In this paper, high-order compact finite volume schemes on the unstructured grids based on the variational reconstruction are developed to solve the Reynolds averaged Navier-Stokes equations closed by the Spalart-Allmaras one-equation turbulence model. Encouraging progress is made in addressing the following two challenging problems: reducing the numerical errors on the large aspect ratio grids and avoiding the negative turbulent viscosity associated with the high-order methods. On grids with large aspect ratios, a three-step procedure is designed to optimize the functional parameters of variational reconstruction. In addition, an exponential decay procedure is proposed to cure the negative turbulent viscosity problem of the Spalart-Allmaras model. The exponential decay procedure has the advantage of being able to be used with any spatial discretization method and with the implicit temporal discretization. Numerical tests show significant benefits of the high-order schemes in predicting the skin frictions, capturing some important flow structures, and achieving grid-independent solutions. The numerical tests also show that the proposed schemes are sufficiently robust for practical applications.

本文提出了基于变分重构的非结构化网格上的高阶紧致有限体积格式，以求解由 Spalart-Allmaras 一方程湍流模型封闭的 Reynolds 平均 Navier-Stokes 方程。在解决以下两个具有挑战性的问题方面取得了令人鼓舞的进展：减少大纵横比网格上的数值误差和避免与高阶方法相关的负湍流粘度。在具有大纵横比的网格上，设计了一个三步程序来优化变分重建的功能参数。此外，提出了一种指数衰减程序来解决 Spalart-Allmaras 模型的负湍流粘度问题。指数衰减过程的优点是能够与任何空间离散化方法和隐式时间离散化一起使用。数值测试表明，高阶方案在预测皮肤摩擦、捕获一些重要的流动结构和实现与网格无关的解决方案方面具有显着优势。数值测试还表明，所提出的方案对于实际应用具有足够的鲁棒性。