A linearized model for the waves can be used for calculating de total force induced by the potential flow, as the integral of the pressure field.
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Equation of Continuity:
As the fluid considered is a liquid, it can be assumed ρ=constant , and thus:
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Balance of momentum:
Assuming ρ=constant , and neglecting viscous effects (irrotational movement), the previous equations result as:
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For solving these equations the boundary conditions to consider are:
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Velocity normal to the surface equal to zero, and tangent velocity non-zero:
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Calling:
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The momentum equations result:
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A solution deriving from a potential Φ is taken. It must satisfy Laplace equation
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Momentum equation now reduces to Bernoulli equation, taken a constant C as integration constant.
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For integration, imposing the boundary conditions, the problem simplifies:
- Interior:
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- Bottom (z=-d ):
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- Free surface (z=0 ):
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A solution for the Laplace problem of the type Φ=f(z).g(x) can be found:
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With:
- K=L/2π ; wave number
- ω ; angular frequence
- ω 2 = K.g.tanh(K.d)
In our particular case, the expression that seems to suit better for the pressures created by the wave over the ship´s hull is:
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"z " coordinates will be negative in this case (i.e. the reference system origin is the wave surface, with the Z axis pointing downwards).