Data assimilation and flow estimation

Starting from limited measurements of a flow field, data assimilation can be adopted to estimate missing information, for example to augment the resolution of under-resolved measurements or to determine unknown parameters. In this chapter, we introduce various data-assimilation techniques, and demonstrate their application in reconstructing transitional and turbulent channel flows from scarce observations. Depending on whether the observations are sequentially or simultaneously incorporated in the estimation, data assimilation methods can be categorized as filters or smoothers, respectively. We first focus on filters and describe the direct substitution, or assimilation, method, which is one of the most straightforward and computationally efficient approaches. We demonstrate the notion of synchronization in order to show that a simulation can be synchronized to an independent system that generated the observations, even when the flow is chaotic. We then introduce the ensemble Kalman filter (EnKF), which relies on multiple forward simulations to construct an optimal forcing that drives the flow state towards available observations. We specialize the method for flows with homogeneous directions. We explain how the implementation in Fourier space can reduce the computational cost of EnKF and improve the estimation accuracy, with aid of an example where we estimate turbulent channel flow from wall observations. The concept of an ensemble-based assimilation is also adopted in smoothers, specifically the ensemble variational (EnVar) method. EnVar is most suited for optimization of low-dimensional control vectors because the size of the ensemble is proportional to the number of dimensions to explore. For high-dimensional control vectors, we discuss adjoint-variational data assimilation (4DVar). In 4DVar, the exact gradient of the cost function with respect to any sized control vector is obtained from a single forward-adjoint loop. This gradient, and possibly also an approximation of the Hessian of the cost, are then adopted to minimize the cost. As an example, we apply 4DVar to reconstruct the initial condition of turbulent channel flow, using sparse velocity observations. Lastly, we introduce physics-informed neural networks (PINNs) and describe their application to assimilation of observation data while also enforcing physics constraints. © 2025 Elsevier Inc. All rights reserved.