Mathematical Sciences: Some Mathematical Problems in Transonic Aerodynamics & in Flows with Vorticity

数值和渐近技术将被用来研究飞机机翼的跨声速流动和具有分布涡度的流动。需要分析的具体跨声速问题包括最优临界轴对称机身的设计、升力、不太细长的跨声速机翼的流动和跨声速面积法则，以及风洞流动。将在一定程度上利用跨音速小扰动公式。涡流问题的研究将包括发展和澄清整合非线性和粘性效应的理论。为了处理复杂的非线性问题，跨音速和涡流问题都需要数值处理和渐近处理之间的相互作用。今天的商用飞机在跨音速状态下巡航。在这种情况下，阻力(与前进运动相反的力)迅速上升，因此人们对了解跨音速流动和减阻机翼的设计感兴趣。对具有分布涡量的流动(感兴趣环境中的涡状运动的某种分布)的研究不仅是流体力学中的基本性质，而且在具体应用中也具有相当大的兴趣。这种应用的一个例子是涡轮机械中空气动力损失和热负荷的预测。研究人员将使用现代应用数学方法来处理拟议研究领域中复杂的问题。

Numerical and asymptotic techniques will be used to investigate transonic flow about aircraft wings and flows with distributed vorticity. Specific transonic problems to be analyzed are design of optimum critical axisymmetric bodies, flows about lifting, not-so-slender transonic wings and the transonic area rule, and wind tunnel flows. Some use will be made of the transonic small disturbance formulation. The investigations of the vortical flow problem will involve development and clarification of theories integrating nonlinear and viscous effects. Both the transonic and vortical flow problems require interaction between numerical and asymptotic treatments in order to deal with complex nonlinearities. Present day commercial aircraft cruise in the transonic regime. In this regime drag (the force opposing forward motion) rises rapidly hence there is interest in understanding transonic flows and in the design of reduced drag wings. The study of flows with distributed vorticity (a certain distribution of whirl-like motion in the environment of interest) is both of a fundamental nature in fluid mechanics, but also of considerable interest in specific applications. One example of such an application is the prediction of aerodynamic losses and heat loads in turbo-machinery. The investigators will use modern methods of applied mathematics to deal with the complexities of the problems in the areas of research proposed.