## 2 Analysis of Resistance Characteristics of Towed Tunnel Ship Navigation
### Analysis of Resistance Characteristics of Ships Navigating Through Towed Navigation Tunnels  
The navigable tunnel represents a specialized type of middle channel, and it can be calculated using the formula for ship resistance in middle channels proposed by [PERSON] et al. [14], as shown in Formula (3):  
\[\mathrm{R}_{t}=1.75v^{1.83}+0.5 Sv^{2}\xi(\mathrm{Fr},\mathrm{n})/\left(\frac {L}{6B}\right)^{3} \tag{3}\]  
The symbols in the formula represent the following: \(\mathrm{R}_{t}\) is the resistance of ships in restricted navigation channels (kN); S represents the wetted surface area of the hull (m\({}^{2}\)), \(S=L_{w}(2T+C_{b}B)\); \(\xi\) is the residual resistance coefficient; \(\mathrm{Fr}\) is the Froude number, \(\mathrm{Fr}=v/\sqrt{gh}\); n is the section coefficient; \(v\) is the ship speed (m/s); g is the acceleration due to gravity (m/s\({}^{2}\)); L is the ship length (m); and \(L_{w}\) is the waterline length (m).  
In Equation (3), under the condition of the same ship type, the total resistance of the ship is related to the residual resistance coefficient and speed, where the residual resistance coefficient is also related to the Froude number and section coefficient.  
The section coefficient is a highly relevant quantity to the navigation characteristics of ships in towed navigation tunnels. It is defined as the ratio of the tunnel's discharge cross-sectional area to the ship's midship transverse sectional immersion area [15]. The section coefficient of the intermediate channel should generally not be less than 3.6, but the towed navigation tunnel adopts a non-self-propelled navigation method, the tunnel width is reduced, and the safe distance between the ship and the side wall is shortened, resulting in a section coefficient of less than 3. Therefore, when the ship is towed through the navigation tunnel, the ship's resistance will increase nonlinearly with the increase in speed and the decrease in section coefficient. The Froude number is related to the ship's motion speed and the channel depth. Under the same water depth, the Froude number increases with the increase of ship speed. The ship's wet area changes due to changes in the flow velocity at the bottom of the hull and changes in the flow field, resulting in a decrease in pressure, causing the hull to sink and the draft depth to increase. The ship's side wet area S increases with the sinking of the hull. Under the overall influence of the section coefficient, Froude number, and ship's wet area, the ship's resistance may increase exponentially.  
When a ship is towed through a navigable tunnel, the narrow width of the tunnel prevents the rapidly dissipating waves generated by the tow from dispersing quickly. This results in the superposition of bow waves and stern waves, leading to an increase in wave resistance [16]. The towing of a ship through a navigable tunnel also causes changes in the flow field around the ship's hull, primarily reflected in an increase in the flow velocity at the bottom of the hull, leading to an increase in viscous resistance. At this time, changes in the ship's posture will be influenced by changes in the flow field, resulting in a further increase in viscous resistance and inevitably increasing frictional resistance.  
When the tunnel is connected to a navigation structure, the flow velocity in the tunnel is relatively low, even at a standstill; when the tunnel is connected to a natural navigation channel, the flow velocity in the tunnel is relatively high, and the impact of the flow velocity in the tunnel on ships needs to be considered during towing navigation. Therefore, both static and dynamic water conditions need to be considered when in the tunnel.