We present a variational assimilation technique (4D-Var) to reconstruct time resolved incompressible turbulent flows from measurements on two orthogonal 2D planes. The proposed technique incorporates an error term associated to the flow dynamics. It is therefore a compromise between a strong constraint assimilation procedure (for which the dynamical model is assumed to be perfectly known) and a weak constraint variational assimilation which considers a model enriched by an additive Gaussian forcing. The first solution would require either an unaffordable direct numerical simulation (DNS) of the model at the finest scale or an inaccurate and numerically unstable large scale simulation without parametrisation of the unresolved scales. The second option, the weakly constrained assimilation, relies on a blind error model that needs to be estimated from the data. This latter option is also computationally impractical for turbulent flow models as it requires to augment the state variable by an error variable of the same dimension. The proposed 4D-Var algorithm is successfully applied on a 3D turbulent wake flow in the transitional regime without specifying the obstacle geometry. The algorithm is validated on a synthetic 3D data-set with full-scale information. The performance of the algorithm is further analysed on data emulating large-scale experimental PIV observations.

我们提出了一种变分同化技术（4D-Var），以从两个正交的2D平面上的测量值重建时间分辨的不可压缩湍流。所提出的技术结合了与流动动力学相关的误差项。因此，这是在强约束同化过程（假定动力学模型被认为是众所周知的）与弱约束变分同化之间的折衷，后者考虑了由加性高斯强迫所丰富的模型。第一个解决方案将需要模型无法承受的最佳直接数值模拟（DNS），或者没有参数化未解析尺度的不精确且数值不稳定的大规模模拟。第二种选择是弱约束同化 依赖于需要从数据中估计的盲错误模型。对于湍流模型，后一种选择在计算上也不切实际，因为它需要通过相同尺寸的误差变量来增加状态变量。提出的4D-Var算法已成功应用于过渡状态下的3D湍流尾流，而无需指定障碍物的几何形状。该算法在具有全面信息的合成3D数据集上进行了验证。该算法的性能将在模拟大规模实验PIV观测值的数据上进一步分析。提出的4D-Var算法已成功应用于过渡状态下的3D湍流尾流，而无需指定障碍物的几何形状。该算法在具有全面信息的合成3D数据集上进行了验证。该算法的性能将在模拟大规模实验PIV观测值的数据上进一步分析。提出的4D-Var算法已成功应用于过渡状态下的3D湍流尾流，而无需指定障碍物的几何形状。该算法在具有全面信息的合成3D数据集上进行了验证。该算法的性能将在模拟大规模实验PIV观测值的数据上进一步分析。