2,737 research outputs found
Mathematical Architecture for Models of Fluid Flow Phenomena
This thesis is a study of several high accuracy numerical methods for fluid flow problems and turbulence modeling.First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step. We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought. Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost. We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. Magnetically conducting fluids arise in important applications including plasma physics, geophysics and astronomy. In many of these, turbulent MHD (magnetohydrodynamic) flows are typical. The difficulties of accurately modeling and simulating turbulent flows are magnified many times over in the MHD case. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations. Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests
Hydrodynamic interactions in polar liquid crystals on evolving surfaces
We consider the derivation and numerical solution of the flow of passive and
active polar liquid crystals, whose molecular orientation is subjected to a
tangential anchoring on an evolving curved surface. The underlying passive
model is a simplified surface Ericksen-Leslie model, which is derived as a
thin-film limit of the corresponding three-dimensional equations with
appropriate boundary conditions. A finite element discretization is considered
and the effect of hydrodynamics on the interplay of topology, geometric
properties and defect dynamics is studied for this model on various stationary
and evolving surfaces. Additionally, we consider an active model. We propose a
surface formulation for an active polar viscous gel and exemplarily demonstrate
the effect of the underlying curvature on the location of topological defects
on a torus
A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations
summary:A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the new error correction algorithm are established. Numerical examples are also presented to verify the theoretical analysis and demonstrate the efficiency of the proposed method
Parallel three-dimensional simulations of quasi-static elastoplastic solids
Hypo-elastoplasticity is a flexible framework for modeling the mechanics of
many hard materials under small elastic deformation and large plastic
deformation. Under typical loading rates, most laboratory tests of these
materials happen in the quasi-static limit, but there are few existing
numerical methods tailor-made for this physical regime. In this work, we extend
to three dimensions a recent projection method for simulating quasi-static
hypo-elastoplastic materials. The method is based on a mathematical
correspondence to the incompressible Navier-Stokes equations, where the
projection method of Chorin (1968) is an established numerical technique. We
develop and utilize a three-dimensional parallel geometric multigrid solver
employed to solve a linear system for the quasi-static projection. Our method
is tested through simulation of three-dimensional shear band nucleation and
growth, a precursor to failure in many materials. As an example system, we
employ a physical model of a bulk metallic glass based on the shear
transformation zone theory, but the method can be applied to any
elastoplasticity model. We consider several examples of three-dimensional shear
banding, and examine shear band formation in physically realistic materials
with heterogeneous initial conditions under both simple shear deformation and
boundary conditions inspired by friction welding.Comment: Final version. Accepted for publication in Computer Physics
Communication
Time integration for diffuse interface models for two-phase flow
We propose a variant of the -scheme for diffuse interface models for
two-phase flow, together with three new linearization techniques for the
surface tension. These involve either additional stabilizing force terms, or a
fully implicit coupling of the Navier-Stokes and Cahn-Hilliard equation. In the
common case that the equations for interface and flow are coupled explicitly,
we find a time step restriction which is very different to other two-phase flow
models and in particular is independent of the grid size. We also show that the
proposed stabilization techniques can lift this time step restriction. Even
more pronounced is the performance of the proposed fully implicit scheme which
is stable for arbitrarily large time steps. We demonstrate in a Taylor flow
application that this superior coupling between flow and interface equation can
render diffuse interface models even computationally cheaper and faster than
sharp interface models
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