Date of Award

Spring 1-1-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

First Advisor

John A. Evans

Second Advisor

Kenneth Jansen

Third Advisor

Carlos Felippa

Fourth Advisor

Peter Hamlington

Fifth Advisor

Yuri Bazilevs

Abstract

In this dissertation, a class of methods which combines divergence-conforming discretizations with residual-based subgrid modeling for large eddy simulation of turbulent flows is introduced. These methods fall within two frameworks: residual-based variational multiscale methods and residual-based eddy viscosities. These methods utilize variationally-consistent formulations for the fine-scale velocities in order to construct subgrid-scale models based on the coarse-scale residual. The result is an LES methodology that responds naturally to spatially- and temporally-varying turbulence. Numerical results demonstrate that these new methods demonstrate proper behavior for homogeneous turbulence and outperform classical LES models for transitional flows and wall-bounded turbulent flows. Furthermore, the resulting formulations contain no ``tunable'' parameters, and thus extend generally across various classes of flow.

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Additionally, a differential variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method is presented and examined. Stability and convergence results for the methodology as applied to the scalar transport problem are established, and it is proven that the method exhibits optimal convergence rates in the SUPG norm and is robust with respect to the P\'{e}clet number if the discontinuous subscale approximation space is sufficiently rich. The method is applied to isogeometric NURBS discretizations of steady and unsteady transport problems, and the corresponding numerical results demonstrate that the method is stable and accurate in the advective limit even when low-order discontinuous subscale approximations are employed. Based on these promising results, a class of differential subgrid vortex models for large eddy simulation of turbulent flows is proposed.

Finally, the underlying discretization utilized by the simulations here offers the opportunity to develop efficient new geometric multigrid linear solvers. In this regard, a

geometric multigrid methodology for the solution of matrix systems associated with isogeometric compatible discretizations of the generalized Stokes and Oseen problems is presented. The methodology provably yields a pointwise divergence-free velocity field independent of the number of pre-smoothing steps, post-smoothing steps, grid levels, or cycles in a V-cycle implementation. The methodology relies upon Scwharz-style smoothers in conjunction with specially defined overlapping subdomains that respect the underlying topological structure of the generalized Stokes and Oseen problems. Numerical results in both two- and three-dimensions demonstrate the robustness of the methodology through the invariance of convergence rates with respect to grid resolution and flow parameters for the generalized Stokes problem as well as the generalized Oseen problem, provided it is not advection-dominated.

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