## 2 Immersed Boundary-Lattice Boltzmann Coupling Scheme  
### Immersed Boundary Method  
In the IB method, the flow field is described on a fixed Eulerian mesh, and the IB is represented by sets of Lagrangian points. The formulation is as follows [4]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{1}\]  
\[\frac{\partial(\rho\mathbf{u})}{\partial t}+\
abla\cdot(\rho\mathbf{u} \mathbf{u})=-\
abla p+v\
abla\cdot[\rho(\
abla\mathbf{u}+(\
abla\mathbf{u})^{ T})]+\mathbf{f} \tag{2}\]  
\[\frac{\partial\mathbf{X}_{l}(s,t)}{\partial t}=\mathbf{U}(\mathbf{X}_{l},t)= \int_{\Omega_{f}}\mathbf{u}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X}_{l}(s,t ))\mathrm{d}\mathbf{x} \tag{3}\]  
\[\mathbf{f}(\mathbf{x},t)=\int_{\Gamma_{b}}\mathbf{F}(s,t)\delta(\mathbf{x}- \mathbf{X}_{l}(s,t))ds \tag{4}\]\[\mathbf{F}(s,t)=\mathbf{F}_{e}(s,t)-\mathbf{F}_{d}(s,t) \tag{5}\]  
where \(\mathbf{x}\), \(p\), \(\mathbf{u}\), and \(\mathbf{f}\) are the Eulerian spatial coordinate, flow pressure, fluid velocity, and external force density, respectively; \(\mathbf{X}_{l}\), \(\mathbf{F}\), and \(\mathbf{U}\) are the Lagrangian IB position, IB force density, and IB moving speed, respectively; and \(\delta(r)\) is the delta function, which can be written as:  
\[\delta_{h}\big{(}x,y\big{)}=h^{-2}\phi\big{(}\frac{x}{h}\big{)}\phi\big{(} \frac{y}{h}\big{)} \tag{6}\]  
where \(h\) is the mesh spacing, and \(\phi(r)\) is normally selected as:  
\[\phi(r)=\left\{ \begin{array}{ll}\frac{1}{4}\big{(}1+\cos(\frac{\pi r}{2}) \big{)},&\text{if }|\ r\mid\leq 2\\ 0,&\text{otherwise}.\end{array} \right. \tag{7}\]  
Equations (1) and (2) are the N-S equations with external force \(\mathbf{f}\) in Eulerian form for the fluid flow in domain \(\Omega_{f}\), whereas Equations (3)-(5) are the dynamic equations in Lagrangian form for the boundary \(\Gamma_{b}\). Equations (3) and (4) describe the interaction between the IB and the fluid flow. The IB imposes the flow velocity on the boundary to obtain IB velocity \(\mathbf{U}\) and the fluid flow spreads the Lagrangian IB force to the fluid to obtain the Eulerian force. \(\mathbf{F}\) in Equation (5) is the reaction force of IB, which is contributed to by the elastic force \(\mathbf{F}_{e}\) and the inertial force \(\mathbf{F}_{d}\), and given by:  
\[\mathbf{F}_{e}(s,t) =\mathbf{F}_{s}(s,t)+\mathbf{F}_{b}(s,t)+\mathbf{F}_{f}(s,t) \tag{8}\] \[=\frac{\partial}{\partial s}\Big{[}K_{b}\frac{\partial\mathbf{X }}{\partial s}\Big{]}-K_{b}\frac{\partial^{4}\mathbf{X}}{\partial s^{4}}+K_{ f}(\mathbf{Z}-\mathbf{X})\]  
\[\mathbf{F}_{d}(s,t)=m_{s}\frac{\partial\mathbf{U}(s,t)}{\partial t} \tag{9}\]  
The right-hand side in Equation (8) consists of a tension force \(\mathbf{F}_{s}\), a bending force \(\mathbf{F}_{b}\), and a fastening force \(\mathbf{F}_{f}\), which represent the stretching/compression effect, the bending/torsion effect, and prescribing speed for certain structure points, respectively. The discrete coefficients \(K_{s}\), \(K_{b}\), and \(K_{f}\) are the tension stiffness, bending rigidity, and fastening stiffness (an artificial coefficient), respectively. According to the literature [19], Equation (8) is discretized as follows:  
\[(\mathbf{F}_{s})_{l} =\frac{K_{s}}{(\Delta s)^{2}}\sum_{m=1}^{n_{f}-1}(|\mathbf{X}_{m +1}-\mathbf{X}_{m}|-\Delta s)\frac{\mathbf{X}_{m+1}-\mathbf{X}_{m}}{|\mathbf{ X}_{m+1}-\mathbf{X}_{m}|}(\delta_{ml}-\delta_{m+1,l}) \tag{10}\] \[(\mathbf{F}_{b})_{l} =\frac{K_{b}}{(\Delta s)^{4}}\sum_{m=2}^{n_{f}-1}(\mathbf{X}_{m+ 1}+\mathbf{X}_{m-1}-2\mathbf{X}_{m})(2\delta_{ml}-\delta_{m+1,l}-\delta_{m-1, l})\] (11) \[(\mathbf{F}_{f})_{l} =K_{f}(\mathbf{X}_{l}-\mathbf{Z}_{l}) \tag{12}\]  
Here, \((\mathbf{F}_{s})_{l}\), \((\mathbf{F}_{b})_{l}\), and \((\mathbf{F}_{f})_{l}\) are the three Lagrangian force densities associated with the node \(l\) (\(l=1,2,\cdots,n_{f}\) is the serial number of the Lagrange points), \(\mathbf{Z}_{l}(s,t)\) denotes the position of the fixed point. \(\delta_{ml}\) is the Kronecker symbol, which is defined as:  
\[\delta_{ml}=\left\{