r9_hydroforces - 13.42 Spring 2005 13.42 Design Principles...

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13.42 Spring 2005 13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. Forces on Large Structures For discussion in this section we will be considering bodies that are quite large compared to the wave amplitude and thus the inertial component of force dominates over the viscous forces. Typically we can neglect the viscous force when it is less that 10% of the total force, except near sharp edges and separation points. We must be careful to consider wave diffraction when the wavelength is less that 5 cylinder diameters. If we assume that the viscous effects can be neglected and we consider the case of irrotational flow, then we can write the velocity field in terms of the potential function, φ ( x ,,, yz t ) . φφ  ∂ ∂∂ Vx y z ) , ( =∇ = x , z (1) y The governing equation of motion is given by the Laplace equation 2 2 2 2 0 ∇= + + = . (2) x 2 y 2 z 2 Given a body in the presence of the wave field we much consider the relevant boundary conditions on the free surface, the seafloor and the body. Boundary conditions are, on the bottom, n ± 0 ⋅= =, (3) n and, on the body, ± n n ± = V B = V B n . (4) n At the free surface the kinematic and dynamic boundary conditions must be satisfied. The version 1.0 updated 3/29/2005 -1- 2004, aht
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13.42 Spring 2005 linearized free surface boundary conditions are both taken about z = 0 as we are accustomed to, given that a λ /< < 1 , 2 φ + g = 0 (5) t 2 z 1 η = − . (6) g t In the case of a free floating body we must also take into consideration the wave field generated by the body motion alone. At some distance far away from the body, the potential function must take into consideration the waves radiating from the body. 1.1. The Total Wave Potential Figure 1. Boundary conditions for the total potential must be met at three places: sea floor (3), free surface (2), structure surface (4). The continuity equation must be satisfied within the fluid (1). Due to the nature of potential flow and linear waves it is possible to sum multiple potential functions to obtain the total potential representative of the complete flow field. Each component of the total potential must also satisfy the appropriate boundary conditions. For linear waves incident on a floating body the total potential is a sum of the undisturbed incident waved potential, ( x ,,, yz t ) , the diffraction potential, ( ) , due to the x yz I D version 1.0 updated 3/29/2005 -2- 2004, aht
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13.42 Spring 2005 presence of the body when it is motionless, and the radiation potential, φ ( ,,, ) , x yz t R representing the waves generated (radiating outwards) by a moving body. For a permanently fixed body the radiation potential is non-existent (ie. ( xy z = 0). ) R ( ) + ( x yz ) R = ( ) + ( ) (7) I D It is good here to note the important conditions on each component of the total potential.
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r9_hydroforces - 13.42 Spring 2005 13.42 Design Principles...

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