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

2019
Views:

222

Pages:

14
Author:

tusjm5f

SOLUTION OF WAVE-BODY INTERACTION PROBLEMS TWO TYPES OF WAVE BODY INTERACTION PROBLEMS ARE ENCOUNTERED FREQUENTLY IN APPLICATIONS AND SOLVED BY THE METHODS DESCRIBED IN THIS SECTION ZERO-SPEED LINEAR WAVE BODY INTERACTIONS IN THE FREQUENCY DOMAIN IN 2D AND 3D FORWARD-SPEED SEAKEEPING PROBLEMS IN THE FREQUENCY OR TIME DOMAIN IN THREE DIMENSIONS (LINEAR & NONLINEAR) A CONSENSUS HAS BEEN REACHED OVER THE PAST TWO DECADES THAT THE MOST EFFICIENT AND ROBUST SOLUTION METHODS ARE BASED ON GREEN'S THEOREM USING EITHER A WAVE-SOURCE POTENTIAL OR THE RANKINE SOURCE AS THE GREEN FUNCTION. THE NUMERICAL SOLUTION OF THE RESULTING INTEGRAL EQUATIONS IN PRACTICE IS IN ALMOST ALL CASES CARRIED OUT BY PANEL METHODS. FREQUENCY-DOMAIN RADIATION-DIFFRACTION. U=0 BOUNDARY-VALUE PROBLEM: [Diagram: A 3D coordinate system with a curved surface (S) submerged in a fluid. Vectors x, y, and z are labeled. The surface is labeled "S" and the fluid domain is labeled "V".] R = (x²+y²+z²)½ OUTGOING WAVES AS R → ∞ φ = V V = {n_i, i=1, 6; RADIATION -∂φ/∂n_i; DIFFRACTION GREEN'S THEOREM GENERATES A BOUNDARY INTEGRAL EQUATION FOR THE COMPLEX POTENTIAL Φ OVER THE BODY BOUNDARY S_B FOR THE PROPER CHOICE OF THE GREEN FUNCTION: IN ORDER TO DETERMINE φ(x) ON THE BODY BOUNDARY S_B, SIMPLY ALLOW x_e → S_B IN WHICH CASE THE SPHERE S_e BECOMES A ½ SPHERE AS E→0: [Diagram: A 3D coordinate system with a curved surface (S_B) and a small sphere (S_e) centered around a point on the surface.] * NOTE THAT x_e IS A FIXED POINT WHERE THE POINT SOURCE IS CENTERED AND x IS A DUMMY INTEGRATION VARIABLE MOVING OVER THE BODY BOUNDARY S_B. * THE REDUCTION OF GREEN'S THEOREM DERIVED ABOVE SURVIVES ALMOST IDENTICALLY WITH A FACTOR OF 1/2 NOW MULTIPLYING THE S_e INTEGRAL SINCE ONLY 1/2 OF THE S_e SURFACE LIES IN THE FLUID DOMAIN IN THE LIMIT AS E→0 AND FOR A BODY SURFACE WHICH IS SMOOTH. IT FOLLOWS THAT: