Improving the Downwind Sail Design Process by Means of a Novel FSI Approach

Journal of Marine Science and Engineering

Article Improving the Downwind Sail Design Process by Means of a Novel FSI Approach

Antonino Cirello 1, Tommaso Ingrassia 1,*, Antonio Mancuso 1, Vincenzo Nigrelli 1 and Davide Tumino 2

1 Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, 90127 Palermo, Italy; [email protected] (A.C.); [email protected] (A.M.); [email protected] (V.N.) 2 Facoltà di Ingegneria e Architettura, Università degli Studi di Enna Kore, Cittadella Universitaria, 94100 Enna, Italy; [email protected] * Correspondence: [email protected]

Abstract: The process of designing a sail can be a challenging task because of the difficulties in predicting the real aerodynamic performance. This is especially true in the case of downwind sails, where the evaluation of the real shapes and aerodynamic forces can be very complex because of turbulent and detached flows and the high-deformable behavior of structures. Of course, numerical methods are very useful and reliable tools to investigate sail performances, and their use, also as a result of the exponential growth of computational resources at a very low cost, is spreading more and more, even in not highly competitive fields. This paper presents a new methodology to support sail designers in evaluating and optimizing downwind sail performance and manufacturing. A new weakly coupled fluid–structure interaction (FSI) procedure has been developed to study downwind

sails. The proposed method is parametric and automated and allows for investigating multiple kinds of sails under different sailing conditions. The study of a gennaker of a small sailing yacht is

Citation: Cirello, A.; Ingrassia, T.; presented as a case study. Based on the numerical results obtained, an analytical formulation for Mancuso, A.; Nigrelli, V.; Tumino, D. calculating the sail corner loads has been also proposed. The novel proposed methodology could Improving the Downwind Sail represent a promising approach to allow for the widespread and effective use of numerical methods Design Process by Means of a Novel in the design and manufacturing of yacht sails. FSI Approach. J. Mar. Sci. Eng. 2021, 9, 624. https://doi.org/10.3390/ Keywords: ﬁnite element method; computational ﬂuid dynamics; FSI; sail design; gennaker; sail loads jmse9060624

Academic Editor: Apostolos Papanikolaou 1. Introduction In recent years, the exponential growth of computational resources at a very low cost Received: 29 April 2021 has allowed for the widespread use of numerical methods in the nautical ﬁeld as well Accepted: 31 May 2021 Published: 4 June 2021 as at an industrial level, not only for scientiﬁc research or highly competitive purposes. Nevertheless, to date, advanced numerical methods are commonly used in the nautical

Publisher’s Note: MDPI stays neutral field only for competitive applications, but it is conceivable that in the future these methods with regard to jurisdictional claims in could also be used in other application fields. Recent racing yachts, as demonstrated by published maps and institutional affil- the last edition of the America’s Cup, have achieved very high performances thanks to the iations. massive improvements in yacht and sail design, materials, and fabrication. The design of the hulls, sails, and rigging of competitive racing yachts needs more and more detailed research and development that can be obtained by combining advanced computational resources and experimental studies. In particular, with regard to the design of sails, based on traditional and empirical manufacturing processes, nowadays, the best sail designers Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. aim to use more new research and development tools [1]. This article is an open access article To improve sail design and to better understand sail aerodynamics, numerical sim- distributed under the terms and ulations and experimental testing are the most used and reliable methods [2–5]. These conditions of the Creative Commons methods have led to a better comprehension of the complex phenomena and the physics Attribution (CC BY) license (https:// underlying sailing and, consequently, through them, a remarkable increase of the perfor- creativecommons.org/licenses/by/ mances of yacht sails has been obtained in recent years [6,7]. Experimental full-scale testing 4.0/). is the most accurate and reliable method, but, because of cost constraints, its use is only

J. Mar. Sci. Eng. 2021, 9, 624. https://doi.org/10.3390/jmse9060624 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, 624 2 of 19

recommended for a few exceptional cases [7,8] or to validate numerical methods [2,9,10]. Instead, numerical simulations are cheaper than experimental methods, but in order to obtain reliable and accurate results, they need long setup times by highly qualified and experienced users and they require high-performing computational resources. Among the numerical methods, computational fluid dynamics (CFD) simulations are the most widely used in the design process of high-performance sailing yachts [8]. CFD analyses can be effectively used to design high-performance hulls [11] and to simulate both upwind and downwind sail configurations [12–17]. Usually, the flow in upwind sailing is mostly attached and the turbulence effect is quite limited; the flow regime around a downwind sail can be highly turbulent [7] and, consequently, simulating real sailing conditions can be a very difficult task. For this reason, in the past, many studies have concentrated on upwind sails [6], and in recent years, many researchers have focused on simulating the complex downwind sailing conditions [6,18,19]. The first numerical studies on downwind sails frequently treated the topic in a simplified numerical way, considering only the flow effects around a rigid structure [17]. However, the physics of downwind sails is by far more complex than the upwind ones, not only because of the highly turbulent flow, but also because these kinds of sails have an inherent unsteadiness, even in quite stable conditions [7]. This happens mainly because downwind sails are made of very light and flexible cloth, and are attached to the yacht’s structure at three points, namely, the head, the clew, and the tack [20]. The shape of a downwind sail is formed by self-generated aerodynamic forces that are strongly affected by the sail shape itself [21]. Moreover, the shape can change remarkably when sailing, depending on the trim settings and wind conditions [7]. All of these aspects must be taken into account when simulating the real sailing conditions of these kinds of sails, and the sail shape must be accurately measured in the flying condition [10,21]. For this purpose, nowadays, several authors focus on the issue of predicting the flying shape a sail develops under the impact of flow forces. In this context, it is well known that the fluid–structure interaction (FSI) is a key feature to study these kinds of problems [11,16,22,23]. Today, sail designers use specific software to define the constructed (molded) sail shape based on their experience, but they do not have any certainty about the real flying shape [6]. The possibility of predicting the real flying shape of a downwind sail could allow designers to improve the sailing performances of yachts [5,7,24,25], for instance, by maximizing the driving forces. However, knowing the real flying shape could also allow for optimizing the mechanical characteristics or the manufacturing process of the sails, in order to improve their stability, to reduce their weight, and to optimize their mechanical behavior [18,22,26,27]. Many papers have focused on evaluating the real shape of sails only to predict sailing performances [20,22,25,28]. However, to the best of our knowledge, no relevant papers have tried to measure the flying shape and the loads of a sail, taking into account both the point of view of sailors, whose primary interest is the sail performance, and of the sailmakers, who are also responsible for suitably dimensioning and manufacturing the sail. In this work, a new approach based on the perspective of both sailors and sailmakers is proposed. In particular, the authors propose calculating the flying shape and the loads on downwind sails, not only to estimate propulsive forces [2,10], but also to help sailmakers in dimensioning and defining the best arrangement of the sail reinforcements. For this purpose, a weakly coupled FSI procedure [16] is used to find the real flying shape and to estimate the loads on the head, the clew, and the tack of the sail when the sailing conditions change. On the basis of the numerical results obtained, an analytical formulation for the calculation of the forces on the sail is also proposed.

2. Materials and Methods In this work, to determine the ﬂying shape and to evaluate the loads on downwind sails, a numerical FSI approach was used. Starting from a previous work of the authors [16], a new weakly coupled method was developed and tested. The new procedure is parametric and fully automated, and it only needs some input data to calculate the ﬂying shape and J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 3 of 19

2. Materials and Methods In this work, to determine the flying shape and to evaluate the loads on downwind J. Mar. Sci. Eng. 2021, 9, 624 sails, a numerical FSI approach was used. Starting from a previous work of the authors3 of 19 [16], a new weakly coupled method was developed and tested. The new procedure is parametric and fully automated, and it only needs some input data to calculate the flying shape and the loads on the sail. As it is known, to perform a numerical FSI analysis, two the loads on the sail. As it is known, to perform a numerical FSI analysis, two embedded embedded problems, aerodynamic and structural, need to be solved together [29]. In this problems, aerodynamic and structural, need to be solved together [29]. In this study, to study, to solve the aerodynamic and structural problems, the commercial CFD solver An- solve the aerodynamic and structural problems, the commercial CFD solver Ansys CFX sys CFX and the Ansys Static Structural solver were used, respectively. JavaScript and a and the Ansys Static Structural solver were used, respectively. JavaScript and a python- python-based programming language were used to develop the algorithm that managed based programming language were used to develop the algorithm that managed the whole the whole procedure, including the data exchange between the workbench and spread- procedure, including the data exchange between the workbench and spreadsheet, which sheet, which was developed ad hoc. was developed ad hoc.

2.1. FSI Analysis:Analysis: SetupSetup of CFD EnvironmentEnvironment The proposed procedureprocedure waswas developeddeveloped toto solvesolve thethe FSIFSI problemproblem ofof aa downwinddownwind sailsail but, with the the aim aim to to simulate simulate the the flow flow as asaccurately accurately as aspossible possible,, and and so a so mainsail a mainsail was wasalso alsocons consideredidered in the in CFD the CFDsimulations. simulations. As mentioned, As mentioned, the procedure the procedure is parametric is parametric,, in order in orderto study to studydifferent different sailing sailing configurations configurations in a very in a simple very simple and fast and way. fast way.The m Theain mainCFD CFDparameters parameters that must that mustbe first be set first to set run to the run FSI the analysis FSI analysis are reported are reported in Table in Table 1. 1.

TableTable 1.1. ComputationalComputational ﬂuidfluid dynamicsdynamics (CFD)(CFD) analysisanalysis parameters.parameters.

ParameterParameter Unit Unit Apparent wind speed (AWS) m/s Apparent wind speed (AWS) m/s Apparent windApparent angle (AWA) wind angle (AWA) deg deg Mainsail boomMainsail angle (α boomm) angle (αm) deg deg Downwind sail trim angle (∆αg) deg Downwind sail trim angle (Δαg) deg

The downwind sail sail trim trim angle, angle, Δα∆gα, gis, isa parameter a parameter that that has hasbeen been introduced introduced to sim- to simulateulate different different sail sail trim trims.s. In Inparticular, particular, the the sheet sheet angle angle of of the the downwind downwind sail, sail, that is thethe angle betweenbetween thethe tack–clewtack–clew lineline andand thethe centerlinecenterline ofof thethe boatboat ((ααgg inin FigureFigure1 1),), is is deﬁned defined α α α α as αg = αg_designg_design + Δα+ ∆g; wherg; wheree αg_designg_design is theis design the design sheet sheet angle angle of the of downwind the downwind sail. sail. In this In thisway, way, once once the specific the speciﬁc sail sailtype type and andits design its design sheet sheet angle angle have have been been chosen, chosen, different different sail α sailtrim trimss can canbe simulated be simulated by simply by simply changing changing the thevalue value of parameter of parameter Δαg∆. g.

FigureFigure 1.1. MainMain CFDCFD parametersparameters andand sailingsailing geometricgeometric variables.variables.

The domaindomain usedused for for the the CFD CFD simulation simulation was was made made parametric parametric by settingby setting the positionsthe posi- oftions the of main the main boundary boundary surfaces surfaces (inlet, (inlet, outlet, outlet, and top)and astop a) functionas a function of the of maximum the maximum sail chord,sail chord, Cm, C asm deﬁned, as defined in Table in Table2. 2.

Table 2. Distance of the boundary surfaces in function of Cm [m].

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Table 2. Distance of the boundary surfaces in function of Cm [m]. J. Mar. Sci. Eng. 2021, 9, 624 4 of 19 Inlet Outlet Top 5 × Cm 10 × Cm 1 × Cm

FollowingFollowing aa largely used used approach, approach, the the boundary boundary conditions conditions at the at theinlet inlet surfaces surfaces (Fig- (Figureure 2) were2) were set following set following the formulation the formulation proposed proposed by Richards by Richards and Hoxey and [30]. Hoxey Concern- [ 30]. Concerninging the other the boundary other boundary surfaces, surfaces, the outlet the was outlet set was as “opening set as “opening condition” condition” [16], the [16 bottom], the bottomsurface surface was set was as a setrough as a wall rough, and wall, the and top thesurface top surfacewas set wasas the set inlet. as the The inlet. flow The velocity ﬂow velocityat the top at thesurface top surface was calculated was calculated using usingthe same the sameformulation formulation of the of velocity the velocity profile proﬁle pro- proposedposed in in[30]. [30 The]. The k-ω k- SSTω SST formulation formulation [31] [31 was] was chosen chosen as asthe the turbul turbulenceence model. model.

FigureFigure 2. 2.CFD CFD domain: domain: boundary boundary surfaces surfaces (top (top is is fully fully transparent, transparent, bottom bottom is white),is white) and, and inner inner (grey) (grey) and outer regions. and outer regions.

RegardingRegarding the the shapes shapes of of the the sails, sails, the the geometry geometry of of the the mainsail mainsail was was fixed fixed while while the the downwinddownwind sail sail was was updated updated at eachat each simulation simulation according according to the to deformedthe deformed shape shape calculated calcu- usinglated theusing FEM the analysis. FEM analysis. To optimize To optimize the computational the computational costs, thecosts, domain the domain was divided was di- intovided an into inner an regioninner region (grey (grey in Figure in Figure2), meshed 2), meshed using using a mixed a mixed hexa-tetrahedral hexa-tetrahedral mesh mesh withwith prismatic prismatic layers layers on on the the surfaces surfaces of of the the sails, sails, and and an an outer outer region, region where, where hexahedral hexahedral elementselements were were used. used. Mesh Mesh refinements refinements were were applied applied to to the the cells cells adjacent adjacent and and close close to to the the sailsail surfaces. surfaces. The The height height of of the the first first prismatic prismatic element element attached attached to to the the surfaces surfaces of of the the sails sails waswas set set to to achieve achieve the the condition condition of of y y + +≈ ≈ 1.1. AA convergence convergence study study was was performed performed to to quantify quantify the the influence influence of of the the grid grid resolution resolution √3 onon the the CFD CFD results. results. The The ratio ratio between between the the distances distances of of the the grid grid nodes nodes is is equal equal to to 3 √2.2 Four. Four similarsimilar grids grids with with about about 1.5 1.5 M M (base), (base), 0.75 0.75 M, M, 0.38 0.38 M, M, and and 0.18 0.18 M M cells cells were were investigated. investigated. q 33 N푁base푏푎푠푒 TheThe relative relative step step ratio ratio of of the thei -thi-th grid grid is ish iℎ== √ N ,, with withN Nbeing being the the number number of of cells, cells, 푖 푁i resulting in h = 1.00 (base), 1.26, 1.59, and 2.00, respectively.푖 The convergence trend is resulting in ℎ = 1.00 (base), 1.26, 1.59, and 2.00, respectively. The convergence trend is approximated with the following equation [32]: approximated with the following equation [32]: S(h ) = S + C × h p 푝 (1) 푆i (ℎ푖) =0푆0 + 퐶 ×iℎ푖 (1)

TheThe values values of of coefﬁcients coefficientsS S0,0,C C,, and andp pare are computed computed with with the the least-squares least-squares method. method. FigureFigure3 3and and Table Table3 show3 show the the drive drive force force computed computed for for the the different different analyzed analyzed grids grids and and thethe results results of of the the uncertainty uncertainty estimation. estimation. AA monotonic monotonic convergence convergence can can be be observed observed in in FigureFigure3. 3 The. The value value of of the the estimated estimated uncertainty, uncertainty, U Ug,g for, forh h= = 1 1 (corresponding (corresponding to to 1.5 1.5 M M cells cells grid)grid) is is 2.1%; 2.1%; this this value value can can be be considered considered acceptable. acceptable.

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50.0 50.0 49.5 49.5 Fit Fit 49.0 49.0 DriveDrive Force Force 48.5 48.5 UGUG 48.0 48.0 47.5 47.5 47.0 47.0

Drive Force [N] Force Drive 46.5 Drive Force [N] Force Drive 46.5 46.0 46.0 45.5 45.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 h h

Figure 3. Drive force vs. relative grid size. Figure 3. Drive force vs.Figure relative 3. gridDrive size force. vs. relative grid size.

Table 3. Estimated uncertainty for different grids. Table 3. EstimatedTable 3. uncertaintyEstimated uncertaintyfor different for grids different. grids.

Number of Cells Drive Force [N] h UG [N] UG% NumberNumber of Cells of Cells Drive ForceForce [N [N]] h h UGU [N]G [ N]UUG% G% 1,500,000 47.05 1.00 0.99 2.1% 1,500,0001,500,000 47.0547.05 1.001.00 0.99 0.99 2.1% 2.1% 750,000 47.28 1.26 1.44 3.1% 750,000 750,000 47.2847.28 1.261.26 1.44 1.44 3.1% 3.1% 375,000 48.21 1.59 2.54 5.5% 375,000 375,000 48.2148.21 1.591.59 2.54 2.54 5.5% 5.5% 187,500 187,50048.87 48.87 2.00 2.00 3.25 3.257.0% 7.0% 187,500 48.87 2.00 3.25 7.0%

Basing on the grid verification results and the good performance hardware available, Basing on theBasing grid onverification the grid verification results and resultsthe good and performance the good performance hardware available hardware, available, we used the 1.5 M cells grid. As a result of the unavailability of experimental data, it is not we used thewe 1.5 used M cells the grid 1.5 M. As cells a result grid. of As the a unavailability result of the unavailability of experimental of experimentaldata, it is not data, it is possible, to date, to present a complete validation of the numerical model based on a com- possible, tonot date, possible, to present to date, a complete to present validation a complete of the validation numerical of model the numerical based on modela com- based on a parison of specific CFD results with corresponding experimental data. However, in Sec- parison of comparisonspecific CFD of results specific with CFD corresponding results with correspondingexperimental data. experimental However, data. in Sec- However, in tion 4.0, the results obtained with the proposed method are compared with others in the tion 4.0, theSection results4, obtained the results with obtained the proposed with the method proposed are method compared are comparedwith other withs in the others in the literature. A good level of agreement between the numerical and the experimental results literature. Aliterature. good level A goodof agreement level of agreementbetween the between numerical the numericaland the experimental and the experimental results results has been noted; this demonstrates that the proposed method is able to give sufficiently has been noted;has been this noted;demonstrates this demonstrates that the proposed that the method proposed is methodable to give is able sufficiently to give sufficiently reliable results. Of course, a complete validation will be carried out in future work. reliable results.reliable Of results.course, Ofa complete course, a validation complete validationwill be carried will out be carried in future out work. in future work. The meshed domain and the details of mesh around the sail are shown in Figure 4. The meshedThe domain meshed and domain the details and theof mesh details around of mesh the around sail are the shown sail arein Figure shown 4 in. Figure4.

Figure 4. CFD meshed domain and enlarged views of the mesh around the sail. FigureFigure 4. CFD 4. CFDmeshed meshed domain domain and enlarged and enlarged views views of the of mesh the mesh around around the sail the. sail. The CFD analyses have been set with these stop conditions: The CFD analysesThe CFD have analyses been haveset with been these set withstop theseconditions: stop conditions: • values of the residuals of the continuity and momentum equations lower than 1.0 × • values of the residuals of the continuity and momentum equations lower than 1.0 × −5 10−5; • values of the residuals of the continuity and momentum equations lower than 1.0 × 10 ; 10−5; • maximum• numbermaximum of iterations number equal of iterations to 200. equal to 200. • maximum number of iterations equal to 200. For all of the analyzedFor all of configurations the analyzed, it conﬁgurations, was observed itthat was the observed residuals thatrapidly the de- residuals rapidly For all of the analyzed configurations, it was observed−5 that the residuals rapidly decreased under decreasedthe threshold under value the threshold(1.0 × 10− value5), and (1.0 the× aerodynamic10 ), and the quantities aerodynamic (forces, quantities (forces, creased under the threshold value (1.0 × 10−5), and the aerodynamic quantities (forces,

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pressure,pressure, etc.)etc.) convergedconverged towardstowards stablestablevalues valuesafter after aa reduced reduced number number of of iterations. iterations. As As ◦ ◦ ananpressure, example,example, etc.) for converge AWA = 90°d 90 towards andand Δαg∆ αstable g= =0°, 0 aftervalue, after 45s after 45iterations iterations, a reduced, the thevalues number values of theof of iterations.driving the driving force As forceremainedan example, remained stable for stableAWA around around= 90° a value and a Δαg value of about = 0°, of aboutafter 47 N; 45 47moreover, iterations N; moreover,, the valuesstandard the standard of thedeviation driving deviation of force the ofvaluesremained the values of the stable ofdriving the around driving force a cvalue forcealculated calculatedof about in the 47 last in N; the 30 moreover, lastiterations 30 iterations thewas standard lower was than lower deviation 0.5 than%. Figures 0.5%.of the ◦ ◦ Figures5values–10 show 5of–10 the someshow driving results some force obtained results calculated obtained for AWAin the for =last AWA90° 30 and iterations = 90AWAand = was 115 AWA °.lower In = particular, 115 than. In0.5 particular,% Figures. Figures 5 Figuresand5–10 6 show show5 and some the6 show velocity results the velocityobtainedstreamlines streamlines for thatAWA develop = that90° and developfrom AWA the from inlet= 115 theto°. Inthe inlet particular, outflow to the outflowboundary Figures 5 ◦ boundarysurface.and 6 show For surface. theAWA velocity = For 90°, AWA streamlinesthe flow = 90 was, that the quite flowdevelop clean was aroundfrom quite the cleanthe inlet body aroundto theof the outflow the gennaker, body boundary of even the gennaker,ifsurface. some divergencesFor even AWA ifsome = 90°occur, divergencesthered flow near was the occurred quite head clean and near thearound theleech head the, indicating body and theof theflow leech, gennaker, separation indicating even in ◦ flowtheseif some separation areas. divergences For inAWA these occur = areas.115°,red nearthe For flow AWAthe headwas = 115 slightlyand, thethe less flowleech regular was, indicating slightly than AWAflow less regularseparation = 90°, and than ina ◦ AWAlargetheser = areas.recirculation 90 , and For aAWA larger region = recirculation 115°, could the be flow evidenced region was couldslightly at the be lessdownstream. evidenced regular at than the downstream.AWA = 90°, and a larger recirculation region could be evidenced at the downstream.

Figure 5. Velocity streamlines for AWA = 90°. Figure 5. Velocity streamlines for AWA = 90◦. Figure 5. Velocity streamlines for AWA = 90°.

Figure 6. Velocity streamlines for AWA = 115°. FigureFigure 6. 6.Velocity Velocity streamlines streamlines for for AWA AWA = = 115 115◦.°. This phenomenon is also supported by the horizontal velocity contour plots (Figures 7 andThis 8) at phenomenon the half downwind is also supportedsail height. by In the fact, horizontal for AWA velocity = 115° (Figure contour 8) plots, a larger (Figures area with7 and a 8) low at the flow half velocity downwind at the sail downstream height. In fact, was for observed AWA = compared115° (Figure with 8), aAWA larger = area 90° (Figurewith a low7). flow velocity at the downstream was observed compared with AWA = 90° (Figure 7).

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FigureFigure 7. 7.Velocity Velocity contour contour plots plots at at the the mid-height mid-height section section for for AWA AWA = = 90 90◦.°. Figure 7. Velocity contour plots at the mid-height section for AWA = 90°.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 8 of 19 Figure 8. Velocity contour plots at the mid-height section for AWA = 115° FigureFigure 8.8.Velocity Velocity contour contour plotsplots at at the the mid-height mid-height section section for for AWA AWA = = 115 115◦.° To better visualize the flow structures, the second invariant of the velocity gradient [4], Q,To was better used. visualize Figure sthe 9 and flow 10 structures, show the isothe- surfacessecond invariantof Q = 500 of s −the2. In velocity both cases gradient, it can −2 [4],a strong Q, wasly turbulentused. Figure fluids 9 at and the 10 downstream show the iso can-surfaces be observed of Q =; for500 AWA s . In = both 115° cases (Figure, it can10), aa stronglarge flowly turbulent vorticity fluid can atalso the be downstream seen at the uppercan be partobserved of the; forsail. AWA = 115° (Figure 10), a large flow vorticity can also be seen at the upper part of the sail.

FigureFigure 9. 9.Iso-surfaces Iso-surfaces of of Q-criterion Q-criterion for for AWA AWA = = 90 90°◦..

Figure 10. Iso-surfaces of Q-criterion for AWA = 115°.

2.2. FSI Analysis: Setup of FEM Environment As a result of the assumption of a rigid mainsail, FEM analysis was performed only on the downwind sail. The FEM model is parametric; the main parameters are the sail geometry, the Young’s modulus, the Poisson ratio, and the sail thickness. The geometry of the sail was automatically meshed using eight-node shell elements (Shell 281). This kind of shell element has six degrees of freedom at each node, and it is characterized by a mathematical formulation suitable for simulating thin structures (like sails). Suitable mesh refinements were applied on the head, the clew, and the tack. A convergence study was performed with five structural meshes, with the following average sizes for the shell elements: 200, 100, 5, 30, and 20 mm. The different analyzed meshes had about 1540, 5700, 9200, 12,870, and 18,574 elements, respectively. The result of the convergence study is shown in Figure 11, where the value of the maximum displacement is plotted as a function of the average element size.

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J. Mar. Sci. Eng. 2021, 9, 624 8 of 19 Figure 9. Iso-surfaces of Q-criterion for AWA = 90°.

Figure 10. Iso-surfaces of Q-criterion for AWA = 115°. Figure 10. Iso-surfaces of Q-criterion for AWA = 115◦.

2.2. FSIThis Analysis: phenomenon Setup of isFEM also Environment supported by the horizontal velocity contour plots (FiguresAs 7a andresult8) atof thethe halfassumption downwind of a sailrigid height. mainsail, In fact,FEM foranalysis AWA wa = 115s performed◦ (Figure only8) , aon larger the downwind area with a sail. low The flow FEM velocity model at theis parametric; downstream the was main observed parameters compared are the with sail ◦ AWAgeometry = 90, the(Figure Young’s7). modulus, the Poisson ratio, and the sail thickness. The geometry of theTo sail better was visualize automatically the flow meshed structures, using the eight second-node invariant shell elements of the velocity(Shell 281 gradient). This kind [4], − Q,of wasshell used. element Figures has six9 and degrees 10 show of freedom the iso-surfaces at each node of Q, =and 500 it sis characterized2. In both cases, by ita canmath- a ◦ stronglyematical turbulent formulation fluid suitable at the downstream for simulating can thin be observed;structures for (like AWA sails). = 115 Suitable(Figure mesh 10), re- a largefinements flow vorticitywere applied can also on the be seenhead, at the the clew upper, and part the of tack. the sail. A convergence study was performed with five structural meshes, with the following 2.2. FSI Analysis: Setup of FEM Environment average sizes for the shell elements: 200, 100, 5, 30, and 20 mm. The different analyzed meshesAs ahad result about of the 1540, assumption 5700, 9200, of 12,870 a rigid, and mainsail, 18,574 FEM elements, analysis respectively. was performed The result only onof the downwindconvergence sail. study The is FEM shown model in Figure is parametric; 11, where the the main value parameters of the maximum are the saildis- geometry,placement the is plotted Young’s as modulus, a function the of Poissonthe average ratio, element and the size. sail thickness. The geometry of the sail was automatically meshed using eight-node shell elements (Shell 281). This kind of shell element has six degrees of freedom at each node, and it is characterized by a mathematical formulation suitable for simulating thin structures (like sails). Suitable mesh refinements were applied on the head, the clew, and the tack. A convergence study was performed with five structural meshes, with the following average sizes for the shell elements: 200, 100, 5, 30, and 20 mm. The different analyzed meshes had about 1540, 5700, 9200, 12,870, and 18,574 elements, respectively. The result of J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW the convergence study is shown in Figure 11, where the value of the maximum9 of displacement19

is plotted as a function of the average element size.

195 190 185 180 175 170 165 160 155 150 220 200 180 160 140 120 100 80 60 40 20 [mm] displacement Max Average element size [mm]

Figure 11Figure. Plot of 11. thePlot maximum of the maximum displacement displacement vs. average vs. element average size element. size.

From Figure 11, it can be observed that convergence is obtained (relative differences between two consecutive meshes lower than 5%) for values of the element size lower than 100 mm. Given the result of this convergence study, the mesh with an average size of 50 mm was chosen. This represents a good compromise between the quality of the results and the computing time. The chosen mesh, composed of about 9200 elements, is shown in Figure 12.

Figure 12. FEM mesh of the sail.

To calculate the loads on the rig, the fluid-dynamic pressure distribution calculated at each step by the CFD module was applied over both sail surfaces. The following boundary conditions were applied on the head and the tack (corners “B” and “C”, respectively, in Figure 12): the displacements along the x, y, and z directions were fully constrained and only the rotations along the x, y, and z axes were allowed. The clew (corner “A” in Figure 12) was joined by a spherical joint to a link element, represented by a black dashed line in Figure 12, which simulates a rope. The end of the rope connected to the sail can move freely along the x, y, and z directions; the other end, instead, can only rotate, but no displacement is allowed. A multipoint constraint approach [16] was used to distribute the

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195 190 185 180 175 170 165 160 155 150 220 200 180 160 140 120 100 80 60 40 20 [mm] displacement Max Average element size [mm] J. Mar. Sci. Eng. 2021, 9, 624 9 of 19

Figure 11. Plot of the maximum displacement vs. average element size.

FromFrom FigureFigure 11 11,, itit cancan bebe observedobserved thatthat convergenceconvergence isis obtainedobtained (relative(relative differencesdifferences betweenbetween twotwo consecutiveconsecutive meshesmeshes lowerlower thanthan 5%)5%) forfor valuesvalues ofof thethe elementelement sizesize lowerlower thanthan 100100 mm.mm. GivenGiven thethe resultresult ofof thisthis convergenceconvergence study,study,the the meshmesh withwith anan averageaverage sizesize ofof 5050 mmmm waswas chosen. chosen. This This represents represents a gooda good compromise compromise between between the qualitythe quality of the of results the results and theand computing the computing time. time. TheThe chosenchosen mesh,mesh, composedcomposed ofof aboutabout 92009200 elements,elements,is isshown shownin in Figure Figure 12 12. .

FigureFigure 12.12. FEMFEM meshmesh ofof thethe sail.sail.

ToTo calculate calculate the the loads loads on on the the rig, rig the, the ﬂuid-dynamic fluid-dynamic pressure pressure distribution distribution calculated calculated at eachat each step step by by the the CFD CFD module module was was applied applied over over both both sail sail surfaces. surface Thes. T followinghe following boundary bound- conditionsary conditions were we appliedre applied on theon the head head and and the the tack tack (corners (corners “B” “B” and and “C”,“C”, respectively,respectively, inin FigureFigure 1212):): the the displace displacementsments along along the the x, x,y, y,and and z directions z directions we werere fully fully constrained constrained and andonly only the rotations the rotations along along the x, the y, and x, y, z and axes z we axesre wereallowed. allowed. The clew The (corner clew (corner “A” in “A” Figure in Figure12) was 12 joined) was joinedby a spherical by a spherical joint to joint a link to element a link element,, represented represented by a black by a dashed black dashed line in lineFigure in Figure12, which 12, whichsimulate simulatess a rope. a The rope. end The of end the ofrope the connected rope connected to the tosail the can sail move can movefreelyfreely alongalong the x,the y, and x, y, z and directions z directions;; the other the other end, end,instead, instead, can only can only rotate rotate,, but butno dis- no displacementplacement is isallowed allowed.. A Amultipoint multipoint constraint constraint approach approach [16] [16] was used toto distributedistribute thethe constraints over many nodes of the head, tack, and clew regions so to avoid peaks of stress due to the application of a load in a single node.

2.3. FSI Procedure The ﬂow chart of the developed procedure is shown in Figure 13. To launch the procedure, in addition to the CFD and FEM parameters, the following input parameters must be preliminarily deﬁned:

- the sail design shape, Sd; - the number of steps, Ns, that the FSI analysis is subdivided into; - the displacement threshold of the current deformed shape, Td. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 10 of 19

constraints over many nodes of the head, tack, and clew regions so to avoid peaks of stress due to the application of a load in a single node.

2.3. FSI Procedure The flow chart of the developed procedure is shown in Figure 13. To launch the procedure, in addition to the CFD and FEM parameters, the following input parameters must be preliminarily defined: J. Mar. Sci. Eng. 2021, 9, 624 - the sail design shape, Sd; 10 of 19 - the number of steps, Ns, that the FSI analysis is subdivided into; - the displacement threshold of the current deformed shape, Td.

FigureFigure 13.13. FlowFlow chartchart ofof thethe developeddeveloped ﬂuid–structurefluid–structure interactioninteraction (FSI)(FSI) procedure.procedure.

SSdd represents the shape of the sail asas defineddefined byby thethe designer.designer. TheThe geometrygeometry (shape)(shape) filefile cancan bebe loadedloaded inin differentdifferent filefile formatsformats (iges,(iges, sat,sat, dwg,dwg, etc.).etc.). AsAs “classic”“classic” weakweak coupledcoupled proceduresprocedures couldcould bebe veryvery unstableunstable becausebecause ofof thethe largelarge deformationsdeformations ofof thethe sailsail underunder thethe windwind load,load, in the new new proposed proposed procedure procedure,, the the wind wind velocity velocity was was gradually gradually increased increased in indifferent different subsequent subsequent steps steps,, until until the thedesign design value value was wasreached reached.. In this In way, this the way, sail the shape sail shapegradually gradually change changedd as the aswind the speed wind speedincreased increased and, consequently, and, consequently, no convergence no convergence prob- problemslems of the of structural the structural simulations simulations arose. arose. The parameter The parameter Ns is usedNs isto useddefine to the define prelimi- the preliminary number of steps and influences the wind velocity, V , imposed at each step. nary number of steps and influences the wind velocity, Vs, imposeds at each step. The value The value of the velocity at the i-th step, V , is calculated as follows: of the velocity at the i-th step, Vs, is calculateds as follows: 2 Vs = AWSV {1s −=0.5 AWS[(Ns {1−i)/(N− 0.5[(s−1)]N2s}.− i)/(Ns − 1)] }. (2)(2)

Moreover,Moreover, toto betterbetter simulatesimulate thethe progressiveprogressive deformationdeformation ofof thethe sail,sail, thusthus avoidingavoiding anyany abruptabrupt changechange inin geometrygeometry betweenbetween twotwo subsequentsubsequent steps,steps, aa checkcheck onon thethe maximummaximum valuevalue ofof thethe displacementdisplacement waswas alsoalso introducedintroduced atat thethe endend ofof eacheach step.step. ForFor thisthis purpose,purpose, aa threshold parameter, Td, has been deﬁned. Td represents the admissible maximum value of the displacement of the deformed shape measured at each step. A simple user interface developed in MS Excel (Figure 14) allows for setting the

numerical parameters and launching the FSI analysis. As soon as it is started, the implemented procedure works iteratively in the following way: the sail geometry, which during the ﬁrst iteration is the designed one (Sd), subjected to Vs < AWS, is analyzed through the CFD module. The calculated ﬂuid-dynamic pressure distribution is used as the boundary condition for the subsequent structural simulation performed by the FEM module. When the FEM simulation is completed, the maximum value of the total displacement of the sail, Dmax, is calculated and compared with the threshold value, Td. If Dmax is higher than Td, it means that the sail is deformed excessively, J. Mar. Sci. Eng. 2021, 9, 624 11 of 19

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so the current velocity is decreased and the procedure restarts from the CFD simulation. If Dmax is lower than Td, the deformed shape of the sail is extracted from the FEM solver and isthreshold used as theparameter, updated T inputd, has geometrybeen defined for. aT newd represents CFD analysis. the admissible Before starting maximum the value new CFDof the simulation, displaceme thent of current the deformed wind velocity, shape measured Vs, is increased at each by step. an amount that depends on theA value simple of userNs. Wheninterface Vs =developed AWS, the in procedure MS Excel stops (Figure and 14 the) allows ﬂying for shape setting ofthe the sail nu- ismerical obtained. parameters and launching the FSI analysis.

Figure 14. MS Excel user interface. Figure 14. MS Excel user interface.

2.4. CaseAs soon Study as it is started, the implemented procedure works iteratively in the following way:A the common sail geometry, all-purpose which gennaker during the typically first iteration used tois equipthe design smalled sizeone sailing(Sd), subject boats,ed liketo V dinghies,s < AWS, is was analyzed studied. through The sail the areaCFD wasmodule. 14.2 T mhe2, calculated the luff edge fluid length-dynamic was pressure 5.95 m, distribution is used as the boundary condition for the subsequent structural simulation and the maximum chord (Cm) length was 3.20 m. A nylon typically used for this kind of applications,performed by characterized the FEM module. by a Young When modulus the FEM of simulation 5500 MPa, is was completed, chosen for the this maximum sail. As mentionedvalue of the in totalSection displacement1, the aim of of this the paper sail, D ismax not, i onlys calculated to present and a newcompared FSI approach with the tothreshold measure value, the flying Td. If D shapemax is ofhigher a downwind than Td, it sail, means but that also the to sail test is the deformed proposed excessively, method toso evaluatethe current the velocity structural is decreased loads. In particular,and the procedure it was investigated restarts from how the theCFD loads simulation. on the clew,If Dmax the is head,lower andthan the Td, tackthe deform vary dependinged shape of of the different sail is sailingextracted conditions from the and,FEM on solver the basisand is of used the numerical as the updated results input obtained, geometry an attempt for a wasnew madeCFD analysis. to find a possibleBefore starting analytical the formulationnew CFD simulation, that would the allow current for estimatingwind velocity, loads Vs as, is the increased wind speed by an varied. amount Based that onde- thepends typical on the sailing value conditions of Ns. When of V theses = AWS, types the of procedure sails, we chosestops and to study the flying the gennaker shape of the at differentsail is obtained. values of AWA ranging between 90◦ and 130◦, with 5◦ increments. Moreover, for each analyzed AWA value, we decided to investigate different configurations of the ◦ ◦ ◦ gennaker2.4. Case S bytudy varying its trim angle (∆αg) between 0 and 20 (with increments of 5 ), in orderA to common find the bestall-purpose trim depending gennaker on typically the wind used angle. to Thisequip resulted small size in about sailing 60 boats, different like configurationsdinghies, was studied. to simulate. The sail For area this was reason, 14.2to m limit2, the theluff computationaledge length was time, 5.95 am reduced, and the windmaximum velocity chord was (C chosenm) length (AWS was =3. 320 m/s). m. A nylon This choice typically allowed used for this investigating kind of applica- all of thetions, sailing characterized configurations by a Young in a reduced modulus time of 5500 using MPa, a Dual was Intel chosen Xeon for E5-2670 this sail. processor As men- (12tioned cores) in workstationSection 1.0, the with aim 128 of GBthis RAM. paper Theis not mainsail only to boom present angle a new (αm FSI) was approach set to be to ◦ constantmeasure andthe equalflying to shape 35 . Theof a values downwind of all ofsail the, but parameters also to test are the presented proposed in Tablemethod4. to evaluate the structural loads. In particular, it was investigated how the loads on the clew, Tablethe head 4. Values, and ofthe the tack parameters vary depending used for the of case different study. sailing conditions and, on the basis of the numerical resultsParameter obtained, an attempt was made to find a possible Value analytical formulation that would allow for estimating loads as the wind speed varied. Based on the typical Displacement threshold value (T ) 0.035 m sailing conditions of these types of sails,d we chose to study the gennaker at different values Number of steps (Ns) 5 of AWA rangingSail between thickness 90° and 130°, with 5° increments. Moreover, 0.7 mm for each analyzed AWA value, weYoung’s decided modulus to investigate different configurations 5500 of the MPa gennaker by varying its trim angle (ΔαPoissong) between ratio 0° and 20° (with increments of 5°), 0.25in order to find the best trim dependingApparent on wind the speedwind(AWS) angle. This resulted in about 60 different 3 m/s configurations to Apparent wind angle (AWA) 90◦ ÷ 120◦ simulate. For this reason, to limit the computational time, a reduced◦ wind velocity was Mainsail boom angle (αm) 35 chosen (AWS = 3 m/s). This choice allowed for investigating all◦ of the◦ sailing configura- Downwind sail trim angle (∆αg) 0 ÷ 20 tions in a reduced time using a Dual Intel Xeon E5-2670 processor (12 cores) workstation with 128 GB RAM. The mainsail boom angle (αm) was set to be constant and equal to 35°. 3. Results The values of all of the parameters are presented in Table 4. 3.1. Flying Shapes TableFigures 4. Values 15 of and the 16parameters show the used flying for the shapes case study of the. gennaker obtained for two different sailing configurations. In particular, the presented results are the ones obtained for ◦ ◦ Parameter ◦ ◦ Value AWA = 90 (∆αg = 0 ) and AWA = 110 (∆αg = 15 ), respectively. The design shapes are Displacement threshold value (Td) 0.035 m Number of steps (Ns) 5 Sail thickness 0.7 mm

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 12 of 19 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 12 of 19

Young’s modulus 5500 MPa Young’s modulus 5500 MPa Poisson ratio 0.25 Poisson ratio 0.25 Apparent wind speed (AWS) 3 m/s Apparent wind speed (AWS) 3 m/s Apparent wind angle (AWA) 90° ÷ 120° Apparent wind angle (AWA) 90° ÷ 120° Mainsail boom angle (αm) 35° Mainsail boom angle (αm) 35° Downwind sail trim angle (Δαg) 0° ÷ 20° Downwind sail trim angle (Δαg) 0° ÷ 20° 3. Results 3. Results 3.1. Flying Shapes J. Mar. Sci. Eng. 2021, 9, 624 3.1. Flying Shapes 12 of 19 Figures 15 and 16 show the flying shapes of the gennaker obtained for two different Figures 15 and 16 show the flying shapes of the gennaker obtained for two different sailing configurations. In particular, the presented results are the ones obtained for AWA sailing configurations. In particular, the presented results are the ones obtained for AWA = 90° (Δαg = 0°) and AWA = 110° (Δαg = 15°), respectively. The design shapes are colored = 90° (Δαg = 0°) and AWA = 110° (Δαg = 15°), respectively. The design shapes are colored in greencolored and in the green flying and ones the flying are blue ones. The are maps blue. of The the maps 3D deviations of the 3D deviationsof the flying of shapes the flying in green and the flying ones are blue. The maps of the 3D deviations of the flying shapes fromshapes the design from theones design are also ones presented are also in presented Figures 15 in and Figures 16. Different 15 and point16. Different of views, point two of from the design ones are also presented in Figures 15 and 16. Different point of views, two fromviews, the leech two from and thethe leechluff sides andthe and luff one sides frontal, and oneare frontal,presented are in presented order to in better order detect to better from the leech and the luff sides and one frontal, are presented in order to better detect thedetect differences the differences between the between real and the the real flying and the shape flying. shape. the differences between the real and the flying shape.

FigureFigure 15. 15.DesignDesign shape shape (green) (green),, flying ﬂying shape shape (blue) (blue),, and and map map of the of the 3D 3D deviations deviations [m] [m] for for AWA AWA = 90° = 90 and◦ and Δα∆g α= 0°=. 0◦. Figure 15. Design shape (green), flying shape (blue), and map of the 3D deviations [m] for AWA = 90° and Δαg = g0°.

Figure 16. Design shape (green), flying shape (blue), and map of the 3D deviations [m] for AWA = 110° and◦ Δαg = 15°. ◦ FigureFigure 16. 16.DesignDesign shape shape (green) (green),, flying ﬂying shape shape (blue) (blue),, and and map map of the of the3D 3Ddeviations deviations [m] [m] for forAWA AWA = 110° = 110 andand Δα∆g =α g15°=.15 .

It can be noticed that the largest positive displacements occur for AWA = 110◦ and mainly occur along the leech edge. Moreover, it can be observed that for AWA = 90◦, the largest positive displacements mainly involve the upper part, while for AWA = 110◦, the

lower part of the gennaker is the most deformed.

3.2. Loads on the Corners of the Sail: An Analytical Formulation For all of the analyzed conﬁgurations, the driving force and the loads on the corners of the sail, which are the head, the tack, and the clew, were calculated during all of the steps of the FSI simulation. In this way, it was possible to know how these forces varied as the wind speed increased until the chosen value of AWS. To better investigate the structural loads on the head, the tack, and the clew of the sail, for each of the analyzed conﬁgurations, the polynomial trendlines of the corner loads and their analytical formulations were extracted J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 13 of 19

It can be noticed that the largest positive displacements occur for AWA = 110° and mainly occur along the leech edge. Moreover, it can be observed that for AWA = 90°, the largest positive displacements mainly involve the upper part, while for AWA = 110°, the lower part of the gennaker is the most deformed.

3.2. Loads on the Corners of the Sail: An Analytical Formulation For all of the analyzed configurations, the driving force and the loads on the corners of the sail, which are the head, the tack, and the clew, were calculated during all of the steps of the FSI simulation. In this way, it was possible to know how these forces varied J. Mar. Sci. Eng. 2021, 9, 624 as the wind speed increased until the chosen value of AWS. To better investigate the13 struc- of 19 tural loads on the head, the tack, and the clew of the sail, for each of the analyzed configurations, the polynomial trendlines of the corner loads and their analytical formulations wereby a speciﬁcextracted tool by of a MSspe Excel.cific tool For of example, MS Excel. Figure For example,17 shows Figure the plots 17 ofshows the corner the plots loads of ◦ ◦ theover corner the wind loads speed over for the AWA wind = 120speedand for∆ αAWAg = 20 = .120° The and polynomial Δαg = 20°. trendlines The polynomial and their trendlinesanalytical formulationsand their analytical are also formulations reported. are also reported.

◦ ◦ Figure 1 17.7. Plots of the corners loads vs. AWS for AWA == 120120° and ∆Δαg == 20° 20 .

As soon as the analytical formulations of the trendlines were calculated for all of the configurations,configurations, aa carefulcareful analysis analysis of of the the values values of of the the first first and and second second order order coefficients coefficients was wasperformed. perform Fromed. From this analysis,this analysis it emerged, it emerged that, that in all, in of all the of analyzedthe analyzed configurations, configurations, the thefirst first order order coefficient coefficient was was much much smaller smaller than than the secondthe second order order one. one. In particular, In particular, it was it wascalculated calculated that thethat first the orderfirst order coefficient coefficient was, onwas average,, on average, 3.7% of3.7% the of second the second order coeffi-order coefficient.cient. By taking By taking this into this accountinto account andin and order in order to find to afind simplified a simplified analytical analytical formulation formu- lationfor the for calculation the calculation of the loadsof the on loads the corners,on the corners, we neglected we neglect the firsted the order first coefficient. order coeffi- For cient.this reason, For this the reason, following the following equation hasequation been preliminarilyhas been preliminarily proposed proposed for the calculation for the cal- of culationthe load of F onthe the load corners: F on the corners: 2 F = CC·AWS (3) F = CC∙AWS2 (3) where CC [kg/m] is the load coefficient that has to be defined. where CC [kg/m] is the load coefficient that has to be defined. To determine the most suitable value of CC to make the Equation (3) usable in different sailingTo conditions, determine the secondmost suitable order coefficients value of CC were to make analyzed the Equation for all of (3 the) usable configurations. in differ- J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 14 of 19 entFigure sailing 18 shows conditions, the values the second of the second order coefficients order coefficients were analyzed calculated for for all the of clew, the configu- the tack, rations.and the Figure head in 18 all shows of the the analyzed values configurations.of the second order coefficients calculated for the clew, the tack, and the head in all of the analyzed configurations.

Figure 18. Plots of second order coefﬁcients vs. AWA at different values of ∆αg. Figure 18. Plots of second order coefficients vs. AWA at different values of Δαg. It canIt becan seen be seen that that for each for each corner corner (clew, (clew, tack, tack and, head),and head) the, coefﬁcient the coefficient values values change change α ◦ as AWAas AWA and ∆ andg vary. Δαg vary. For example, For exam it canple, beit can noted be that noted for that the clewfor the corner clew at corner AWA =at 100 AWA = 100° (red circle in Figure 18), the highest value of the coefficient was calculated when Δαg = 0°, while for the head corner at AWA = 120° (blue circle in Figure 18), the highest value was obtained when Δαg = 20°. To simplify the data analysis and to determine the most suitable values of CC to be used in Equation (2), we considered only the maximum values of the second order coefficients at different values of AWA. Figure 19 shows the plots of the maximum values of the second order coefficients for the clew, the head, and the tack at different values of AWA.

Figure 19. Plots of maximum value of second order coefficients vs. AWA.

From Figure 19, it can be observed that the trends of the maximum values of the second order coefficients are substantially constant up to the value of AWA≈110°; beyond this value, a constant decrease can be. Based on this data, adopting a conservative approach, it was decided to use the average value of the second order coefficients calculated between AWA = 90° and AWA = 110° as the load coefficient CC for Equation (3). Table 5 reports the values of the second order coefficients and their average values calculated for the clew, the head, and the tack.

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Figure 18. Plots of second order coefficients vs. AWA at different values of Δαg. J. Mar. Sci. Eng. 2021, 9, 624 14 of 19 It can be seen that for each corner (clew, tack, and head), the coefficient values change as AWA and Δαg vary. For example, it can be noted that for the clew corner at AWA = 100° (red circle in Figure 18), the highest value of the coefficient was calculated when Δαg (red circle in Figure 18), the highest value of the coefficient was calculated when ∆α = 0◦, = 0°, while for the head corner at AWA = 120° (blue circle in Figure 18), the highestg value while for the head corner at AWA = 120◦ (blue circle in Figure 18), the highest value was was obtained when Δα◦g = 20°. To simplify the data analysis and to determine the most obtained when ∆αg = 20 . To simplify the data analysis and to determine the most suitable suitable values of CC to be used in Equation (2), we considered only the maximum values values of C to be used in Equation (2), we considered only the maximum values of the of the secondC order coefficients at different values of AWA. Figure 19 shows the plots of second order coefficients at different values of AWA. Figure 19 shows the plots of the the maximum values of the second order coefficients for the clew, the head, and the tack maximum values of the second order coefficients for the clew, the head, and the tack at at different values of AWA. different values of AWA.

FigureFigure 19. 19.Plots Plots of of maximum maximum value value of of second second order order coefﬁcients coefficients vs. vs. AWA. AWA.

FromFrom FigureFigure 19 19,, it it can can be be observed observed that that the the trends trends of of the the maximum maximum values values of of the the secondsecond order order coefficients coefficients are are substantially substantially constant constant up up to to the the value value of of AWA AWA≈110°≈110◦;; beyond beyond thisthis value, value a, constanta constant decrease decrease can can be. Basedbe. Based on this on data,this data, adopting adopting a conservative a conservative approach, ap- itproach, was decided it was tode usecided the to average use the value average of thevalue second of the order second coefficients order coefficients calculated calculated between ◦ ◦ AWAbetween = 90 AWAand AWA= 90° =and 110 AWAas the = 110° load as coefficient the load CcoefficientC for Equation CC for (3). Equation Table5 reports (3). Table the 5 valuesreports of the the values second of order the second coefficients order and coefficients their average and their values average calculated values for calculated the clew, the for head,the clew, and the head tack., and the tack.

Table 5. Second order coefﬁcients: single and average values.

AWA [◦] Clew Tack Head

90 2.69 4.53 5.53 95 2.67 4.51 5.54 100 2.66 4.56 5.59 105 2.62 4.50 5.48 110 2.58 4.47 5.40 Average values (CC) 2.64 4.51 5.51

Following the proposed approach, the loads on the head, the tack, and the clew could be estimated using the following analytical formulations:

2 2 Fclew = CC_clew·AWS = 2.64·AWS (4)

2 2 Ftack = CC_tack·AWS = 4.51·AWS (5) 2 2 Fhead = CC_head·AWS = 5.51·AWS (6) The values of the loads evaluated through CFD simulations and by Equations (4)–(6) are plotted in Figure 20. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 15 of 19

Table 5. Second order coefficients: single and average values.

AWA [°] Clew Tack Head 90 2.69 4.53 5.53 95 2.67 4.51 5.54 100 2.66 4.56 5.59 105 2.62 4.50 5.48 110 2.58 4.47 5.40 Average values (CC) 2.64 4.51 5.51

Following the proposed approach, the loads on the head, the tack, and the clew could be estimated using the following analytical formulations:

Fclew = CC_clew∙AWS2 = 2.64∙AWS2 (4)

Ftack = CC_tack∙AWS2 = 4.51∙AWS2 (5)

Fhead = CC_head∙AWS2 = 5.51∙AWS2 (6) J. Mar. Sci. Eng. 2021, 9, 624 15 of 19 The values of the loads evaluated through CFD simulations and by Equations (4)–(6) are plotted in Figure 20.

Figure 20. PlotsPlots of of the the loads loads on on the the sai saill corners corners calculated calculated byby the the CFD CFD simulations simulations and and by E byqua- Equa- tions (4)–(6)(4)–(6) for AWS == 3 m/s and and different different values values of of AWA AWA..

◦ It can be seen that for AWAAWA << 110110°,, thethe valuesvalues ofof thethe loadsloads evaluatedevaluated throughthrough thethe proposed analytical formulations are quite similar to thethe correspondingcorresponding onesones calculatedcalculated ◦ using thethe CFDCFD simulations.simulations. For For values values of of AWA AWA >110 > 110°, as, expected,as expected, the the proposed proposed equations equa- returnedtions return slightlyed slightly overestimated overestimated values value for thes for loads, the but,load considerings, but, considering that a correct that a correct design approachdesign approach should takeshould into take account into theaccount highest the values highest of values the loads of the on aloads structure, on a thisstructure, could bethis acceptable. could be acceptable. 4. Discussion and Conclusions 4. Discussion and Conclusions The study of the ﬂuid–structure interaction is still one of the most interesting topics in The study of the fluid–structure interaction is still one of the most interesting topics order to predict the performances of sails. A sail, in fact, is usually made of thin plastic in order to predict the performances of sails. A sail, in fact, is usually made of thin plastic material or fabric-based composite, and it forms a three-dimensional camber shape with material or fabric-based composite, and it forms a three-dimensional camber shape with wind pressure. The shape of the sail camber can remarkably change as the wind conditions vary, and this could result in a varied lift and drag performance by the sail [26]. These phenomena are relevant above all in the case of downwind sails, which are only attached to the yacht’s structure at three points—the head, the clew, and the tack. In this work, the issue of the FSI problem of downwind sails has been addressed

by developing and testing a new weakly coupled procedure. Regarding the presented procedure, one of the main innovative features is related to the possibility of simulating the progressive deformation of the sail, through the gradual application of aerodynamic loads. This feature allows for reducing convergence problems and to avoid solution instabilities, in order to overcome the typical drawbacks of classic weak coupled procedures [19,33]. Another key feature of the new procedure is that it is parametric and fully automated. In fact, both CFD and FEM modules have been setup in such a way as to automatically update the models and the boundary conditions when simulation parameters change. Of course, the procedure is parametric within certain limits; if the type of sail to be analyzed changes remarkably, it is necessary to manually create the preliminary setup of the numerical models. Concerning the automation of the procedure, thanks to a speciﬁcally developed algorithm that fully manages the process and the data exchange between the CFD and the FEM modules, different sailing conditions can be simulated by simply introducing some parameters in a MS Excel user-interface. The new procedure has been used to study a gennaker of a small sailing boat to evaluate its ﬂying shape and to calculate the forces on the corners of the sail. The FSI study has been completed with no interruption or convergence problems within the considered J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 16 of 19

wind pressure. The shape of the sail camber can remarkably change as the wind conditions vary, and this could result in a varied lift and drag performance by the sail [26]. These phenomena are relevant above all in the case of downwind sails, which are only attached to the yacht’s structure at three points—the head, the clew, and the tack. In this work, the issue of the FSI problem of downwind sails has been addressed by developing and testing a new weakly coupled procedure. Regarding the presented procedure, one of the main innovative features is related to the possibility of simulating the progressive deformation of the sail, through the gradual application of aerodynamic loads. This feature allows for reducing convergence problems and to avoid solution instabilities, in order to overcome the typical drawbacks of classic weak coupled procedures [19,33]. Another key feature of the new procedure is that it is parametric and fully automated. In fact, both CFD and FEM modules have been setup in such a way as to automatically update the models and the boundary conditions when simulation parameters change. Of course, the procedure is parametric within certain limits; if the type of sail to be analyzed changes remarkably, it is necessary to manually create the preliminary setup of the numerical models. Concerning the automation of the procedure, thanks to a specif- ically developed algorithm that fully manages the process and the data exchange between the CFD and the FEM modules, different sailing conditions can be simulated by simply introducing some parameters in a MS Excel user-interface. J. Mar. Sci. Eng. 2021, 9, 624 16 of 19 The new procedure has been used to study a gennaker of a small sailing boat to evaluate its flying shape and to calculate the forces on the corners of the sail. The FSI study has been completed with no interruption or convergence problems within the considered sailingsailing conditions.conditions. This This demonstrat demonstrateses that, that, for for the the specific specific analyzed analyzed test test case, case, the thenew new pro- procedurecedure is effective is effective and and stable. stable. AnAn extensiveextensive literatureliterature searchsearch waswas carriedcarried out out to to find find experimental experimental data data to to validate validate thethe numerical numerical proposed proposed methodology. methodology. Unfortunately, Unfortunately, to to the the best best of of our our knowledge, knowledge, in in the the literature,literature, therethere areare no no experimental experimental studiesstudies onon sails sails identical identical or or very very similar similar to to the the one one studiedstudiedhere. here. However,However, somesome experimental experimental studiesstudies performedperformed onon downwind downwind sailssails show show thatthat thethe results results obtained obtained with with the the proposed proposed methodology methodology can can be be considered considered sufficiently sufficiently reliable.reliable. As evidence of of this, this, the the values values of of the the drive drive force force coefficient, coefficient, CFx, CFx, have have bee beenn cal- calculatedculated at atdifferent different AWA AWAss using using the the numerical numerical method proposed herehere (Figure(Figure 21 21)) and and comparedcompared with with the the values values measured measured experimentally experimentally by by Motta Motta et et al. al. [ 34[34].]. Some Some similarities similarities cancan be be evidenced. evidenced.

0.8

0.6

CFx 0.4

0.2

0 90 100 110 120 130 AWA [deg]

FigureFigure 21. 21.Drive Drive force force coefﬁcients coefficients calculated calculated with with the the numerical numerical proposed proposed method. method.

TheThe trendstrends of of the the coefficient coefficient CF CFXXas as thethe AWAAWA varies varies are are very very similar; similar; in in fact, fact, they they are are almostalmost constant constant with with a slighta slight increase increase around around 100 100°◦–110–110°.◦. Moreover, Moreover the, orderthe order of magnitude of magni- oftude the of coefficients the coefficients is comparable, is comparable, with 0.73with being0.73 being the average the average value value calculated calculated with with the numericalthe numerical methodology methodology proposed proposed here here and aboutand about 0.58 being0.58 being the average the average value value measured meas- experimentally.ured experimentally. Of course, Of course, as the as compared the compared sails are sails different, are different, it is not it possibleis not possible to have to identical or very similar values of CFX, but considering that the order of magnitude is identical, it could be deduced that the numerical results are sufficiently reliable. With reference to the loads calculated numerically with the proposed methodology, good levels of agreement were found with other case studies in the literature. In particular, it can be observed that the ratios of the loads on the sail corners and the trends of the forces, when AWA varies, are very similar to those found in other studies [2,20,34]. For example, the force ◦ ratios Fclew/Fhead and Ftack/Fhead calculated at AWA = 90 using the new procedure are about 48% and 81%, respectively; the same ratios have been experimentally measured by Deparday et al. [2] for a similar case study and are equal to about 51% and 85%, respectively. Regarding the trends of the loads, it can be observed that, similarly to what was found with the numerical procedure proposed here, as well as in other experimental studies on similar kinds of sails [2,34], a decrease in the forces was detected when AWA > 110◦. This good level of agreement between the numerical data presented and other experimental data, even if it cannot be considered a direct validation of the numerical results, demonstrates the proposed method could give enough reliable results and can be used as a useful tool for sail design. With regard to the analytical formulation for calculating the sail corner loads discussed in Section 3.2, the proposed Equation (3) of course cannot be used to calculate the corner loads for all types of sails. It applies only for the specific kind of gennaker; however, in our opinion, the proposed approach could be interesting because it could be effectively repeated to find other formulations for different sail types. Still, nowadays, in fact, most sailmakers design the reinforcements to apply on the head, the tack, and the clew, based on their experience. Analytical formulations, grouped by similar types of sails, could J. Mar. Sci. Eng. 2021, 9, 624 17 of 19

be an inexpensive and easy to use method to suitably dimension the reinforcements on the sail corners. Moreover, knowledge of the loads of the sail corners could allow for a better comprehension on wind/rig/sail interaction [2,10]. Such an approach could allow for remarkably improving the design and manufacturing process of sails, not just for competitive sailing yachts, but also for cruising ones. For future work, the authors believe that further developments could be addressed to the setup and validation of a database of analytical formulations in order to evaluate the loads on the sail corners and to the improvements of the numerical models. A database of equations for an initial evaluation of the sail loads could be developed by analyzing numerous sails different in shape and size; in this way, sailmakers could use a simple tool to design the most suitable reinforcements for the sails, based not only on their experience, but also on numerical data. CFD simulations could also be improved by investigating unsteady sailing conditions, while FEM models could be enhanced by introducing additional information on different types of sail materials regarding structure non linearities, and by evaluating the rig deformation. In this way, very accurate and complete information could be obtained and the sail performance could be optimized. In fact, as the developed procedure is based on a weak coupled FSI method, the FEM module could be effectively interfaced with an optimization module to ﬁnd the best structural parameters for the sails (e.g., optimal thickness of different regions of the sail and most suitable layout of composite materials). In conclusion, in our opinion, the novel proposed methodology is surely promising, and further enhancements could allow for getting more information on the dynamics of downwind sails and could lead to the widespread use of this procedure in the design and manufacturing of yacht sails.

Author Contributions: Conceptualization, A.C., T.I., A.M., V.N. and D.T.; data curation, A.C.; methodology, A.C., T.I., A.M., V.N. and D.T.; software, A.C. and T.I.; validation, A.C.; writing— original draft, A.C. and T.I.; writing—review and editing, A.C., T.I., A.M., V.N. and D.T. 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: Not applicable. Acknowledgments: The authors gratefully acknowledge ANSYS Inc. for the support and the issue of academic licenses. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations The following abbreviations are used in this manuscript: CFD Computational ﬂuid dynamics FSI Fluid–structure interaction AoA Angle of attack AWA Apparent wind angle AWS Apparent wind speed TWA True wind angle TWS True wind speed Cm Maximum sail chord αm Mainsail boom angle ∆αg Downwind sail trim angle αg_design Design sheet angle of the downwind sail CFX Drive force coefﬁcient J. Mar. Sci. Eng. 2021, 9, 624 18 of 19

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