This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang [J. Comput. Phys., 328 (2017), pp. 301-343], which was adapted to multicomponent flows by Du and Yang [J. Comput. Phys., 469 (2022), pp. 111548] in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes, resulting in a different dissipation coefficient. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also detail how to account for boundary conditions and incorporate previously developed techniques to reduce spurious pressure oscillations into the positivity-preserving framework. Furthermore, potential issues associated with the Lax-Friedrichs-type viscous flux function in the case of zero species concentrations are discussed and addressed. The resulting formulation is compatible with curved, multidimensional elements and general quadrature rules with positive weights. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction.

翻译：本文致力于为具有复杂热力学的多组分化学反应可压缩Navier-Stokes方程，发展一种完全守恒、保持正性且熵有界的间断Galerkin格式。具体而言，我们将先前提出的用于化学反应Euler方程的完全守恒、保持正性且熵有界的间断Galerkin方法推广至粘性流动情形。该格式的一个重要组成部分是Zhang [J. Comput. Phys., 328 (2017), pp. 301-343] 设计的保持正性的Lax-Friedrichs型粘性通量函数，该函数后被Du和Yang [J. Comput. Phys., 469 (2022), pp. 111548] 以将无粘和粘性通量视为单一通量的方式，推广至多组分流动。在此，我们同样将上述通量函数推广至多组分流动，但将无粘通量与粘性通量分离，从而得到一个不同的耗散系数。这种通量分离允许使用其他无粘通量函数，并且基于物理和数学原理，能够仅对演化状态的对流贡献部分实施熵有界约束。我们还详细阐述了如何在保持正性的框架内处理边界条件，并纳入先前发展的技术以减少虚假压力振荡。此外，讨论并解决了在物种浓度为零的情况下与Lax-Friedrichs型粘性通量函数相关的潜在问题。所得格式与弯曲多维单元以及具有正权重的一般求积规则兼容。计算了多种多组分粘性流动，范围从一维激波管问题到多维爆轰波以及激波/混合层相互作用。