An IB Method for Non-Newtonian-Fluid Flexible-Structure Interactions in Three-Dimensions

Abstract

Problems involving fluid flexible-structure interactions (FFSI) are ubiquitous in engineering and sciences. Peskin's immersed boundary (IB) method is the first framework for modeling and simulation of such problems. This paper addresses a three-dimensional extension of the IB framework for non-Newtonian fluids which include power-law fluid, Oldroyd-B fluid, and FENE-P fluid. The motion of the non-Newtonian fluids are modelled by the lattice Boltzmann equations (D3Q19 model). The differential constitutive equations of Oldroyd-B and FENE-P fluids are solved by the D3Q7 model. Numerical results indicate that the new method is first-order accurate and conditionally stable. To show the capability of the new method, it is tested on three FFSI toy problems: a power-law fluid past a flexible sheet fixed at its midline, a flexible sheet being flapped periodically at its midline in an Oldroyd-B fluid, and a flexible sheet being rotated at one edge in a FENE-P fluid.

Preface: Simulation of Fluid-Structure Interaction Problems

“…Numerical results of flows around an impulsively started circular cylinder were presented in the contribution confirmed the efficiency, for Reynolds numbers 1000 and 3000 [Bouchon, Dubois and James (2019)]. An extension of the IB framework for FSI problems was proposed for non-Newtonian fluids with power-law, Oldroyd-B, and FENE-P fluids in three dimensions in the manuscript [Zhu (2019)]. The motions of the non-Newtonian fluids were modelled in the contribution with the lattice Boltzmann equations (D3Q19 model) [Zhu (2019)].…”

“…An extension of the IB framework for FSI problems was proposed for non-Newtonian fluids with power-law, Oldroyd-B, and FENE-P fluids in three dimensions in the manuscript [Zhu (2019)]. The motions of the non-Newtonian fluids were modelled in the contribution with the lattice Boltzmann equations (D3Q19 model) [Zhu (2019)]. The multi-scale hybrid-mixed (MHM) method was applied to the numerical approximation of two-dimensional matrix fluid flow in porous media with fractures in the manuscript [Zhang (2019)].…”

An IB-LBM for FSI Problems Involving Viscoelastic Fluids and Complex Geometries

“…Here, the lattice Boltzmann method is utilized to solve the Navier-Stokes equations and the constitutive equations of the viscoelastic model [18,19,16,15,17]. It should be noted that a diffusion parameter κ is normally involved in this method (in the transport equations of the conformation tensor C) to improve the stability of simulations, and more details can be found in Refs.…”

“…In addition, the accuracy of this method was confirmed by different validation cases (2D Oldroyd-B flow over a cylinder, a 3D spherical particle rotation in an Oldroyd-B shear flow, and a 3D spherical particle sedimentation in Newtonian fluid under gravity). More recently, LBM and IBM were combined to study the motion of a massless flexible sheet in viscoelastic flows by Zhu [15,17], where the LBM originally proposed by Malaspinas et al [18] was utilized to solve the viscoelastic flow, and the FSI was enforced by IBM. In this study, it was reported that the motion of the sheet may be hindered with the inclusion of the fluid viscoelasticity.…”

Efficient split-step schemes for fluid-structure interaction involving incompressible generalised Newtonian flows

“…Although that FSI has been a major research topic in the past decades, approaches specifically targeting generalised Newtonian fluid flow are still scarcely found in literature [87,[95][96][97][98], even more so, when putting emphasis on efficient coupling schemes and solution procedures. The fluid flow problem, especially if treated semi-implicitly, has to be handled with special care.…”