# Design and Evaluation of Novel Submerged Floating Tunnel Models Based on Dynamic Similarity
## 2 Similarity Theory of the Scale Model
### Dynamic Similarity  
The ratios of all the relevant forces in the model and prototype are equal. Assuming that the similarity ratio of the force is \(\lambda_{F}\), \(F_{m}\) is the resultant force at any point in the model, and \(F_{p}\) is the resultant force at the corresponding point in the prototype, then Gravity \(G\), viscous force \(T\), pressure \(P\), elastic force \(F_{E}\), surface tension \(S\), and inertial force \(I\) should adhere to the following similarity relationship:  
\[\lambda_{F}=\frac{F_{m}}{F_{p}}=\frac{G_{m}}{G_{p}}=\frac{T_{m}}{T_{p}}=\frac {P_{m}}{P_{p}}=\frac{F_{Em}}{F_{Ep}}=\frac{S_{m}}{S_{p}}=\frac{I_{m}}{I_{p}} \tag{7}\]where:  
\[G=mg=\rho l^{3}g \tag{8}\]  
\[F_{E}=EA=El^{2} \tag{9}\]  
\[I=ma=\rho l^{3}u^{2}/l=\rho l^{2}u^{2} \tag{10}\]  
Dynamic similarity is the most complex and difficult to achieve. In the hydrodynamic model experiment, the relevant forces include gravity, viscous force, surface tension, elasticity, inertial force, etc. Due to the varied mechanisms of these forces, maintaining the same similarity ratio for each force is impossible, leading to the scale effect. Although the scale effect is inevitable, its impact can be mitigated by appropriately selecting force similarity criteria based on the experimental purpose and structural characteristics, so that the scale of some of the key forces can be consistent.  
The SFT scale model will be applied to the wave tank experiment, so the model should be able to accurately reflect the interaction between the structure and the wave. Considering that the influence of gravity on wave propagation is more significant than other forces such as surface tension and viscous force, the Froude similarity principle is widely used in wave-related experiments; the model and the prototype maintain an identical Froude number, which is a dimensionless number that indicates the relative effects of inertial forces compared to gravitational forces within a fluid flow. In addition, an SFT is mainly used to cross wide and deep waters. The substantial ratio of length to diameter in SFTs leads to pronounced elastic deformation under environmental loads on the macro scale. The Cauchy similarity principle correlates inertial forces to elastic forces so that the model can reflect the elastic deformation of the structure under the action of waves, buoyancy, and other loads. Incorporating both Froude and Cauchy similarity principles is a distinguishing feature and advantage of this model. The hydroelastic scale model can simulate elastic deformation and complex hydrodynamic phenomena, such as the fluid-structure interaction and vortex-induced vibration, thereby more accurately revealing the hydrodynamic responses of SFTs under environmental excitations.  
The model satisfies both the Froude similarity principle and the Cauchy similarity principle. The dynamic similarity relationship of the model can be explained using Equation (11), which can be divided into three equations.  
\[\lambda_{F}=\frac{G_{m}}{G_{p}}=\frac{F_{Em}}{F_{Ep}}=\frac{I_{m}}{I_{p}} \tag{11}\]