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Issue 33 - 2008 : Flying On Wings: Supported By Numbers
It seems unlikely that the infuriating number game you linger over in the Sunday paper and a Falcon Jet hold much in common. And yet both Sudoku and Dassault high-mach winglet owe much to the same Eighteenth Century mathematician the Swiss-born Leonhard Euler.
Although Euler (pronounced "oiler") lacks the household recognition of great mathematicians like Archimedes and Newton, he is often cited as the most prolific contributor to mathematics ever known. Indeed, his work touches everything from music and astronomy to engineering and algebra.
As one French mathematician put it to his students: "... He is our master in everything."
2007 marked the 300th anniversary of Euler's birth, and as a result, the math great of the European Enlightenment is finally getting some play in the popular discussion of modern marvels. As with so many other scientific discoveries, some very contemporary aeronautical principles took root in Euler equations.
Euler is commonly acknowledged as the founder of rational fluid dynamics, forever coupling the study of geometry and algebra. Expanding on the theories of Isaac Newton, he furthered the ability to model and analyze the motion of rigid bodies (from a fixed point in space) subjected to force and motion, paving the way for future discoveries in aircraft and space design.
Unknowingly, the man credited with the discovery of pi has affected the design of recent Dassault models like the Rafale fighter.
"Euler equations have been the basis of this new branch of aerodynamics called computational fluid dynamics," Bruno Stoufflet, Dassault Falcon director of future planning and scientific strategy, explains. "The capacity of solving the equations of flows by computer has revolutionized aerodynamic design by providing a tool to optimize the shapes of aircraft."
"Whereas in the past, aerodynamic studies of an airplane were based primarily on wind tunnel experiments and simple engineering rules, the extraordinary increase of computing capabilities has opened, in these last 20 years, a new era in this field," Stoufflet says. "It is now possible to numerically solve the equations of fluid dynamics."
Using grids to solve Euler's equations, Dassault engineers can design and test aircraft parts - analyzing flow visualization over a wing, for example, before it is observed under wind tunnel conditions. Shapes can be constructed and optimized in 3-D before ever leaving the proverbial drawing board.
Trial and Error
Until the mid-seventies, aerodynamic design relied predominantly on the trial-and-error approach to selecting a shape that satisfied performance requirements, Stoufflet explains. The emergence of the computer drove the development of computational fluid dynamics, enabling engineers to directly solve the set of equations of fluid motion and to derive efficient design tools.
As computing power grew throughout the 1980's, so too did the scientific community's ability to tackle greater challenges in fluid mechanics. Euler's equations became the most popular and widely-used model, Stoufflet notes.
"Dassault Aviation developed an original approach, based on unstructured tetrahedral descretization of the space around the aircraft, which is still in use today," he says. "This Euler solver has been called Eugenie."
An Innate Gift
While today such calculations are carried out with the aid of computer, Euler tackled number theory in the 1700 armed with nothing but pencil, paper, and brainpower. And yet, his sheer output in the area of scientific publication is unrivaled. In all, he averaged about 800 printed pages a year and won the Paris Academy prize 12 times during his life.
The academy, an appendage of the French crown, was an academic circle in which prominent scientists could conduct experiments and share findings. While members ultimately acted as permanent consultants to the king, the society functioned as a think tank, designed to further mathematics and sciences through collective involvement and government backing. The institution held a prize competition, first annually and then bi-annually, soliciting submissions that tackled the day's leading scientific challenges.
French physicist and astronomer, Franis Arago, described Euler's unique gift as innate:
"He calculated just as men breathe, as eagles sustain themselves in the air." But Euler, often cited for his human qualities and "good-natured sarcasm," sloughed off such suggestions of brilliance, replying "his pencil seemed to surpass him in intelligence."
Blessed with a near photographic memory and a seemingly natural gift for mental computation, Euler entered the University of Basel at 14-years-old, marked for a religious career. He studied theology and Hebrew before catching the attention of one of the preeminent mathematicians of the time, Johann Bernoulli.
Bernoulli was the first to conceive fluid dynamics as a mathematical-physical science. Bernoulli's Law explained that the faster speed of the air along the top of a wing leads to reduced air pressure, thus producing lift. His work was later advanced by his son Daniel.
Euler received private lessons in mathematics from Bernoulli while still studying to fulfill the religious ambitions mapped out for him by his father, a Protestant minister. At 17, Euler completed his master's degree and began publishing on mathematics at the tender age of 18.
Assisted by Daniel Bernoulli, Euler is the physicist who established the set of laws describing the dynamics of flows from Newton's classical mechanics, Stoufflet explains. And just as Euler expanded on Newtonian physics, so Dassault Aviation has built upon Euler's findings.
"[We have] been the first company to solve the Euler equations on an industrial basis. The outputs provided by this numerical code have taken a significant part in the design of our recent aircraft," Stoufflet says.
Today's Applications
Such progress enabled Dassault to unveil the Falcon 2000LX, added to its line of large-cabin business jets in 2007.
When choosing its winglet shape, engineers from Dassault and design partner Aviation Partners International (API) were able to quickly evaluate the drag reduction which could be achieved using Euler solver computations, Stoufflet notes. Throughout the aircraft's design process, engineers were able to simultaneously optimize aerodynamics, wing structure, materials, maintenance and retrofit needs, resulting in better range and boosting fuel efficiency.
The Rafale fighter and Falcon 7X also owe much to Dassault's "Eugenie," Stoufflet explains.
"The Dassault Aviation Euler solver has been a decisive aerodynamic tool, [used] to design the intricate and innovative air inlets of the Rafale fighter, which have a very original position on the aircraft," Stoufflet says.
The Rafale features a delta wing with close-coupled canards. Dassault's in-house research in computational fluid dynamics has shown the specific benefits of close coupling between the wings and the canards.
The same tool was also the basis for the early design of the Falcon 7X wing "in order to select the overall planform, the sweep and dihedral angles," Stoufflet says. Engineers also employed the Euler solver when testing the 7X aeroelasticity and aeroacoustics, he says.
"... Dassault Aviation has always committed to deliver aircraft with matchless performance, and can do so thanks to the strong attention put on design. It is why aerodynamic capacities have continuously increased in order to stay at the forefront of the state of the art."
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