197 research outputs found

    Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations

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    Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered. The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored. The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd

    Variational multiscale approximation of the one-dimensional forced Burgers equation: the role of orthogonal subgrid scales in turbulence modeling

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    This is the accepted version of the following article: [Bayona C, Baiges J, Codina R. Variational multi-scale approximation of the one-dimensional forced Burgers equation: the role of orthogonal subgrid scales in turbulence modeling. Int J Numer Meth Fluids. 2018;86:313–328. https://doi.org/10.1002/fld.4420], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.4420/fullA numerical approximation for the one-dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS-VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite-dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS-VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS-VMS method without any modification of the continuous problem.Peer ReviewedPostprint (author's final draft
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