Multi-Resolution Lattice Green's Function Method for High Reynolds Number External Flows

This work expands the state-of-the-art computational fluid dynamics (CFD) methods for simulating three-dimensional, turbulent, external flows by further developing the immersed boundary (IB) Lattice Green's function (LGF) method. The original IB-LGF method applies an exact far-field boundary condition using fundamental solutions on regular Cartesian grids and allows active computational cells to be restricted to vortical flow regions in an adaptive fashion as the flow evolves. The combination of spatial adaptivity and regular Cartesian structure leads to superior efficiency, scalability, and robustness, but necessitates uniform grid spacing. However, the scale separation associated with thin boundary layers and turbulence at higher Reynolds numbers favors a more flexible distribution of elements/cells, which is achieved in this thesis by developing a multi-resolution LGF approach that permits block-wise grid refinement while maintaining the important properties of the original scheme. We further show that the multi-resolution LGF method can be fruitfully combined with the IB method to simulate external flows around complex geometries at high Reynolds numbers. This novel multi-resolution IB-LGF scheme retains good efficiency, parallel scaling as well as robustness (conservation and stability properties). DNS of bluff and streamlined bodies at Reynolds numbers O(104) are conducted and the new multi-resolution scheme is shown to reduce the total number of computational cells up to 99.87%. We also extended this method to large-eddy simulation (LES) with the stretched-vortex sub-grid-scale model. In validating the LES implementation, we considered an isolated spherical region of turbulence in free space. The initial condition is spherically windowed, isotropic homogeneous incompressible turbulence. We study the spectrum and statistics of the decaying turbulence and compare the results with decaying isotropic turbulence, including cases representing different low wavenumber behavior of the energy spectrum (i.e. k2 versus k4). At late times the turbulent sphere expands with both mean radius and integral scale showing similar time-wise growth exponents. The low wavenumber behavior has little effect on the inertial scales, and we find that decay rates follow Saffman's predictions in both cases, at least until about 400 initial eddy turnover times. The boundary of the spherical region develops intermittency and features ejections of vortex rings. These are shown to occur at the integral scale of the initial turbulence field and are hypothesized to occur due to a local imbalance of impulse on this scale.