Abstract: In order to explore valid methods for three-dimensional supersonic aerodynamic inverse problem， well-posedness of three-dimensional supersonic streamline pressure inverse problem was explored based on the concept of Bicharacteristics and Butler algorithm. For ensuring the uniqueness of the solution of this inverse problem， a statement that the one-to-one mapping relationship between wall pressure and three-dimensional coordinates was existent with a local constrained expansion or compression direction was proposed. According to the numerical solution of the mapping relationship， an abbreviated iMOC-3D solver for the pressure inverse problem based on Bicharacteristics method was established. The accuracy order of expansion and compression progress of the scheme was tested by comparing numerical solutions with analytical results of Prandtl-Meyer expansion wave and Busemann flow. The reliability of iMOC-3D solution was proved to some extent by the order of relative error being 1×10-4. In order to further verify the reliability and controllability of the design method， supersonic nozzles with different inlets such as rectangle and triangle were designed， and an axisymmetric nozzle with prescribed pressure distribution was designed， which was verified by CFD numerical simulations. Research has shown that for three dimensional streamlines， there are multiple flow directions to meet the downstream pressure condition that has been preset. So the solution of this inverse problem is not unique. Thus， inverse problem for three-dimensional supersonic flow is not well-posed. Comparing simulations between CFD and the solver， it would be found that the solver’s numerical solutions agree well with CFD simulations and the maximum error of flow field is 1%. Thus， the proposed method would be reliable and might provide new ideas for three-dimensional supersonic aerodynamic design.

Key words: Three-dimensional pressure inverse problem；Bicharacteristics method；Three-dimensional supersonic aerodynamic design；Numerical method

摘要： 为了进一步探索三维超声速气动反问题的求解方法，基于双特征线理论及Butler解法，研究了三维超声速流线压力反问题的适定性。为了确保解的唯一性，提出在限定壁面膨胀/压缩方向时存在壁面压力与三维坐标的一一映射关系。基于该映射关系，提出了三维压力反问题的双特征线求解技术(iMOC-3D 求解器)。采用Prandtl-Meyer膨胀波、Busemann进气道的理论解，对iMOC-3D求解方法的膨胀、压缩过程进行了精度评估，误差均为1×10-4量级。为了进一步验证设计方法的可靠性和易控性，设计了进口为矩形和三角形的超声速喷管；通过设计壁面压力分布，完成了均匀膨胀的轴对称喷管设计，并将设计结果与数值模拟进行对比验证。研究表明:预设三维流线下游未知点的压力值，存在多个流动方向满足该压力条件，即该问题的解不唯一，因此三维超声速流线压力反问题是非适定的。对比验证表明:所设计的流场与CFD计算得到的等值线符合得较好，流场参数的最大误差为1%。因此，所提出的双特征线解法具有一定的可靠性，有望为三维超声速流道设计提供新思路。

关键词: 三维压力反问题；双特征线方法；三维超声速气动设计；数值方法