Convergence of the Immersed Boundary Method for an Elastically Bound Particle Immersed in a 2D Navier-Stokes Fluid Fluid

TL;DR

This work proves convergence of the immersed boundary method for a co-dimension 2 immersed elastically bound particle in a nonlinear 2D Navier–Stokes fluid by leveraging the mollified delta kernel with finite width $c$. The authors formulate the coupled IB–Navier–Stokes problem, design a stable implicit-explicit numerical scheme on a fixed grid, and establish a Lax–Richtmyer–style convergence result under the condition $h^2 \propto Δt$, with velocity and particle-position errors bounded by $O(Δt)$. Theoretical results are complemented by simulations (e.g., vortex formation behind a tethered cylinder) that corroborate the predicted convergence and illustrate the method’s behavior in nonlinear FSI. The approach addresses the singular forcing challenge intrinsic to IB methods by using a physically meaningful mollification, paving the way for rigorous analysis and potential extensions to more complex co-dimension scenarios.

Abstract

The immersed boundary (IB) method has been used as a means to simulate fluid-membrane interactions in a wide variety of biological and engineering applications. Although the numerical convergence of the method has been empirically verified, it is theoretically unproved because of the singular forcing terms present in the governing equations. This paper is motivated by a specific variant of the IB method, in which the fluid is 2 dimensions greater than the dimension of the immersed structure. In these co-dimension 2 problems the immersed boundary is necessarily mollified in the continuous formulation. In this paper we leverage this fact to prove convergence of the IB method as applied to a moving elastically bound particle in a fully non-linear fluid.

Figures (3)

• Figure 1: The function $\varphi$ defined by conditions 1-5. The support of the function is defined over a closed set--- the closure of the set of points at which the function is non-zero. This is useful here because +3 and -3 are therefore within the support and they are indeed the points at which $\varphi$ is only $C^{3}$
• Figure 2: Results of a numerical experiment. Top: the x and y coordinates of the centre of the cylinder as a function of time. The different colours represent the analogous results for various grid refinements, with yellow representing the coarsest mesh and blue representing the finest grid. The cylinder’s x coordinate appears to reach a an equilibrium as the mean flow balances with the restoring spring force. This equilibrium breaks down at about t= 4s. This timing coincides with the moment that the vortex sheet in the wake of the cylinder becomes unstable, breaking into a vortex street. This vortex street then induces a force that pulls the cylinder further from the centre. Bottom: A colourmap of the fluid vorticity. Note the upward trend of the vortex street. This is because of the choice of initial fluid velocity (chosen to break symmetry).
• Figure 3: These data show the results of an empirical convergence analysis of the experiment described above. Each measurement is made by comparing a simulation to the same simulation on a coarser grid (double the gridwidth and quadruple the timestep size). Both simulations were run to the final time $T=8$ s and compared. The norm of the difference is then plotted on this graph. The blue data point is the discrete $L_2$ norm of the difference in the fluid velocity between the coarse and fine grids. The red data is the distance between the centres of the cylinders for the fine and coarse simulations at the final time.

Theorems & Definitions (11)

• Theorem : Convergence
• Lemma 1: Consistency
• Lemma 2: Discrete Hemholtz Decomposition
• Lemma 3: Stability bound on deviation from constraint subspace
• Lemma 4: Iterative inequality
• Corollary 1
• proof : Proof of the convergence theorem
• proof : Proof of Lemma \ref{['Consistency']}
• proof : Proof of Lemma \ref{['Decomp']}
• proof : Proof of Lemma \ref{['ConstraintBound']}