Immersed boundary method

In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions.[1] Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations (the elastic boundary changes the flow of the fluid and the fluid moves the elastic boundary simultaneously). In the immersed boundary method the fluid is represented in an Eulerian coordinate system and the structure is represented in Lagrangian coordinates. For Newtonian fluids governed by the Navier–Stokes equations, the fluid equations are

${\displaystyle \rho \left({\frac {\partial {u}({x},t)}{\partial {t}}}+{u}\cdot \nabla {u}\right)=-\nabla p+\mu \,\Delta u(x,t)+f(x,t)}$

and if the flow is incompressible, we have the further condition that

${\displaystyle \nabla \cdot u=0.\,}$

The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by ${\displaystyle \Gamma }$. Each fiber can be viewed as a parametric curve ${\displaystyle X(s,t)}$ where ${\displaystyle s}$ is the Lagrangian coordinate along the fiber and ${\displaystyle t}$ is time. The physics of the fiber is represented via a fiber force distribution function ${\displaystyle F(s,t)}$. Spring forces, bending resistance or any other type of behavior can be built into this term. The force exerted by the structure on the fluid is then interpolated as a source term in the momentum equation using

${\displaystyle f(x,t)=\int _{\Gamma }F(s,t)\,\delta {\big (}x-X(s,t){\big )}\,ds,}$

where ${\displaystyle \delta }$ is the Dirac δ function. The forcing can be extended to multiple dimensions to model elastic surfaces or three-dimensional solids. Assuming a massless structure, the elastic fiber moves with the local fluid velocity and can be interpolated via the delta function

${\displaystyle {\frac {\partial X(s,t)}{\partial t}}=u(X,t)=\int _{\Omega }u(x,t)\,\delta {\big (}x-X(s,t){\big )}\,dx,}$

where ${\displaystyle \Omega }$ denotes the entire fluid domain. Discretization of these equations can be done by assuming an Eulerian grid on the fluid and a separate Lagrangian grid on the fiber. Approximations of the Delta distribution by smoother functions will allow us to interpolate between the two grids. Any existing fluid solver can be coupled to a solver for the fiber equations to solve the Immersed Boundary equations. Variants of this basic approach have been applied to simulate a wide variety of mechanical systems involving elastic structures which interact with fluid flows.

Since the original development of this method by Peskin, a variety of approaches have been developed to simulate flow over complicated immersed bodies on grids that do not conform to the surface of the body. These include methods such as the immersed interface method, the Cartesian grid method, the ghost fluid method and the cut-cell method. Mittal and Iaccarino[2] refer to all these (and other related) methods as Immersed Boundary Methods and provide various categorizations of these methods. From the point of view of implementation, they categorize immersed boundary methods into continuous forcing and discrete forcing methods. In the former, a force term is added to the continuous Navier-Stokes equations before discretization, whereas in the latter, the forcing is applied (explicitly or implicitly) to the discretized equations. Under this taxonomy, Peskin's original method is a continuous forcing method whereas Cartesian grid, cut-cell and the ghost-fluid methods are discrete forcing methods.

For simulations of viscoelastic fluids, curved fluid interfaces, microscopic biophysical systems (proteins in lipid bilayer membranes, swimmers), and engineered devices, further variants and extensions of the immersed boundary method have been developed, such as the Stochastic Immersed Boundary Methods of Atzberger, Kramer, and Peskin [3] [4] , Stochastic Eulerian Lagrangian Methods of Atzberger [5] [6] [7], Massed Immersed Boundary Methods of Moria [8] , and Rotational Immersed Boundary Methods of Olson, Lim, Cortez [9].

In general, for immersed boundary methods and related variants, there is an active research community that is still developing new techniques and related software implementations and incorporating related techniques into simulation packages and CAD engineering software.

Notes

1. ^ Peskin, Charles S (1972-10-01). "Flow patterns around heart valves: A numerical method". Journal of Computational Physics. 10 (2): 252–271. Bibcode:1972JCoPh..10..252P. doi:10.1016/0021-9991(72)90065-4. ISSN 0021-9991.
2. ^
3. ^ Atzberger, Paul J. (2011). "Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations". Journal of Computational Physics. 230 (8): 2821–2837. arXiv:1009.5648. Bibcode:2011JCoPh.230.2821A. doi:10.1016/j.jcp.2010.12.028. S2CID 6067032.
4. ^ Rower, David A.; Padidar, Misha; Atzberger, Paul J. (April 2022). "Surface fluctuating hydrodynamics methods for the drift-diffusion dynamics of particles and microstructures within curved fluid interfaces". Journal of Computational Physics. 455: 110994. arXiv:1906.01146. doi:10.1016/j.jcp.2022.110994.
5. ^ Atzberger, Paul J. (2011). "Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations". Journal of Computational Physics. 230 (8): 2821–2837. arXiv:1009.5648. Bibcode:2011JCoPh.230.2821A. doi:10.1016/j.jcp.2010.12.028. S2CID 6067032.
6. ^ Atzberger, Paul (2013), "Incorporating Shear into Stochastic Eulerian Lagrangian Methods for Rheological Studies of Complex Fluids and Soft Materials", Physica D, 265: 57–70, arXiv:2212.10651, doi:10.1016/j.physd.2013.09.002
7. ^ Atzberger, Paul (2016). "Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes". Soft Matter, The Royal Society of Chemistry. 12: 6685–6707. arXiv:1601.06461. doi:10.1039/C6SM00194G.
8. ^ Moria, Yoichiro; Peskin, Charles S. (2008). "Implicit Second-Order Immersed Boundary Methods with Boundary Mass". Computer Methods in Applied Mechanics and Engineering. 197 (25–28): 2049–2067. Bibcode:2008CMAME.197.2049M. doi:10.1016/j.cma.2007.05.028.
9. ^ Olson, S.; Lim, S.; Cortez, R. (2013). "Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation". Journal of Computational Physics. 238. doi:10.1016/j.jcp.2012.12.026.