Data-driven turbulence modelling is becoming common practice in the field of fluid mechanics. Complex machine learning methods are applied to large high fidelity data sets in an attempt to discover relationships between mean flow features and turbulence model parameters. However, a clear discrepancy is emerging between complex models that appear to fit the high fidelity data well a priori and simpler models which subsequently hold up in a posteriori testing through CFD simulations. With this in mind, a novel error quantification technique is proposed consisting of an upper and lower bound, against which data-driven turbulence models can be systematically assessed. At the lower bound is models that are linear in either the full set or a subset of the input features, where feature selection is used to determine the best model. Any machine learning technique must be able to improve on this performance for the extra complexity in training to be of practical use. The upper bound is found by the stable insertion of the high fidelity data for the Reynolds stresses into CFD simulation. Three machine learning methods, Gene Expression Programming, Deep Neural Networks and Gaussian Mixtures Models are presented and assessed on this error quantification technique. We further show that for the simple canonical cases often used to develop data-driven methods, lower bound linear models can provide very satisfactory accuracy and stability with limited scope for substantial improvement through more complex machine learning methods.

数据驱动的湍流建模正在成为流体力学领域的常见做法。复杂的机器学习方法应用于大型高保真数据集，试图发现平均流特征和湍流模型参数之间的关系。然而，在似乎很好地先验地拟合高保真数据的复杂模型和随后在后验中成立的简单模型之间出现了明显的差异。通过 CFD 模拟进行测试。考虑到这一点，提出了一种由上限和下限组成的新型误差量化技术，据此可以系统地评估数据驱动的湍流模型。下限是在完整集或输入特征子集中线性的模型，其中特征选择用于确定最佳模型。任何机器学习技术都必须能够提高这种性能，以使训练中的额外复杂性具有实际用途。上限是通过将雷诺应力的高保真数据稳定插入 CFD 模拟而找到的。介绍了三种机器学习方法，基因表达编程、深度神经网络和高斯混合模型，并对这种误差量化技术进行了评估。