Model form uncertainty arises from physical assumptions made in constructing models either to model physical processes that are not well understood or to reduce the physical complexity. Understanding these uncertainties is important for both quantifying prediction uncertainty and unraveling the source and nature of model errors. Physics-based uncertainty quantification utilizes inherent physical model assumptions to estimate and ascertain the sources of model form uncertainty or error. Compared to data-based approaches, physics-based approaches can be extrapolated beyond available data and go beyond strictly uncertainty estimation. In this work, an implied models approach is developed where the transport equation for the model error is derived by taking the difference between the exact transport equation for a quantity of interest and the transport equation implied by a particular model form. The implied models approach is then specifically applied to the modeling of the Reynolds stresses by the Boussinesq eddy viscosity model. Budgets of the model error transport are analyzed to better understand the sources of error in two-equation Reynolds-averaged Navier-Stokes models focusing on the relative contributions from the Boussinesq hypothesis and the specific form of the eddy viscosity in turbulent channel flow at various friction Reynolds numbers. The results indicate that the errors are largely due to the misalignment of the mean strain rate tensor and the Reynolds stress tensor as well as the high degree of anisotropy near the wall, with errors in the shear component being dominant. An exploration of the $k\text{−}\varepsilon$ and $k\text{−}\omega$ models reveals that both models benefit from error cancellation. In particular, the improved results of the $k\text{−}\omega$ model over the $k\text{−}\varepsilon$ model are shown to be the direct result of this error cancellation. An exploration of the effect of friction Reynolds number on the error budgets reveals that the errors saturate with increasing Reynolds number owing to the relative decrease of anisotropy.

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基于物理的不确定性量化的湍流模型的隐含模型方法

模型形式的不确定性来自于构建模型时所做出的物理假设，这些物理假设要么用于建模尚未充分理解的物理过程，要么用于降低物理复杂性。理解这些不确定性对于量化预测不确定性以及揭示模型误差的来源和性质都是很重要的。基于物理的不确定性量化利用固有的物理模型假设来估计和确定模型形式不确定性或误差的来源。与基于数据的方法相比，基于物理的方法可以推断出可用数据之外的范围，也可以超出严格的不确定性估计范围。在这项工作中，提出了一种隐含模型方法，其中通过获取感兴趣量的精确传递方程与特定模型形式隐含的传递方程之间的差来导出模型误差的传递方程。然后，隐式模型方法专门用于通过Boussinesq涡流粘度模型对雷诺应力进行建模。分析模型误差传输的预算以更好地理解两方程雷诺平均Navier-Stokes模型中的误差源，重点是来自Boussinesq假设的相对贡献以及在各种摩擦下湍流通道中涡流粘度的特定形式雷诺数。结果表明，误差主要归因于平均应变率张量和雷诺应力张量的不对准以及壁附近的高度各向异性，其中剪切分量的误差占主导。探索$ķ\text{-}\varepsilon$ 和 $ķ\text{-}\omega$模型显示这两个模型都可以从错误消除中受益。特别是，改进后的结果$ķ\text{-}\omega$ 在模型 $ķ\text{-}\varepsilon$模型被证明是这种误差消除的直接结果。对摩擦雷诺数对误差预算的影响的探索表明，由于各向异性的相对减小，误差随着雷诺数的增加而饱和。