Mesh-free methods for turbulent reacting flows: the next generation of DNS

湍流及其相关现象（如燃烧）的直接数值模拟（DNS）仍然是数学和物理科学中的重大挑战之一。涉及的长度和时间尺度的范围是显著的，具有极快的，结构精细的，多物种的反应过程与大尺度湍流运动非线性相互作用。高阶数值方法可以提供有效的分辨率，但目前的方法是基于高阶有限差分或（伪）谱方法，这些方法不太适合复杂的几何形状，因此对实际湍流应用的研究非常有限。发展非平凡几何的高阶数值格式对数值模拟和燃烧界有明显的好处。该项目的目的是开发一个有前途的新的无网格框架，用于任意复杂几何形状的湍流反应流的高阶DNS。最近发展的一种高阶无网格方法-局部各向异性基函数方法(LABFM) -显示出提供相当大的几何灵活性和高阶精度（例如空间的10阶）的希望。这项可行性研究的目的是在无网格框架中为下一代DNS代码奠定基础，从而实现迄今为止不可能的火焰-湍流-结构相互作用的模拟。从长远来看，这将导致对湍流反应流的基础和应用的更好理解，并将支持相关技术的发展，如大规模制氢、清洁燃烧和低热量气体利用。

The direct numerical simulation (DNS) of turbulence and related phenomena, e.g. combustion, remains one of the great challenges in the mathematical and physical sciences. The range of length- and time-scales involved is significant, with extremely fast, finely structured, many-species reaction processes interacting non-linearly with large-scale turbulent motions. High-order numerical methods can provide efficient resolution but current approaches are based on high-order finite differences or (pseudo-)spectral methods, which are not well suited to complex geometries, and hence the study of real turbulence applications is very limited. The development of high-order numerical schemes for non-trivial geometries is of clear benefit to the numerical modelling and combustion communities. The aim of this project is to develop a promising new mesh-free framework for high-order DNS of turbulent reacting flows in arbitrarily complex geometries. Very recent developments of a high-order mesh-free method - the Local Anisotropic Basis Function Method (LABFM) - show promise in providing considerable geometric flexibility as well as high-order accuracy (e.g. 10th order in space). The intention of this feasibility study is to lay the foundations for the next generation of DNS codes, in a meshless framework, enabling simulations of flame-turbulence-structure interactions hitherto impossible. In the longer term this will result in a greater understanding, both fundamental and in application, of turbulent reacting flows, and will support the development of related technologies, such as hydrogen production at scale, cleaner combustion, and low calorific gas utilisation.