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Sep 2005

Article selected from our quarterly magazine dedicated to the largest and most luxurious boats with information, interviews, technical articles, images and yachting news



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Article by
Francesca Mura and Marco Mulas, Gruppo di Computational Fluid Dynamics, CRS4, Sardegna, Italia



The aeroelastic problem of sails was mentioned in a previous article. Under the aerodynamic load the sail fabrics strain. Such deformations change the sail shape, which in turn determines a modification of the pressure distribution on the sails, changing the sails performance in terms of thrust and lateral forces.



This is a so-called fluid-structure interaction problem which requires the coupled solution of both the sail aerodynamic and the sail structural problems. This is done generally with an iterative procedure where the aerodynamic and the structural problems are repeatedly solved one after the other until the changes between two successive iterations, both in the material deformation and in the aerodynamic forces, become smaller than a predefined tolerance.

The first component of the iterative procedure is the aerodynamic analysis carried out with CFD (Computational Fluid Dynamics) codes, which determines the pressure distributions on the two sides of the sail. The second component is the structural analysis which determines the material strain under the known aerodynamic load. For such an analysis, computer codes implementing the Finite Element Method are used. The importance of the aeroelastic effect on the sail performance depends on the elastic properties of the sail fabrics. Clearly, different materials show different stress-strain relations and the deformations may be somehow limited by choosing suitable, more expensive, materials. The sails fabrics are generally made of a weave of warp, fill and bias ribbons. A light but rugged polyester taffeta backs the tri-axial configuration. Warp, fill and diagonal axis ribbons are often made of Dacron (material imposed by racing regulations). From an elastic point of view, the global behaviour of such fabrics, called cruising laminates, is orthotropic, which means that the elastic matrix [C], relating the stresses {s} to the strains {e}, given by:

{s} = [C] {e}

has 9 independent elements (elastic constants) which characterize the elastic behaviour of the material. In the above relation, the stresses and the strains are represented by arrays of dimension 6x1, and the 6x6 matrix [C] is symmetric with, at most, 21 independent elements. In the simplest case of an isotropic material, the independent elastic constants are only two.

The objective of the structural analysis is the determination of the displacement, strain and stress fields given the applied forces and the constraints. The analytical model consists of three equilibrium equations, six compatibility equations (imposing the continuity of the material) and the stress-strain relation, known as Hooke's law. Discrete Finite Element Method (FEM) analysis is generally used to obtain an approximate solution to engineering structural problems, and it is very accurate and efficient for complex and irregular structures with arbitrary loads and constraints. FEM is based on subdividing the domain in an arbitrary number of sub-domains (elements), the unknowns are calculated in a number of nodes distributed in each elements.

The error of the numerical solution depends on the number of elements and nodes used for the simulation and there is a trade-off between accuracy and the computational resources in terms of the required computer time and memory. If the numerical solution obtained using a more refined discrertization does not change, such a solution is called grid-independent and represents the sought solution to the engineering problem.

  • Other errors (the difference between the discrete solution obtained by computer simulation and the actual state of stress and deformation of the real problem) are determined by the problem complexity. In the case of sail deformation due to the aerodynamic load, such a complexity derives from the following, most important, points:

  • orthotropic elastic characteristics of the sail material, which need to be known or determined by experimental analysis;
  • the sail is constructed by joining many small pieces which are to be glued or sewed together; each single piece presents a different warp orientation and the elastic properties may present some degree of discontinuities at the junctions;
  • the sail thickness is very small (often smaller than one millimeter), however, near the head and the clew of the sail, many layers of fabric are place one upon the other, locally increasing the overall thickness and changing the elastic properties;
  • battens are used to increase the sail stiffness;
  • the sail is constrained in a complex way, mainly at the luff and the leech;
  • tension loads are applied along the luff and the leech and they can be arbitrarily varied depending on the wind.

The second part of the article will follow on the next issue of Superyacht.