| One of the long standing problems of fluid mechanics has been and still is, to obtain the spatio-temporal solution of the velocity field for general boundary conditions. The complexity of the Navier-Stokes equations makes obtaining the solution to general problems intractable except for some very special cases. Under such circumstances, one is left with either running a series of experiments in a wind tunnel or perform numerical simulations. For obtaining the spatio-temporal evolution of a 3D turbulent velocity field, the preferred method of choice is numerical simulations.In this thesis, a third possible approach is considered, namely, modelling the numerical simulations themselves and combining the models with sensor information to produce approximations to the velocity fields. In order for the models to gain a significant advantage over conventional numerical simulations, the models must be able to operate at much lower computational cost and should have comparable accuracy to conventional numerical simulations. The focus of this thesis is to look at a variety of strategies that could be used to construct different types of models. A central concept that will be found to be common to all the models developed in this work is low-dimensionality. The idea that a complicated turbulent flow can be broken down into different scales of motions, or structures, plays a key role in reducing the number of degrees of freedom on which the models operate.Measurement models are constructed using different methods to correlate the sensor measurements to the velocity fields. State-space models using radial basis functions are constructed to mimic the numerical simulations. The nonlinear Kalman filter is studied as a possible means of optimally combining the two models to produce dynamically consistent solutions. In this regard, a novel approach called "Episodic-POD" is developed that uses the two models to produce accurate representations to the velocity fields. Furthermore, extensions of episodic-POD are introduced that enable long-term prediction and the capability of handling extremely high-dimensional data. The models are shown to be robust in the presence of noise and are capable of handling sparse data.Direct and inverse problems of contaminant dispersion problems are solved using Eulerian and Lagrangian approaches based on the developed models. It is demonstrated that the developed methods can reliably predict the dispersion of the contaminant as well as estimate the location of the contaminant release. |
