化学反应纳维-斯托克斯方程的守恒间断Galerkin离散
A Conservative Discontinuous Galerkin Discretization for the Chemically Reacting Navier-Stokes Equations
摘要：
我们提出了一个详细的描述和验证的间断Galerkin有限元方法（DG）的多组分化学反应可压缩纳维-斯托克斯方程，保留了理想的性能DG，即离散守恒和高阶精度在光滑区域的流动。通过对热力学模型和所得到的弱形式的一致评价以及节点基的适当选择，保持相邻单元之间的压力平衡。因此，离散化不会在流动的平滑区域或温度连续的材料界面处产生非物理压力振荡。此外，我们提出了一个hp-自适应DG方法求解系统的常微分方程，DGODE，这是用来解决由于僵硬的化学反应的物种浓度的时间演变。耦合求解器适用于几个具有挑战性的测试问题，包括多组分冲击流以及化学反应爆炸，爆燃，剪切流与详细的动力学。我们证明，离散化不会产生非物理的压力振荡，并在适用时，我们验证它保持离散守恒。求解器还示出再现预期的温度和物种分布在整个爆轰以及预期的二维细胞爆轰结构。我们还表明，求解器可以产生准确的，高阶的，近似的温度和物种的配置文件，而无需人工稳定的情况下，一维预混火焰。最后，高阶解决方案的二维和三维多组分化学反应剪切流，计算没有任何额外的稳定，提出。
英文摘要：
We present a detailed description and verification of a discontinuous Galerkin finite element method (DG) for the multi-component chemically reacting compressible Navier-Stokes equations that retains the desirable properties of DG, namely discrete conservation and high-order accuracy in smooth regions of the flow. Pressure equilibrium between adjacent elements is maintained through the consistent evaluation of the thermodynamics model and the resulting weak form, as well as the proper choice of nodal basis. As such, the discretization does not generate unphysical pressure oscillations in smooth regions of the flow or at material interfaces where the temperature is continuous. Additionally, we present an hp-adaptive DG method for solving systems of ordinary differential equations, DGODE, which is used to resolve the temporal evolution of the species concentrations due to stiff chemical reactions. The coupled solver is applied to several challenging test problems including multi-component shocked flows as well as chemically reacting detonations, deflagrations, and shear flows with detailed kinetics. We demonstrate that the discretization does not produce unphysical pressure oscillations and, when applicable, we verify that it maintains discrete conservation. The solver is also shown to reproduce the expected temperature and species profiles throughout a detonation as well as the expected two-dimensional cellular detonation structure. We also demonstrate that the solver can produce accurate, high-order, approximations of temperature and species profiles without artificial stabilization for the case of a one-dimensional pre-mixed flame. Finally, high-order solutions of two- and three-dimensional multi-component chemically reacting shear flows, computed without any additional stabilization, are presented.
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