Verification of the coupled‐momentum method with Womersley's Deformable Wall analytical solution

Abstract

In this paper, we perform a verification study of the Coupled‐Momentum Method (CMM), a 3D fluid‐structure interaction (FSI) model which uses a thin linear elastic membrane and linear kinematics to describe the mechanical behavior of the vessel wall. The verification of this model is done using Womersley's deformable wall analytical solution for pulsatile flow in a semi‐infinite cylindrical vessel. This solution is, under certain premises, the analytical solution of the CMM and can thus be used for model verification. For the numerical solution, we employ an impedance boundary condition to define a reflection‐free outflow boundary condition and thus mimic the physics of the analytical solution, which is defined on a semi‐infinite domain. We first provide a rigorous derivation of Womersley's deformable wall theory via scale analysis. We then illustrate different characteristics of the analytical solution such as space‐time wave periodicity and attenuation. Finally, we present the verification tests comparing the CMM with Womersley's theory.

A reduced unified continuum formulation for vascular fluid-structure interaction

“…Since its introduction, CMM has been implemented in the open-source blood flow simulation software packages SimVascular [28,29] and CRIMSON [30] and extensively used in clinical applications ranging from interventions for coronary artery disease [31,32,33] and aortic coarctation [34] to single-ventricle physiology [35], and Alagille syndrome [36,37]. It has also been validated against experimental measurements from compliant in vitro phantom models [38,39] and Womersley's analytical solution for axisymmetric flow in a thin, linear elastic pipe subject to an oscillating pressure gradient [40,41]. While the studies found good agreement for pressure, flow, pulse wave propagation, and wall displacement, Filonova et al [41] documented large errors in radial velocity.…”

An efficient, localised approach for the simulation of elastic blood vessels using the lattice Boltzmann method

“…The analytical derivations of Womersley 19 , 20 represents some of the seminal work in fluid flow through pipes. These two papers focus on idealised results for pulsatile flow through rigid and flexible pipes and provide a reference case that has been widely used as a verification model for computational fluid mechanics 11 , 21 , 22 . For a full derivation we refer to the source works or that of Figueroa 21 or Filonova et al 22 but here we highlight equations of particular relevance to the later development of our elastic wall model.…”

“…These two papers focus on idealised results for pulsatile flow through rigid and flexible pipes and provide a reference case that has been widely used as a verification model for computational fluid mechanics 11 , 21 , 22 . For a full derivation we refer to the source works or that of Figueroa 21 or Filonova et al 22 but here we highlight equations of particular relevance to the later development of our elastic wall model. These field equations describe properties at a given radial, r , and axial, z , position in time, t , as a combination of steady and oscillatory components of flow.…”

“…The purpose of the following section is to build upon this to examine how the new and existing boundary conditions are able to replicate the elastic wall conditions. In 21 , 22 , the problem of flow through the carotid artery is examined as a test case for an elastic wall model. We demonstrate how our model is able to capture the essence of elastic wall flow in a similar domain whilst not losing computational performance compared to a rigid wall model.…”

Modeling of Fluid Flow in a Flexible Vessel with Elastic Walls

“…Compared with the classical works [19][20][21] of J.R. Womersley (for an alternative description of the above works see [5]), our formulation of problem has much in common, both of them involve momentless shell theory for modeling the elastic wall. In Wormerley's works, axisymmetric pulsative blood flow in a vessel with circular isotropic elastic wall is found as a perturbation of the steady Poisseulle flow.…”