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Development of safer more fuel efficient passenger aircraft relies heavily of the numerical calculation of airflow around a subsonic airplane flying at close to the speed of sound. The air flow relative to existing commercial aircraft moves at subsonic speeds -- less than the speed of sound -- and hence is governed by an elliptic set of partial differential equations (PDEs). In some regions, such as a pocket over the wings, the air moves at supersonic speeds -- greater than the speed of sound. This flow is governed by a set of hyperpolic PDEs. These two types of equations model different behaviours and require an integrated solution method that is specifically devised. Aerodynamic flows which are governed by different types of equations in different regions are called transonic flows. Thin shock waves and slip lines separate the different regions.

Another complicating factor is the existence of very thin boundary layers over the surface of the airplane. These layers are present because of the air's viscosity. In an ideal flow over a streamlined airplane these boundary layers remain thin, only a few millimetres thick, and are unimportant. They are however - surprisingly perhaps, crucial to the generation of the smooth streamlined flow in the first place. Boundary layer separation occurs at high angles of attack, in the landing configuration of wings, and in poor wing-body-engine designs.

This increases the drag force acting to slow down the airplane thereby decreasing fuel efficiency. If the boundary layers separate inappropriately from the airplane, the drag can be too high and cause vehicle control problems which are dangerous. The numerical simulation of flow past aircraft and other objects is a formidable task. The presence of thin boundary layers, flow separation, shock lines and slip lines require the use of simplifying approximations and sophisticated mathematical and numerical methods.

Today, numerical simulations help in the design of new aircraft, automobiles and ships. Their use goes beyond computing fluid flow, and now includes simulations of jet engines and structural and thermal analysis of components. With recent advances in computational methods and computational resources, numerical simulations are often cheaper than experimental tests. Future advances will enable numerical computations to play a more prominent role.

Examples of ongoing projects in Computational Mathematics