lecture13 - 2.20 - Marine Hydrodynamics, Spring 2005...

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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 U(t) x 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 Check: = U ( t ) cos θ r = a 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20
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±± ² ³ ´ ² ³ ² ³ ´ ± Hydrodynamic force: ∂φ 1 2 F x = ρ + |∇ φ | n x dS ∂t 2 B On r = a, ´ ´ = ˙ a 3 1 ˙ U cos θ | = Ua cos θ ´ 2 r 2 r = a 2 r = a 1 1 1 φ | = , , = U cos θ, U sin 0 r = a ∂r r ∂θ r sin θ∂ϕ 2 |∇ φ | 2 ´ = U 2 cos 2 θ + 1 U 2 sin 2 θ 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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± ± ² ³ ´ µ ( ) 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 · ¸¹ º 2 2 4 0 · ¸¹ º · ¸¹ º n x ∂φ 2 |∇ φ | ∂t π π F x = U ˙ ( ρa 3 ) π d θ sin θ cos 2 θ +( ρU 2 ) 2 sin θ cos θ cos 2 θ + 1 sin 2 θ 4 0 0 · ¸¹ º · ¸¹ º 2/3 = 0, D’alembert revisited 2 F x = U ˙ ( t ) ρ 3 ·¸¹º ·¸¹º ·¸¹º 3 · ¸¹ º 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 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 : MU ˙ F = F H + F B = U ˙ m a + F B ·¸¹º ·¸¹º ·¸¹º ·¸¹º Body mass Hydrodynamic force All other forces on body Fluid mass
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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lecture13 - 2.20 - Marine Hydrodynamics, Spring 2005...

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