# Smoothed Profile Method: Error Analysis, Verification and Application in Dielectric Problems

## Overview

Title
Smoothed Profile Method: Error Analysis, Verification and Application in Dielectric Problems
Contributors
Liu, Xiaohe (creator)
Brown University. Engineering: Fluid, Thermal, and Chemical Processes (sponsor)
Doi
10.7301/Z0KD1WB3
2017
Abstract
We study the smoothed profile method (SPM) first proposed by Nakayama & Yamamoto (2005) and extended by Luo et al. (2009), which is a numerical method for particle-laden flows. SPM represents each particle with a smoothed indicator function, and solves the Navier-Stokes equations for incompressible flow on the whole domain on a fixed computational mesh, while applying a local momentum impulse at each time step of the iterative solver to adjust the velocity field in the particle domain so that it matches a target velocity field for rigid-body motion. First we use SPM as a direct forcing method, and quantify the accuracy of SPM for several prototype flows including steady Couette flow and unsteady oscillating Stokes layer flow. Further verification of SPM is presented by simulations of a spherical particle settling in a channel at finite Reynolds number. We found that the modeling error of SPM depends on the ratio of $\sqrt{\nu \Delta t}/\xi$, and the most accurate results occur at $\sqrt{\nu \Delta t}/\xi$ equals to $0.75$ to $1$; the exact optimum ratio depends on the time stepping scheme. Comparisons with the analytic solutions showed that SPM is resolving accurately the far-field flows and the particle forces, while allowing some error locally at the particle-fluid interfaces. Subsequently, we extended SPM to simulate electro-rheology flows allowing for spatially varying dielectric coefficient. We solve the Poisson equation in electrostatics for the whole domain and verify the method for prototype problems of dielectric beads in non-conducting fluid, where the modeling error is quantified. We also proposed an alternative indicator function based on a fourth order polynomial, which showed similar effect in simulations as the traditional tanh indicator function, but since it has a more definite start point and end point of the smoothed layer, it can provide better control of the smoothed thickness.
Keywords
Fluid Dynamics
Computational fluid dynamics
Two-phase flow
Notes
Thesis (Sc. M.)--Brown University, 2017
Extent
x, 49 p.

## Citation

Liu, Xiaohe, "Smoothed Profile Method: Error Analysis, Verification and Application in Dielectric Problems" (2017). Fluid, Thermal, and Chemical Processes Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.7301/Z0KD1WB3

• ## Fluid, Thermal, and Chemical Processes Theses and Dissertations

Theses and Dissertations for the Fluid, Thermal, and Chemical Processes department.