Guderley-Mach反射的有限强度灵敏度与Euler-UTSD对应关系
Finite-Strength Sensitivity and Euler--UTSD Correspondence for Guderley--Mach Reflection
AI总结：
本研究针对近掠入射的弱激波反射，推导了等熵势流问题的一阶有限强度扰动，结合数值方法得到固定正则入射角的灵敏度系数，通过链式法则转换为固定λ路径的系数，经自相似欧拉研究验证了 leading-order 的Euler-UTSD对应关系， cubic-order对应关系仍未解决。
AI中文摘要：
近掠入射的弱激波反射在跨声速弱激波标度后，由自相似非定常跨声速小扰动（UTSD）自由边界问题支配。我们沿固定正则入射角a=α/δ的路径推导并微分了相应等熵势流问题的一阶有限强度扰动，其中μ=δ²=2(M²-1)。二阶回流自适应有限体积法、精确离散切线与伴随算子，以及受Rankine-Hugoniot约束的拟合主前沿，给出固定a的正则灵敏度H₂,a^PF(0.5;1.4)=-0.217±0.012；完全微分的物理逆映射给出诊断用固定a系数K₂,a^PF≃-0.266。我们推导了将这些量转换为特殊固定λ路径（λ=(M-1)/α²）的精确链式法则，明确表明该转换需要对领先UTSD分支的独立入射角导数，因此无法仅从固定a的计算中推断。强度相关加密的匹配边界自相似欧拉研究包含21个合格非线性状态和252次常见前沿函数族评估，耦合外推得到g₀^Eul=0.510±0.006和物理领先角系数G₀^Eul=0.256±0.004，与激波拟合的UTSD极限一致。使用18个欧拉状态的单独等强度相位 bracket 审核显示，捕获激波的亚格子相位与所需三次信号相当。所得有限分辨率欧拉割线与势流修正兼容，但它们的ρ=h_η/√μ→0外推不具有模型稳定性。因此，领先阶Euler-UTSD对应关系得到数值验证，而三次阶欧拉对应关系仍未解决。
英文摘要：
Weak shock reflection at nearly glancing incidence is governed, after the transonic weak-shock scaling, by a self-similar unsteady transonic small-disturbance (UTSD) free-boundary problem. We derive and differentiate the first finite-strength perturbation of the corresponding isentropic potential-flow problem along paths of fixed canonical incidence $a=α/δ$, where $μ=δ^2=2(M^2-1)$. A second-order refluxed adaptive finite-volume method, exact discrete tangents and adjoints, and a Rankine--Hugoniot-constrained fitted principal front give the fixed-$a$ canonical sensitivity $H_{2,a}^{\PF}(0.5;1.4)=-0.217\pm0.012$; a fully differentiated physical back-map gives the diagnostic fixed-$a$ coefficient $K_{2,a}^{\PF}\simeq-0.266$. We derive the exact chain rule that converts these quantities to the distinguished fixed-$λ$ path $λ=(M-1)/α^2$, showing explicitly that the conversion requires the independent incidence derivative of the leading UTSD branch and therefore cannot be inferred from the fixed-$a$ calculation alone. A matched-boundary self-similar Euler study with strength-dependent refinement contains 21 qualified nonlinear states and 252 evaluations of a common front-functional family. Coupled extrapolation gives $g_0^{\Eul}=0.510\pm0.006$ and the physical leading-angle coefficient $G_0^{\Eul}=0.256\pm0.004$, consistent with the shock-fitted UTSD limits. A separate same-strength phase-bracket audit using 18 Euler states shows that the captured-shock subcell phase is comparable to the desired cubic signal. The resulting finite-resolution Euler secants are compatible with the potential-flow correction, but their $ρ=h_η/\sqrtμ\to0$ extrapolation is not model-stable. Thus leading-order Euler--UTSD correspondence is numerically verified, whereas cubic-order Euler correspondence remains unresolved.
