Accurate prediction of supersonic and hypersonic turbulent flows is essential to the design of high-speed aerospace vehicles. Such flows are mainly predicted using the Reynolds-Averaged Navier–Stokes (RANS) approach in general, and in particular turbulence models using the effective viscosity approximation. Several terms involving the turbulent kinetic energy (k) appear explicitly in the RANS equations through the modelling of the Reynolds stresses in such approach, and similar terms appear in the mean total energy equation. Some of these terms are often ignored in low, or even supersonic, speed simulations with zero-equation models, as well as some one- or two-equation models. The omission of these terms may not be appropriate under hypersonic conditions. Nevertheless, this is a widespread practice, even for very high-speed turbulent flow simulations, because many software packages still make such approximations. To quantify the impact of ignoring these terms in the RANS equations, two linear two-equation models and one non-linear two-equation model are applied to the computation of five supersonic and hypersonic benchmark cases, one 2D zero-pressure gradient hypersonic flat plate case and four shock wave boundary layer interaction (SWBLI) cases. The surface friction coefficients and velocity profiles predicted with different combinations of the turbulent kinetic energy terms present in the transport equations show little sensitivity to the presence of these terms in the zero-pressure gradient case. In the SWBLI cases, however, comparisons show that inclusion of k in the mean flow equations can have a strong effect on the prediction of flow separation. Therefore, it is highly recommended to include all the turbulent kinetic energy terms in the mean flow equations when dealing with simulations of supersonic and hypersonic turbulent flows, especially for flows with SWBLIs. As a further consequence, since k may not be obtained explicitly in zero-equation, or certain one-equation, models, it is debatable whether these models are suitable for simulations of supersonic and hypersonic turbulent flows with SWBLIs.

中文翻译：

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超音速和高超音速湍流的 RANS 方程的公式

准确预测超音速和高超音速湍流对于高速航空航天器的设计至关重要。这种流动主要使用雷诺平均纳维-斯托克斯 (RANS) 方法进行预测，特别是使用有效粘度近似的湍流模型。涉及湍动能的几个项（ķ) 通过在这种方法中对雷诺应力建模，明确出现在 RANS 方程中，并且类似的项出现在平均总能量方程中。在使用零方程模型以及一些一或二方程模型的低速甚至超音速速度模拟中，这些术语中的一些经常被忽略。在高超声速条件下，省略这些条款可能不合适。尽管如此，这是一种普遍的做法，即使对于非常高速的湍流模拟也是如此，因为许多软件包仍然进行这种近似。为了量化在 RANS 方程中忽略这些项的影响，将两个线性二方程模型和一个非线性二方程模型应用于计算五个超音速和高超声速基准案例，一个二维零压力梯度高超声速平板案例和四个冲击波边界层相互作用（SWBLI）案例。用输运方程中存在的湍动能项的不同组合预测的表面摩擦系数和速度分布对零压力梯度情况下这些项的存在几乎不敏感。然而，在 SWBLI 案例中，比较表明，包含ķ在平均流量方程中，对流量分离的预测有很大的影响。因此，在处理超音速和高超声速湍流模拟时，强烈建议在平均流动方程中包含所有湍流动能项，特别是对于带有 SWBLI 的流动。作为进一步的结果，由于ķ可能无法在零方程或某些一方程模型中明确获得，但这些模型是否适用于使用 SWBLI 模拟超音速和高超声速湍流仍存在争议。