Numerical study of aeroelasticity of sails Alessandro Leone, Luciano Teresi* SMFM@DiS, Università Roma Tre, Roma, Italy * Corresponding author: Dip. of Studies on Structures, via Corrado Segre 6, 00146 Roma, Italy, email teresi@uniroma3.it Abstract: Aim of our paper is to present a simple, yet reliable, method for parametric sail design under many different upwind sailing conditions. By using script language, we can generate a family of sail shapes, and then investigate both aerodynamics and structural issues. We implement our simulations using the Incompressible Navier-Stokes mode coupled with models from the Structural Mechanics Module of COMSOL Multiphysics. We start by assessing computational capabilities in evaluating 2D flows: we compare the results for a zero-thickness airfoil section with those obtained with a sequence of airfoils whose thickness decreases to zero. Then, we analyze the influence of sails shape on their performances using a parameterized geometry to generate sails having different curvatures, leading and trailing edges. In particular, we investigate the maximum lift force that a given sail is able to generate for a large range of the angle of attack; we also evaluate the pressure coefficient, the laminar and turbulence flow areas, and the stall conditions. Finally, we propose a parametric generation for 2D sails, and study a 3D airflow around it, presenting some of the characteristic results that our model can provide. Keywords: Parametric sail design, upwind sailing, aeroelasticity of sails. 1. Introduction Sail design is a classical problem of solidfluid interaction; actually, sail thickness is negligible with respect to the overall dimensions, and we are prompted to use the so-called structural theories to model a sail. Thus, we consider a 1-dimensional sail, modeled as a flexible beam, immersed in a 2- dimensional fluid flow, or, a 2-dimensional sail, modeled as an elastic shell, surrounded by a 3- dimensional fluid flow. In both cases, we consider a steady flow of an incompressible, linearly viscous fluid, that we model using the Incompressible Navier-Stokes application mode. For the sail, we use the Inplane Euler Beam or the Shell application mode of the Structural Mechanics Module, for the 1-D or the 2-D sail, respectively. Unknowns of our problem are the velocity and the pressure of the fluid, and the displacement of the sail. We do not consider this as the suitable place to present and discuss the aforementioned models, and we like better to present their usage for the present problem; thus, we refer to the Comsol Documentation for any further detail. We focus on upwind sailing for a mainsail having low camber and operating in the range of efficient lift-to-drag ratios. For such conditions, sail shape is determined by its natural configuration and by the trimming, whereas wind has minor effects on its further displacements. Thus, we use a one-way coupling between solid and fluid, accounting for the aerodynamic force acting as a load on the sail-- which is, in turn, elastically deformed--but neglecting the effect of sail displacement on the fluid flow. We refer to the monograph by Marchaj [1] for extensive discussions on the subject, and the paper [2] for a review of more sophisticated computational techniques. 2. Model Definition We begin by illustrating main modeling issues in a two dimensional settings. To tackle the problem of a 1D elastic structure-the sailembedded in a 2D fluid-the air, we adopt a virtual wind tunnel divided in two superimposed regions, with the sail in between, and we use two distinct Incompressible Navier Stokes modes, each active in just one domain [Fig.1]. On the portion of the common boundary that represents the sail, we pose no slip conditions; on the remaining part, we require the continuity of the velocity and the pressure field; we were able to set both dynamics-the pressure-and kinematics-the velocity-conditions on the same portion of a boundary by exploiting the possibility of a fine editing of the application mode equations, typical of the Comsol Multiphysics modeling environment.
On the boundary of the tunnel, we set the flow velocity at the inlet (left and bottom boundaries), a zero pressure at the outlet (right boundary) and a neutral condition at the top boundary. It is then possible to evaluate separately the stress fields on the upper and the lower surface of the airfoil generated from the flow past the sail section; finally, by summing up the two contributions, the bulk load acting on the sail can be computed. subdomain 1 Geometry Model Neutral Sail profile subdomain 2 No Slip Inlet Velocity V =5 ms -1 p1=p2 u1=u2,v1=v2 Outlet Pressure P=0 Figure 1. The virtual wind tunnel, consisting of two superimposed regions with the sail section in between. To capture relevant details, we drew sub regions in specific areas of interest, and used the Mesh mode commands to generate unstructured meshes with different element size, ranging from a coarse mesh on the boundary to a finer one around the airfoil, and a very fine one on it [Fig.2]. isotropic diffusion (δ id =0.5) as stabilization technique. We used the Comsol Script to run many different parametric analyses, testing various sail shapes and inflow conditions, and measuring performances under different attack angle (the angle between airfoil chord and wind direction). 3. Sail versus Thick Airfoil In this section we compare, from a purely fluid dynamics point of view, a zero thickness airfoil our sail, and a sequence of thick airfoils, whose thickness decreases to zero. Our study consists in analyzing the Lift and Drag coefficients of the two airfoils. 3.1 Geometry The airfoils sections have been generated with the NACA Four-Digit Series algorithm [6]. The first digit specifies the maximum camber in percentage of the chord (airfoil length), the second indicates the position of the maximum camber in tenths of chord, and the last two numbers provide the maximum thickness of the airfoil in percentage of chord, see Figure 3. As the thickness tends to zero, the airfoil reduces to the main line, which we assume to coincide with our sail profile. Camber 10% Thickness Upper Line Main Line Draft 40% Lower Line Chord Length (1 m) Figure 3. Four-Digit NACA airfoil; its main line is identified with the sail profile. 3.2 Results Figure 2. Unstructured meshes; elements size can be tuned separately in each different region. The nonlinear steady problem has been solved using the direct solver UMFPACK, with We used the Integration Coupling Variables feature to compute the resultant force of the pressure distribution along the sail; the component parallel to the wind direction represents the lift L, the orthogonal component the drag D. Let V be the inflow velocity, ρ the air density and As the sail area. The lift (Cl) and the drag (Cd) coefficients are defined by: Cl = 2L/(ρ As V 2 ), Cd = 2D/(ρ As V 2 ).
Denoting with Cl th, Cd th the corresponding coefficients for the thick airfoil, we computed the ratios (Cl-Cl th )/Cl, (Cd-Cd th )/Cd, for a thickness ranging from 10% to 1% of the chord length [Fig.5], and from 10 to 5 [Fig.6]. For this analysis, we choose a main line camber of 10%, a 40% draft, and an angle of attack of 18 degrees, values representative of the average sail conditions that we used in following analysis. We found the zero-thickness sail to have higher values of Cl and Cd with respect the thick airfoil; as thickness tends to zero, lift and drag coefficients increases their values and smoothly tends to the limit ones. By analyzing the graphs [Fig. 4], we can identify, at least qualitatively, two different regions: the first for a thickness in the range 10% to 2% of the chord length and the second in the range 2% to 5. For high values of the thickness, both ratios decrease quite slowly, and drag appears to be much less sensible to thickness variations [Fig. 4, top]. For a thickness close to zero, differences goes to zero faster, and lift and drag ratios have approximately the same trend [Fig. 4, bottom]. This analysis has been made using an angle of attack at 18 degrees; however, we noticed that drag coefficient is much more sensible to the variations of attack angle than the lift coefficient is. 4. The 1-D Sail We present a parametric design study of a 1D sail immersed in a 2D airflow, with the aim of assessing the maximum lift coefficient that can be generated from a given profile. We define sail sections by adopting a method similar to that used by North Sails New Zealand Limited, and consisting in generating an arc-wise Bezier curve, controlled by some key geometric parameters; despite its simplicity, this method covers the typical range of most common mainsails [4]. 4.1 Parametric design The sail profile is realized by connecting two Bezier curves of 2 nd order, whose shape is controlled by four parameters: camber, draft, leading edge angle, and trailing edge angle, defined as follows (see Fig. 5): Camber = Ymax/C, where Ymax is the greatest perpendicular distance between the sail and the chord line; Draft = Xd/C, where Xd is the position of Ymax on the chord length; Leading edge angle = La, is the slope of the tangent to the sail at the leading edge; Trailing edge angle = Ta, is the slope of the tangent to the sail at the leading edge. The two Bezier curves are connected each other in the point of max depth (Xd,Ymax); here, they share the same (horizontal) tangent. Points 1, 5 fix the chord, point 2, 4 control the tangents to the leading and the trailing edges, respectively, and point 3 set the draft and the camber. The whole procedure has been coded in a script file. Parametric Design 2 Bezier 2nd-order Sail shape Control points 2 3 4 la Camber ta 1 Draft Chord Lenght 5 Figure 5. Sail profile generated by Bezier curves. 4.2 Performance (Lift & Drag coefficient) Figure 4. Lift (red line) and Drag (blue line) coefficient ratios VS airfoil thickness, measured as % (top) and (bottom) of the chord. As first step, we analyze the dependence of lift and drag coefficients on the angle of attack a, for
a sail profile generated using the values in following table. Camber 12% Draft 40% La 33 Ta 17 Table 1. Parameter used for sail design. We use a parametric analysis with a spanning the range (0, 30) degrees, with a 1 degree step, mantaining a fixed value for V (V =5 m/s). Lift coefficient appears to be a concave function of a, increasing up to a maximum, and then decreasing, while drag is a monotone increasing function of a. For the present sail, we found a maximum lift at about a=19 ; at this angle, Cl=1.85 and Cd=4.69, corresponding to a lift-todrag ratio of 4.69 [Fig. 6]. zone at the trailing edge. This is an important issue that is at the base of sails functioning, especially in upwind conditions: as much the airflow remains attached to the whole sail length, as more lift can be generated, and thus, more sail thrust [5]. It is interesting to compare this result with the case of fully attached flow [Fig. 8]. For an angle of attack a=7, streamlines run smoothly around the leading edge, and pressure distribution is much more regular. In this case, we have lower values for Cl and Cd with respect to the previous ones. Figure 7. Velocity field (color map ranges from blue (0 m/s) to red (10 m/s)), streamlines and pressure distribution (black arrows) for angle of attack a=19. Figure 6. Lift (red line) and drag (green line) coefficients versus angle of attack (degrees). Furthers considerations derive from the analysis of the streamlines around the sail and the pressure distribution on it, represented with red lines and black arrows, respectively, in following figures. Figure 7 shows the case with a=19 ; on the upper face we notice an attached flow only in the initial part, and a separation, with the consequent generation of turbulence, near the trailing edge; moreover, flow separation is a clear sign of stall triggering. On the lower face, streamlines touch the airfoil slightly behind the leading edge; therefore, we have a small area of strong negative pressure that generates the high values of pressure at the leading edge on the upper surface. Let us remark that a headsail-mainsail system may help in reducing the recirculation Figure 7. Velocity field (color map ranges from blue (0 m/s) to red (7 m/s)), streamlines and pressure distribution (black arrows) for angle of attack a=7. Let us introduce the pressure coefficients Cp 1, Cp 2, defined by Cp 1 =P 1 /(ρ As V 2 ), Cp 2 =P 2 /(ρ As V 2 ),
where P 1 and P 2 represent the pressure fields at the upper and the lower face, respectively. Plots of the pressure coefficients versus chord coordinate show main differences [Fig. 9]. Upper surface Figure 10. Effect of camber on lift (red line) and drag (blue line). Lower surface Figure 9. Pressure coefficient versus chord, at different angles of attack: a=7 (red), a=19 (blue). For a=7 (red line), Cp 1 and Cp 2 attain almost the same values at leading edges, and increases smoothly up to 40% chord length. On the contrary, for a=19 (blue line), Cp 1 and Cp 2 have very different values at the leading edge; Cp 1, after a sudden decrease, follows closely the previous trend between 30% and 60% of the chord. Then, it remains almost flat until 90% or more, where another pronounced decrement, foreboding a stall, is present. After studying sail performances for a given profile at varying angles of attack, we now analyze the effects of different geometries for a given air flow conditions. The idea is to retain the flow features and the leading and trailing edge angles fixed as in the previous analysis (V =5 m/s, a=7, La =33, Ta=17 ). Then, we investigate the dependence of the lift and drag coefficients to the position of the camber and the draft. The first parametric analysis concerns the camber, with values ranging from 6% to 15% of the chord, thus preventing an extreme sail shape. As we can see [Fig. 10], the lift increases its values up to a maximum of Cl=1.27 for a camber value of 14% of the chord, and then it starts decreasing. This is due to the lack of necessary curvature that provides the flow acceleration, with consequent high negative pressure values on the upper surface; conversely, drag rises for any camber value. Draft variation yields very different trend of Cl and Cd, as shown in the Figure 11. The draft varies between 35% and 45% of chord length, 5% above and below the value used in our previous analysis (draft at 40%). Cl has an increasing trend in the range between 38% and 42%, the central part of the graph, and presents very irregular and oscillating behaviour both before and after that interval (see Fig. 11, top); Cd is more regular and appears to be an almost decreasing function of the draft (see Fig. 11, bottom). Figure 11. Effect of draft on lift (red line) and drag (blue line). 5.3 Elastic analysis We model the sail as a flexible beam, using the In-plane Euler Beam application mode of Comsol Multiphysics; unknown fields of the elastic problem are displacement of the axis and rotation of the cross section. As stated in the introduction, we consider a one-way coupling between solid and fluid, that is, aerodynamic forces acts as a load on the sail and deforms it,
but the effects of sail displacement on the fluid flow are neglected. Desired value of bending and axial stiffness have been obtained by considering a square cross section with side length l=0.002m, section area A=4*10-6 (m 2 ), and moment of inertia I=1.3*10-12 (m 4 ); we assume an elastic modulus E=2*10 9 MPa and a density ρ=1150 kg/m 3, corresponding to Nylon. Loads are defined as the total boundary force per area (measured in N/m 2, and given by the solution of the fluid flow equations), times the beam section width, in order to get the correct bulk load (measured in N/m) as required from the beam implementation. Using coupling capabilities, we solve simultaneously for all variables, thus obtaining in one run all desired results. Following plots show displacements [Fig. 12], bending moment and axial force [Fig. 13] for a sail profile corresponding to Table 1, and flow conditions V =5 m/s, a=7. Figure 12. Total displacement (top) and normal and tangential displacement (bottom) along the chord. 6. The 2-D Sail We design our 2D sail using a sequence of 1D profiles; each profile is generated with the algorithm given in section 4 and constitutes the skeleton for the generation of the two dimensional surface representing our sail. Finally, a surface is obtained via extrusion and/or lofting commands. As done for the 1D sail, the whole procedure has been implemented using a script language; it is then possible to test in a fast and easy way a large numbers of different designs. In particular, we focus on a sail with a 12m height, 4m large, under upwind sailing conditions at 13 knots of true wind speed, 41 true wind angle and 5.9 knots of boat speed [6]. 6.1 Parametric design We consider a box containing the airfoil with an 18m height, a 24x24 meter square base; the top and bottom face are located 3m above and below the head and the foot of the sail, respectively. 3D simulations must take into account the logarithmic boundary layer developed over the sea surface. The True Wind Speed (TWS) can be modelled as TWS=uu/k*log(z/z 0 ), where z is the height above the sea surface, z 0 =0.01 is a surface roughness length, u=0.380 is the friction velocity and k=0.42 is a tuning factor. This boundary layer profile generates a twisted flow, and both the Apparent Wind Speed (AWS) and the Apparent Wind Angle (AWA), which coincides with the angle of incidence, changes with z. As consequence of this, head and foot sail sections may experience quite different flows regimes that must be accounted for during the design. Typical values are: (Foot sail section) TWS = 10 knots, AWA = 23.86 degrees. (Head sail section) TWS = 12.8 knots, AWA = 23.22 degrees. Figure 13. Bending moment (top) and axial force (bottom) along the chord. The whole sail has been generated using a skeleton composed of 13 different profiles, equally spaced along vertical direction, starting from the foot section at height 0 m, till the head section 12 m above (see Fig. 14 and table 2);
each profile has its own design parameters, twist included. In order to keep constant the angle of attack in the whole sail, we use a linear function of height for the twist angle Twa: Twa=AWA(0)+(AWA(z)-AWA(0))*z/L. Twist Angle Head Section Twist Angle Vertical plane through foot section Twist Angle Leech Luff 2D Parametric Cross Section Figure 15. 3D flow channel, 2D sail and sub domain regions to control mesh generation. 6.2 3D simulations Foot Section Figure 14. Sketch of sail generation. Height Leading Trailing Twist rc Camber Draft [m] Angle Angle angle 0 m 0% 10% 47% 33% 17% 0,00 1m 0% 10% 47% 33% 17% 1,44 2m 0% 10% 47% 33% 17% 2,18 3m 0% 10% 47% 33% 17% 2,66 4m 1% 10% 47% 33% 17% 3,00 5m 1% 10% 46% 33% 17% 3,30 6m 1% 10% 46% 33% 17% 3,52 7m 2% 10% 46% 33% 17% 3,71 8m 2% 10% 46% 33% 17% 3,87 9m 5% 10% 45% 33% 17% 4,00 10m 10% 10% 45% 33% 17% 4,14 11m 20% 10% 45% 33% 17% 4,25 12 m 40% 10% 45% 33% 17% 4,36 Table 2. Parameters used for 2D sail design; rc represents the chord reduction ratio. Once generated, the sail is positioned inside a virtual 3D flow channel, split in two adjacent regions. Then, to control meshing, we add further domains (see Fig. 15 and Fig. 2). Again, we use two distinct Incompressible Navier Stokes modes, each active in just one domain; boundary conditions on the sail, on the interface between the two sub domains, and on the boundary of channel are the same as for the 2D case. We model the sail as a flexible shell, using the Shell application mode of Comsol Multiphysics; unknown fields of the elastic problem are displacement and rotation, totaling 6 degrees of freedom. Geometric and material properties follow those used for elastic beam, with minor obvious adjustment; shear factor has the default value 1.2. Loads are defined directly in terms of the total boundary force per area (N/m 2 ) computed when solving fluid equations. 3D simulations are quite demanding; moreover, post-processing results is often a major task. Our typical simulations involve 15k elements, 3.4k boundary elements, and 0.5k edge elements. These values yield an overall element quality of 0.26, with 80K degrees of freedom; solution time is about 4 hours on a 64-bit workstation. We conclude by showing some results concerning the flow field around the sail, the aerodynamic load on it and the consequent stress distribution; sail under investigation is that generated using the values in table 2, flow conditions are V =5 m/s and a=7 at foot section. We slice the computational domain to plot the magnitude of the velocity field, and we use streamlines to capture the overall flow behaviour. On the sail, we plot the von Mises stress distribution [Fig. 16]. Next figures show in better detail the field of the aerodynamic load acting on the sail and the von Mises stress, using a vertical [Fig. 17] and a transversal perspective [Fig. 18].
8. References 1. C.A. Marchaj, Aero-Hydrodynamics of Sailing, Dodd, Mead & Company, New York (1979). 2. P. Conti, M. Argento, M. Scarponi, Unconventional Sail Design, Fluent News, Spring Issue, (2006). 3. Aerospaceweb.org, http://www.aerospaceweb.org/question/airfoils/q 0041.shtml Figure 16. Flow field around 2D sail. Velocity color map on slices ranges from blue (0 m/s) to red (7 m/s). Von Mises stress is about 2*10 5 Pa. 4. S.J. Collie, P.S. Jackson, M. Gerritsen, J.B. Fallow, Two-Dimensional CFD-based Parametric analysis of Downwind sail Designs, RINA Annual Report and Transactions (2004). 5. P. Bogataj, How Sails Work, North Sail-One Design Technical Paper (2001). 6. R. Ranzenbach, Z. Xu, Aero-Structures: Studying Primary Load Paths and Distortion, Proc. of The 17th CHESAPEAKE SAILING YACHT SYMPOSIUM (2005). Figure 17. Aerodynamic load and von Mises stress; vertical perspective. Figure 18. Aerodynamic load and von Mises stress; transversal perspective.