Abstract: In order to explore valid methods for three-dimensional supersonic aerodynamics inverse problems， the characteristics equations and compatible equations of the reference plane method were deduced based on the method of descending three-dimensional characteristics. And a method for solving the inverse problem of three-dimensional curved shock wave was proposed. The design method could cut the three-dimensional flow field into multiple transverse planes， and solve characteristic equations and the compatibility equations in the reference plane which had a same form of two-dimensional MOC. The integral iteration method was applied in the direction of perpendicular to the reference plane in order to determine the cross derivations of solution point， which could insure the second-order accuracy. The accuracy order of the reference plane method is tested by comparing numerical solutions with analytical results of Taylor-Maccoll flow and curved shock theory. By contrast， it could be found that the pressure maximum relative error is 1.3×10-3 and 6.1×10-5， respectively. In order to further verify the reliability of the method， a three-dimensional wall surface generating a conical shock wave at 3 degrees of attack angle was design. And the reliability of the design result is verified using commercial software Fluent. It could be found that the numerical solutions of the solver are good in agreement with those of CFD numerical simulations， and the maximum error of flow field is no more than 2%. Therefore， the proposed method would be reliable and might provide new ideas for three-dimensional supersonic aerodynamic design.

Key words: Inverse problem of curved shock wave；Reference plane method；Three-dimensional supersonic aerodynamics design；Characteristics method；Conical shock wave

摘要： 为了进一步探索三维超声速气动反问题的求解方法，从降维的三维特征线方法出发，推导出适用于参考平面法的特征方程和相容方程，并提出曲面激波反问题的求解方法。该设计方法可以把三维流场切为多个横截平面。在参考平面内，通过局部迭代求解准二维的特征方程和相容方程；在垂直于参考平面方向，通过引入整体迭代法来确定解面上的交叉导数，使结果达到二阶精度。采用泰勒-麦科尔流动和斜激波关系式验证了该设计方法的精度，其中压力的相对误差分别为1.3×10-3，6.1×10-5。为了进一步验证设计方法的可靠性，设计了在3°来流攻角情况下，产生圆锥激波的三维型面，并采用商用软件Fluent验证所设计的三维构型。对比结果表明:所设计的流场与CFD数值模拟结果符合得较好，且流场参数最大误差不超过2%。

关键词: 曲面激波反问题；参考平面法；三维超声速气动设计；特征线法；圆锥激波