可压缩流动通常具有复杂的流动结构，如激波、流动转捩以及激波/湍流相互作用等，因此对于可压缩复杂流动的数值模拟就要求数值格式既能够无振荡地捕捉激波，又具有高精度、低耗散的性质，以识别流场中的小尺度涡结构。加权基本无振荡（Weighted essentially non-oscillatory scheme，简称WENO）格式作为近些年来最具有代表性的高精度激波捕捉格式之一，自提出以来得到了广泛应用和持续发展。本文针对WENO格式在临界点精度降阶的问题，发展了高精度、低耗散的WENO格式，并将发展的方法应用于翼型气动力计算以及激波/复杂流动干扰的数值模拟。主要的工作和研究成果如下：

（1）发展了一致最优精度收敛的WENO格式。本文分析了几种WENO格式（如Jiang & Shu提出的WENO-JS格式、Borges等提出的WENO-Z格式以及Baeza等提出的OWENO格式）在临界点降阶的原因，在此基础上利用OWENO格式中的调和平均的思想，设计了新的高阶整体光滑因子，实现了在各阶临界点处均满足五阶收敛的充分条件，同时引入了自适应函数来控制数值耗散，构造了一致最优精度的五阶WENO格式。数值算例表明，新的五阶WENO格式在各阶临界点处均能实现五阶精度收敛，同时在复杂流动的模拟中具有更低的数值耗散和较强的鲁棒性。

（2）构造了基于四点模板的间断识别方法及三阶混合基本无振荡算法。与五阶WENO格式相比，三阶WENO格式所用节点少，具有更好的稳定性和更强的鲁棒性。但正是由于三阶格式仅使用了三个节点，因而在临界点难以有效的提高精度从而产生较大的数值耗散。本文通过引入下游节点，设计了四点间断识别方法，并利用该方法构造了三阶WENO混合格式，数值算例显示混合格式能够准确地识别间断，在光滑区域（含临界点）实现一致的三阶精度收敛，同时新的三阶混合格式具有更低的数值耗散。

（3）激波/湍流边界层干扰过程中涉及到激波、转捩、流动分离和激波/湍流边界层干扰等不同类型的复杂流动结构，此类问题的数值模拟对数值格式的精度和耗散性质要求较高。本文利用前文发展的五阶优化WENO格式对来流马赫数$Ma = 2.9$，$24^\circ$压缩折角构型的超声速激波/湍流边界层干扰问题进行了直接数值模拟。数值结果表明，本文发展的WENO格式在激波/湍流边界层干扰问题的模拟中具有较高的计算精度和较低的耗散，计算结果与实验数据和已有的DNS数据较好吻合，计算分辨率与七阶WENO-JS格式相当，计算效率有明显提升。

For the numerical simulation of the compressible flows, which have complex flow structures, such as shock waves, transition, and shock wave/turbulence interaction. It is required that the  used numerical algorithms can capture shock waves without oscillations and also resolve the small-scale structures with high accuracy and low dissipation. The weighted essentially non-oscillatory (WENO) scheme, as one of the outstanding capturing methods in recent years, has been extensively applied and improved since it was put forward. Based on the analysis of the accuracy of WENO scheme at high-order critical points, this paper develops a series of WENO schemes with uniform convergence rate even at high-order critical points. Many numerical examples, including the direct numerical simulation of shock/turbulent boundary layer interation,  are tested to confirm the performances of these new schemes.

（1）A WENO scheme with the uniform convergence rate of fifth-order at high-order critical points is developed. First, we analyze the convergence rate of several fifth-order WENO scheme，including the WENO-JS scheme, the WENO-Z scheme, and the OWENO scheme. Then, a new high-order global stencil smoothness indicator is constructed by using the harmonic averaging. The resulting weights meet the sufficient condition for fifth-order convergence. In addition, an adaptive function generated by the sub-stencil smoothness indicators is introduced to control the numerical dissipation. Numerical examples show that the new WENO scheme can achieve fifth-order convergence rate at high-order critical points, and has low dissipation and strong robustness for simulating the complex flow fields.

（2）A discontinuity-detecting method with a four-point stencil  and a third-order hybrid WENO scheme are proposed. Compared to the fifth-order schemes, the third-order WENO scheme uses less grid nodes and has better stability and robustness. However, it is because the third-order scheme only uses three nodes that it is difficult to effectively improve accuracy at critical points and also generates excessive numerical dissipation. This article designs a four point discontinuity-detecting method by introducing a downstream node. And then a hybrid scheme of the upwind scheme and the third-order WENO scheme is constructed. Several numerical results show that the discontinuity-detecting method is effective, and the hybrid scheme can obtain uniform third-order accuracy in smooth regions including critical points and has low dissipation.

（3）The flows of shock wave/turbulent boundary layer interaction (SWTBLI) contain complex flow structures, such as shock waves, transition, flow separation, and the interaction between shock waves and the turbulent boundary layer. Hence, it is required that the used numerical schemes have high accuracy and low dissipation. This paper employs the uniformly fifth-order WENO scheme developed in the article to simulate the SWTBLI problems with Ma=2.9 and 24°compression corner configuration. The numerical results obtained by the new scheme is in agreement with the experimental data and existed DNS data. The new fifth-order WENO scheme is comparable with the seventh-order WENO-JS scheme, but the computational efficiency is significantly improved.