This work presents an anisotropic formulation for data assimilation (DA) of turbulent flows using the continuous adjoint system. This DA scheme serves as a tool to complement flow measurement data (the observations), which are usually limited in spatial range and measurable quantities, with the help of a predictive model in the framework of the Reynolds-averaged Navier–Stokes approach (the estimator). Herein, the measured data profiles at several locations are used as observations. In the estimator, a forcing term \({\varvec{F}}\) is added to the momentum equation to compensate for the contribution of the anisotropic eddy viscosity, whilst the isotropic part of the eddy viscosity is determined by conventional turbulence models. \({\varvec{F}}\) is optimised using the continuous adjoint equations, driving the predicted flow quantities towards the observations. Similar treatments are conducted for the scalar prediction. Three test cases are used for assessment and validation of the present DA scheme. The results of a circular jet demonstrate that the mean flow and scalar fields can be perfectly reproduced from the observations due to the \({\varvec{F}}\) optimisation. The flow over a blunt plate demonstrates the ability of the present DA scheme to reconstruct global flow fields from several measured velocity profiles. The flow and heat transfer on a ribbed wall reveal that once the global flow field is well recovered, the wall heat transfer characteristics can be correctly determined with a limited number of observations. The present anisotropic DA approach is quite practical for complex flow applications, serving as a complement to our previous isotropic adjoint-based data assimilation for further DA work.

Graphic abstract

中文翻译：

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各向异性公式对湍流平均流和标量场的数据同化

这项工作提出了一种使用连续伴随系统的湍流数据同化（DA）的各向异性公式。该DA方案可作为工具，对通常在空间范围和可测量数量方面受限的流量测量数据（观测值）进行补充，并借助雷诺平均Navier-Stokes方法（估算器）的预测模型）。在此，将在多个位置处的测量数据轮廓用作观察值。在估计器中，将强迫项\（{\ varvec {F}} \）添加到动量方程中，以补偿各向异性涡粘度的贡献，而涡粘度的各向同性部分则由常规湍流模型确定。\（{\ varvec {F}} \）使用连续伴随方程优化，将预测流量推向观测值。对标量预测进行了类似的处理。三个测试用例用于评估和验证当前​​的DA方案。圆形射流的结果表明，由于\（{\ varvec {F}} \），可以从观测值中完美地再现平均流场和标量场。优化。钝板上的流动证明了本发明的DA方案能够从几个测得的速度分布图中重建全局流场的能力。带肋壁上的流动和传热表明，一旦整体流场得到了很好的恢复，就可以通过有限的观察来正确确定壁的传热特性。当前的各向异性DA方法对于复杂的流动应用非常实用，可以作为我们以前基于各向同性的基于同伴的数据同化的补充，以进行进一步的DA工作。

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