AI 中文总结
该研究针对零压力梯度可压缩湍流边界层，提出基于映射的精确积分壁面摩擦变换，改进了经典范·德里斯特理论，基于VIPL的变换在宽马赫数和热输入范围内预测误差低至3.07%。
AI 中文摘要
长期以来，在零压力梯度可压缩湍流边界层中，高效表面阻力预测的一条途径是将壁面摩擦系数$C_f$和动量厚度雷诺数$Re_\theta$映射到它们的“不可压缩”对应量。对大量直接数值模拟（DNS）数据库的重新评估表明，即使变换后的数据表现出更好的汇聚性，现有公式也无法一致地恢复参考不可压缩壁面摩擦行为。我们将映射后的“不可压缩”状态定义为物理可压缩边界层的常物性对应量，并从规定的平均速度和壁面法向坐标映射中推导变换因子。这种以定义为先的方法将壁面摩擦缩放与底层速度变换的全层精度联系起来，并揭示了继承的外层误差。范·德里斯特（Van Driest）理论被重构为有限雷诺数下的精确积分形式，经典的vD I和vD II变换作为主导阶渐近简化被恢复。我们量化了它们在有限雷诺数下的局限性，并将vD II的历史成功归因于截断误差的偶然抵消。该精确积分形式随后产生了改进的变换，通过先验缩放和从规定的宏观量及壁面热输入对$C_f$进行独立后验预测来评估。基于VIPL的改进变换表现出最佳的整体性能。在$0.30 \leq M_\infty \leq 13.64$和$-0.55 \leq \varTheta \leq 2.85$的范围内，其预测误差保持在11%以下，平均误差为3.07%。总体而言，该分析将壁面摩擦变换置于基于映射的精确积分基础上，将它们直接与规定的平均流映射相关联，同时避免了限制经典范·德里斯特理论在有限雷诺数下应用的主导阶渐近截断。
英文摘要
A long-standing route to efficient surface-drag prediction in zero-pressure-gradient compressible turbulent boundary layers is to map the skin-friction coefficient $C_f$ and momentum-thickness Reynolds number $Re_θ$ onto their `incompressible' counterparts. Reassessment against an extensive DNS database shows that existing formulations do not consistently recover the reference incompressible skin-friction behaviour, even when transformed data exhibit improved collapse. We define the mapped `incompressible' state as a constant-property counterpart of the physical compressible boundary layer and derive the transformation factors from prescribed mean-velocity and wall-normal-coordinate mappings. This definition-first approach links skin-friction scaling to the full-layer accuracy of the underlying velocity transformation and exposes inherited outer-layer errors. Van Driest's theory is recast in a finite-Re exact-integral form, with the classical vD I and II transformations recovered as leading-order asymptotic reductions. Their limitations at finite Reynolds numbers are quantified, and the historical success of vD II is traced to a fortuitous cancellation of truncation errors. The exact-integral formulation then yields modified transformations assessed through a priori scaling and standalone a posteriori prediction of $C_f$ from prescribed macroscopic and wall-thermal inputs. The VIPL-based modified transformation gives the best overall performance. Across $0.30 \leq M_\infty \leq 13.64$ and $-0.55 \leq \varTheta \leq 2.85$, its prediction errors remain below $11\%$, with a mean error of $3.07\%$. Overall, the analysis places skin-friction transformations on a mapping-based exact-integral footing, relating them directly to prescribed mean-flow mappings while avoiding the leading-order asymptotic truncations that limit classical van Driest theory at finite Reynolds numbers.
Comments43 pages, 18 figures
