## 2 Numerical Method
### The IB Method for Fluid-Structure Interaction and Heat Transfer  
The pIB method developed by [PERSON] and [PERSON] [45] is used to handle the no-slip boundaries between the rigid body and the fluid. The interaction force between the fluid and the structure can be determined by the feedback law [45; 46]:\[\mathbf{F}=\alpha\int_{0}^{t}(\mathbf{U}_{ib}-\mathbf{U})dt+\beta(\mathbf{U}_{ib}-\mathbf{U}) \tag{16}\]  
where \(\mathbf{U}_{ib}\) is the boundary velocity obtained by interpolation at the IB, \(\mathbf{U}\) is the structure velocity, and \(\alpha\) and \(\beta\) are large positive free constants. The force acting on the Lagrange structure from the ambient fluid can be taken as a concentrated force acting on the corresponding nodes; thus, it can be added to the body force \(\mathbf{f}\) in Equation (7). Compared to the sharp-interface method [47; 48; 49; 50], the pIB method is particularly suitable here, since in this approach all the grid points within the computational domain are treated with a unified equation.  
The transformation between the Euler and Lagrange variables can be realized by the Dirac delta function. The interpolation of velocity and the spreading of the Lagrange force to the adjacent grid points are expressed as  
\[\mathbf{U}_{ib}(s,t)=\int_{V}\mathbf{u}(x,t)\delta_{h}(\mathbf{X}(s,t)-\mathbf{x})d\mathbf{x} \tag{17}\]  
\[\mathbf{f}(\mathbf{x},t)=-\int_{\Gamma}\mathbf{F}(s,t)\delta_{h}(\mathbf{X}(s,t)-\mathbf{x})ds \tag{18}\]  
where \(\mathbf{u}\) is the fluid velocity, \(\mathbf{X}\) is the coordinates of structural nodes, \(\mathbf{x}\) is the coordinates of fluid, \(s\) is the arc coordinate, and \(V\) is the fluid domain, and \(\Gamma\) is the structure domain.  
The position of the filament is updated explicitly by  
\[\mathbf{X}^{n+1}=\mathbf{X}^{n}+\mathbf{U}_{ib}\Delta t \tag{19}\]  
where \(\mathbf{U}_{ib}\) is calculated using (17). The internal force density of the filament can be calculated using (14) and (15). The details of the computation of the inertial force defined in (13) can be found in [17]. The \(\mathbf{F}_{f}\) can then be calculated by Equation (13). The force \(\mathbf{F}\) acting on the fluid in Equation (18) is the reaction force of \(\mathbf{F}_{f}\) defined in Equation (13). Equation (18) is used to distribute the force to the ambient fluid.  
The smooth function \(\delta_{h}\) is used to approximate the Dirac delta function  
\[\delta_{h}(x,y)=\frac{1}{h^{2}}\lambda(\frac{x}{h})\lambda(\frac{y}{h}). \tag{20}\]  
In this paper, the four-point delta function introduced by [PERSON] [51] is used  
\[\lambda(r)=\begin{cases}\frac{1}{8}(3-2|r|+\sqrt{1+4|r|-4|r|^{2}}),&0\leq|r|<1 \\ \frac{1}{8}(5-2|r|-\sqrt{-7+12|r|-4|r|^{2}}),&1\leq|r|<2\\ 0,&2\leq|r|.\end{cases} \tag{21}\]  
The heat transferred from the immersed boundary to the fluid can be written as [52]  
\[q=\int_{\Gamma}Q(s,t)\delta_{h}(\mathbf{X}(s,t)-\mathbf{x})ds \tag{22}\]  
where \(Q\) is the virtual boundary heat flux. When the temperature boundary is used, then \(Q\) can be calculated using the penalty immersed-boundary method, expressed as  
\[Q=\alpha_{T}(T_{ib}-T_{0}) \tag{23}\]  
where \(\alpha_{T}\) is a large factor, \(T_{0}\) is the boundary condition on the immersed boundary, and \(T_{ib}\) is the temperature of the virtual boundary, interpolated by  
\[T_{ib}(s,t)=\int_{V}T(x,t)\delta_{h}(\mathbf{X}(s,t)-\mathbf{x})d\mathbf{x}. \tag{24}\]