Wave load on a navigation lock sliding gate

Abstract The wave load on a navigation lock sliding gate equipped with a ballast tank is investigated assuming the linear wave theory is valid and using the eigenfunction expansion matching method to solve the governing equations. The correctness and the limits of the solution have been checked by comparing the results with those of a numerical code which solve the full non linear Navier-Stokes equations. The results relating to the effect of the geometrical parameters and wave characteristics on the load acting on the gate are presented and discussed. The ballast tank is found to increase the complexity of the phenomenon in relation to the wave interaction with the gate. The results indicate that the peak value of the vertical force occurs for wave numbers mostly dependent on the tank length and on the tank position along the depth, while the thickness has a smaller influence. The ballast tank has also a significant effect on the horizontal dynamic force acting on the gate, which vanishes when the wave number takes particular values. Finally, the moment applied to the gate shows a dependency on the geometrical and hydrodynamic parameters similar to that of the forces.


Formulation of the problem and analytical solution
The flow generated by a progressive water wave propagating towards a 90 navigation lock is here considered. Figure 1 shows a sketch of the problem. The origin of the reference system is placed at the intersection between the 92 still water level and the vertical wall. The x axis points in the direction of the 93 incoming waves, while the z axis points upwards. It is assumed that the waves 94 propagate in the direction orthogonal to the gate and that the latter have 95 constant geometrical characteristics along the y axis of the reference system. 96 Therefore, considering a structure infinitely extended in the y direction, the 97 flow is two dimensional. It is assumed that the incoming wave is characterized 98 by an amplitude H/2 much smaller than the wavelength L such that the 99 linear potential wave theory can be applied (Mei et al., 2005). Denoting the 100 velocity potential by the symbol Φ, the mathematical problem is posed by 101 the Laplace equation plus appropriate boundary condition as reported below: ∂Φ ∂n = 0 at the rigid boundaries, where g is the gravity acceleration, t is the time and n is the unit vector 103 orthogonal to the boundary. Assuming that the incoming wave is sinusoidal 104 with angular frequency σ, the potential can be written as Φ = Re(ϕe iσt ) 105 where i is the imaginary unit and ϕ is a new potential function depending on 106 x and z. In terms of ϕ, the governing equations 1-3 can be written as follows ∂ϕ ∂n = 0 at the rigid boundaries, −σ 2 ϕ + g ∂ϕ ∂z = 0 at z=0, After ϕ has been determined, the pressure p is computed as follows: The mathematical problem posed by equations 4-6 has been solved by means while region Π 3 is located above the ballast tank. 113 In region Π 1 the potential ϕ is given by the sum of three terms as follows The first term is the potential of the incoming wave, the second term is 115 the potential due to the wave reflected by the gate and finally the third term 116 is the potential of vanishing wave modes. The wavenumbers k n satisfy the 117 following dispersion relations: 118 σ 2 = gk n tanh(k n h) n = 1 (9) σ 2 = −gk n tan(k n h) n > 1 (10) It is worth to highlighting that equation 10 has infinite k n solutions. In 119 region Π 2 the potential ϕ can be written as follows: where the wavemunbers β n are given by the relations: where h 1 = h − (d + s) (see Figure 1).

122
In region Π 3 the potential has the following expression: where the wavenumbers λ n satisfy the dispersion relations: The potentials ϕ 1 , ϕ 2 and ϕ 3 given by equations 8, 11 and 13 respectively 125 already satisfy the boundary conditions at the rigid walls (eq. 5), except at 126 the vertical face of the ballast tank, and at the free surface (eq. 6).
The coefficient A ji , that appear in equations 8, 11 and 13, are expansion coefficients whose values must be determined by matching the solutions per- imposed by means of the following equations: The matching of the solutions between the regions Π 1 and Π 2 and between 142 the regions Π 1 and Π 3 at x = −b is expressed by the following equations: Analogously, the solution in region Π 3 , provides the following expression 161 for the force acting on the upper face of the ballast tank: The horizontal force is the sum of the following components: the force 163 acting on the surface that lies below the ballast tank denoted as f h2 , the 164 force acting on the vertical face of the ballast tank denoted as f h1 and the 165 force acting on the surface that lies above the ballast tank, denoted as f h3 .

166
These forces have the following expressions: When analyzing the gate stability under the action of the wave load, the 168 moment produced by the pressure acting on the gate must also be considered.

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In the specific case, such moments can be distinguished into moments due view of their importance as concern the gate stability and also because these 179 moments depend on the geometrical and flow parameters in a complex way.
In the previous equations the number N of eigenfunctions that appear in

Validation and limits of the model
In order to validate the model, a first comparison was made with the 190 results of Wu et al. (1998). Figure 2 shows the amplitude of the dimensionless        On the contrary, the horizontal force acting on the lower gate portion Figure 11, has a much larger os- As regards the moment of the vertical forces, Figure 13 shows the am-