Second-order curved shock theory is developed and applied to planar and axisymmetric curved shock flow fields. Explicit equations are given in an influence coefficient format, relating the second-order gradients of pre-shock and post-shock flow parameters to shock curvature gradients. Two types of applications are demonstrated. First, the post-shock flow fields behind known curved shocks are solved using the second-order curved shock equations. Compared with the first-order curved shock equations, the second-order equations give better agreement with solutions obtained using the method of characteristics. Second, the second-order theory is applied to capture the curved shock shape with limited flow field information. In terms of the residual sum of squares of the curved shock, the second-order curved shock equations give a value one order of magnitude better than those given by the Rankine–Hugoniot equations and the first-order equations. This improved accuracy makes the second-order theory a good candidate for solving shock capture problems in computational fluid dynamics algorithms.

中文翻译：

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二阶弯曲激波理论

二阶曲线激波理论被开发并应用于平面和轴对称曲线激波流场。显式方程以影响系数格式给出，将冲击前和冲击后流动参数的二阶梯度与冲击曲率梯度相关联。演示了两种类型的应用程序。首先，使用二阶曲线激波方程求解已知曲线激波背后的激波后流场。与一阶曲线激波方程相比，二阶方程与使用特征法得到的解更加吻合。其次，应用二阶理论来捕捉流场信息有限的弯曲激波形状。就弯曲激波的残差平方和而言，二阶曲线激波方程给出的值比 Rankine-Hugoniot 方程和一阶方程给出的值好一个数量级。这种改进的准确性使二阶理论成为解决计算流体动力学算法中冲击捕获问题的良好候选者。