湍流流体统计量的网格效应分析
Analysis of Mesh Effects on Turbulent Flow Statistics
AI总结：
本文研究了网格对湍流流体统计量的影响，通过分析Smagorinsky模型的能量耗散率，得出了在低分辨率网格下能量耗散的上界，并指出湍流边界层比Kolmogorov微尺度更为重要。
摘要：
湍流模型，如本文中的Smagorinsky模型，用于表示由于能量级联（即非线性）导致从解析到未解析尺度的能量损失。最近对几个湍流模型的能量耗散率的分析估计已经出现，但尚无研究将能量耗散限制在解析尺度上，即在空间离散化后$ h > $微尺度。本文针对Smagorinsky模型进行了此类研究。对于低分辨率网格$h$，推导了计算的时均能量耗散率$\langle \varepsilon (u^h) angle$的上界。对于粗网格大小$\mathcal{O}(\mathcal{Re}^{-1}) < h < L$，证明了$$ \langle \varepsilon (u^h) angle\leq ig[ ( rac{C_s\, δ}{h})^2+ rac{L^5}{(C_s δ)^4\,h}+ rac{L^{ rac{5}{2}}}{(C_s\, δ)^{4}}\, {h^{ rac{3}{2}}}ig]\, rac{U^3}{L}, $$其中$U$和$L$是全局速度和长度尺度，$C_s$和$δ$是模型参数。该上界与高雷诺数下的粘度无关，符合湍流的尺度理论。该估计表明对于任何$C_s>0$和$δ>0$，都会出现过耗散，这与不考虑壁面阻尼函数的模型粘度的数值证据一致。此外，分析表明湍流边界层比Kolmogorov微尺度对剪切流更为重要。
英文摘要：
Turbulence models, such as the Smagorinsky model herein, are used to represent the energy lost from resolved to under-resolved scales due to the energy cascade (i.e. non-linearity). Analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared, but none (yet) study energy dissipation restricted to resolved scales, i.e. after spacial discretization with $h >$ micro scale. We do so herein for the Smagorinsky model. Upper bounds are derived on the \textit{computed} time-averaged energy dissipation rate, $\langle \varepsilon (u^h)\rangle$, for an under-resolved mesh $h$ for turbulent shear flow. For coarse mesh size $ \mathcal{O}(\mathcal{Re}^{-1}) < h < L $, it is proven, $$ \langle \varepsilon (u^h)\rangle\leq \big[ (\frac{C_s\, δ}{h})^2+ \frac{L^5}{(C_s δ)^4\,h}+\frac{L^{\frac{5}{2}}}{(C_s\, δ)^{4}}\, {h^{\frac{3}{2}}}\big]\, \frac{U^3}{L}, $$ where $U$ and $L$ are global velocity and length scale and $C_s$ and $δ$ are model parameters. This upper bound is independent of the viscosity at high Reynolds number, is in accord with the scaling theory of turbulent. This estimate suggests over-dissipation for any of $C_s>0$ and $δ>0$, consistent with numerical evidence on the effects of model viscosity (without wall damping function). Moreover, the analysis indicates that the turbulent boundary layer is a more important length scale for shear flow than the Kolmogorov microscale.
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