## 2 Mathematical Model and Numerical Method
### The Local DFD Method  
A self-developed immersed boundary method, called the local DFD method [22], is employed to deal with the moving cylinder boundaries. In this method, all the used unstructured triangular meshes are obtained by dividing the rectangular cells of the Cartesian meshes (see Figure 3) and Equations (1) and (2) are discretized at all the nodes in the fluid phase. However, the discrete form at an interior node near the wall may involve some exterior nodes. The key point of the local DFD method is how to calculate the flow variables at an exterior-dependent node, which will be described briefly here.  
A reference point in the fluid phase is first defined for the calculation of the flow variables at a given exterior-dependent node. As depicted in Figure 3, the reference point  
Figure 3: Sketch map of the IB model.  
Figure 2: Three-dimensional schematic diagram of the investigated model. (**A**): an isolated cylinder; (**B**): two inline cylinders.  
for the exterior-dependent node \(D\) is defined to be the intersection between the wall-normal line and the interior cell edge nearest to the wall. The two vertices of the interior cell edge are interior nodes. For easy programming, the point located on the wall-normal line and with a constant distance away from the wall is set to be the corresponding reference point in the codes. The user-defined constant distance is about one local mesh interval, which makes the vertices of the host triangle the reference point located in all interior nodes, like \(F^{\prime}\) depicted in Figure 3.  
In the hybrid-Cartesian immersed boundary (HCIB) method proposed by [PERSON] and his coworkers [28; 29], a reference point is also needed to reconstruct solutions at the interior nodes in the immediate vicinity of the solid boundary. The distance of this defined reference point away from the solid wall is about twice the local mesh interval. The reference points defined both in the local DFD method and in the HCIB method are regarded as the first-layer mesh nodes. Therefore, for the same computational case, the near-wall spacing of the mesh used by the HCIB should be twice smaller than that of the mesh used by the local DFD.  
The flow variables at the reference point \(F^{\prime}\) are calculated via linear interpolation over its host triangle, i.e., \(ABC\) in Figure 3. Then, the Cartesian velocity components at node \(D\) are obtained by linear extrapolation along the wall-normal direction with the no-slip boundary condition enforced:  
\[u_{i,\;D}=\frac{|F^{\prime}D|U_{i,\;W}-|WD|u_{i,\;F^{\prime}}}{|F^{\prime}W|}, \tag{3}\]  
where \(U_{i,\;W}\) (\(i=1,2\)) represents the velocity components of the surface at \(W\) and \(u_{i,\;D}\) and \(u_{i,\;F^{\prime}}\) are the velocity components at \(D\) and \(F^{\prime}\), respectively. Solving a simplified momentum equation in the wall-normal direction and using the non-penetration at \(W\), the pressure at D can be obtained:  
\[p_{D}=p_{F^{\prime}}+\left|F^{\prime}D\right|\left(\frac{d\overset{\sim}{V}}{ dt}\right)_{W}, \tag{4}\]  
where \(\overset{\sim}{V}\) is the normal velocity of the body motion and \(p_{F^{\prime}}\) is the pressure at \(F^{\prime}\).