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lecture13 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

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Lecture 13 - Marine Hydrodynamics Lecture 13 3.18 Unsteady Motion - Added Mass D’Alembert : ideal, irrotational, unbounded, steady. Example Force on a sphere accelerating ( U = U ( t ), unsteady) in an unbounded fluid that is at at rest at infinity. θ ϕ n ˆ U(t) 3D Dipole x r a U(t) ∂φ K.B.C on sphere: = U ( t ) cos θ ∂r r = a Solution: Simply a 3D dipole (no stream) 3 a φ = U ( t ) cos θ 2 r 2 ∂φ Check: ∂r = U ( t ) cos θ r = a 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20
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Hydrodynamic force: ∂φ 1 2 F x = ρ + |∇ φ | n x dS ∂t 2 B On r = a, ∂φ = ˙ a 3 1 ˙ U cos θ | = Ua cos θ ∂t 2 r 2 r = a 2 r = a ∂φ 1 ∂φ 1 ∂φ 1 φ | = , , = U cos θ, U sin θ, 0 r = a ∂r r ∂θ r sin θ ∂ϕ 2 |∇ φ | 2 = U 2 cos 2 θ + 1 U 2 sin 2 θ ; ˆ n = e ˆ r , n x = cos θ r = a 4 π dS = ( adθ ) (2 πa sin θ ) B 0 x a θ θ ad θ sin a 2
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( ) Finally, ˙ 1 2 F x = ( ρ ) 2 πa 2 π (sin θ ) cos θ 1 Ua cos θ + U cos 2 θ + 1 U 2 sin 2 θ 2 2 4 0 n x ∂φ 2 |∇ φ | ∂t π π F x = U ˙ ( ρa 3 ) π sin θ cos 2 θ + ( ρU 2 ) πa 2 sin θ cos θ cos 2 θ + 1 sin 2 θ 4 0 0 2/3 = 0, D’alembert revisited 2 F x = U ˙ ( t ) ρ πa 3 3 Hydrodynamic Force Acceleration Fluid Density Volume = 1 / 2 sphere Thus the Hydrodynamic Force on a sphere of diameter a moving with velocity U ( t ) in an unbounded fluid of density ρ is given by F x = U ˙ ( t ) ρ 2 πa 3 3 Comments: If U ˙ = 0 F x = 0, i.e., steady translation no force (D’Alembert’s Condition ok). F x U ˙ with a ( ) sign, i.e., the fluid tends to ‘resist’ the acceleration. [ · · · ] has the units of ( fluid ) mass m a Equation of Motion for a body of mass M that moves with velocity U : M U ˙ = Σ F = F H + F B = U ˙ m a + F B Body mass Hydrodynamic force All other forces on body Fluid
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