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

2019
Views:

218

Pages:

30
Author:

zrnje9b3

EXCITING FORCES AND HYDRODYNAMIC COEFFICIENTS THERE REMAINS TO DERIVE THE BOUNDARY VALUE PROBLEMS LEADING TO THE DEFINITION OF THE EXCITING FORCES Xj(w) AND THE IMPEDANCE HYDRODYNAMIC COEFFICIENTS Aij(w), Bij(w); i,j=1,3,4 THE STEPS FOLLOWED IN TWO DIMENSIONS AND THE RELATIONS THAT FOLLOW EXTEND ALMOST WITH NO CHANGE IN THREE DIMENSIONS. EXCITING FORCES [Diagram: A diagram shows a fixed body in a fluid domain. Incident waves are shown as sinusoidal waves approaching the body. Scattered waves are shown radiating away from the body.] Φ₁ = Re{ ∑ i ς_A e^(ikz - ikx - iωt)} k = w/g ξ = A cos(ωt - kx) = A cos ωt, x=0 = A Re{ e^{iωt} } PLANE PROGRESSIVE WAVES WITH POTENTIAL Φ₁ ARE INCIDENT TOWARDS A FIXED 2D BODY. THE CORRESPONDING WAVE ELEVATION IS DEFINED ABOVE AND HAS A ZERO PHASE RELATIVE TO x=0. ● ALL OTHER SEAKEEPING QUANTITIES HAVE A PHASE MEASURED RELATIVE TO THAT OF ξ AT x=0. LET φ be THE "DIFFRACTION" VELOCITY POTENTIAL DEFINED AS FOLLOWS { φ = Re{ φ_T e^{iωt} } ∂n(Φ₁ + φ) = 0, on SB (-ω² + g∂z) φ_T = 0, on z=0 } EXPRESSING IN COMPLEX FORM: F3(t) = - Re{ [-w² A33(w) ξ3 + iω B33(w) ξ3] x e^{iωt} } EQUATING THE TWO DEFINITIONS OF F3(t) AND KEEPING THE COMPLEX TERMS: w² A33(w) - iω B33(w) = -iω ∫∫ φ3 n3 ds SB THIS EQUATION DEFINES THE HEAVE ADDED MASS AND DAMPING COEFFICIENTS IN TERMS OF Ψ3, OBTAINED FROM THE SOLUTION OF THE FREE-SURFACE BOUNDARY-VALUE PROBLEM NOTING THAT: ∂ψ3/∂n = iωn3, on SB WE MAY ALSO USE: w² A33(w) - iω B33(w) = -ρ ∫∫ Ψ3 ∂ψ3/∂n ds SB