Mathematical Sciences: Analysis & Calculation of Solutions to the Acoustics Equations

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The principal investigator will continue his research into the numerical solution of the acoustic equations that govern the generation and propagation of noise by aircraft. Specifically he will study two major problem areas: the development and implementation of numerical codes that can resolve the acoustic equations in the presence of vorticity, and the posing of physically correct and numerically consistent boundary conditions for flow problems in unbounded domains. The vorticity that is generated near sharp edges like the engines or the landing gear creates difficulties for conventional numerical codes, and so the principal investigator is developing a finite element code that obviates many of the problems associated with vorticity. With regard to the posing of proper boundary conditions it is necessary to set boundary conditions on an artificial interface, in order to make the computations for a problem in an infinite domain tractable. The problem of aircraft noise during landings and takeoffs is of growing concern to many communities and local governments. One obvious solution of this problem is to design airplanes whose engines and exterior surfaces operate at minimal noise levels. As one can imagine easily this is a complicated phenomenon, and so mathematicians and aerodynamicists often resort to modelling the complex sets of equations with simpler ones, and then solving the approximate equations numerically. However this can be tricky too, since aircraft generate vorticity, which influences the noise pattern and which often defeats conventional numerical schemes. The principal investigator is working on numerical algorithms that can handle the vorticity and the unboundedness of the flow domain in a computationally efficient manner.