POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Magnetic and Acoustic Investigations of Turbulent Spherical Couette Flow

Title of dissertation: MAGNETIC AND ACOUSTIC INVESTIGATIONS OF TURBULENT SPHERICAL COUETTE FLOW Matthew M. Adams, Doctor of Philosophy, 2016 Dissertation directed by: Professor Daniel Lathrop Department of Physics This dissertation describes experiments in spherical Couette devices, using both gas and liquid sodium. The experimental geometry is motivated by the Earth's outer core, the seat of the geodynamo, and consists of an outer spherical shell and an inner sphere, both of which can be rotated independently to drive a shear flow in the fluid lying between them. In the case of experiments with liquid sodium, we apply DC axial magnetic fields, with a dominant dipole or quadrupole component, to the system. We measure the magnetic field induced by the flow of liquid sodium using an external array of Hall effect magnetic field probes, as well as two probes inserted into the fluid volume. This gives information about possible velocity patterns present, and we extend previous work categorizing flow states, noting further information that can be extracted from the induced field measurements. The limitations due to a lack of direct velocity measurements prompted us to work on developing the technique of using acoustic modes to measure zonal flows. Using gas as the working fluid in our 60~cm diameter spherical Couette experiment, we identified acoustic modes of the container, and obtained excellent agreement with theoretical predictions. For the case of uniform rotation of the system, we compared the acoustic mode frequency splittings with theoretical predictions for solid body flow, and obtained excellent agreement. This gave us confidence in extending this work to the case of differential rotation, with a turbulent flow state. Using the measured splittings for this case, our colleagues performed an inversion to infer the pattern of zonal velocities within the flow, the first such inversion in a rotating laboratory experiment. This technique holds promise for use in liquid sodium experiments, for which zonal flow measurements have historically been challenging.

Inverse Spectral Methods in Acoustic Normal Mode Velocimetry of High Reynolds Number Spherical Couette Flows

Experimental geophysical fluid dynamics often examines regimes of fluid flow infeasible for computer simulations. Velocimetry of zonal flows present in these regimes brings many challenges when the fluid is opaque and vigorously rotating; spherical Couette flows with molten metals are one such example. The fine structure of the acoustic spectrum can be related to the fluid’s velocity field, and inverse spectral methods can be used to predict and, with sufficient acoustic data, mathematically reconstruct the velocity field. The methods are to some extent inherited from helioseismology. This work develops a Finite Element Method suitable to matching the geometries of experimental setups, as well as modelling the acoustics based on that geometry and zonal flows therein. As an application, this work uses the 60-cm setup Dynamo 3.5 at the University of Maryland Nonlinear Dynamics Laboratory. Additionally, results obtained using a small acoustic data set from recent experiments in air are provided.