Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels

TL;DR

This paper develops a theory and performs simulations to understand elastoinertial rectification in oscillatory flows within deformable 2D channels, revealing how flow inertia and deformation produce steady streaming effects.

Contribution

It extends elastoinertial rectification theory to 2D channels with elastic deformation, validated by numerical simulations using an ALE FSI approach.

Findings

Excellent agreement between theory and simulations for pressure and deformation.

Cycle-averaged pressures and displacements predicted accurately by the theory.

Flow inertia enhances streaming through nonlinear flow-structure coupling.

Abstract

A slender two-dimensional (2D) channel bounded by a rigid bottom surface and a slender elastic layer above deforms when a fluid flows through it. Hydrodynamic forces cause deformation at the fluid--solid interface, which in turn changes the cross-sectional area of the fluidic channel. The nonlinear coupling between flow and deformation, along with the attendant asymmetry in geometry caused by flow-induced deformation, produces a streaming effect (a nonzero cycle-average despite time-periodic forcing). Surprisingly, flow inertia provides another nonlinear coupling, tightly connected to deformation, that enhances streaming, termed ``elastoinertial rectification'' by Zhang and Rallabandi [J.\ Fluid Mech.\ \textbf{996}, A16 (2024)]. We adapt the latter theory of how two-way coupled fluid--structure interaction (FSI) produces streaming to a 2D rectangular configuration, specifically taking care to capture the deformations of the nearly incompressible slender elastic layer via the combined foundation model of Chandler and Vella [Proc.\ R.\ Soc.\ A \textbf{476}, 20200551 (2020)]. We supplement the elastoinertial rectification theory with direct numerical simulations performed using a stabilized, conforming arbitrary Lagrangian--Eulerian (ALE) FSI formulation, implemented via the open-source computing platform FEniCS. We examine the axial variation of the cycle-averaged pressure as a function of key dimensionless groups of the problem: the Womersley number, the elastoviscous number, and the compliance number. Assuming a small compliance number, we find excellent agreement between theory and simulations for the leading-order pressure and deformation across a range of conditions. At the next order, the cycle-averaged pressures agree well. Finally, the theory also predicts nontrivial cycle-averaged vertical and horizontal displacements, in agreement with the simulations.