用数值积分矩法求解湍流燃烧的PDF方程的研究

湍流燃烧是发动机中化学能释放的主要形式。概率密度函数（probability density function, PDF）输运方程模型是一种能精确封闭方程中化学源项的湍流燃烧模型，PDF方程和大涡模拟（LES）相结合的LES-PDF方法是湍流燃烧计算研究的一个前沿热点。PDF方程是关于时间、空间和组分空间的高维方程，数值求解较为困难。目前研究中主要采用随机的蒙特卡罗方法来解决高维PDF方程的计算可行性问题，但随机方法在模拟高马赫数湍流燃烧时受到统计涨落误差和激波数值误差相互干扰使计算精度和计算稳定性恶化的困扰。本项目拟采用确定性的数值积分矩方法求解高马赫数下流湍流燃烧PDF方程，研究基于有限个delta函数近似PDF的标准积分矩法和半离散积分矩法，构造矩的高精度保可实现性离散格式，发展从积分矩反算或从演化方程求解积分权和组分坐标点的有效算法，为有效求解湍流燃烧PDF输运方程提供新型的数值方法。

Turbulent combustion is the main form of chemical energy release in engines. The probability density function (PDF) transport equation is a versatile turbulent combustion model that can exactly close the filtered chemical source terms in the equation, and its combination with LES (LES-PDF approach) is a hot front in the computational study of turbulent combustion. The PDF equation is a high-dimensional equation in time, space and chemical composition space, and is very difficult to solve numerically. Current main numerical methods for the PDF equation are stochastic Monte Carlo methods which can solve the equation efficiently, but the interactions between statistic fluctuations of stochastic methods and numerical errors near shocks deteriorate the accuracy and stability severely for high Mach number cases. This project will adopt deterministic quadrature based moment methods to solve the PDF equation for high Mach number, research on the quadrature method of moments and semi-discrete quadrature method of moments based on finite number of delta functions approximate PDF, construct high order realizable discretization scheme for the moments, and develop effective inverse algorithm from the moments or direct solving method from evolution equations for the weights and composition abscissas. The aim of the project is to provide effective solution algorithms for the PDF equation of turbulent combustion at high Mach number.