潜流层的数学问题

The investigator and colleagues propose to study the Navier-Stokes-Darcy-heat equations and related systems which model coupled surface water-groundwater interaction，hyporheic flow in particular, together with thermal effects. The mathematical models that the PI proposes for the study of hyporheic flow embody a couple of challenges: the strong nonlinearity associated with the Navier-Stokes flow in the river, the enhanced nonlinearity via the inclusion of the physically and biologically important thermal influence, the substantial disparity of spatial and temporal scales between flows in the river and in surrounding porous media (small Darcy number), the uncertainty associated with the geometric form and the permeability of the riverbed and riverbank, and the different physics in different parts of the physical domain. Although these difficult issues have been studied separately before for some subsystems of the models under consideration here, the physical and biological need of knowledge of flow and transport in hyporheic zone requires us to investigate these issues in a coupled fashion. This is a challenge that has not been well-addressed so far. The PI and collaborators plan to investigate the models from several different angles. Firstly, the PI will conduct mathematical analysis of the models in terms of their well-posedness and their asymptotic behavior at the physically important small Darcy number regime. Secondly, the PI will design, and analyze accurate and efficient decoupled numerical methods for the models. Thirdly, the PI will use our model to assess the validity of various simplifications utilized by the water resources community in hyporheic flow studies. Tools from partial differential equations, functional analysis, asymptotic analysis, numerical analysis and computation will be combined to investigate these problems.

申请人准备研究Navier-Stokes-Darcy热模型及其相关系统。这些模型是研究地表水-地下水耦合作用与热效应，特别是潜流层的数学模型。这些模型非常具有挑战性，难点包括：Navier Stokes流带来的强非线性性，引入物理和生物上重要的热影响所带来的更强的非线性性，时空尺度的巨大差异，以及问题的多物理性。虽然对一些子问题已经有了不少工作， 由于物理和生物学需求我们必须考虑耦合的问题。这是迄今为止尚未解决的一个挑战。申请人和合作者计划从几个不同角度研究这些模型。首先，申请人将对模型进行数学分析，考虑模型的适定性和物理上重要的在小Darcy数假设下系统的渐近行为。其次，申请人准备设计和分析精准高效数值格式。第三，申请人将利用我们的模型来评估水资源领域在潜流层研究中一些常用的简化的有效性。我们准备综合利用偏微分方程，泛函分析，渐近分析，数值分析和计算来研究这些问题。

本项目研究了Navier-Stokes-Darcy热模型及其相关系统行为并在几个方面取得重要进展。这些模型是研究地表水-地下水耦合作用与热效应，特别是潜流层的重要数学模型。这些模型非常具有挑战性，难点包括：Navier Stokes流带来的强非线性性，引入物理和生物上重要的热影响所带来的更强的非线性性，时空尺度的巨大差异，以及问题的多物理性。取得的成果包括如下几个方面。首先，本项目对模型进行了数学分析，研究了模型的适定性和物理上重要的在小Darcy数假设下系统的渐近行为；其次，本项目解决了流体力学界和地下水研究领域中常用的几个自由流和多空介质流之间相互矛盾的界面条件之间的关系； 第三，设计和分析了多个精准高效数值格式；第四，提供了地下水研究中半解耦方法的理论依据并指出了该方法的局限性和改进方法。