Numerical modelling of unsteady mixed flows

This thesis concerns mixed flows (which are characterized by the simultaneous occurrence of free-surface and pressurized flow in sewers, tunnels, culverts or under bridges), and contributes to the improvement of the existing numerical tools for modelling these phenomena. The classic Preissmann slot approach is selected due to its simplicity and capability of predicting results comparable to those of a more recent and complex two-equation model, as shown here with reference to a laboratory test case. In order to enhance the computational efficiency, a local time stepping strategy is implemented in a shock-capturing Godunov-type finite volume numerical scheme for the integration of the de Saint-Venant equations. The results of different numerical tests show that local time stepping reduces run time significantly (between −29% and −85% CPU time for the test cases considered) compared to the conventional global time stepping, especially when only a small region of the flow field is surcharged, while solution accuracy and mass conservation are not impaired. The second part of this thesis is devoted to the modelling of the hydraulic effects of potentially pressurized structures, such as bridges and culverts, inserted in open channel domains. To this aim, a two-dimensional mixed flow model is developed first. The classic conservative formulation of the 2D shallow water equations for free-surface flow is adapted by assuming that two fictitious vertical slots, normally intersecting, are added on the ceiling of each integration element. Numerical results show that this schematization is suitable for the prediction of 2D flooding phenomena in which the pressurization of crossing structures can be expected. Given that the Preissmann model does not allow for the possibility of bridge overtopping, a one-dimensional model is also presented in this thesis to handle this particular condition. The flows below and above the deck are considered as parallel, and linked to the upstream and downstream reaches of the channel by introducing suitable internal boundary conditions. The comparison with experimental data and with the results of HEC-RAS simulations shows that the proposed model can be a useful and effective tool for predicting overtopping and backwater effects induced by the presence of bridges and culverts.

Studies on Wilson loops, correlators and localization in supersymmetric quantum field theories

In this thesis we study at perturbative level correlation functions of Wilson loops (and local operators) and their relations to localization, integrability and other quantities of interest as the cusp anomalous dimension and the Bremsstrahlung function. First of all we consider a general class of 1/8 BPS Wilson loops and chiral primaries in N=4 Super Yang-Mills theory. We perform explicit two-loop computations, for some particular but still rather general configuration, that confirm the elegant results expected from localization procedure. We find notably full consistency with the multi-matrix model averages, obtained from 2D Yang-Mills theory on the sphere, when interacting diagrams do not cancel and contribute non-trivially to the final answer. We also discuss the near BPS expansion of the generalized cusp anomalous dimension with L units of R-charge. Integrability provides an exact solution, obtained by solving a general TBA equation in the appropriate limit: we propose here an alternative method based on supersymmetric localization. The basic idea is to relate the computation to the vacuum expectation value of certain 1/8 BPS Wilson loops with local operator insertions along the contour. Also these observables localize on a two-dimensional gauge theory on S^2, opening the possibility of exact calculations. As a test of our proposal, we reproduce the leading Luscher correction at weak coupling to the generalized cusp anomalous dimension. This result is also checked against a genuine Feynman diagram approach in N=4 super Yang-Mills theory. Finally we study the cusp anomalous dimension in N=6 ABJ(M) theory, identifying a scaling limit in which the ladder diagrams dominate. The resummation is encoded into a Bethe-Salpeter equation that is mapped to a Schroedinger problem, exactly solvable due to the surprising supersymmetry of the effective Hamiltonian. In the ABJ case the solution implies the diagonalization of the U(N) and U(M) building blocks, suggesting the existence of two independent cusp anomalous dimensions and an unexpected exponentation structure for the related Wilson loops.