Mathematical Sciences: Mathematical and Numerical Problems in Incompressible Fluids

9303779 E调查员研究流体流动中的两个问题。(1)。二维湍流的统计行为。这一领域存在两个主要问题。第一个与空间(尺度)行为有关，第二个与时间尺度行为有关。经典理论试图通过科尔莫戈罗夫式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组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2D湍流的另一种图景。(2)。边界条件下不可压缩流动的数学和数值问题。特别研究了边界层的分离以及分离前对N-S方程的逼近。最近，他证明了在存在不利压力梯度的情况下，稳定的边界层必须以自相似的方式分离和分离。现在，他采用了非定常普朗特方程，检验了它在描述零粘极限边界层方面的有效性。FIFE ET之前的结果。艾尔仅限于满足某种一致边界(与粘度无关)的解的类别。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，该问题抓住了整个问题的本质，并且是显式可计算的。E和Liu已经对投影法进行了这样的程序设计。人们可以将这两个问题视为研究湍流的准备。在现实中，大多数湍流是通过摩擦在流固边界产生的。大规模计算为研究这类问题提供了另一个维度。但要有效地利用这一新的维度，还需要强大的数值方法。目前，只要有边界，就连建立数值方法的基本问题都悬而未决。如果引入简单的代表性模型，洞察力可以得到极大的提高。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中的一个中心问题，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. ***