143_add_Added Masses of Ship Structures.pdf - 132 5 Added Masses of Bodies Moving Close to a Free Surface and analogously from the remaining two

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132 5 Added Masses of Bodies Moving Close to a Free Surface and, analogously, from the remaining two equations of motion we get v y = − ∂y p t ρ ; (5.4) v z = − ∂z p t ρ , (5.5) where the variable p t = τ 0 p dt , called the pressure momentum, has finite value as τ 0. The formulas ( 5.3 )–( 5.5 ) imply the existence of potential ϕ such that v x = ∂ϕ/∂x , v y = ∂ϕ/∂y , v z = ∂ϕ/∂z . Due to ( 5.3 )–( 5.5 ) we get ϕ = − p t ρ . (5.6) Applying Eq. ( 5.6 ) to the free surface we get the boundary condition ϕ = 0 . (5.7) Since on a free surface the pressure is always constant and to equal p 0 , and τ 0, we see that p t = 0 . 5.1.2 Boundary Conditions on a Free Surface under Periodic Oscillations of a Floating Body Periodic oscillations of a body floating close to the free surface of a fluid cause periodic wave motion. Denote the amplitude of the waves by r , the wavelength by λ (the wave number is k 0 = 2 π/λ ) and the circular frequency by σ . Consider the Navier–Stokes equations ( 5.1 ). Introducing the following dimensionless variables [100]: t 0 = σt,

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• Fall '09
• Fundamental physics concepts, free surface

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