Section: New Results

Tracking, Data assimilation and model-data coupling

Optimal control techniques for the coupling of large scale dynamical systems and image data

Participants : Mohamed Yacine Ben Ali, Pranav Chandramouli, Dominique Heitz, Etienne Mémin.

In this axis of work we are exploring the use of optimal control techniques for the coupling of Large Eddies Simulation (LES) techniques and 2D image data. The objective is to reconstruct a 3D flow from a set of simultaneous time resolved 2D image sequences visualizing the flow on a set of 2D plans enlightened with laser sheets. This approach is experimented on shear layer flows and on wake flows generated on the wind tunnel of Irstea Rennes. Within this study we aim to explore techniques to enrich large-scale dynamical models by the introduction of uncertainty terms or through the definition of subgrid models from the image data. This research theme is related to the issue of turbulence characterization from image sequences. Instead of predefined turbulence models, we aim here at tuning from the data the value of coefficients involved in traditional LES subgrid models. A 4DVar assimilation technique based on the numerical code Incompact3D has been implemented for that purpose to control the inlet and initial conditions in order to reconstruct a turbulent wake flow behind an unknown obstacle. We are actually extending this first data assimilation technique to control the subgrid parameters. This study is performed in collaboration with Sylvain Laizet (Imperial College). In another axis of research, in collaboration with the CSTB Nantes centre and within the PhD of Yacine Ben Ali we will explore the definition of efficient data assimilation schemes for wind engineering. The goal will be here to couple Reynolds average model to pressure data at the surface of buildings. The final purpose will consist in proposing improved data-driven simulation models for architects.

Ensemble variational data assimilation of large-scale dynamics with uncertainty

Participant : Etienne Mémin.

This study is focused on the coupling of a large scale representation of the flow dynamics built from the location uncertainty principle with image data of finer resolution. The velocity field at large scales is described as a regular smooth component whereas the complement component is a highly oscillating random velocity field defined on the image grid but living at all the scales. Following this route we have assessed the performance of an ensemble variational assimilation technique with direct image data observation. Good results have been obtained for simulation under location uncertainty of 1D and 2D shallow water models [26]. This open the way to the definition of efficient data assimilation schemes for the coupling of high resolution data with large scale dynamical system.

Reduced-order models for flows representation from image data

Participants : Mamadou Diallo, Dominique Heitz, Cédric Herzet, Etienne Mémin, Valentin Resseguier.

During the PhD thesis of Valentin Resseguier we proposed a new decomposition of the fluid velocity in terms of a large-scale continuous component with respect to time and a small-scale non continuous random component. Within this general framework, an uncertainty based representation of the Reynolds transport theorem and Navier-Stokes equations can be derived, based on physical conservation laws. This physically relevant stochastic model has been applied in the context of POD-Galerkin methods. This uncertainty modeling methodology provides a theoretically grounded technique to define an appropriate subgrid tensor as well drift correction terms. The pertinence of this stochastic reduced order model has been successfully assessed on several wake flows at different Reynold number. It has been shown to be much more stable than the usual reduced order model construction techniques. Beyond the definition of a stable reduced order model, the modeling under location uncertainty paradigm offers a unique way to analyse from the data of a turbulent flow the action of the small-scale velocity components on the large-scale flow [25]. Regions of prominent turbulent kinetic energy, direction of preferential diffusion, as well as the small-scale induced drift can be identified and analyzed to decipher key players involved in the flow. This study has been published in the journal of fluid mechanics. Note that these reduced order models can be extended to a full system of stochastic differential equation driving all the temporal modes of the reduced system (and not only the small-scale modes). This full stochastic system has been evaluated on wake flow at moderate Reynolds number. For this flow the system has shown to provide very good uncertainty quantification properties as well meaningful physical behavior with respect to the simulation of the neutral modes of the dynamics. This study described in the PhD of Valentin Resseguier will be soon submitted to a journal paper.

On the other hand, in the following of several approaches proposed by the team [50], [54], we continued our investigations on the estimation of deterministic reduced order model from partial observations. In this line of research, we proposed a Bayesian framework for the construction of reduced-order models from image data. Our framework combines observation and prior information on the system to reduce the model and takes into account the uncertainties on the parameters of the model. The proposed approach reduces to some well-known model-reduction techniques for complete observations (i.e., the observation operator can be inverted). A theoretical analysis of our methodology has been investigated in a simplified context (namely, the observations are supposed to be noiseless linear combinations of the state of the system). This result provides worst-case guarantees on the reconstruction performance which can be achieved by a reduced model built from the data. These contributions have led to publications in a journal [18] and a conference [33].