lecture14 - 2.20 - Marine Hydrodynamics, Spring 2005...

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±±± ± ± ±± Lecture 14 - Marine Hydrodynamics Lecture 14 3.20 Some Properties of Added-Mass Coefficients 1. m ij = ρ · [function of geometry only] F, M = U, Ω] [linear function of m ij ] × [function of instantaneous U, ˙ not of motion history 2. Relationship to fluid momentum. F(t) where we define Φ to denote the velocity potential that corresponds to unit velocity U = 1. In this case the velocity potential φ for an arbitrary velocity U is φ = U Φ. The linear momentum L ± in the fluid is given by L ± = vdV = ρ φdV = + ρφ ˆ ρ± ndS V V Green’s B theorem ²³´µ φ 0a t L x ( t = T )= ρU Φ n x dS = U ρ Φ n x dS B B The force exerted on the fluid from the body is F ( t ( m A U ˙ m A U ˙ . 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20
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±±± ²³ ´µ ±± ²³´ µ T T ± ± Newton’s Law ˙ T dt [ F ( t )] = m A Udt = m A U ] 0 = L x ( t = T ) L x ( t =0)= U ρ Φ n x dS 0 0 m A U B Therefore, m A = total fluid momentum for a body moving at U = 1 (regardless of how we get there from rest) = fluid momentum per unit velocity of body. Φ K.B.C. ∂φ = φ · n ˆ=( U, 0 , 0) · n ˆ= Un x , = x ∂U Φ = x = n x ∂n Φ m A = ρ Φ dS B For general 6 DOF: Φ j m ji = ρ Φ i n j dS = ρ Φ i dS = j fluid momentum due to ²³´µ ²³´µ i body motion j force/moment B potential due to body B i direction of motion moving with U i =1 3. Symmetry of added mass matrix m ij = m . · Φ j m = ρ Φ i dS = ρ Φ i ( Φ j · n ˆ) dS = ρ ∇· i Φ j ) dV B B Divergence V Theorem = ρ Φ i ·∇ Φ j i 2 Φ j dV V =0 Therefore, m = ρ Φ i Φ j dV = m ij V 2
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±²³´ µµµ 4. Relationship to the kinetic energy of the fluid. For a general 6 DoF body motion U i =( U 1 ,U 2 ,...,U 6 ), φ = U i Φ i i = potential for U i =1 notation 1 ¶¶¶ K.E. = ρ φ ·∇ φdV = 2 1 ρ U i Φ i · U j Φ j dV 2 V V 1 = 2 ρU i U j Φ i Φ j dV = 1 2 m ij U i U j V K.E. depends only on m ij and instantaneous U i . 5. Symmetry simplifies m ij . From 36 21 ‘?’. Choose such coordinate system symmetry that some m ij = 0 by symmetry. Example 1 Port-starboard symmetry. m 11 m 12 0 0 0 m 16 F x 0 0 0 F y F z M x 12 independent coefficients m 22 m 26 0 m 33 m 34 m 35 m ij = 0 m 44 m 45 0 M y m 55 m 66 M z U 1 U 2 U 3 Ω 1 Ω 2 Ω 3 3
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Example 2 Rotational or axi-symmetry with respect to
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lecture14 - 2.20 - Marine Hydrodynamics, Spring 2005...

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