid the smearing of the incoming water surface which is the mean source of numerical ventilation [18]. Appl. Sci. 2021, 11, 6442 8 of 19

Appl. Sci. 2021, 11, x FOR PEER REVIEW to improve the resolution to capture the water surface. Note that a uniform cell height8 of 19

should be adopted at the free surface to avoid the smearing of the incoming water surface which is the mean source of numerical ventilation [18]. •• TheThe surrounding surrounding fuselage fuselage zone. zone. In In order order to to capture capture the the complex complex flow flow near near the the hull, hull, especiallyespecially thethe sprayspray flowflow at chine and and the the separation separation flow flow at at step, step, the the grids grids that that en- encasedcased the the immersed immersed surfaces surfaces of of seaplane seaplane in in the the overset overset region region were were refined. refined. •• The Kelvin wave zone. For capturing the Kelvin wave pattern induced by the seaplane, The Kelvin wave zone. For capturing the◦ Kelvin wave pattern induced by the sea- aplane, triangular a triangular zone with zone a cuspwith anglea cusp of angle 39 was of 39° refined was refined downstream downstream of the seaplane.of the sea- Otherwise, low-resolution grids would smear the wave patterns, resulting in an plane. Otherwise, low-resolution grids would smear the wave patterns, resulting in underestimation of wave resistance. an underestimation of wave resistance. • The overlapping interface. A critical factor in successfully applying overset meshes is • The overlapping interface. A critical factor in successfully applying overset meshes to have sufficient cells with uniform size and topology across the overlapping zone is to have sufficient cells with uniform size and topology across the overlapping zone between background and overset regions. To improve the accuracy of interpolation, between background and overset regions. To improve the accuracy of interpolation, the overlapping interface in the background should be refined to match the size and the overlapping interface in the background should be refined to match the size and the distribution of the grid at the interface of overset region, and the overlapping zone the distribution of the grid at the interface of overset region, and the overlapping must contain at least 4–5 cell layers in both background and overset meshes. zone must contain at least 4–5 cell layers in both background and overset meshes.

FigureFigure 5. 5.Grids Grids around around the the seaplane. seaplane.

3.5.3.5. Time Time Step Step and and Iteration Iteration Number Number UnsteadyUnsteady RANS RANS equationsequations werewere solvedsolved using implicit and and iterative iterative solver solver in in order order to tofind ﬁnd the the field ﬁeld of of all all unknown unknown quantities quantities in ineach each time time step. step. The The transient transient terms terms were were dis- discretizedcretized using using 1st 1st order order temporal temporal scheme. scheme. Th Thee time time step step used used in inthe the simulations simulations was was de- decidedcided by by two two rules: rules: First, First, in in order order to to reduce reduce the interpolation errorerror atat thethe oversetoverset interface, interface, thethe time time step step should should be be limited limited to to ensure ensure the the overset overset mesh mesh moves moves less less than than half half the the smallest small- cellest sizecell relativesize relative to the to overlapping the overlapping zone within zone within a time step.a time Second, step. Second, the time the step time should step satisfyshould the satisfy following the following equation: equation:

∆tΔ𝑡= 0.01 = 0.01∼ ~ 0.0050.005L/⁄V 𝐿 𝑉 (10)(10)

SinceSince a a steady steady state state is is expected, expected, increasing increasing the the number number of of iterations iterations would would bring bring little little differencedifference in in final final results. results. Weighting Weighting computational computational stability stability and and efficiency, efficiency, five five iterations iterations werewere applied applied in in each each time time step. step. All All simulations simulations were were performed performed in in parallel parallel using using 24 24 Cores Cores ofof Intel Intel Xeon Xeon E5-2670 E5-2670 type type CPU CPU clocked clocked at at 2.3 2.3 GHz. GHz.

4. Validation Study 4.1. Validation with F6 Wing Body The validation study towards aerodynamic analysis was performed based on the benchmark data of F6 wing body offered by Second AIAA Drag Prediction Workshop [19]. One of our main concerns of the aerodynamic simulation is the applicability of trimmed Cartesian mesh in describing the complicated air flow surrounding the wing. A Appl. Sci. 2021, 11, 6442 9 of 19

4. Validation Study 4.1. Validation with F6 Wing Body The validation study towards aerodynamic analysis was performed based on the Appl. Sci. 2021, 11, x FOR PEER REVIEWbenchmark data of F6 wing body offered by Second AIAA Drag Prediction Workshop9 [of19 19]. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 19 One of our main concerns of the aerodynamic simulation is the applicability of trimmed Cartesian mesh in describing the complicated air flow surrounding the wing. A grid refinement study was conducted using 3 systematic refined grids as demonstrated in gridgrid refinementrefinement studystudy was was conducted conducted using using 3 systematic3 systematic refined refined grids grids as demonstrated as demonstrated in in6 Figure6. The simulations were operated under the condition of Ma = 0.75, Re = 3 × 10 6, FigureFigure 6. The simulations were were operated operated under under the the condition condition of ofMa Ma = 0.75, = 0.75, Re =Re 3 ×= 103 ×6, 10 , and lift coefficient of 0.5 to inspect the change in drag coefficient with the cells number. As andand liftlift coefficient ofof 0.50.5 to to inspect inspect the the change change in in drag drag coefficient coefficient with with the thecells cells number. number. shown in Figure7, an increase in cells number has the effect of lowering the predicted drag AsAs shownshown in Figure 7,7, an an increase increase in in cells cells number number has has the the effect effect of loweringof lowering the thepredicted predicted coefficient values. The deviation between the numerical results and experimental data dragdrag coefficientcoefficient values.values. TheThe deviation deviation betw betweeneen the the numerical numerical results results and and experimental experimental increased with increasing cell number, and the fine grid underpredicted the drag coefficient datadata increasedincreased withwith increasingincreasing cell cell number, number, and and the the fine fine grid grid underpredicted underpredicted the thedrag drag by 2.3%. Consequently, the numerical results of the trim grids were in accord with the ones coefficientcoefficient by 2.3%. Consequently,Consequently, the the numerica numerical resultsl results of ofthe the trim trim grids grids were were in accord in accord of multiblock structured grids and unstructured tetrahedral grids. withwith thethe ones of multiblockmultiblock structured structured grids grids and and unstructured unstructured tetrahedral tetrahedral grids. grids.

(a) Coarse (b) Medium (c) Fine (a) Coarse (b) Medium (c) Fine Figure 6. Systematic refined trimmed Cartesian grids of F6 wing body. FigureFigure 6.6. SystematicSystematic reﬁnedrefined trimmedtrimmed CartesianCartesian gridsgrids ofof F6F6 wingwing body.body.

Figure 7. Grid convergence result of drag coefficient. Figure 7. Grid convergence result of drag coefficient. FigureFurther, 7. Grid convergencethe medium resultgrid with of drag 4.28 coefficient. million cells was simulated at the angle of attack ranging from −5–2 deg, corresponding to lift coefficient ranging from −0.1–6. The polar Further, the medium grid with 4.28 million cells was simulated at the angle of attack curvesFurther, of numerical the medium result gridand experimental with 4.28 million data are cells compared was simulated in Figure at 8. the It anglecan be of seen attack rangingrangingthat the fromnumericalfrom −−5–2 result deg, correspondingarecorresponding in good agreemen toto liftlifttcoefficient withcoefficient the experimental rangingranging fromfrom data− − 0.1–6.with0.1–6. an TheThe aver- polarpolar curvescurvesage error ofof numericalnumericalin predicted resultresult drag and andcoefficient experimentalexperimental of 2.81%, datadata which areare comparedsatisfiescompared the in inaccuracy FigureFigure8 .requirement8. It It can can be be seen seen thatthatof aerodynamic thethe numericalnumerical analysis. result result are are in in good good agreement agreemen witht with the the experimental experimental data data with with an averagean aver- errorage error in predicted in predicted drag drag coefficient coefficient of 2.81%, of 2.81%, which which satisfies satisfies the the accuracy accuracy requirement requirement of aerodynamicof aerodynamic analysis. analysis.

Figure 8. Comparison of the polar curves between numerical result and experimental data.

Figure 8. Comparison of the polar curves between numerical result and experimental data. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 19

grid refinement study was conducted using 3 systematic refined grids as demonstrated in Figure 6. The simulations were operated under the condition of Ma = 0.75, Re = 3 × 106, and lift coefficient of 0.5 to inspect the change in drag coefficient with the cells number. As shown in Figure 7, an increase in cells number has the effect of lowering the predicted drag coefficient values. The deviation between the numerical results and experimental data increased with increasing cell number, and the fine grid underpredicted the drag coefficient by 2.3%. Consequently, the numerical results of the trim grids were in accord with the ones of multiblock structured grids and unstructured tetrahedral grids.

(a) Coarse (b) Medium (c) Fine Figure 6. Systematic refined trimmed Cartesian grids of F6 wing body.

Figure 7. Grid convergence result of drag coefficient.

Further, the medium grid with 4.28 million cells was simulated at the angle of attack ranging from −5–2 deg, corresponding to lift coefficient ranging from −0.1–6. The polar curves of numerical result and experimental data are compared in Figure 8. It can be seen Appl. Sci. 2021, 11, 6442 that the numerical result are in good agreement with the experimental data with an10 aver- of 19 age error in predicted drag coefficient of 2.81%, which satisfies the accuracy requirement of aerodynamic analysis.

Appl. Sci. 2021, 11, x FOR PEER REVIEWFigure 8. Comparison of the polar curves between numerical result and experimental data.10 of 19 Figure 8. Comparison of the polar curves between numerical result and experimental data. 4.2. Validation with Fridsam Hull 4.2. ValidationThe validation with Fridsam study Hull towards hydrodynamic analysis was carried out using the benchmarkThe validation experiment study of Fridsma.towards Thehydrodynamic Fridsma hull analysis of load was coefficient carried ofout 0.608, using deadrise the anglebenchmark of 20 deg, experiment and the of length Fridsma. beam The ratio Fridsma of 4 was hull modeled of load coefficient for hydrodynamic of 0.608, deadrise simulations. Theangle center of 20 ofdeg, gravity and the of thelength hull beam is located ratio of 60% 4 was of length modeled behind for hydrodynamic the bow in longitudinal simula- andtions. 29.4% The center of beam of gravity abovethe of the keel hull in vertical.is located The 60% detailed of length geometry behind the of bow the hullin longitu- is shown indinal [20]. and 29.4% of beam above the keel in vertical. The detailed geometry of the hull is shownThe in girds[20]. with 1.9 million cells were generated following the rules reported in SectionThe 3.4 girds. The with simulations 1.9 million were cells executedwere generated at five following Froude numberthe rules reported based on in beam section rang- ing3.4. fromThe simulations 0.59–1.78 to were compare executed the resistance,at five Froude trim, number and sinkage based on with beam experimental ranging from data, Savitsky0.59–1.78 method,to compare and the previous resistance, numerical trim, and works sinkage of Mousaviraadwith experimental [21] data, and O’SheaSavitsky [ 22]. Asmethod, shown and in Figureprevious9, thenumerical model works underestimated of Mousaviraad the trim [21] with and O’Shea an average [22]. errorAs shown of 6.5% inin comparisonFigure 9, the withmodel the underestimated experimental the data. trim But with at an high average Fr, the error numerical of 6.5% errorin compari- in trim is significantlyson with the experimental smaller than data. the other But at numerical high Fr, the works. numerical Besides, error wein trim accurately is significantly predicted thesmaller maximum than the trim other at Frnumerical= 1.19, works. which Besi wasdes, not we achieved accurately by Savitskypredicted andthe maximum the other nu- mericaltrim at works.Fr = 1.19, The which calculated was not resultsachieved of byh/ SavitsB matchedky and the the experimental other numerical data works. well, The which wascalculated slightly results underestimated of ℎ/𝐵 matched at low the Fr experimental and over-predicted data well, at which high was Fr.For slightly resistance, under- we over-predictedestimated at low the Fr valueand over-predicted at all Froude at number high Fr. with For anresistance, average we error over-predicted of 8.2%. The the error increasedvalue at all with Froude speed number which with reaches an average the maximum error of 13.9%8.2%. The at a error Froude increased number with of 1.78. speed Frus- tratingly,which reaches the hump the maximum of resistance 13.9% at Froudeat a Froude number number of 0.89 of 1.78. shown Frustratingly, in the experimental the hump data wasof resistance not predicted at Froude by any number numerical of 0.89 methodshown in or the Savitsky experimental method. data In was sum, not the predicted numerical deviationsby any numerical of the dynamic method two-phaseor Savitsky flow method. solver In are sum, acceptable the numerical at the speedsdeviations ranging of the from thedynamic displacement two-phase to theflow planing solver are stage. acceptable With the at anisotropicthe speeds ranging refinements, from ourthe displace- model used lessment efforts to the to planing achieve stage. desired With accuracy, the anisotro particularlypic refinements, in trim predictionour model whichused less is criticalefforts for theto achieve performance desired of accuracy, a seaplane. particularly in trim prediction which is critical for the performance of a seaplane.

(a) Trim angle (b) Resistance (c) Heave Figure 9. Comparisons of resluts among numerical studies, experimental data and Savitsky method. Figure 9. Comparisons of resluts among numerical studies, experimental data and Savitsky method. 5. Results and Discussion The grid of 6.5 million cells was generated for the simulations of the seaplane after a grid sensitivity analysis. The seaplane planning at twelve speeds ranging from 0 m/s to 14 m/s were simulated to examine the variations of heave and trim angle and their inherent relationship with aerodynamic performance and hydrodynamic performance. In particular, a detailed analysis of the resistance characteristics is performed. Additionally, a convergence study was performed to examine the influence of initial attitude and damping factors on the convergence time of 2DOF motion.

5.1. Heave During the takeoff process, the seaplane is lifted by the aerodynamic lift acting on the wings and the buoyancy and hydrodynamic lift acting on the fuselage. The changes of buoyancy, aerodynamic lift, and hydrodynamic lift are shown in Figure 10. The heave of seaplane is essentially determined by the displacement volume of the fuselage that reflects the contribution of buoyancy. Appl. Sci. 2021, 11, 6442 11 of 19

5. Results and Discussion The grid of 6.5 million cells was generated for the simulations of the seaplane after a grid sensitivity analysis. The seaplane planning at twelve speeds ranging from 0 m/s to 14 m/s were simulated to examine the variations of heave and trim angle and their inherent relationship with aerodynamic performance and hydrodynamic performance. In particular, a detailed analysis of the resistance characteristics is performed. Additionally, a convergence study was performed to examine the inﬂuence of initial attitude and damping factors on the convergence time of 2DOF motion.

5.1. Heave During the takeoff process, the seaplane is lifted by the aerodynamic lift acting on the wings and the buoyancy and hydrodynamic lift acting on the fuselage. The changes of buoyancy, aerodynamic lift, and hydrodynamic lift are shown in Figure 10. The heave of Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 19 seaplane is essentially determined by the displacement volume of the fuselage that reﬂects the contribution of buoyancy.

FigureFigure 10.10. VariationsVariations ofof liftlift componentscomponents againstagainst speed.speed.

InIn general,general, asas thethe velocityvelocity increases,increases, thethe increasesincreases ofof hydrodynamichydrodynamic liftlift andand aerody-aerody- namicnamic liftlift wouldwould reducereduce thethe contributioncontribution ofof buoyancybuoyancy leadingleading toto anan increaseincrease inin heaveheave asas shownshown inin FigureFigure 1111.. While,While, itit cancan bebe seenseen thatthat thethe seaplaneseaplane experiencedexperienced aa slightslight sinkagesinkage soonsoon afterafter thethe seaplaneseaplane startedstarted moving.moving. AsAs indicatedindicated inin FigureFigure 1010,, atat lowlow speedspeed regimeregime thethe weightweight ofof thethe seaplaneseaplane waswas mainlymainly supportedsupported byby buoyancybuoyancy actingacting as a displacement hull. TheThe forwardforward motion motion of of the the seaplane seaplane caused caused a lowa low pressure pressure field field on theon bottomthe bottom surface, surface, creatingcreating a negative a negative hydrodynamic hydrodynamic lift that lift suckedthat sucked the fuselage the fuselage deeper deeper into the into water, the known water, asknown squat as effect. squat effect. Figure 12 displays the disturbed water surface and pressure distribution at the bottom at speeds of 2 m/s, 4 m/s, and 6 m/s. It can be observed that the water interacted the fuselage at the stagnation line resulting in a pressure jump at the line. And part of the water rose up and blended into a thin sheet water flowing forward and outboard along the bow surface which is the source of spray. Figure 13 demonstrates pressure distribution along the keel line changes with speed. The peak of pressure at the stagnation line increased with speed, while slight decreases of pressure were observed behind the stagnation line. For the afterbody, the water separated at step and re-attached to the afterbody causing a second stagnation line which is apparently weaker than the one of forebody. After the speed of 4 m/s, the hydrodynamic lift increased rapidly and significantly heaved the seaplane. At high speeds, the heave further increased due to the growing aerodynamic lift.

Figure 11. Heave curve against speed. Figure 12 displays the disturbed water surface and pressure distribution at the bottom at speeds of 2 m/s, 4 m/s, and 6 m/s. It can be observed that the water interacted the fuselage at the stagnation line resulting in a pressure jump at the line. And part of the water rose up and blended into a thin sheet water flowing forward and outboard along the bow surface which is the source of spray. Figure 13 demonstrates pressure distribution along the keel line changes with speed. The peak of pressure at the stagnation line increased with speed, while slight decreases of pressure were observed behind the stagnation line. For the afterbody, the water separated at step and re-attached to the afterbody causing a second stagnation line which is apparently weaker than the one of forebody. After the speed of 4 m/s, the hydrodynamic lift increased rapidly and significantly heaved the seaplane. At high speeds, the heave further increased due to the growing aerodynamic lift. Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 19

Figure 10. Variations of lift components against speed.

In general, as the velocity increases, the increases of hydrodynamic lift and aerodynamic lift would reduce the contribution of buoyancy leading to an increase in heave as shown in Figure 11. While, it can be seen that the seaplane experienced a slight sinkage soon after the seaplane started moving. As indicated in Figure 10, at low speed regime the weight of the seaplane was mainly supported by buoyancy acting as a displacement hull. The forward motion of the seaplane caused a low pressure field on the bottom surface, Appl. Sci. 2021, 11, 6442 12 of 19 creating a negative hydrodynamic lift that sucked the fuselage deeper into the water, known as squat effect.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 19

FigureFigure 11. 11.Heave Heave curve curve against against speed. speed. Figure 12 displays the disturbed water surface and pressure distribution at the bottom at speeds of 2 m/s, 4 m/s, and 6 m/s. It can be observed that the water interacted the fuselage at the stagnation line resulting in a pressure jump at the line. And part of the water rose up and blended into a thin sheet water flowing forward and outboard along the bow surface which is the source of spray. Figure 13 demonstrates pressure distribution along the keel line changes with speed. The peak of pressure at the stagnation line increased with speed, while slight decreases of pressure were observed behind the stagnation line. For the afterbody, the water separated at step and re-attached to the afterbody causing a second stagnation line which is apparently weaker than the one of forebody. After the speed of 4 m/s, the hydrodynamic lift increased rapidly and significantly heaved the seaplane. At high speeds, the heave further increased due to the growing aerodynamic lift.

FigureFigure 12. 12.Disturbed Disturbed water water surface surface and and pressure pressure distribution distribution at theat the bottom bottom at 2at m/s, 2 m/s, 4 m/s,4 m/s, and and 6 6m/s. m/s.

Figure 13. Pressure distribution along the keel at speeds ranging from 1 m/s to 6 m/s.

5.2. Trim Angle The trim angle plays an important role in both aerodynamic performance and hydrodynamic performance. Figure 14 shows the variation of pitching moments acting on CG produced by air, water and propeller thrust. And the trim curve against velocity is plotted in Figure 15. Since the propeller was high-mounted to avoid the water spray, it brought an unexpected bow-down pitching moment that lowered the trim angle. As illustrated in Figure 13, the increase of pressure at stagnation line moved the pressure center forward, creating a bow-up pitching moment that drove the trim angle to increase at low speed. Note that the maximum trim angle was limited by the afterbody, since the higher trim angle, the larger wetted surface area, and hence the stronger hydrodynamic bow-down Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 19

Appl. Sci. 2021, 11, 6442 13 of 19 Figure 12. Disturbed water surface and pressure distribution at the bottom at 2 m/s, 4 m/s, and 6 m/s.

Figure 13. Pressure distribution along the keel at speeds ranging from 1 m/s to 6 m/s. Figure 13. Pressure distribution along the keel at speeds ranging from 1 m/s to 6 m/s.

5.2.5.2. TrimTrim Angle Angle TheThe trim trim angleangle playsplays anan importantimportant rolerole inin bothboth aerodynamicaerodynamic performanceperformance andand hydro-hydro- dynamicdynamic performance.performance. FigureFigure 14 14 shows shows thethe variationvariation ofof pitching pitching moments moments acting acting on on CG CG producedproduced by by air, air, water water and and propeller propeller thrust. thrust. And And the the trim trim curve curve against against velocity velocity is is plotted plotted inin Figure Figure 15 15.. SinceSince thethe propellerpropeller waswas high-mountedhigh-mounted toto avoidavoid thethe waterwater spray,spray, it it brought brought anan unexpectedunexpected bow-down bow-down pitching pitching moment moment that that lowered lowered the the trim trim angle. angle. As As illustrated illustrated in in FigureFigure 13 13,, thethe increaseincrease ofof pressurepressure at at stagnation stagnation line line moved moved thethe pressure pressure center center forward,forward, Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 19 creatingcreating aa bow-upbow-up pitchingpitching momentmoment thatthat drovedrove thethe trimtrim angleangle toto increaseincrease atat lowlow speed.speed. NoteNote thatthat thethe maximummaximum trimtrim angleangle waswas limitedlimited byby thethe afterbody,afterbody, sincesince the the higher higher trim trim angle,angle, thethe largerlarger wettedwetted surface surface area, area, and and hence hence the the stronger stronger hydrodynamic hydrodynamic bow-down bow-down pitchingpitching momentmoment producedproduced on on thethe afterbodyafterbody prevented prevented further further increase increase in in trim trim angle. angle. TheThe trimtrim angleangle reachedreached thethe peak peak of of 7 7 deg deg at at a a velocity velocity of of 6 6 m/s m/s whenwhen thethe afterbodyafterbody waswas nearlynearly entirelyentirely unwetted.unwetted.

FigureFigure 14.14. VariationsVariations ofof pitchingpitching momentmoment componentscomponents againstagainst speed. speed. Then, the seaplane started to plane on the single forebody. As the increasing hydrodynamic lift signiﬁcantly reduced the displacement volume and wetted surface area of forebody, the pressure center moved backwards that decreased the trim angle. Moreover, as the water load reduced by the growing aerodynamic lift, it required a lower trim angle to create equal pitching moment by moving the pressure center forward. At high speeds, the aerodynamic pitching moment has been growing rapidly that it increased the trim angle preparing for a successful takeoff.

Figure 15. Trim curve against speed.

Then, the seaplane started to plane on the single forebody. As the increasing hydrodynamic lift significantly reduced the displacement volume and wetted surface area of forebody, the pressure center moved backwards that decreased the trim angle. Moreover, as the water load reduced by the growing aerodynamic lift, it required a lower trim angle to create equal pitching moment by moving the pressure center forward. At high speeds, the aerodynamic pitching moment has been growing rapidly that it increased the trim angle preparing for a successful takeoff.

5.3. Resistance Resistance characteristics, including water resistance and air resistance, are the main concern of a seaplane design. Especially, the magnitude and corresponding speed of the resistance hump are critical in determining take-off performance and the requirement for propulsion. It can be seen from Figure 16 that there are two humps of total resistance: the first hump occurred at the speed of 4 m/s when the seaplane was transiting from the displacement to the planing stage, and the second hump occurred at the speed of 14 m/s around the corner of the takeoff. At the first hump, the hydrodynamic resistance contributed 97% of the total resistance and the residual thrust was about 15%. For the second hull, the hydrodynamic resistance contributed 65% of the total resistance while the residual thrust was only 7%. Since the propeller thrust decreased with speed, it is surprising to Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 19

pitching moment produced on the afterbody prevented further increase in trim angle. The trim angle reached the peak of 7 deg at a velocity of 6 m/s when the afterbody was nearly entirely unwetted.

Appl. Sci. 2021, 11, 6442 14 of 19

Figure 14. Variations of pitching moment components against speed.

FigureFigure 15.15. TrimTrim curvecurve againstagainst speed.speed.

5.3. ResistanceThen, the seaplane started to plane on the single forebody. As the increasing hydro- dynamicResistance lift significantly characteristics, reduced including the displacement water resistance volume and and air resistance, wetted surface are the area main of concernforebody, of the a seaplane pressure design. center moved Especially, backward the magnitudes that decreased and corresponding the trim angle. speed Moreover, of the resistanceas the water hump load are reduced critical by in the determining growing aero take-offdynamic performance lift, it required and the a lower requirement trim angle for propulsion.to create equal It canpitching be seen moment from Figureby moving 16 that the therepressure are center two humps forward. of totalAt high resistance: speeds, thethe firstaerodynamic hump occurred pitching at moment the speed has of been 4 m/s gr whenowing therapidly seaplane that wasit increased transiting the from trim theangle displacement preparing for to a the successful planing takeoff. stage, and the second hump occurred at the speed of 14 m/s around the corner of the takeoff. At the first hump, the hydrodynamic resistance Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 19 contributed5.3. Resistance 97% of the total resistance and the residual thrust was about 15%. For the secondResistance hull, the characteristics, hydrodynamic including resistance water contributed resistance 65% and of air the resistance, total resistance are the while main theconcern residual of a thrustseaplane was design only. 7%. Especially, Since the the propeller magnitude thrust and decreasedcorresponding with speed speed, of it the is surprisingfindresistance that the hump to second find are that humpcritical the secondwas in determining a more hump critical was take-off a point more performance in critical a takeoff point comparedand in athe takeoff requirement with compared the first for withhump,propulsion. the which first It hump,was can notbe which seen concerned from was notFigure in concernedprevious 16 that studies inthere previous are [14,15]. two studies humps [14 of,15 total]. resistance: the first hump occurred at the speed of 4 m/s when the seaplane was transiting from the displacement to the planing stage, and the second hump occurred at the speed of 14 m/s around the corner of the takeoff. At the first hump, the hydrodynamic resistance contributed 97% of the total resistance and the residual thrust was about 15%. For the second hull, the hydrodynamic resistance contributed 65% of the total resistance while the residual thrust was only 7%. Since the propeller thrust decreased with speed, it is surprising to

Figure 16. Variations of resistance components against speed. Figure 16. Variations of resistance components against speed.

As shown in Figure 1616,, it is no doubt th thatat hydrodynamic resistance played the dominant role role in in total total resistance. resistance. Next, Next, the the hydrodynamic hydrodynamic resistance resistance characteristics characteristics will will be fur- be furtherther analyzed. analyzed. Based Based on the on planing the planing theory, theory, the hydrodynamic the hydrodynamic resistance resistance can be can divided be di- videdinto the into pressure the pressure component component and friction and friction component component that fundamentally that fundamentally satisfy satisfy the rela- the 2 2 relationshiptionship Rw R= w△=tan4τ tan+ 0.5τρ+v0.5SwρCvf. SThewC fpressure. The pressure resistance, resistance, including including the wave the wave drag drag and andviscous viscous drag drag due dueto the to theseparation separation of ofboundary boundary layer, layer, is is proportional proportional to to the the water load and trim angle. And the friction resistance could be evaluated with the dynamic pressure, wetted surface area, and skinskin frictionfriction coefﬁcient.coefficient. With CFD-post technique, the pressure component and friction component acting on forebody and afterbody could be calculated directly as shown in Figure 17. At low-speed regime, the main source of forebody pressure resistance was the generation of water waves at bow. Also, the water flow separating at step caused turbulence behind the step, which added to the pressure resistance. In theory, the pressure resistance is supposed to increase as the trim rises with speed. Surprisingly, we noticed that the pressure resistance started to decrease at a velocity of 5 m/s when the trim angle was still rising. This phenomenon is related to the convex surface of the fuselage. At low speeds, the water surface encountered with the convex bow making the local trim angle of the stagnation line higher than the trim angle measured from the keel, which produced a larger backward component of pressure force that increased the resistance. As the seaplane heaved, the stagnation line moved backward to the straight section, and the local trim angle has been reduced, resulting in the decrease of the pressure resistance. It can be seen from the Figure 18a that the wetted surface area of forebody gradually decreased that limited the increase of friction resistance. Additionally, we found the reversed flow in the spray area provided a forward shear force that could slightly decrease the overall friction resistance.

Appl. Sci. 2021, 11, 6442 15 of 19

With CFD-post technique, the pressure component and friction component acting on forebody and afterbody could be calculated directly as shown in Figure 17. At low-speed regime, the main source of forebody pressure resistance was the generation of water waves at bow. Also, the water ﬂow separating at step caused turbulence behind the step, which added to the pressure resistance. In theory, the pressure resistance is supposed to increase as the trim rises with speed. Surprisingly, we noticed that the pressure resistance started to decrease at a velocity of 5 m/s when the trim angle was still rising. This phenomenon is related to the convex surface of the fuselage. At low speeds, the water surface encountered with the convex bow making the local trim angle of the stagnation line higher than the trim angle measured from the keel, which produced a larger backward component of pressure force that increased the resistance. As the seaplane heaved, the stagnation line moved backward to the straight section, and the local trim angle has been reduced, resulting in the decrease of the pressure resistance. It can be seen from the Figure 18a that the wetted Appl. Sci. 2021, 11, xsurface FOR PEER area REVIEW of forebody gradually decreased that limited the increase of friction resistance. 15 of 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 19 Additionally, we found the reversed ﬂow in the spray area provided a forward shear force that could slightly decrease the overall friction resistance.

(a) Forebody (b) Afterbody (a) Forebody (b) Afterbody FigureFigure 17. 17. VariationsVariations of of pressure pressure resistance resistance and and friction resistance on (a) forebodyforebody andand ((bb)) afterbody.afterbody. Figure 17. Variations of pressure resistance and friction resistance on (a) forebody and (b) afterbody.

(a) (b) (a) (b) Figure 18. Variations of friction resistance and wetted surface area on (a) forebody and (b) afterbody. Figure 18. Variations of friction resistance and wetted surface area on (a) forebody and (b) afterbody. Figure 18. Variations of friction resistance and wetted surface area on (a) forebody and (b) afterbody. For the afterbody, due to the upward stern, the pressure acting on the afterbody al- For the afterbody,waysFor provided the due afterbody, to a the forward upward due componentto stern,the upward the that pressure stern, reduced the acting pressurethe ontotal the acting hydrodynamic afterbody on the afterbody resistance. al- always providedwaysSince a forward providedthe fuselage component a forwardis narrower thatcomponent toward reduced the that the stern, totalreduced the hydrodynamic afterbody the total sinked hydrodynamic resistance. farther into resistance. the water Since the fuselageSinceas the is the narrower trim fuselage increased, toward is narrower significantly the stern, toward the increasing afterbody the stern, the sinkedthe wetted afterbody farther surface sinked into thearea farther water and afterbodyinto the water fric- as the trim increased,astion the resistance. trim significantly increased, Taken increasing significantlytogether, the the wetted increasingincrease surface of thepressure area wetted and resistance surface afterbody area at frictionforebody and afterbody and friction fric- resistance. Takentionresistance resistance. together, at afterbody the Taken increase together, determined of pressure the increase that resistance th ofe totalpressure at hydrodynamic forebody resistance and at re frictionforebodysistance andreached friction the resistance at afterbodyresistancefirst hump determined at at afterbody the velocity that determined the of total4 m/s. hydrodynamic that the total resistance hydrodynamic reached re thesistance first reached the hump at the velocityfirst humpAt of rest, 4 at m/s. the the step velocity was entirelyof 4 m/s. submerged under the water surface. As shown in Figure At rest, the12, stepwithAt rest, wasthe thewater entirely step flowing was submerged entirely over the submerged under step, thethe un waterlow-pressureder the surface. water region Assurface. shown formed As inshown behind in the Figure step Figure 12, with12,essentially the with water the forced flowingwater theflowing over water the over surface step, the the step,to low-pressuresag the an low-pressured allowed region air regionto formed be sucked formed behind in. behind Air proceeded the step the step essentiallyessentiallydownward forced forced along the water thethe waterstep surface from surface to the sag tochine andsag anto allowed dthe allowed keel air until toair be tothe sucked be step sucked was in. Airin.completely Air proceeded venti- proceeded downwarddownwardlated atalong a velocity along the stepthe of 3 fromstep m/s. from the Then, chine the the chine to afterbody the to keel the until keelacted the until as step an thewas additional step completely was planingcompletely surface venti- in latedthe disturbed at a velocity wake of of3 m/s.the forebody.Then, the Theafterbody water actedseparating as an fromadditional the step planing and re-attached surface in theto thedisturbed afterbody wake created of the an forebody. air-filled The cavity water that separating reduced thefrom wetted the step surface and re-attached area of the toafterbody. the afterbody The air created cavity ankept air-filled extending cavity towa thatrds reducedthe stern theuntil wetted the afterbody surface entirelyarea of thegot afterbody.out of the waterThe air at cavity a velocity kept ofextending 6 m/s. towards the stern until the afterbody entirely got out ofSince the water then, atthe a seaplanevelocity ofhas 6 m/s.been planing with the single forebody. As shown in Fig- ure 18,Since the then, wetted the surface seaplane area has slightly been planing grew with with speed,the single while forebody. the friction As shown resistance in Fig- in- urecreased 18, the rapidly wetted and surface overtakook area slightly the pressure grew rewithsistance speed, to whilebecome the the friction major resistance source of hy-in- creaseddrodynamic rapidly resistance. and overtakook Moreover, the thepressure results re showedsistance tothat become the hydrodynamic the major source resistance of hy- drodynamiccharacteristics resistance. were sensitive Moreover, to the the change results of showed trim angle. that Thethe hydrodynamichydrodynamic resistanceresistance characteristicscharacteristics wereof fixed sensitive trim angles to the ranging change from of trim 3–6 angle.deg at Thea velocity hydrodynamic of 10 m/s areresistance plotted characteristicsin Figure 19. Since of fixed the trimwetted angles surface ranging area fromvaries 3–6 inversely deg at a with velocity an exponential of 10 m/s are power plotted of intrim Figure [9], the 19. frictionSince the resistance wetted surfaceincreased area rapi vadlyries as inversely the trim withangle an decreased. exponential Whereas, power the of trimpressure [9], the resistance friction resistanceincreased increasedlinearly with rapi dlythe astrim the angle.trim angle Hence, decreased. there is Whereas,an optimum the pressuretrim angle resistance that minimizes increased the sumlinearly of pressure with the and trim friction angle. components Hence, there for is each an optimumspeed. To trimavoid angle the substantialthat minimizes hydrodynamic the sum of pressureresistance and at the friction second components hump, the for trim each angle speed. should To avoidbe close the to substantial the optimum hydrodynamic trim angle. resistance at the second hump, the trim angle should be close to the optimum trim angle. Appl. Sci. 2021, 11, 6442 16 of 19

ventilated at a velocity of 3 m/s. Then, the afterbody acted as an additional planing surface in the disturbed wake of the forebody. The water separating from the step and re-attached to the afterbody created an air-ﬁlled cavity that reduced the wetted surface area of the afterbody. The air cavity kept extending towards the stern until the afterbody entirely got out of the water at a velocity of 6 m/s. Since then, the seaplane has been planing with the single forebody. As shown in Figure 18, the wetted surface area slightly grew with speed, while the friction resistance increased rapidly and overtakook the pressure resistance to become the major source of hydrodynamic resistance. Moreover, the results showed that the hydrodynamic resistance characteristics were sensitive to the change of trim angle. The hydrodynamic resistance characteristics of ﬁxed trim angles ranging from 3–6 deg at a velocity of 10 m/s are plotted in Figure 19. Since the wetted surface area varies inversely with an exponential power of trim [9], the friction resistance increased rapidly as the trim angle decreased. Whereas, the pressure resistance increased linearly with the trim angle. Hence, there is an optimum trim angle that minimizes the sum of pressure and friction components for each speed. To Appl. Sci. 2021, 11, x FOR PEER REVIEWavoid the substantial hydrodynamic resistance at the second hump, the trim angle should16 of 19

be close to the optimum trim angle.

FigureFigure 19.19. VariationsVariations ofof resistanceresistance characteristicscharacteristics against against trim trim angle angle at at a a velocity velocity of of 10 10 m/s. m/s.

However,However, itit isis difﬁcultdifficult toto operateoperate thethe seaplaneseaplaneto toplane plane withwiththe the optimumoptimum trimtrim angleangle atat eacheach moment moment during during the the takeoff takeoff to achieve to achieve the optimum the optimum takeoff takeoff performance. performance. As shown As inshown Figure in 19Figure, actually 19, actually the hydrodynamic the hydrodynamic resistance resistance increased increased slowly inslowly a range in a within range 2within deg around 2 deg thearound optimum the optimum trim angle. trim Hence, angle. it Hence, is more it practical is more topractical operate to the operate seaplane the planningseaplane atplanning trim angle at trim within angle the within optimum the range,optimum typically range, between typically 4–6 between deg, to 4–6 achieve deg, to a satisﬁedachieve a takeoff satisfied performance. takeoff performance.

5.4.5.4. ConvergenceConvergence StudyStudy AnotherAnother purposepurpose of of this this paper paper is tois analyzeto analyze the the factors factors that that determine determine the convergence the conver- timegence of time the 2DOFof the 2DOF motion. motion. It is time-consuming It is time-consuming to wait to forwait the for 2DOF the 2DOF motion motion to reach to reach the equilibriumthe equilibrium state, state, especially especially for for the the simulations simulations with with millions millions of of cells. cells. Setting Setting a aproper proper initialinitial attitudeattitude oror addingadding additionaladditional dampingdamping termsterms intointo thethe rigidrigid motionmotion equationsequations areare consideredconsidered asas anan efficientefficient wayway toto savesave convergenceconvergence timetime [[23],23], butbut havehave notnot beenbeen verifiedverified withwith aa systematicsystematic study.study. Therefore,Therefore, aa convergenceconvergence studystudy waswas performedperformed toto discussdiscuss thethe influenceinfluence of of initial initial attitude attitude and and damping damping terms terms on on the the convergence convergence time time through through a series a series of simulationsof simulations at aat velocity a velocity of 5of m/s. 5 m/s. First,First, differentdifferent damp damp factors factors were were introduced introduced into into the the 2DOF 2DOF equations, equations, and and the the histor- his- icaltorical curves curves of trimof trim and and heave heave are are compared compared in in Figure Figure 20 20.. It It can can be be seen seen that that the the damping damping forceforce andand moment moment observably observably weaken weaken the the excessive excessive oscillation oscillation and and reduce reduce the convergence the conver- time. But the stronger the damping factors, the longer the oscillatory period. We recom- gence time. But the stronger the damping factors, the longer the oscillatory period. We recommend that the damp factor should be decided to ensure that one oscillatory period could be observed in the historical curve.

(a) Historical curves of trim angle (b) Historical curves of heave Figure 20. Historical curves of (a) trim angle and (b) heave using different damping factors.

Then, the influence of the initial attitude was investigated by releasing the hull at the different trim angles of 2 deg, 4 deg, and 6 deg, respectively. As demonstrated in Figure Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 19

Figure 19. Variations of resistance characteristics against trim angle at a velocity of 10 m/s.

However, it is difficult to operate the seaplane to plane with the optimum trim angle at each moment during the takeoff to achieve the optimum takeoff performance. As shown in Figure 19, actually the hydrodynamic resistance increased slowly in a range within 2 deg around the optimum trim angle. Hence, it is more practical to operate the seaplane planning at trim angle within the optimum range, typically between 4–6 deg, to achieve a satisfied takeoff performance.

5.4. Convergence Study Another purpose of this paper is to analyze the factors that determine the convergence time of the 2DOF motion. It is time-consuming to wait for the 2DOF motion to reach the equilibrium state, especially for the simulations with millions of cells. Setting a proper initial attitude or adding additional damping terms into the rigid motion equations are considered as an efficient way to save convergence time [23], but have not been verified with a systematic study. Therefore, a convergence study was performed to discuss the influence of initial attitude and damping terms on the convergence time through a series of simulations at a velocity of 5 m/s. First, different damp factors were introduced into the 2DOF equations, and the his- Appl. Sci. 2021, 11, 6442 17 of 19 torical curves of trim and heave are compared in Figure 20. It can be seen that the damping force and moment observably weaken the excessive oscillation and reduce the convergence time. But the stronger the damping factors, the longer the oscillatory period. We mend that the damprecommend factor shouldthat the be damp decided factor to should ensure be that decided one oscillatory to ensure period that one could oscillatory period be observed incould the historical be observed curve. in the historical curve.

(a) Historical curves of trim angle (b) Historical curves of heave Figure 20. Historical curves of (a) trim angle and (b) heave using different damping factors. Appl. Sci. 2021, 11, Figurex FOR PEER 20. Historical REVIEW curves of (a) trim angle and (b) heave using different damping factors. 17 of 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 17 of 19 Then, the inﬂuenceThen, ofthe the influence initial attitude of the initial was attitude investigated was investigated by releasing by the releasing hull at the hull at the the different trimdifferent angles trim of 2angles deg, 4of deg, 2 deg, and 4 deg, 6 deg, and respectively. 6 deg, respectively. As demonstrated As demonstrated in in Figure Figure 21, the difference21, the difference between thebetween initial the trim initial angle trim and an thegle ﬁnal and equilibrium the final equilibrium trim angle trim angle de- decides the amplitude21,cides the thedifference of amplitude oscillation, between of but oscillation, the barely initial affects buttrim barely theangle convergence andaffects the thefinal time. convergence equilibrium But when time. trim angleBut when de- damping termscidesdamping were the added, amplitudeterms the were free of added, motionsoscillation, the were free but allmotions barely converged wereaffects within all the converged convergence one period within and time. one Butperiod when and little differencedampinglittle in convergence difference terms werein time convergence added, is observed the time free in Figure ismotions observed 22. were In in conclusion, Figureall converged 22. for In conclusion, a dynamicwithin one for period a dynamic and motion endinglittlemotion with difference a ending stable in state,with convergence addinga stable damping state,time is adding observed terms damping is in the Figure most terms 22. effective Inis conclusion,the waymost toeffective for a dynamic way to accelerate themotion convergenceaccelerate ending the regardless convergencewith a stable of the regardless state, initial adding attitude, of the damping initial which attitude, couldterms saveis which the more most could than effective save more way than to 50% physical time.accelerate50% physical the convergencetime. regardless of the initial attitude, which could save more than 50% physical time.

(a) Historical curves of trim angle (b) Historical curves of heave (a) Historical curves of trim angle (b) Historical curves of heave Figure 21. Historical curves of (a) trim angle and (b) heave released at different initial trim angel. Figure 21. Historical curves of (a) trim angle and (b) heave releasedreleased atat differentdifferent initialinitial trimtrim angel.angel.

(a) Historical curves of trim angle (b) Historical curves of heave (a) Historical curves of trim angle (b) Historical curves of heave Figure 22. Historical curves of (a) trim angle and (b) heave with medium damping factors released at different initial trim Figure 22. Historical curves of (a) trim angle and (b) heave with medium damping factors released at different initial trim angel. Figure 22. Historical curves of (a) trim angle and (b) heave with medium damping factors released at angel. different initial trim angel. 6. Conclusions 6. Conclusions This paper aims to investigate the take-off performance of a seaplane by numerically modelingThis paper the air-water aims to investigate two-phase the flow take-off around performance a seaplane of using a seaplane RANS by equations numerically with modelingVOF method. the air-water In order totwo-phase determine flow the aroundequilibrium a seaplane trim angle using and RANS heave equations of the seaplane with VOFat different method. speeds, In order the to free determine motion the in trimequili andbrium heave trim in angleresponse and toheave aerodynamic of the seaplane forces, athydrodynamic different speeds, forces, the hydrostaticfree motion forces,in trim and and propeller heave in thrustresponse was to realized aerodynamic by DFBI forces, mod- hydrodynamicule and overset forces, mesh hydrostatictechnique. Theforces, numeri and calpropeller model thrustwas validated was realized based by on DFBI the bench-mod- ulemark and model overset of meshthe F6 technique. wing body The and numeri the Fridsamcal model hull. was validated based on the benchmark Themodel seaplane of the F6planning wing body at speeds and the ranging Fridsam from hull. the standstill to the takeoff were simulated.The Basedseaplane on theplanning numerical at speeds results, ranging we in fromvestigated the standstill the variations to the takeoffof heave were and sim- trim ulated.angle andBased their on inherentthe numerical relationship results, withwe in thevestigated aerodynamic the variations performance of heave and andhydrody- trim anglenamic and performance. their inherent The relationshipresults prove with that thehydrodynamic aerodynamic performance performance plays and a hydrody- dominant namicrole in performance. governing the The take-off results performance, prove that hydrodynamic and the hydrodynamic performance performance plays a dominant was es- rolesentially in governing determined the take-offby the pressure performance, distributi andon the and hydrodynamic the location performanceof the stagnation was line.es- sentiallyBesides, determined the results turnedby the pressureout that heavedistributi couldon andbe treated the location as a main of the parameter stagnation that line. re- Besides,flects the the contribution results turned of buoyancy. out that heave could be treated as a main parameter that reflects theThe contribution resistance characteristics of buoyancy. are the main concern in this study. There are two humpsThe ofresistance total resistance: characteristics the first arehump the occurred main concern at the transitionin this study. from displacementThere are two to humpsplaning, of andtotal the resistance: second hump the first happened hump occurred at the high at thespeed transition planing from stage displacement near the takeoff. to planing,The results and showthe second the hydrodynamic hump happened resistance at the high contributed speed planing nearly stage all of near the resistancethe takeoff. at The results show the hydrodynamic resistance contributed nearly all of the resistance at Appl. Sci. 2021, 11, 6442 18 of 19

6. Conclusions This paper aims to investigate the take-off performance of a seaplane by numerically modeling the air-water two-phase flow around a seaplane using RANS equations with VOF method. In order to determine the equilibrium trim angle and heave of the seaplane at different speeds, the free motion in trim and heave in response to aerodynamic forces, hydrodynamic forces, hydrostatic forces, and propeller thrust was realized by DFBI module and overset mesh technique. The numerical model was validated based on the benchmark model of the F6 wing body and the Fridsam hull. The seaplane planning at speeds ranging from the standstill to the takeoff were simulated. Based on the numerical results, we investigated the variations of heave and trim angle and their inherent relationship with the aerodynamic performance and hydrodynamic performance. The results prove that hydrodynamic performance plays a dominant role in governing the take-off performance, and the hydrodynamic performance was essentially determined by the pressure distribution and the location of the stagnation line. Besides, the results turned out that heave could be treated as a main parameter that reflects the contribution of buoyancy. The resistance characteristics are the main concern in this study. There are two humps of total resistance: the first hump occurred at the transition from displacement to planing, and the second hump happened at the high speed planing stage near the takeoff. The results show the hydrodynamic resistance contributed nearly all of the resistance at the first hump. And the increase of pressure resistance at forebody and friction resistance at afterbody were the major causes of the first hump of hydrodynamic resistance. We considered that the stagnation line located at convex surface at bow created a backward pressure component that considerably increased the pressure resistance at forebody. And the increase of friction resistance at afterbody is related to the afterbody sinking farther into the water as the trim increased. Furthermore, we found that the hydrodynamic resistance at high speed was significantly affected by the trim angle. The trim angel of a seaplane should be operated in an optimum trim range, typical between 4–6 deg, to minimize the hydrodynamic resistance. Otherwise, the hydrodynamic resistance would significantly increase as the trim angle deviates from the optimum trim range, resulting in a substantial second hump of resistance. Note that as the propeller thrust decreased with speed, more attention should be paid to the second hump of which the residual thrust might be less than the one at the first hump. The convergence study concludes that the utilization of damping terms is the most effective way to accelerate the convergence of the 2DOF motion ending with a quasi-static state, while the offset between the initial trim angle and the final equilibrium trim angle only decides the amplitude of oscillation but barely affect the convergence time. This paper provides a numerical approach to predict the resistance and attitude of a seaplane taking off in calm water. The investigation on take-off performance gives a better understanding of the variation of aerodynamic performance, hydrodynamic performance, and hydrostatic performance during the takeoff, which would provide some guidance in the integrated design of aerodynamics and hydrodynamics of the seaplanes.

Author Contributions: Conceptualization, Y.G. and D.M.; methodology, Y.G. and D.M.; software, Y.G.; validation, Y.G. and X.L.; formal analysis, Y.G.; investigation, Y.G. and X.L.; resources, D.M.; data curation, Y.G.; writing—original draft preparation, Y.G.; writing—review and editing, M.Y. and X.L.; visualization, Y.G.; supervision, D.M. and M.Y.; project administration, D.M. and M.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Appl. Sci. 2021, 11, 6442 19 of 19

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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