13.42 LECTURE 16: FROUDE KRYLOV EXCITATION FORCE SPRING 2003

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1 13.42 LECURE 16: FROUDE KRYLOV EXCIAION FORCE SPRING 2003 c A. H. ECHE & M.S. RIANAFYLLOU 1. Radiation and Diffraction Potentials he total potential is a linear superposition of the incident, diffraction, and radiation potentials, φ = (φ I + φ D + φ R ) e iωt. he radiation potential is comprised of si components due to the motions in the si directions, φ j where j = 1,2,3,4,5,6. Each function φ j is the potential resulting from a unit motion in j th direction for a ody floating in a quiescent fluid. he resulting ody oundary condition follows from lecture 15: (1.1) (1.2) φ j n φ j n = iωn j ; (j = 1,2,3) = iω( r n) j 3 ; (j = 4,5,6) (1.3) r = (,y,) (1.4) n = n j (j = 1,2,3) = (n,n y,n ) In order to meet all the oundary conditions we must have waves that radiate away from the ody. hus φ j e ik as ±. For the diffraction prolem we know that the derivative of the total potential (here the incident potential plus the diffraction potential without consideration of the radiation potential) normal to the ody surface is ero on the ody: φ n = 0 on S B, where φ = φ I + φ D Spring

2 2 SPRING 2003 c A. H. ECHE & M.S. RIANAFYLLOU φ I n = φ D n ; on S B We have so far talked primarily aout the incident potential. he formulation of the incident potential is straight forward from the oundary value prolem (BVP) setup in lecture 15. here eist several viale forms of this potential function each are essentially a phase shifted version of another. he diffraction potential can also e found in the same fashion using the BVP for the diffraction potential with the appropriate oundary condition on the ody. his potential can e approimated for a long wave condition. his long wave approimation assumes that the incident wavelength is very long compared to the ody diameter and thus the induced velocity field from the incident waves on the structure can e assumed constant over the ody and approimated y the following equation: φ D i ω [ φi φ 1 + φ I y φ2 + φ ] I φ 3 Further eplanation of this approimation can e found in Newman (p. 301). Ultimately, if we assume the ody to e sufficiently small as not to affect the pressure field due to an incident wave, then we can diffraction effects can e completely ignored. his assumption comes from the Froude-Krylov hypothesis and assures a resulting ecitation force equivalent to the froude-krylov force: (t) = ρ φ t I n ds

3 FROUDE-KRYLOV FORCES 3 2. Vertical Froude-Krylov Force on a Single Hull Vessel B Deep water incident wave potential is: φ I = aω k ek Re {i e i(ωt k) he force in the vertical direction is found from the incident potential using eq. 1.5 along the ottom of the vessel. Here the normal in the -direction, n, is negative: n = 1, so the force per unit length in the -direction is (2.1) = Re { B/2 B/2 ρ i ω iaω k e k e i(ωt k) d (2.2) (2.3) { ρoω 2 = Re k 2 a e k e iωt ( e ikb/2 e ikb/2) = Re {2ρ ω2 k 2 a e k e iωt sin(kb/2) Recall that sin = ei e i 2i. Using the vertical velocity we can rewrite the force in terms of the velocity.

4 4 SPRING 2003 c A. H. ECHE & M.S. RIANAFYLLOU (2.4) (2.5) (2.6) w(t) = Re {aω e k i e i(ωt k) ẇ(t) = Re { aω 2 e k e i(ωt k) ẇ( = 0, = 0,t) = Re { a ω 2 e iωt Now we can write the force in the vertical direction as a function of the vertical (heave) acceleration, { 2ρ F = Re k 2 e k sin(kb/2) ẇ(0,0,t). Let s look at the case where ω 0 the wavenumer, k = ω 2 /g 0, also goes to ero and the following simplifications can e made: (2.7) e kt 1 k (2.8) sin(kb/2) kb/2 to yield a simplified heave force. (2.9) (2.10) F FK Re {2ρ ω2 { Re ρ g ab a (1 k) (kb/2) eiωt k2 ) e iωt (1 ω2 g If we look at the case where ω 0 and consider the heave restoring coefficient, C 33 = ρ g B, and the free surface elevation, η(,t) = Re { a e i(ωt k) we can rewrite this force as Re {C 33 η( = 0,t)

5 FROUDE-KRYLOV FORCES 5 3. Horiontal Froude-Krylov Force on a Single Hull Vessel he horiontal force on the vessel aove can e found in a similar fashion to the vertical force. (3.1) (3.2) (3.3) F = ρ φ S B t I n ds { = Re ρ i ω iωa 0 [ e k d e i(ωt kb/2) e i(ωt+kb/2)] k = Re {iρ aω2 ( 1 e k ) e iωt 2 sin(kb/2) k As frequency approaches ero similar simplifications can e made like aove for the vertical force: (3.4) (3.5) (3.6) F (t) Re {iρ aω2 k (K) eiωt 2 kb/2 u(t) = Re {aω e k e i(ωt k) u(t) = Re {i a ω 2 e k e i(ωt k) F (t) Re {ρb u( = 0, = 0,t) Where ρb = ρ, and is the vessel volume such that we are left with the surge force (3.7) F ρ u (3.8) F C 33 η + ρ ẇ

6 6 SPRING 2003 c A. H. ECHE & M.S. RIANAFYLLOU 4. Multi Hulled Vessel -B/2 B/2 Again, let s make a few asic assumptions: (/λ << 1), (B/λ 1), (a < ), and ( ). Let s look at the force in the -direction: ρ u( = B/2, = 0,t) + ρ u( = B/2, = 0,t) (4.1) η(,t) = acos(ωt k) (4.2) u(,,t) = a ω 2 e k sin(ωt k) (4.3) (4.4) ρ ( a ω 2 ) {sin(ωt + kb/2) + sin(ωt kb/2) (4.5) 2ρ (a ω 2 ) cos(kb/2) sin(ωt) Note that when kb/2 = π/2 (or B = λ/2) then (t) = 0.

7 FROUDE-KRYLOV FORCES 7 -B/2 B/2 c 4.1. Multi Hulled Vessel with additional pontoon. Use the same assumptions from aove to find the -force adjusted for the additional pontoon etween the two hulls. 2 ρ (a ω 2 ) cos(kb/2) sin(ωt) +c p( = B/2 + /2, = 0,t) c p( = B/2 /2, = 0,t) he last two terms are the adjustment to the force for the addition of the pontoon, δf FK (t). Pressure is found from the incident potential: p(,,t) = ρg a e k cos(ωt k). δ ( ) k = 2 ρ g a sin(ωt) sin (B ) 2 For B >> using g = ω 2 /k we get a force: F FK (t) 2 ρ a ω 2 sin(ωt) { cos(kb/2) + δ/2 sin(kb/2)