AI 中文总结
本研究通过直接数值模拟，探究壁面热边界条件对马赫2.5全粗糙湍流边界层的影响，提出等效沙粒粗糙度并验证粗糙度修正项，揭示雷诺类比失效及冷壁温度变换局限。
AI 中文摘要
我们研究了在匹配的$Re_\tau=784$下，马赫2.5湍流边界层在正弦粗糙度上的行为。我们考虑了三种壁面温度，$T_w/T_r=$ 1.0、0.7和0.4。每种壁面温度分别对应光滑和粗糙表面，其中粗糙表面采用有效斜率为0.5且匹配$k^+=79.1$的三维正弦轮廓。总共完成了六次直接数值模拟。对平均壁面剪切力和传热的分析详细说明了光滑与粗糙案例之间壁面摩擦和传热系数的增强，导致粗糙壁面上雷诺类比经典失效。我们还表明，尽管压力分量占主导，但不同壁面温度下剪切应力的差异由粘性分量驱动。粗糙度子层被发现为$R_{RSL}=3k-6k$，与文献一致。此外，动量边界层和热边界层的壁面偏移量均为$d=d_\Theta\approx0.5k$，表明该偏移量代表半峰谷高度而非“平均粗糙度高度”。对于平均动量边界层，我们发现当前条件通过半局部粗糙度雷诺数$k^*$匹配时，恢复了不可压缩粗糙度函数$\Delta u_1^+$。因此，提出了等效沙粒粗糙度$k_s^*\approx3.7k^*$。现有的可压缩变换无论壁面温度或粗糙度如何均成立，这与近期相反的主张相矛盾。广义雷诺类比适用于光滑壁面，但在粗糙壁面上失效；我们提出并验证了粗糙度修正项。可压缩平均温度变换在冷壁面情况下失效，尤其是当$(T_w-T_e)/(T_r-T_e)<0$时。鉴于变换奇异性，热粗糙度函数$\Delta\Theta^+$的报道需谨慎。
英文摘要
We investigate turbulent boundary layers at Mach 2.5 over sinusoidal roughness at matched $Re_τ=784$. We considered three wall temperatures, $T_w/T_r=$ 1.0, 0.7, and 0.4. Each wall temperature was repeated with a smooth and rough surface, the latter following a three-dimensional sinusoidal profile with effective slope 0.5 and matched $k^+=79.1$. In total six direct numerical simulations were completed. Analysis of mean wall shear and heat transfer detailed the augmentation in skin friction and heat transfer coefficient between smooth and rough cases; resulting in the classical failure of the Reynolds analogy for rough walls. We also show that differences in shear stress across wall temperatures are driven by the viscous component even though the pressure component dominates. The roughness sublayer was found to be $R_{RSL}=3k-6k$, consistent with the literature. Moreover, the wall offset $d=d_Θ\approx0.5k$ for both momentum and thermal boundary layers, signifying the offset represents the half peak-to-valley height rather than the "mean roughness height." For the mean momentum boundary layer, we find the present conditions recover the incompressible roughness function $Δu_1^+$ when matched via the semi-local roughness Reynolds number $k^*$. For this reason, an equivalent sand-grain roughness of $k_s^*\approx3.7k^*$ is proposed. The existing compressible transformations hold regardless of wall temperature or roughness - contradicting recent claims otherwise. The generalized Reynolds analogy works for smooth walls but fails for rough walls; a roughness correction term is proposed and validated. Compressible mean temperature transformations fail for cold walls, especially when $(T_w-T_e)/(T_r-T_e)<0$. The thermal roughness function, $ΔΘ^+$, is reported with caution given transformation singularities.
