Title: An analysis of the spatio-temporal resolution of the immersed boundary method with direct forcing

Abstract

The immersed boundary method (IBM) with direct forcing is very popular in the simulation of rigid particulate flows. In the IBM, an interaction force is introduced at the interface between fluid and particle in order to approximate the no-slip boundary condition. The interaction force is calculated through dividing the velocity difference (or error) between fluid and particle at the interface by the time step. Here, a dynamic equation for the velocity difference is derived. Additionally, analyses on the dynamic equation provide a few new findings: (i) The interaction force is the solution of a least-squares error problem, with the direct implication that the Lagrangian marker distribution has no effect on the large scale flow structure once the distribution of Lagrangian markers become saturated along the interface (i.e., each marker remains properly correlated with all its neighbors); (ii) The Lagrangian volume-weight is a relaxation factor to control how fast the velocity error decays to the ideal value of zero; (iii) The optimal choice of the Lagrangian volume-weight is the largest value permissible by a stability condition. A comprehensive convergence analysis with regard to the spatial and temporal resolution is presented for the velocity error and also for the shear-stress and surfacemore » pressure. In three simple canonical problems, it is analytically and numerically shown that the IBM results converge to the theoretical solutions obtained with precise imposition of no-slip and no-penetration boundary conditions. It is observed that it is not necessary to match the Lagrangian marker volume-weight to that of the local Eulerian cell volume and in fact this matching leads to lower than optimal computational efficiency. However, it is found that extremely high Eulerian grid resolution and small time step have to be used to obtain high precision simulation results. Especially, the time step should be inversely proportional to the particle Reynolds number for low Reynolds number flows. For high frequency oscillation problems, the grid size needs to be reduced by a factor of the square root of the frequency, and the time step to be reduced by a factor of the frequency. The theoretical findings here can be used to alleviate the technical difficulties in simulating non-spherical particles by not requiring the Lagrangian marker distribution to match the Eulerian grids and also in the implementation of IBM on non-uniform Eulerian grids. The present work also provides simple practical guidance on the choice of temporal and spatial resolution so as to control the simulation error a priori.« less

Authors:

Zhou, Kun  ^[1];

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Balachandar, S. ^[2]

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+ Show Author Affiliations

1. Wuhan Univ. of Science and Technology, Wuhan (China). The State Key Lab. of Refractories and Metallurgy; Univ. of Florida, Gainesville, FL (United States). Dept. of Mechanical and Aerospace Engineering
2. Univ. of Florida, Gainesville, FL (United States). Dept. of Mechanical and Aerospace Engineering

Publication Date:

Wed Sep 23 00:00:00 EDT 2020

Research Org.:

Univ. of Florida, Gainesville, FL (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA); National Natural Science Foundation of China (NSFC); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:

1850325

Alternate Identifier(s):

OSTI ID: 1775914

Grant/Contract Number:  

NA0002378; 11572274; 1197233; N00014-16-1-2617

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Volume: 424; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computer Science; Physics; Immersed boundary method; Grid resolution; Lagrangian volume-weight; Particle laden flows