论文标题
二维稳定放热反应欧拉在有限喷嘴中的跨性别冲击
Transonic Shocks for 2-D Steady Exothermically Reacting Euler Flows in a Finite Nozzle
论文作者
论文摘要
本文涉及存在跨气体反应的透射冲击，使欧拉在几乎平坦的喷嘴中流动，而放热反应的速率很小。关键点之一是定量确定冲击锋中的冲击锋的位置。我们专注于平面喷嘴扰动的贡献以及通过比较$σ$和$κ$的订单来确定冲击阵线位置的放热反应的贡献，其中$σ$代表平面喷嘴的扰动和$κ$的尺寸。为此，提出了线性化反应的欧拉系统的自由边界问题，以捕获冲击阵线的近似位置以及相关的近似冲击解决方案，通过该解决方案，可以通过非线性迭代方案建立靠近它的冲击溶液的存在。关键步骤之一是求解从冲击前后亚音速流的域中椭圆形子问题的溶解性条件得出的非线性方程，这决定了提议的问题的自由边界。分析了四种典型情况，描述了喷嘴边界几何形状与放热反应之间的可能相互作用。结果还表明，放热反应对喷嘴中的跨声音冲击具有稳定作用。
This paper concerns the existence of transonic shocks for steady exothermically reacting Euler flows in an almost flat nozzle with the small rate of the exothermic reaction. One of the key points is to quantitatively determine the position of the shock front in the nozzle. We focus on the contributions of the perturbation of the flat nozzle and the exothermic reaction in determining the position of the shock front by comparing the orders of $σ$ and $κ$, where $σ$ represents the scale of the perturbation of the flat nozzle and $κ$ the rate of the exothermic reaction. To this end, a free boundary problem for the linearized reacting Euler system is proposed to catch an approximating position of the shock front as well as the associated approximating shock solution, with which the existence of a shock solution close to it can be established via a nonlinear iteration scheme. One of the key steps is solving the nonlinear equation derived from the solvability condition for the elliptic sub-problem in the domain of the subsonic flow behind the shock front, which determines the free boundary of the proposed problem. Four typical cases are analyzed which describe possible interactions between the geometry of the nozzle boundary and the exothermic reaction. The results also manifest that exothermic reaction has a stabilization effect on transonic shocks in the nozzles.
