University:
Massachusetts Institute of Technology
Course:
2.24 | Ocean Wave Interaction with Ships and Offshore Energy Systems (13.022)
Academic year:

2019
Views:

177

Pages:

11
Author:

Elise Donaldson

ANALYTICAL SOLUTIONS OF THE WAVE-BODY PROBLEM FORMULATED ABOVE ARE RARE. THE FEW EXCEPTIONS WHICH FIND FREQUENT USE IN PRACTICE ARE: * WAVEMAKER THEORY (STUDIED) * DIFFRACTION BY A VERTICAL CIRCULAR CYLINDER (STUDIED BELOW) * LONG-WAVELENGTH APPROXIMATIONS (STUDIED NEXT) LONG-WAVELENGTH APPROXIMATIONS VERY FREQUENTLY THE LENGTH OF AMBIENT WAVES λ IS LARGE COMPARED TO THE DIMENSION OF FLOATING BODIES. FOR EXAMPLE THE LENGTH OF A WAVE WITH PERIOD T=10 sec IS λ = gT²/2π = 150m THE BEAM OF A SHIP WITH LENGTH L=100m CAN BE 20m AS IS THE CASE FOR THE DIAMETER OF THE LEG OF AN OFFSHORE PLATFORM. GI TAYLOR'S FORMULA [Diagram: A body (B) is shown immersed in a fluid flow. The fluid velocity (U) and pressure (P) are indicated.] BODY FIXED IN SPACE U(x,t): VELOCITY OF AMBIENT UNIDIRECTIONAL FLOW P(x,t): PRESSURE CORRESPONDING TO U(x,t) λ ~ |U| / |dU/dt| >> B = BODY CHARACTERISTIC DIMENSION IN THE ABSENCE OF VISCOUS EFFECTS AND TO LEADING ORDER FOR λ >> B : Fx = - (V + A₁₁) ∂P/∂x |x=x₀ Fx : FORCE IN X-DIRECTION V : BODY DISPLACEMENT A₁₁ : SURGE ADDED MASS {Proof IN MH} F) HEAVE EXCITING FORCE ON A SURFACE PIERCING SECTION IN LONG WAVES, THE LEADING ORDER EFFECT IN THE EXCITING FORCE IS THE HYDROSTATIC CONTRIBUTION: X₃ ~ ρgA_w A WHERE A_w IS THE BODY WATER PLANE AREA IN 2D OR 3D. A IS THE WAVE AMPLITUDE. THIS CAN BE SHOWN TO BE THE LEADING ORDER CONTRIBUTION FROM THE FROUDE-KRYLOV FORCE X₃ = ρgA_w ∫∫ e^(kz-ikx) ds SB USING THE TAYLOR SERIES EXPANSION: e^(kz-ikx) = 1 + (kz-ikx) + O(kB)² IT IS EASY TO VERIFY THAT: X₃ → ρgA_w A. THE SCATTERING CONTRIBUTION IS OF ORDER KB. FOR SUBMERGED BODIES: X₃ = O(kB)