Advances in Engineering Software 37 (2006) 491–501 www.elsevier.com/locate/advengsoft 3D solid fin model construction from 2D shapes using non-uniform rational B-spline surfaces Dave Carswell, Nick Lavery * Materials Research Centre, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom Received 29 July 2005; received in revised form 15 December 2005; accepted 10 January 2006 Available online 28 February 2006 Abstract A computer aided design (CAD) tool has been specifically developed for rapid and easy design of solid models for surfboard and sailboard fins. This tool simplifies the lofting of advanced fin cross-sectional foils, in this instance based upon the family of standard airfoil series set by the National Advisory Committee for Aeronautics (NACA), whilst retaining a basic parametric description at each cross-section. This paper describes the way in which non-uniform rational B-spline (NURBS) surfaces are created from 2D profile splines, and are then used to generate 3D geometrical surfaces of the fins, which can be imported directly into commercial software packages for finite element stress analysis (FEA) and computational fluid dynamics (CFD). Pressure distributions, lift and drag forces are determined from a CFD flow analysis for various fins designed with this tool, and the results suggest that the incorporation of advanced foils into surfboard fins could indeed lead to increased performance over fins foiled using current standard techniques.  2006 Elsevier Ltd. All rights reserved. Keywords: Computer aided design (CAD); NACA airfoils; NURBS surfaces; Computational fluid dynamics (CFD) 1. Introduction Surfing has expanded rapidly over the last few years, and in the UK estimates have been suggested of over 300,000 surfers taking to the water every summer, increasing tourism income into costal towns such as Newquay by over £40M every year. From the board manufacturing perspective, there has been a natural progression towards more mechanical forms of surfboard and surfboard fin manufacturing to cope with increased demand, inevitably accompanied by computerisation of design. While resisted by many in the surfing community, this progress has opened the doors to the type of advancement of design previously reserved for more affluent aquatic sports such as sailing and power-boat racing. * Corresponding author. Tel.: +44 (0) 1792 295850; fax: +44 (0) 1792 295244. E-mail address: n.p.lavery@swansea.ac.uk (N. Lavery). 0965-9978/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2006.01.002 The primary objective of the current work is to understand and improve surfboard fin design by using advanced computer aided engineering (CAE) software such as finite element analysis (FEA) for stress fields and computational fluid dynamics (CFD) for fluid flow and lift/drag forces. The fin(s) on a surfboard lie on the rear underside of the board and serve the dual functionality of providing stability (like a keel on a boat) and directionality (like a rudder). To this extent, the underlying hypothesis of this work is that there may be an optimal set of foils for a particular set of fins which will give best performance with minimal drag, and that the answer may lie in an appropriate selection of NACA foils along various sections. However, a static foil may not be sufficient and may lead to only minor improvements, however if coupled with polymer-composite materials of variable elasticity distributions could lead to foils which can adaptively enhance their profile through the range of angles of attack, thus optimising performance. 492 D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 To this extent a combination of CFD and finite element stress analysis will play an increasing role in the design. Established in 1915, the National Advisory Committee for Aeronautics (NACA) had a primary interest in the development of better airfoils, and the labelling techniques they introduced to describe their airfoils have now become a standard. In particular, a family of airfoils known as the NACA 4-series [1] were developed in 1933. Improvements to the 4-series later produced the 6-series [1,2], and the aim of these foils was to reduce drag by increasing the extent of the laminar-boundary layer and hence reduce the skin friction drag. It was found early on in the project that it was difficult to generate parametrically defined solid models of fins from commercial CAD software in a way which would retain the defining parameters for foiled sections, and which would also create the surface and volumetric geometry required for a CAE analysis. The basic problem is that commercial CAD packages do not have native support for the NACA foil series of equations. To remedy this problem, a tool has been specifically developed for rapid and easy design of solid models for surfboard and sailboard fins, based on simple and intuitive parameters that fin designers can use on a daily basis, such as length, height, and rake, and which relate to the existing empirical-based design criteria. However, in order to empower the designer, more scientifically advanced design parameters have also been made available in the form of foils which can be defined parametrically according to the NACA standards at varying cross-sections. In this paper, the underlying geometrical techniques used for the tool are presented, as these may be equally of interest for CAD development in other applications such as aircraft wings, turbine blades, automotive spoilers and yacht keel designs. Key to the usability of this software is the development of 3D non-uniform rational B-spline (NURBS) surfaces from 2D profile splines, and the way in which these surfaces are put together is described in detail. Finally, to demonstrate the application of the tool, CFD flow results for pressure, drag and lift forces are presented for fins designed with a specific 4-series, 5-series and 6-series NACA base foil and compared at various angles of attack. 2. NACA foil series The NACA foils are a combination of a mean line and a thickness distribution defined for the upper and lower surfaces of the foil as shown in Fig. 1, and given by the following separate equations for the upper surface: xU ¼ x  y t sin h y U ¼ y c þ y t cos h And for the lower surface: xL ¼ x þ y t sin h y L ¼ y c  y t cos h Fig. 1. NACA 4-series foils as created in the CAD tool. where yt is the thickness distribution and yc is the mean line calculated by different equations and techniques depending on the series of the foil in use [1–5]. The NACA 0012 and NACA 4312 foils are shown in Fig. 1 and were created by feeding in the maximum camber and its location, these being the only two parameters required for the creation of a NACA 4-series aerofoil in the current tool, as the third parameter (the thickness) is predefined based on parameters derived from frontal and vertical profile definitions. The generation of the 6-series foil is based upon an iterative technique using conformal mappings of the foil in complex planes, and which lies beyond the scope of this paper but which can readily be found in [3]. Of interest is that in the derivation of the 6-series foil, the guiding parameter is a theoretical lifting coefficient based upon a desired lift characteristic of the foil. 3. Non-uniform rational B-spline surfaces Non-uniform rational B-spline (NURBS) surfaces are used in many applications and are supported by most common CAD file formats including IGES [6]. The requirements for a NURBS surface are a set of control points, a set of weights and two sets of knot vectors and are parameterised in the u- and v-directions. The knots are both a vector of non-decreasing real numbers given as U ¼ fu0 ; . . . ; um g V ¼ fv0 ; . . . ; vn g The spline also has a degree in both the u- and v-directions usually designated the variable names p and q, respectively. In order for the spline to start exactly at the first control point and end exactly at the last control point it is necessary to set the first p + 1 (or q + 1) knots to some constant, a, and the last p + 1 knots to some constant, b. Before the surface equations are given, it is necessary to define the basis function of the spline which is defined recursively by  1 if ui 6 u < ui þ 1 N i;0 ðuÞ ¼ 0 otherwise u  ui uiþpþ1  u N i;p ðuÞ ¼ N i;p1 ðuÞ þ N iþ1;p1 ðuÞ uiþpþ1  uiþ1 uiþp  ui The weights are a two-dimensional array of real numbers, wi,j, that specify how much each control point affects the D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 493 path of the spline. In the application described here each control point has an equal weight of 1 and can, therefore, be ignored in the calculation. The NURBS surface, given a set of control points, Pi,j, is defined as smooth surfaces, however more recently the STEP format (Standard for the Exchange of Product Model Data) has been found to be more reliable. Pn Pm i¼0 j¼0 N i;p ðuÞN j;q ðvÞwi;j P i;j Sðu; vÞ ¼ Pn Pm ; j¼0 N i;p ðuÞN j;q ðvÞwi;j i¼0 5. Geometrical algorithms 0 6 u; v < 1 4. IGES overview The initial graphics exchange specification (IGES) has been in development since the late 1970s and aimed to provide a common file format in which different CAD vendors could share CAD drawings [7]. The file format itself consists of five main sections: • The start section may be used as the vendor sees fit and is often used to provide a description of the file. • The global section describes the parameters of the drawing and integer and real number characteristics. • The directory entries describe each drawing entity in the parameters section and record things such as whether the entity is visible, the line style and starting line number in the IGES parameter section. • The parameters section contains the individual entity data. • The final section records the total number of lines each of the above sections occupies. The data in the IGES file consists of lines 80 characters long where the first 72 contain the actual data and the remaining eight serve to identify the section and the section line number [6]. As is often the case when entity data does not fit into the 72-character segment, it may span multiple lines but single integers or real numbers must not span multiple lines. For example the number 20.86156 (located at the end of the first line) in the IGES extract in Table 1 spans multiple lines, and should look like the data in Table 2. This may lead to importation problems with other software, and needs to be addressed carefully. IGES has been the format of choice for many finite element applications, as it is non-propriety and provides The current CAD design tool has been developed to be able to deal with the geometries of surfboard and sailboard fins; however, it could easily be used for other applications such as aircraft wings, turbine blades, automotive spoilers and yacht keel designs. Essentially, any application which requires cross-sectional foils lofted along a finite length, and also tapered/foiled in the vertical as well as horizontal directions. Thus, the ability of the tool to export in a CAD format capable of being imported by other applications was deemed important right from the initial phases of the development, and in a first instance, the IGES file format was selected as it was able to use NURBS surfaces, although other formats such as STEP and ACIS will also be considered in the future. Essentially, the designer starts with nothing and has to build the fin from basic shapes such as lines, radii and splines. While the side view of the fin (the profile) is easy to create in this manner, the cross-sections can be a bit trickier as each one needs to be manipulated individually to create the desired shape. The tool provides the user with three views of the fin: the profile (y–x plane), the projected front (y–z plane) and the foils (x–y planes) at various cross-sections of the profile. Some typical fin shapes are shown in Fig. 2 where the hollow squares represent the control points as seen by the user, known as the geometrical control points. Associated with each geometrical point are two virtual control points that serve to shape the Bezier curve, which can also be manipulated by the user but are not shown in the figures. Table 1 Incorrect line span for the IGES file format 0.,120.,20.86156,3.000003,71.52499,20.86156,6.000005,23.0 4191,20.8615 6,6.000005,130.1506,54.15784,0.,122.8559,41.72313; Table 2 Correct line span for the IGES file format 0.,120.,20.86156,3.000003,71.52499,20.86156,6.000005,23.04191, 20.86156,6.000005,130.1506,54.15784,0.,122.8559,41.72313; Fig. 2. Basic shapes that define the fin. 494 D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 5.1. Inserting additional geometrical points for existing foils in x–z planes The user is allowed to insert foils at any vertical crosssection along the profile and each foil can be defined using a different number of control points. It is the uncertainty in the number of control points present on each foiled section which needs to be addressed in the transformation of these points into a NURBS surface. Firstly, the maximum number of control points on both sides of each the foils are found. Next, control points are added to each foil with less than the maximum so that it too has the same number of control points on both sides. New control points are added on the line with the greatest length. The pseudo-code in Table 3 ensures all foils in FoilList contain the same number of control points on the upper and lower surfaces. The distance lists, DistancesUpper and DistancesLower, store an ordered list of the control points starting at the control point where x = 0. The point of insertion must not alter the path of the spline, this is discussed later. 5.2. Inserting additional geometrical points for sections in y–z and x–y planes Since the profile and front sections need to be combined they must have control points at every vertical height where the user has inserted a control point or where a foil section exists. For example, consider the profile and front sections as shown in Fig. 3. The following steps need to be followed: Table 3 Pseudo-code ensuring foils have same number of points on upper and lower surfaces MaxUpper :¼ GetGreatestPoints(UPPER, FoilList); MaxLower :¼ GetGreatestPoints(LOWER, FoilList); for n :¼ 1 to FoilList.TotalFoils do begin Foil :¼ FoilList.Foil[n]; DistancesUpper :¼ CreateDistanceList(UPPER, Foil); DistancesLower :¼ CreateDistanceList(LOWER, Foil); for i :¼ 0 to MaxUpper – 1 do begin x :¼ GetLargestIntersectionX(DistancesUpper); s :¼ GetLineSegment(Foil, x, UPPER); t :¼ GetParameterValue(Foil, s, x); Foil.InsertPoint(s, t); DistanceUpper.Insert(s, x); end; for i :¼ 0 to MaxLower – 1 do begin x :¼ GetLargestIntersectionX(DistancesLower); s :¼ GetLineSegment(Foil, x, LOWER); t :¼ GetParameterValue(Foil, s, x); Foil.InsertPoint(s, t); DistancesLower.Insert(s, x); end; end; Fig. 3. Example foil locations on the profile and front sections. 1. Firstly, a list must be made that contains all unique heights of each control point and each foil. In the example the list, L = {0, 60, 80, 105, 120} correlate to points and foils in Fig. 3. 2. Secondly, for each element in the list a control point must be inserted on the front and rear of the profile for which a control point does not already exist. On the front of the profile, this involves inserting control points at y = 60, 80 and 105. On the rear of the profile only a control point at y = 80 needs to be inserted as all other heights exist. The front section needs control points inserting at y = 60, 80 and 105 on both sides as there are no control points present here. When inserting control points, it is important that the path of the spline is unaffected. This is done by the pseudo-code presented in Table 4. Firstly, the parameter, t, at the point on the spline where the new point is to be inserted needs to be computed. The CAD tool uses Bezier curves defined by a starting point, Point1, an ending point, Point2, and two virtual control points (Fig. 4), ControlPoint1 and ControlPoint2. These four points are sequenced such that the spline starts at Point1, then encounters ControlPoint1, then ControlPoint2 and finally Point2. The implementation of this code in the fin design tool does not require the absolute positions of the virtual control points as they can be calculated using the tangent at the geometric point and a flatness value that defines how far the virtual point is from the point on the spline. Table 4 Bezier curve point insertion pseudo-code p1 :¼ (ControlPoint1  Point1) * t + Point1; p2 :¼ (Point2  ControlPoint2) * t + ControlPoint2; pcp :¼ (ControlPoint2  ControlPoint1) * t + ControlPoint1; ControlPoint1 :¼ p1; ControlPoint2 :¼ p2; p1 :¼ (pcp  p1) * t + p1; p2 :¼ (p2  pcp) * t + pcp; NewPoint :¼ (p2  p1) * t + p1; NewControlPoint1 :¼ p1; NewControlPoint2 :¼ p2; D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 495 Fig. 4. Inserting a geometric control point at t = 0.5. 5.3. Construction of a single NURBS surfaces from 2D shapes A form of transfinite interpolation is then used with the exiting virtual control points to derive missing virtual control points that define the NURBS surface. This is done by combining corresponding points Pi and Pi+1 on adjacent foil sections that bound the surface with points Pj and Pj+1 on the front section. The profile merely serves to define the overall shape of the fin when i = 0 or i + 1 = the back point of a foil section (then the rear part of the profile is used to define the shape). In order that the NURBS surface exactly follows the Bezier curve the knot vectors must be set accordingly, these being U = V = {0, 0, 0, 0, 1, 1, 1, 1}, [8,9]. Fig. 5 shows the control points that can be derived from the 2D shapes, denoted by hollow circles, and the ones that need to be interpolated are denoted by solid circles. Interpolating between the known control points gives the position of the new control points, which must also take into account the tangent angle at each point in both the front and profile views as well as the flatness values that were described earlier. The interpolation of point P1,1, for example, requires the virtual point situated at P1,0 have a tangent as specified by the following equation: x  ðtangent2  tangent1 Þ þ tangent1 tangent ¼ length where tangent1 and tangent2 are the tangents on the front and rear profile sections, respectively, length is the distance between the points at these tangents. The variable x is P1,0’s x-coordinate. Taking the flatness value from the profile section and the newly computed tangent allows the computation of the x and y coordinates for point P1,1. This is the basis of how each of the other unknown points is calculated. The z-coordinate is calculated in much the same way except for the fact that instead of using the profile section to define the tangents and lengths the front section must be used. Fig. 5. B-spline surface showing virtual control points that do not have an exact match on the 2D shapes (solid circles). 5.4. Multiple NURBS surfaces A NURBS surface must be created for every spline that makes up the foil sections. To ensure that the join between two surfaces is clean the next surface uses the last four control points in the v-direction at u = 3, as shown in Fig. 6. As each foil is essentially a closed loop, the last surface uses the first control points of the surface that was created first. A typical wireframe representation of the NURB surfaces are shown in Fig. 13a and b, and a 3D rendering by the CAD tool using OpenGL, is shown in Fig. 12. It is important that the stitching of the multiple NURBS surfaces is also smooth (i.e., that the gradients across the intersection also match), as when a 3D volume is created in the CAE packages, non-smooth intersections are propagated into the computational domain and can lead to severe meshing problems. Fig. 6. New NURBS surfaces use the last points of the previous surface. 496 D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 5.5. CFD results for a standard foiled fin compared to NACA 4- and 6-series foils Three double-foiled (symmetrical) centre fins were designed with different base foils: 1. A standard foil (measured from CAD files supplied by manufacturer). 2. A NACA 4-series base foil (NACA-0007). 3. A NACA 6-series base foil (NACA63-007). For all three fins, the same side and front profiles were used as the one used by the manufacturer (RedX-X1 centre fin), as shown in Fig. 7, so that only the foils along the span-wise direction were different between the fins. All three base foils are compared in Figs. 8 and 9, and as can be seen on a smaller y-range in Fig. 9, the standard foil as used on the fin supplied by the manufacturer is very similar to the 4-series, but slightly thicker towards the trailing edge. More technology has been applied to sailing and windsurfing equipment than to surfing, and borrowing from their terminology, the NACA 4- and 5-series are referred to as convectional section foils, and the NACA 6-series are known as laminar section foils. The conventional sections are characterised by the thickest section occurring further forward towards the leading edge resulting in a slightly blunter looking foil. The IGES files for these fins were imported directly into FLUENT (the commercial CFD software used for this work), and the fins were placed in a 2 · 0.3 · 0.15 m box and flow passed over then corresponding to 30 l/s (kg/s) as a mass-flow inlet boundary condition, which results in inlet velocity values in the range of 0.6–0.7 m/s. The same tetrahedral mesh distribution (1 mm) was used for all fins and box edges (20 mm), and FLUENTs meshing program GAMBIT had no problem recognising volumes from the IGES file provided, furthermore, the meshes had very acceptable mesh skewness distributions indicating that the base surfaces were indeed acceptably smooth and clean. Numerical experiments were performed for angles of attack varying from 0 to 12, in 2 intervals, using a k–e turbulence flow model with 1% turbulence intensities for both k and e, set at the inlet. Pressure distributions halfway up the fin are shown in Figs. 10 and 11 for all three fins at angles of attack of 0 and 8 to demonstrate differences found between the foils. Lower pressure differences along the foil for the 6-series fin at the 0 angle of attack contribute towards the lower drag coefficients encountered. The drag and lift coefficients shown in Figs. 14 and 15 were calculated using the projected area of the fin on the Fig. 7. Side profile of the X1 fin used for all foils. 497 D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 Fig. 8. All base foils used for X1 fin analysis. 0.006 X1 with standard foil X1 with 4-series foil (NACA 0007) X1 with 6-series foil (NACA 63063) 0.004 Y-coordinate (m) 0.002 0 0 0.02 0.04 0.06 -0.002 -0.004 -0.006 x-coordinate (m) Fig. 9. Scaled base foils to maximise differences. 0.08 0.1 498 D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 Pressure along base foils at angle of attack, α = 0° 0.01500 110 0.01000 X1 with standard foil X1 with 4-series foil (NACA 0007) X1 with 6-series foil (NACA 63063) 0.00500 10 Y-coordinate (m) P-Pmax (Pa) 60 0.00000 -40 -0.00500 -90 0 0.02 0.04 0.06 0.08 0.1 x-coordinate (m) Fig. 10. Pressure (P–Pmax) distributions along the base-line foil for all foils at an angle of attack of 0. Pmax is the maximum pressure along the given foil. Pressure along base foils at angle of attack, α = 8° 300 0.01500 250 200 150 0.01000 P-Pmax (Pa) X1 with standard foil X1 with 4-series foil (NACA 0007) X1 with 6-series foil (NACA 63063) 50 0.00500 0 Y-coordinate (m) 100 -50 0.00000 -100 -150 -0.00500 -200 0 0.02 0.04 0.06 0.08 0.1 x-coordinate (m) Fig. 11. Pressure (P–Pmax) distributions along the base-line foil for all foils at an angle of attack of 8. Pmax is the maximum pressure along the given foil. inlet, and demonstrate classical distributions over the range of angles of attack tested, with a deeper ‘‘drag bucket’’ for the 6-series than for the 4-series, and the stalling angle occurring somewhere between 6 and 10. This term has been borrowed from sailing keel design terminology [10], where designers have traditionally kept the 4-series foil on keels and rudders for two reasons, (a) due to the higher angles of attack (‘‘leeway’’) at which yachts are required to operate, specifically because the 4-series tends to be more forgiving and stalls at higher angles of attack, and (b) the D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 499 Fig. 12. Rendered fin with NACA 4-series foils. six series was designed for high Reynolds numbers, as opposed to the lower Reynolds numbers encountered generally in hydrodynamics. These graphs, together with the ratio of lift to drag graph in Fig. 16, clearly demonstrate that for surfboard fins, both the 4-series and the 6-series can give lower drag than the currently used foil, and certainly produce higher lift values over the lower ranges of angles of attack, which are believed to occur in practise for the middle fins of surfboards (not being as extreme as with the outside fins). Even this small improvement should manifest itself in greater manoeuvrability and speed of the surfboard, particularly for larger fins (single fins used on longer boards) or when board speeds are higher, e.g., for waves greater than 3.5 m, where surfer speeds can exceed 7–10 m/s [11]. This would agree with more recent experiences in sail boat design, in that laminar sections (e.g., 6-series) are more suitable for the keel (which encounters lower angles of attack) than the rudder. Fig. 13. Wire frame of fins with (a) NACA 4-series foils and (b) NACA 6series foils. 6. Future work The CAD design tool is currently being released in beta testing mode to fin manufacturers. Work in the immediate future will hinge on computational aspects of the software, such as data management aspects to allow easy storage of multiple sets of fins and an internet database of sample fins, as well as extending the CAD export capabilities to include the STEP format. Additionally, the fin design tool will also be coupled with an existing surfboard design tool, also developed at Swansea University. This will allow designers to visualise the placement of the fins upon the complete surfboard. On the scientific level, work is currently underway examining genetic algorithms as a way of intelligently developing new fin designs based upon computational flow results. This has required a highly efficient in-house flow solver to be developed specifically for the fin, giving detailed pressure distributions to calculate lift and drag forces for each fin, and which can rapidly give this data for many generations of fin design. 500 D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 0.73 0.71 Drag Coefficient, C D 0.69 0.67 0.65 0.63 0.61 0.59 0.57 0.55 0.53 -12 -10 -8 -6 -4 -2 0 2 4 Angle of attack, α (°) X1 with standard foil 6 8 10 12 X1 with 6-series foil (NACA 63063) X1 with 4-series foil (NACA 0007) Fig. 14. Drag coefficient for all fins as a function of angle of attack. 2.8 Lift Coefficient, C L 1.8 0.8 -0.2 -1.2 -2.2 -3.2 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Angle of attack, α (°) X1 with standard foil X1 with 6-series foil (NACA 63063) X1 with 4-series foil (NACA 0007) Fig. 15. Lift coefficient as a function of angle of attack. tank at Swansea University for medium-high Reynolds numbers. Preliminary results are looking promising with a match in both magnitude and trend for the drag and lift forces, and the authors will be publishing this data in the near future. Future CFD work will also take into account variations in profile design, which could have a more dramatic effect on the performance of the fin than the foil design, and to this extent, will follow experimental work already done keel design in yachts examining forward and backward rake, and various elliptical design profiles. Higher flow rates (Reynolds numbers) also need to be examined, as a function of both foil and profile, as the drag and lift characteristics could also change dramatically as the flow rate is increased. 7. Conclusions A computer aided design (CAD) tool, specifically developed for rapid and easy design of solid models for surfboard and sailboard fins, has been described. The surfaces produced are smooth and clean, and result in small IGES files which can be used for a variety of applications, ranging from actual design and production through to importation into CFD and FEA software for performance analysis. To test this principle, fins designed with specific NACA foils have been numerically tested using CFD software to get pressure distributions, lift and drag forces, and the results have shown that the incorporation of advanced foils into surfboard fins could indeed lead to increased performance over fins foiled using current standard techniques. There may be issues such as tolerances in moulding the polymer-composites, day-to-day strength requirements and safety perspectives, which all need to be addressed in the use of NACA 6-series foils such as those used in this study, particularly in areas with thin foil sections close to the trailing edge which result in very sharp edges. 4.8 Acknowledgements 3.8 Lift / Drag 2.8 1.8 0.8 -0.2 -1.2 0 2 4 6 8 10 12 14 The authors would like express their gratitude to the assistance of Mr. Graham Foster for his input and design ideas of the CAD design tool discussed in this paper, as well as to Tom O’Keefe from DAUM tooling for supplying CAD files for the RedX fin used in this paper. They would also like to thank Dr. Steve Brown, Dr. Ian Pearce and Dr. Adam Bere for many invaluable discussions on fluid flow and fin/surfboard design. Angle of attack, α (°) X1 with standard foil X1 with 4-series foil (NACA 0007) References X1 with 6-series foil (NACA 63063) Fig. 16. Ratio of lift over drag as a function of angle of attack. At a more basic level, CFD results are also being validated via experimental tests for fins and plates in a flow [1] Abbott IH, Doenhoff AEv, Stivers LS. Summary of airfoil data, National Advisory Committee for Aeronautics 824, 1945. [2] Jacobs EN. Preliminary report on laminar-flow airfoils and new methods adopted for airfoil and boundary-layer investigations. National Advisory Committee for Aeronautics NACA WR L-345, 1939. D. Carswell, N. Lavery / Advances in Engineering Software 37 (2006) 491–501 [3] Ladson CL, Cuyler J, Brooks W, Hill AS, Sproles DW. Computer program to obtain ordinates for NACA airfoils. NASA Langley Research Center, Technical Memorandum December 1996. [4] Loftin LK. Theoretical and experimental data for a number of NACA 6A-series airfoil sections. National Advisory Committee for Aeronautics 903, 1948. [5] Wadlin KL, Shuford CL, McGehee JR. A theoretical and experimental investigation of the lift and drag characteristics of hydrofoils at subcritical and supercritical speeds. National Advisory Committee for Aeronautics 1232, 1955. [6] Kennicott PR. The initial graphics exchange specification (IGES) Version 5.3, IGES/PDES Organisation 1995. [7] Nagel R, Braithwaite W, Kennicott P. Initial graphics exchange specification IGES (Version 1.0). National Bureau of Standards, Washington (DC) NBSIR 80-1978, 1980. 501 [8] Foster C, Shapirstein Y, Cera C, Regli W. Multi-user modelling of NURBS-based objects. Presented at ASME design engineering technical conferences & computers and information in engineering conference, Pittsburgh, Pennsylvania, 2001. [9] Piegl L, Tiller W. The NURBS book. 2nd ed. Springer-Verlag; 1997. [10] Jeffery CB. Boards and rudders—a scientific view. In: Yacht racing, vol. 13, 1974. [11] Lavery N, Foster G, Carswell D, Brown S. Optimization of surfboard fin design for minimum drag by computational fluid dynamics (Do Glass-on fins induce less drag than boxed fins?), presented at 4th International Surfing Reef Symposium on Natural and Artificial Surfing Reefs, Surf Science and Coastal Management, Manhattan Beach, California, 2005.