13.42 Spring 2005 version 1.0 updated 3/29/2005 -1- 2004, aht 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, ()xyztφ,,, . ()Vxyztxyzφφφφ∂∂∂,,, =∇ = , ,∂∂∂ (1) The governing equation of motion is given by the Laplace equation 22222220xyzφφφφ∂∂∂∇= + + =.∂∂∂ (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, 0nnφφ∂∇⋅= =,∂ (3) and, on the body, BBnnVnVnφφ∂∇⋅= = ⋅= .∂ (4) At the free surface the kinematic and dynamic boundary conditions must be satisfied. The13.42 Spring 2005 version 1.0 updated 3/29/2005 -2- 2004, aht linearized free surface boundary conditions are both taken about 0z = as we are accustomed to, given that 1aλ/<<, 220gtzφφ∂∂+=∂∂ (5) 1gtφη∂=−.∂ (6) 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, ( )Ixyztφ,,, , the diffraction potential, ( )Dxyztφ,,, , due to the13.42 Spring 2005 version 1.0 updated 3/29/2005 -3- 2004, aht presence of the body when it is motionless, and the radiation potential, ( )Rxyztφ,,, , representing the waves generated (radiating outwards) by a moving body. For a permanently fixed body the radiation potential is non-existent (ie. ( ) 0Rxyztφ,,, = ). ()()()()IDRxyzt xyzt xyzt xyztφφφ φ,,, = ,,, + ,,, + ,,, (7) It is good here to note the important conditions on each component of the total potential. The incident potential is formulated from that of a free wave without consideration for the presence of the body. Therefore ( )Ixyztφ,,, satisfies only the free surface boundary conditions and the bottom boundary condition, in addition to the Laplace equation. The diffraction potential, ( )Dxyztφ,,, , must also satisfy the Laplace equation, the free surface and the bottom boundary conditions. In order to compensate for the disturbance of the incident wave around the body by the an additional condition at the body boundary such that the normal gradient of the diffraction potential is equal but opposite in sign to the normal gradient of the incident potential: DInnφφ∂∂=−.∂∂ (8) The radiation potential satisfies the same conditions as the incident potential as well as an additional condition on the moving body boundary in the absence of incoming waves. On the body boundary the normal gradient of the velocity potential must equal the normal velocity of the body. RBVnnφ∂=⋅.∂ (9)13.42 Spring 2005 version 1.0 updated 3/29/2005 -4- 2004, aht 1.2. Complex form of the Wave Potential It is often easiest, for the purpose of systems analysis, to write the wave potential in its complex form: ˆˆ()itID Rxyzt Re a a eωφφφ φ,,, = + + , (10) where a is the wave amplitude and ω is the incoming wave frequency. The complex incident potential is mitIIaeωφφ= . The amplitude of the incident potential is the wave amplitude times mIφ, which is a function of depth and position in space. The diffraction potential takes the same form in order to satisfy the body boundary condition. The radiation potential is not necessarily directly related to the wave amplitude. Since Rφ results from the motion of a floating body in the absence of waves, we must consider the body motion in all six degrees-of-freedom. The vessel motions are prescribed by, jx, where 1 2 3 4 5 6j =,,,,, (surge, sway, heave, roll, pitch, and yaw). It is customary to write the complex radiation potential in the following form: 61jRjjxφφ==∑ (11) where jx is the velocity of the body in the jth direction and jφ is the velocity potential due to a unit motion in the jth direction. 1.2.1. Incident Potential Boundary Conditions The incident potential, ( )Ixyztφ,,, , is considered without knowledge of the presence of any structure. The boundary conditions must be sufficient to arrive at the correct potential far from the structure. 1. 20Iφ∇=; Continuity equation. 2. 220IIztgφφ∂∂∂∂+=; Combined free surface condition.13.42 Spring 2005 version 1.0 updated 3/29/2005 -5- 2004, aht 3. 0Izφ∂∂= ; Bottom boundary condition. 4. Incident potential does not have any knowledge of body in the flow. Rewriting the incident potential in its complex form, itIIRea eωφφ=, we can simplify the boundary conditions even further: 1. 20Iφ∇=; 2. 20IzIgφωφ∂∂−+=; 3. 0Izφ∂∂= ; 4. Incident potential does not have any knowledge of body in the flow. 1.2.2. Diffraction Potential Boundary Condtions The diffraction potential, ( )Dxyztφ,,, , results from the presence of the structure in the flow field. Without the structure, there would be no wave diffraction. This potential accounts for the alteration of the incident wave train by the structure and we must now include