In this dissertation, we present a novel hybrid approach combining discrete exterior calculus (DEC) and finite difference (FD) methods to simulate fully-coupled three-dimensional Boussinesq convection in spherical shells subject to internal and basal heating---relevant to planetary and stellar phenomenon. We employ DEC to compute the surface spherical operators, taking advantage of its unique features, including coordinate system independence to preserve the spherical geometry, while we discretize the radial direction using FD method. The grid employed for this novel method is free of problems like the coordinate singularity, grid non-convergence near the poles, and the overlap regions.
We first developed a semi-parallel in-house code based on the PETSc framework to validate the hybrid DEC-FD formulation and demonstrate convergence. The solution is evolved through a semi-implicit time integration scheme. We have performed a series of numerical tests, which include quantification of the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, simulation of robust convective patterns, and the quantification of Nusselt and Reynolds numbers for basally heated spherical shells.
Further, we developed a second version of the code for large-scale parallelism with the motivation to simulate high Rayleigh number flows. The governing equations are formulated as a non-linear function and solved fully implicitly---the Jacobian of the non-linear function is hand-coded and utilized in the time integration scheme. We verified the code employing the method of manufactured solutions and demonstrated asymptotic local convergence. We validated the fully implicit method by quantifying the critical Rayleigh numbers and simulating a stationary spiral roll from a finite amplitude equatorially symmetric initial condition. We present parallel scaling studies utilizing up to 16k processor cores on Shaheen-3 for non-linear function and Jacobian computation kernels.
Finally, we performed proper orthogonal decomposition and dynamic mode decomposition to extract the characteristic modes of convection at supercritical Rayleigh numbers. We demonstrated the randomized projective compression scheme based on Fast Johnson-Lindenstrauss Transform, proposed by Glazkov and Schmid (2024), on the simulation data, where reductions down to 3.4% of the full degrees of freedom could be reported without perceptible deterioration of the spatial structures.
Biography
Bhargav Mantravadi is a PhD candidate in the Mechanical Engineering program in PSE division, working under the supervision of Prof. Peter Schmid. He was formerly supervised by Prof. Ravi Samtaney. Bhargav completed his MS degree at KAUST in 2019, with his thesis research focused on investigating the behavior of two-fluid plasma in the context of Brio-Wu shock tube Riemann problem. His research interests include fluid dynamics, numerical methods, and high-performance computing.
