# Numerical Simulation of a Submerged Floating Tunnel: Validation and Analysis
## 2 CEL Method for One- and Two-Way Fluid-Structure Coupling Validation
### Numerical Simulation of Circular Cylinder Flow  
The circular cylinder flow problem is an obtuse-body winding problem and thus exhibits the unique phenomenon of vortex shedding into its wake. This phenomenon was first formally proposed by [PERSON] in 1912, and forms a vortex array known as the \"Karman vortex street\". Numerous researchers, including [PERSON] [29], [PERSON] [30], and [PERSON] [20], have studied this phenomenon experimentally, and [PERSON] [20] found that it is related to the Reynolds number, which is defined in Equation (4) as follows:  
\[Re=\frac{\rho VD}{\eta} \tag{4}\]  
where \(\rho\), \(V\), and \(\eta\) represent the density, velocity, and viscosity of the fluid, respectively; \(D\) is the diameter of the cylinder; and \(Re\) is the Reynolds number. As shown in Table 1, when \(15<Re<40\), two Foppl vortices were produced in the flow field upstream and downstream of the rear sides of a cylinder, and, as \(Re\) increased, the tail vortex gradually became longer but remained attached to the rear side of the cylinder. When \(150<Re<300\),000, the two tail vortices were shed alternately, influenced by the downstream fluid propagation, and propagated at a slightly lower speed than the fluid flow, thereby forming the Karman vortex street. As \(Re\) increased, the shedding frequency of the tail vortices increased.  
The calculation parameters of the hypothetical model [32] used in the current study are as follows. The circular rod diameter was 0.02 m and the length of the part of the rod submerged in the water body was 0.14 m. The water body was simulated using Eulerian elements, with a density of 1000 kg/m\({}^{3}\), a viscosity of 0.001 Pa\(\cdot\)s, and a speed of sound of 1480 m/s\({}^{2}\). The upper surface of the water body was a free boundary, and the initial inflow velocity was set for the fluid. An Euler inflow boundary was applied on one side and an Euler outflow boundary on the other side. The inflow boundary had an initial velocity of \(V_{x}\), while the outflow boundary allowed the material to flow freely (Figure 1). A smooth boundary was used at the bottom and on both sides, where \(V_{z}\) (the velocity in the z  
\begin{table}
\begin{tabular}{c c} \hline \hline \(15<Re<40\): A group of small, fixed vortices \\ appeared after a cylinder \\ \hline \(150<Re<300\),000 Periodic alternating turbulent \\ vortex discharge occurred \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relationship between vortex shedding and Reynolds number (\(Re\)) [31].  
direction) and \(V_{y}\) (the velocity in the y direction) were set to zero. The contact between the cylinder and the fluid was defined as a smooth hard contact, where no mutual intrusion was allowed in the normal direction and the tangential friction was zero.  
Figure 2 shows the results of a CEL-based simulation of the circular cylinder flow experiment. When \(Re\) = 20, a long vortex area was generated downstream of the cylinder. When \(Re\) was increased to 200, a Karman vortex street formed, and the volume of the detached vortex increased. When \(Re\) was increased to 2000, the Karman vortex street intensified, and the volume of the detached vortex decreases. When \(Re\) was again increased to 20,000, the Karman vortex street became even more intense and formed a periodic alternating turbulent vortex discharge. These results indicate that the CEL-based simulation method accurately models the Karman vortex street in the circular cylinder flow problem.  
Figure 1: Simplified modeling diagram of cylindrical flow.  
Figure 2: Velocity clouds and vector diagrams of simulated circular cylinder flow at various Reynolds numbers (\(Re\)) (the left and right figures correspond to the same time).  
The above simulation of the Karman vortex street was based on a one-way fluid-structure coupling calculation. However, as SFTs involve two-way fluid-structure coupling, it is important to verify the accuracy of the CEL method for simulating two-way fluid-structure coupling.