Sparse Data-Driven Simulation of Turbulent Flows

Data has always played an important role in the modeling and simulation of turbulent flows. In the last decade, with the advancement and abundance of measurement techniques, computational resources, and mathematical algorithms, data has been systematically used to develop and improve turbulence models. Experimental measurements, which are generally sparse or of low resolution, can be incorporated into models using mathematical techniques such as variational data assimilation to enable data-driven simulation of turbulent flows. In this project, we investigate different techniques to incorporate sparse data into the turbulence models to obtain the most accurate results possible. Such inverse problems are typically severely ill-posed and the solutions are non-unique. Therefore, in addition to sparse measurements, further information, expert knowledge, or physical constraints should be added to the problem to obtain a reconstructed solution that is smooth, accurate, and physical. We, therefore, impose the linear eddy viscosity (LEV) assumption to reduce the degrees of freedom of the problem and the constraint of positivity of the eddy viscosity. The optimization problem is defined as reducing the discrepancy between the LEV RANS model's solution and the sparse measurement data by tuning a corrective parameter field that results in an optimal eddy viscosity field. The discrete adjoint method is implemented in OpenFOAM to compute the gradients. The case study of flow over periodic hills is chosen for the investigation. We see that the LEV assumption without including further information would lead to irregular, jagged velocity profiles, due to the ill-conditioning and non-uniqueness nature of the problem. Regularization is then introduced to promote regularity and smoothness to the solution. The $L_2$, total variation, and Sobolev gradient regularization methods were employed. We find that physical velocity profiles can be reconstructed using these methods. However, we observe that with each of these regularization methods we cannot accurately reconstruct wall shear stresses, which are important quantities of interest, even if sparse wall shear stress measurement data is assimilated. Therefore, we propose a method called piecewise linear dimension reduction (PLDR). In this method, the parameter field is constrained to be piecewise linear to avoid, in a controlled way, any noise and large fluctuations in the gradients. The results suggest that the PLDR method can provide accurate and smooth velocity profiles as well as wall shear stresses. Next, we investigate a scenario where only sparse wall shear stress measurements are available. We observe that incomplete and not uniformly distributed data cannot be assimilated properly resulting in deteriorated solutions in the free shear flow region. Therefore, we augmented the available data by using the solution of a high-fidelity but computationally efficient model. A loosely coupled hybrid LES/RANS method is proposed, in which an under-resolved LES is coupled with a steady-RANS model. We show that such a method is easy to implement and that it provides an accurate velocity field with less accurate wall shear stresses. The solution of the proposed hybrid LES/RANS method in combination with the sparse wall shear stress measurements was successfully assimilated into a RANS model providing both accurate velocities and wall shear stresses. We can conclude that accurate sparse data-driven simulation is possible if the inverse problem is properly defined such that any available knowledge is incorporated, from adding data from a higher fidelity simulation, constraining the Reynolds stress tensor, and imposing a hard constraint on the positivity of the eddy viscosity, to penalizing noise and non-smoothness, etc. The project thus shows a promising path towards using sparse data-driven simulation for practical applications.