In recent years, the exponential growth of computational resources at a very low cost has allowed for the widespread use of numerical methods in the nautical field as well 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 field only for competitive applications, but it is conceivable that in the future these methods could also be used in other application fields. Recent racing yachts, as demonstrated by the last edition of the America’s Cup, have achieved very high performances thanks to the 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 aim to use more new research and development tools [
1].
To improve sail design and to better understand sail aerodynamics, numerical simulations and experimental testing are the most used and reliable methods [
2,
3,
4,
5]. These methods have led to a better comprehension of the complex phenomena and the physics underlying sailing and, consequently, through them, a remarkable increase of the performances of yacht sails has been obtained in recent years [
6,
7]. Experimental full-scale testing is the most accurate and reliable method, but, because of cost constraints, its use is only 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,
13,
14,
15,
16,
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.