Mathematical Sciences: Mathematical and Numerical Problems in Incompressible Fluids

9303779 E 研究人员研究流体流动中的两个问题。 (1). 二维湍流的统计行为。 该领域存在两个主要问题。 第一个涉及空间（缩放）行为，第二个涉及时间缩放行为。 经典理论试图通过类似柯尔莫哥洛夫的论证来回答这些问题。 最近，E 和 Majda 非常详细的数值和分析结果的结合对这一经典图景提出了挑战。 研究人员继续这项研究，以建立二维湍流的替代图像。 （2）。存在边界时不可压缩流动的数学和数值问题。 特别是，研究者研究了边界层的分离以及分离前纳维-斯托克斯方程的近似。 最近他表明，在存在逆压梯度的情况下，稳定边界层必须分离并且自相似地分离。 现在，他采用非定常普朗特方程，检验其在描述零粘度极限下边界层的有效性。 Fife 等人之前的结果。 等人。仅限于满足某些统一边界（与粘度无关）的解决方案类别。 这样一个阶级的存在是值得怀疑的。 在数值方面，研究人员通过引入一个简单的模型问题来研究具有涡度边界条件的数值方法，该模型捕获了整个问题的本质并且可以显式计算。 E和Liu已经针对投影方法执行了这样的程序。 人们可以将这两个问题视为研究湍流的准备。 实际上，大多数湍流是通过摩擦在流固边界产生的。 大规模计算为研究此类问题提供了另一个维度。 但有效利用这个新维度还需要强大的数值方法。 目前，只要存在边界，即使是制定数值方法的基本问题也未得到解决。 如果引入简单的代表性模型，洞察力可以大大提高。 这激发了研究者对数值方法的研究和设计。对湍流的理解是应用数学的一个核心问题，具有广泛的科学和技术影响。 ***

9303779 E The investigator studies two problems in fluid flow. (1). Statistical behavior of two dimensional turbulence. There are two major problems in this area. The first is concerned with the spatial (scaling) behavior and the second the temporal scaling behavior. Classical theory attempts to answer these questions via Kolmogorov-like arguments. Recently this classical picture is challenged by a combination of very detailed numerical and analytical results of E and Majda. The investigator continues this study to establish an alternative picture for 2D turbulence. (2). Mathematical and numerical problems for incompressible flows in the presence of boundaries. In particular, the investigator studies the separation of boundary layers and the approximation to Navier-Stokes equation before the separation. Recently he has shown that in the presence of an adverse pressure gradient the steady boundary layers have to separate and separate self-similarly. Now he takes up the unsteady Prandtl equation, examining its validity in describing the boundary layers in the zero-viscosity limit. Previous results of Fife et. al. are restricted to the class of solutions satisfying some uniform bounds (independent of viscosity). The existence of such a class is in doubt. On the numerical side, the investigator studies numerical methods with vorticity boundary conditions through the introduction of a simple model problem that captures the essence of the full problem and is explicitly computable. Such a program has already been carried out for the projection method by E and Liu. One can view both problems as preparation for studying turbulence. In reality, most turbulence is generated at the fluid-solid boundary through friction. Large-scale computing offers another dimension for studying such problems. But the efficient exploitation of this new dimension also requires powerful numerical methods. At the present time, even the basic issues in formulating a nu merical method are unsettled whenever boundaries are present. Insight can be improved drastically if simple representative models are introduced. This motivates the investigator's study and design of numerical methods. Understanding of turbulence is a central problem in applied mathematics, with broad scientific and technological ramifications. ***