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

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.