A unified constraint formulation of immersed body techniques for coupled fluid-solid motion: continuous equations and numerical algorithms

TL;DR

This work unifies immersed boundary and fictitious-domain approaches by casting fluid–solid interaction in an extended-domain, strong-form framework. Through body-force and stress formulations, it shows how IBM, FDM, velocity-forcing, and fully implicit schemes arise from the same governing equations, enabling cross-method analysis and design. The paper extends the framework to multiphase and interface-tracking contexts, discusses sharp vs diffuse constraint implementations, and tackles practical issues such as force/torque evaluation, leakage, and high density ratios. The resulting perspective provides a rigorous theoretical foundation with actionable guidance for developing robust, efficient FSI solvers across single- and multiphase regimes, including Brownian simulations and complex boundary conditions.

Abstract

Numerical simulation of moving immersed solid bodies in fluids is now practiced routinely following pioneering work of Peskin and co-workers on immersed boundary method (IBM), Glowinski and co-workers on fictitious domain method (FDM), and others on related methods. A variety of variants of IBM and FDM approaches have been published, most of which rely on using a background mesh for the fluid equations and tracking the solid body using Lagrangian points. The key idea that is common to these methods is to assume that the entire fluid-solid domain is a fluid and then to constrain the fluid within the solid domain to move in accordance with the solid governing equations. The immersed solid body can be rigid or deforming. Thus, in all these methods the fluid domain is extended into the solid domain. In this review, we provide a mathemarical perspective of various immersed methods by recasting the governing equations in an extended domain form for the fluid. The solid equations are used to impose appropriate constraints on the fluid that is extended into the solid domain. This leads to extended domain constrained fluid-solid governing equations that provide a unified framework for various immersed body techniques. The unified constrained governing equations in the strong form are independent of the temporal or spatial discretization schemes. We show that particular choices of time stepping and spatial discretization lead to different techniques reported in literature ranging from freely moving rigid to elastic self-propelling bodies. These techniques have wide ranging applications including aquatic locomotion, underwater vehicles, car aerodynamics, and organ physiology (e.g. cardiac flow, esophageal transport, respiratory flows), wave energy convertors, among others. We conclude with comments on outstanding challenges and future directions.

Figures (7)

• Figure 1: A schematic representation of the fluid-structure interaction system. The computational domain boundary $\partial \Omega$ is demarcated by a solid black line. The filled pink region denotes the fluid domain $\Omega_f$, whereas the filled blue region denotes the solid domain $\Omega_s$. The solid boundary $\partial \Omega_s$ is shown by a solid blue line. A representative control volume/pillbox used to derive the jump condition given by Equation \ref{['eq:gov-eqns5']} is shown by light grey color. The unit normal vector $\hat{\bm{n}}$ of the solid surface points outwards into the fluid and is shown by a red arrow.
• Figure 2: Representative scenarios of freely moving bodies in fluids: \ref{['fig_Two_spheres']} a two-body system of rigid spheres that self-propels itself by actuating a spring tethered at their center of mass points; and \ref{['fig_Fish']} a freely-swimming fish that locomotes in water by undulating part of its body.
• Figure 3: A schematic representation of a wave energy converter device oscillating on the air-water interface under the action of incoming water waves. The air phase $\Omega_{\rm air}$ is shown in white, the water phase $\Omega_{\rm water}$ in blue, and the solid phase $\Omega_{\rm solid}$ in brown color, respectively. The flowing/fluid phase domain is $\Omega_f(t) = \Omega_{\rm air}(t) \cup \Omega_{\rm water}(t)$. The density ratio between air and water phase is approximately 1000, whereas between air and solid (mechanical oscillator) phase is approximately 500. The gas-liquid interface is advected in \ref{['fig_wec_vof']}$\Omega_f(t)$ and \ref{['fig_wec_ls']}$\Omega$ regions, respectively.
• Figure 4: Local unit normal ($\hat{\bm{n}}$) and tangent ($\hat{\bm{t}}$ and $\hat{\bm{b}}$) vectors of the interface. A local orthonormal system is chosen to derive the jump conditions.
• Figure 5: Imposing spatially-varying Neumann and Robin boundary conditions on an embedded interface using penalized Poisson Equation \ref{['eq:penalized_poisson']}. Here $\hat{\bm{n}}$ is the unit outward normal of the fluid region which is shown by filled pink region.