Data-driven methods for complex PDEs: data assimilation and Optimal control
File(s)
Author(s)
Ghosh, Souvik
Type
Thesis
Abstract
A robust variational framework for data assimilation of time-averaged mean flow fields using a sparse set of highly noisy measurements (outliers), is presented. The numerical framework is governed by the two-dimensional, incompressible Reynolds-averaged Navier–Stokes equations with an unknown momentum forcing. This forcing, which corresponds to the divergence of the Reynolds stress tensor, is calculated from a direct-adjoint optimization procedure to reduce the deviation between the measured and assimilated velocities. A linear measure operator is used to project the assimilated field onto the low-dimensional subspace of the measurements to calculate the point-wise discrepancy and simultaneously retrieve the adjoint solutions on the high-dimensional subspace of the assimilated field. L2, L1, Huber, and hybrid loss functions are used to represent the point-wise error deviation between the measurements and the predictions. A variety of algorithms are considered to solve the optimization problem with different loss functions and their performances are evaluated. Huber and hybrid loss functions remained robust to the strong outliers in the measurement data set with its L1 contribution and also ensured the convergence to the optimal solution with its L2 contribution. The numerical data assimilation framework is applied to the case of two-dimensional laminar flow around a circular cylinder at Re = 100, and further to the case of two-dimensional laminar flow over a backward-facing step at Re = 500.
The data assimilation procedure is further extended to the case of the two-dimensional turbulent flows, modeled using a one-equation Spalart-Allmaras turbulence closure model obeying the Boussinesq hypothesis. The performance of L2 and the IRLS form of regularization function is evaluated for two different cases of separated and attached turbulent flows, in the vicinity of sparse point-wise measurements.
Lastly, a review study is performed to analyze the performance of different methods for the sensitivity analysis of the chaotic Kuramoto-Sivashinsky equation.
The data assimilation procedure is further extended to the case of the two-dimensional turbulent flows, modeled using a one-equation Spalart-Allmaras turbulence closure model obeying the Boussinesq hypothesis. The performance of L2 and the IRLS form of regularization function is evaluated for two different cases of separated and attached turbulent flows, in the vicinity of sparse point-wise measurements.
Lastly, a review study is performed to analyze the performance of different methods for the sensitivity analysis of the chaotic Kuramoto-Sivashinsky equation.
Version
Open Access
Date Issued
2022-08
Date Awarded
2023-04
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Schmid, Peter
Papageorgiou, Demetrios
Sponsor
European Commission
Grant Number
675008
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)