Sculpting of landforms by fluid flow

流体流动与表面之间的相互作用通常是一个具有挑战性的非线性自由边界问题：流场取决于表面形状，而表面形状又取决于流固边界附近的局部流体输运性质。人们对这些过程很感兴趣，特别是在这种相互作用导致从厘米到几十公里的长度范围内的显著特征和模式的地质背景下。一些具体的例子包括分叉的溪流网络、冰星、通风现象以及浮冰和火星景观下洞穴的表面扇形。这些例子中流动的流体可以是海洋、地表水或地下水，也可以是风，地形的形成可以通过磨损、侵蚀、泥沙输送、降水或溶解来实现。在二维中，假设无粘性流体的无旋流动，并且与流体流动相关的时间尺度相比，表面演化缓慢，上述自由边界问题可以用从具有固定边界的某个正则域到具有表示流固界面的演化边界的物理域的保角映射来表示。然后，数学任务就变成了确定依赖于时间的映射。这个博士项目将结合解析、渐近(例如早/晚)和数值方法(例如将Polubarinova-Galin简化为高阶耦合常微分方程组)来描述和解决一系列问题。各种初始面将被考虑，例如圆形/椭圆形物体、平面；以及强迫类型，例如界面的法向速度与流体压力成正比。脱落和困住的漩涡的影响是特别有趣的：它们在绕任意形状物体的流动中的包含将使用由Trefeten和同事最近开创的类型的拉普拉斯‘快速’解算器来计算。这种方法还能够计算到正则区域的共形映射，从而为计算在涡旋存在的情况下演化的物体提供了可能性。确定被困漩涡的存在是否会像在一些沙漠景观和火星表面上常见的那样，产生边缘锋利的雕刻表面，这将是一件有趣的事情。该项目将引起应用复杂分析、流体力学、涡旋动力学的研究人员以及致力于自由边界问题的应用数学家的兴趣。该项目也可能对地理学家、地球和行星科学家产生跨学科的吸引力。这项研究的主题是EPSRC的“流体动力学和空气动力学”。

The interaction between fluid flow and surfaces which evolve in response to erosion, abrasion, chemical dissolution or freezing/melting effects is, in general, a challenging nonlinear free boundary problem: the flow field depends on the surface shape which, in turn, depends on the local fluid transport properties near the fluid-solid boundary. There is much interest in these processes, especially in a geological context where the interaction leads to striking features and patterns spanning lengthscales from centimetres to tens of kilometres. Some specific examples include ramified stream networks, ice-stars, ventifacts and the surface scalloping of caves, beneath ice floes and the Martian landscape. The flowing fluid in these examples can be either ocean, surface- or ground-water, or wind, and the shaping of the landscape can be through abrasion, erosion, sediment transport, precipitation or dissolution.In two-dimensions, and assuming irrotational flow of inviscid fluid and that the surface evolves slowly in comparison to the timescale associated with the fluid flow, the free boundary problems described above can be formulated in terms of a conformal map from some canonical domain with fixed boundaries to the physical domain with an evolving boundary representing the fluid-solid interface. The mathematical task then becomes one of determining the time-dependent map. It can be shown that the map satisfies a Polubarinova-Galin type equation with forcing dependent on the 'physics' of the erosion process e.g. abrasion, melting.This PhD project will formulate and solve a series of problems using a combination of analytic, asymptotic (e.g. early/late times) and numerical methods (e.g. reducing the Polubarinova-Galin to a high-order system of coupled ODEs). A variety of initial surfaces will be considered e.g. circular/elliptical objects, flat surfaces; and forcing types e.g. normal velocity of the interface being proportional to the fluid pressure.The effect of shed and trapped vortices is of special interest: their inclusion in the flow about arbitrary shaped bodies will be computed using Laplacian 'fast' solvers of the type recently pioneered by Trefethen and co-workers. Such methods are also capable of computing conformal maps to canonical domains, opening up the possibility of computing the evolution of evolving bodies in the presence of vortices. It will be interesting to determine if the presence of trapped vortices gives rise to sculpted surfaces with 'sharp' edges as are commonly observed in some desert landscapes and the Martian surface.The project will be of interest to researchers in applied complex analysis, fluid mechanics, vortex dynamics and applied mathematicians working on free boundary problems. The project is also likely to have interdisciplinary appeal to geographers, Earth and planetary scientists.The research falls under the EPSRC theme 'Fluid dynamics and Aerodynamics'.