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

2019
Views:

55

Pages:

12
Author:

Beckett Gray

NEUMANN-KELVIN LINEARIZATION OF UXO SHIP SEAKEEPING PROBLEM [Diagram: A coordinate system with x_o and y_o axes is shown. A wavy line represents the sea surface.] [Diagram: A coordinate system with x and y axes is shown. A ship is moving to the right with velocity U. A wave with amplitude ξ_0(t) is propagating towards the ship.] ● LET Φ(x_0,y_0,z_0,t) BE THE TOTAL POTENTIAL RELATIVE TO THE INERTIAL COORDINATE SYSTEM x_0 = x + Ut ● LET φ(x,y,z,t) BE THE SAME POTENTIAL EXPRESSED RELATIVE TO THE TRANSLATING FRAME. IT WAS SHOWN BEFORE THAT dΦ/dt = (∂/∂t - U ∂/∂x) φ(x,t) WHERE NOW THE TIME DEPENDENCE OF φ WRT TIME WILL BE OF THE e^(iωt) FORM IN THE SHIP SEAKEEPING PROBLEM. • THE TOTAL POTENTIAL Φ_tot CONSISTS OF THE SUM OF TWO COMPONENTS IN A LINEARIZED SETTING Φ_tot = Φ_bar + Φ WHERE Φ_bar IS THE VELOCITY POTENTIAL DUE TO THE VESSEL FORWARD TRANSLATION WITH CONSTANT SPEED U AND Φ IS THE SEAKEEPING COMPONENT DUE TO VESSEL MOTIONS IN WAVES. • RELATIVE TO THE SHIP FRAME: Φ_bar = Φ_bar(x,y,z) Φ = Φ(x,y,z,t) = Re{φ(x,y,z)e^(iωt)} WHERE ω IS THE ENCOUNTER FREQUENCY AND φ = φ_r + Σ φ_j WITH φ_j, j=1,...,6 BEING THE RADIATION AND φ_r BEING THE DIFFRACTION POTENTIALS. RELATIVE TO THE EARTH-FIXED FRAME THE AMBIENT WAVE VELOCITY POTENTIAL TAKES THE FORM: Φ_i = Re{ψ_i} ψ_i = (igA/w_o) * exp[kz_o - ikX_o cosβ - iky_o sinβ + iω_o t] WHERE IN DEEP WATER: k = w_o²/g LET: X_o = x + Ut Y_o = y Z_o = z IT FOLLOWS THAT: ψ_i = (igA/w_o) * exp[kz - ikx cosβ - iky sinβ] * exp[i(w_o - Uk cosβ)t] LET: ω = w_o - Uk cosβ BE DEFINED TO BE THE ENCOUNTER FREQUENCY WHICH ACCOUNTS FOR THE DOPPLER EFFECT INCLUDED IN THE SECOND TERM IN THE RHS. COMMENTS ON N-K FORMULATION: ● THE SHIP IS ASSUMED TO BE STREAMLINED IN ORDER TO JUSTIFY THE DECOMPOSITION OF THE STEADY & TIME HARMONIC COMPONENTS ● THE VESSEL MOTIONS ARE ASSUMED SHALL AND OF THE SAME ORDER AS THE AMBIENT WAVE AMPLITUDE. TERMS OMITTED ARE OF O(A^2). ● WHEN TAYLOR EXPANDING THE FREE-SURFACE AND BODY-BOUNDARY CONDITION ABOUT Z=0 AND S=0 RESPECTIVELY, THE STEADY FLOW POTENTIAL Φ_0 = 0. ● FOR SHIPS WITH APPRECIABLE THICKNESS A BETTER APPROXIMATION FOR Φ IS THAT OF THE DOUBLE-BODY FLOW DISTURBANCE SUCH THAT Φ_0 = 0 on z = 0 and ∂Φ/∂n = Un_0 on S_0 THIS LEADS TO THE STATE-OF-THE-ART LINEAR 3D STEADY FLOW AND SEAKEEPING FORMULATION DISCUSSED LATER IN CONNECTION WITH PANEL METHODS.