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# Math Modeling

When facing a quantitative problem, whether it be in engineering, earth and environmental sciences, medicine, or sports competition, a standard first step is to find an appropriate mathematical model. This may require the development of new equations, the construction of geometric models, the estimation of several relevant physical quantities under different flow regimes, or the analysis of uncertainty and parameter sensitivity.

A standard second step is to set up efficient and accurate numerical algorithms for the approximation of solutions to the mathematical model. Once the problem is solved by the computer (only rarely can a formula be found for solutions to a complicated mathematical model), the numerical results need to be displayed graphically and validated against experimental results. For sailboat design, this step also includes understanding the level of turbulence around the various yacht components, as well as the transition regions from smooth laminar flow to turbulent flow as parameters, like the direction of sail relative to the wind, are changed.

When dealing with downwind sails, streamlines visualization can be extremely useful in order to detect whether (and where) the air stream smoothly follows the sail surfaces or detaches to generate huge vortices and turbulence. Even if wakes behind downwind sails can be usually accepted (and often cannot be avoided), one of the main goals in order to increase the driving force is to keep the flow stream attached to the sail surface, and this can be obtained by means of a careful sail design and a suitable trimming of the sail sheets as function of the wind conditions (speed, direction, …). Enlarge.

(Image courtesy of CMCS (Chair of Modelling and Scientific Computing-EPFL-Lausanne))

This second step led to mathematical problems consisting of more than 160 million equations and 160 million unknowns. Even though they were working with a very fast computer at EPFL, mathematicians had to develop efficient algorithms to solve such problems quickly. Moreover, the researchers had to solve these equations many times. Parameters characterizing new shapes for the keel or the position of the winglets on the bulb needed to be varied to see which would produce the best results. In the end, the research group was able to run a complete simulation of a new boat configuration in less than 24 hours.

Next comes a crucial third step. The modeling of complicated real-world problems relies on feedback from testing to modeling to further testing to provide a powerful and versatile tool. So once the Alinghi team started training, the researchers were given suggestions on ways to improve the whole design. Due to this interaction, simulations were needed until the very final phase of the competition.

Detecting flow separation is also useful to check the stability properties of sails. In fact, due to the strong non-linearity of the Navier-Stokes equations, small variations of the inlet wind speed and/or direction can significantly change the behavior of the air stream, triggering abrupt flow detachments over some sail regions and reducing separations over some others. Ideal sails should remain reasonably stable under the effect of such perturbations, avoiding large deformations and flutter. Enlarge.

(Image courtesy of CMCS (Chair of Modelling and Scientific Computing-EPFL-Lausanne))

Mathematicians also considered different scenarios for competition and studied the interaction between the two opponent boats. Their analysis led to a decision-based numerical tool that could suggest to the tactician ways to react in different contexts.

The section "More Mathematics" provides more sophisticated details of the topics that were considered in the mathematics and the modeling.

Visualization of turbulent kinetic energy can be an important tool in order to check the size and intensity of the turbulent structures around the sails. This 3D representation can be integrated with other tools like the visualization of the velocity vectors and the streamlines. This comparison helps to reach a deep knowledge about the behavior of the involved physical phenomena and their complexity. Enlarge.

(Image courtesy of CMCS (Chair of Modelling and Scientific Computing-EPFL-Lausanne))