[WAVES] 1. Waves and wave forces. Definition of waves

Transcription

1 1. Waves and forces Defnton of s In the smuatons on ong-crested s are consdered. The drecton of these s (μ) s defned as sketched beow n the goba co-ordnate sstem: North West East South The eevaton can be defned n two was: 1. B means of a user-supped tme trace. B means of a random generated from the spectrum In both cases, the s defned at a user-supped poston (, ) n the goba coordnate sstem. The eevaton can be wrtten as a super poston of near components. ( t,, ) Where, N 1 cos(ω t k ( )sn μ k ( )cosμ) t = (-1)Δt [s] ω = ( 1) ω [rad/s] ω = π N t [rads -1 ] gk tanh( kh) = ω [rad s - ] h = water depth [m] = amptude of -th component [m] Page 1 of 7

2 In case of a user-defned tme trace, the tme step s supposed to be equdstant. Ths tme step can be dfferent from the requred tme step for the generaton of the forces. In that case the s re-samped at the requred tme step wth a Dscrete Fourer Transform. The amptudes and phases are obtaned b means of a Dscrete Fourer Transform of the user-supped sgna or b means of the spectrum and random generated phases, accordng to: S (ω ) ω rand[ π] (Random number between and π ) The forces on a foater depend on the eevaton at the actua poston of the Centre of Gravt of the foater n the goba co-ordnate sstem. Referrng to ths poston as (X, Y ), the eevaton at ths poston on tme eve t can be computed accordng to: ( t,, ) N 1 cos(ω t k ( )sn μ k ( )cosμ) Durng the smuaton, the tota phase of each component, ε, s determned: k ( )sn μ k ( )cosμ Ths adusted phase together wth the amptudes s used to compute the frst and second-order forces. The spectra avaabe n the WAVE MODULE are: JONSWAP Person-Moskowtz Gaussan swe User-defned Reguar s are ncuded as we. The defntons of these spectra can be found at the end of ths document. Interacton between current and s are taken nto account. Formuatons for dampng are not ncuded. Computaton of forces Wave forces are apped to each of the foaters. Both the frst-order forces and the secondorder forces ( drft forces) are computed. The epresson for the computaton of the frst-order forces s as foows: Page of 7

3 (1) N F (t) A cos( t,, 1, ); =1..6 =1: surge force =: swa force =3: heave force =4: ro moment =5: ptch moment =6: aw moment Where: A, = amptude of frst-order transfer functon of mode for frequenc ω. γ, = phase of frst-order transfer functon of mode for frequenc ω. = phase of component, takng nto account the horzonta transaton ε of the Centre of Gravt n the goba co-ordnate sstem The frst-order transfer functon depends on the reatve headng of the foater, μ r, defned accordng to the fgure beow. μ r So: head s: s from port sde: stern s: s from starboard sde: μ r = 18 degrees μ r = 7 degrees μ r = degrees μ r = 9 degrees Page 3 of 7

4 Dependng on the actua reatve headng, a near nterpoaton s carred out on the transfer functons at the avaabe headngs (from the hdrodnamc database). Ths nterpoaton s carred out on the rea and magnar parts of the frst-order transfer functon. Ths method mpes that durng the smuaton, the forces need to be re-evauated. Ths does not need to happen at each tme eve n the smuaton, but on f the horzonta offset of the foater or headng change eceeds a certan threshod vaue (compared to the tme eve where the forces were computed the prevous tme). Beow, an eampe s shown: T=T 1 T=T At t=t the forces are computed wth the foater at poston ((T ), (T ), (T )). At tme eve t=t 1 >T, the poston of the foater s ((T 1 ), (T 1 ), (T 1 )). The offset equas: offset = ( ( T T ) ( T1 )) ( ( T ) ( 1)) Besdes the changng aw ange of the vesse, the earth-fed drecton ma change as we n the smuaton. Therefore, f t s consdered to re-compute the forces, the change n the reatve aw ange must be consdered: drecton change = ) μ( T ) (ψ(t ) μ( )) ψ(t 1 T1 When these vaues eceed a certan threshod mt, the forces need to be re-computed. Ths threshod mt s determned n the course of the deveopment of the WAVE MODULE and s n baance between accurac and computatona tme. The epresson for the computaton of the second-order forces s as foows: Page 4 of 7

5 F () N N 1 1 [ P cos{(ω ω ) t ( ε ε )} Q sn{(ω ω ) t ( ε ε )}] =1: =: =3: =4: =5: =6: surge force swa force heave force ro moment ptch moment aw moment Where, P = n-phase part of second-order transfer functon for mode and frequences ω and ω Q = quadrature part of second-order transfer functon for mode and frequences ω and ω Just as for the frst-order forces, an nterpoaton s carred out on the avaabe headngs to obtan the second-order transfer functon at the present headng. The same offset vaue and drecton change can be used as for the frst-order forces. It s possbe that the sgnfcant heght changes durng the smuaton. However, t s assumed that the spectra shape does not change. That means that the new spectrum s obtaned as foows from the od spectrum: S H1/ 3,new, new S,od H1/ 3,od The amptude of each component s therefore mutped b a factor H H 1/ 3,new 1/ 3,od. Page 5 of 7

6 WAVE SPECTRA Jonswap Spectrum ) -5-4 ep [- ( - /(. ) ] S ( ) =. g.. ep [ -1.5 ( / ) ]. = for a for > b In whch: S ( ) = spectra denst at frequenc [m s] = frequenc [rads -1 ] = spectrum peak frequenc [rads -1 ] g = gravtatona acceeraton [ms - ], g = 9.81 ms - = peakedness factor [-] The vaues of the dmensoness shape parameters a and b are as foows: a =.7 b =.9 The factor s such, that the foowng reaton s fufed: H 4 S ( ) d 4 m 1/3 In whch: H 1/3 = sgnfcant heght [m] m = area beneath the spectrum [m ] Person-Moscowtz spectrum S PM Where: A PM = 1.5 APM 4 t ( ω) e 5 t.315 H ω s p S PM = spectra denst (m s) H s = sgnfcant heght (m) = anguar frequenc (rads -1 ) t p = peak (moda) frequenc (rads -1 ) = / p (dmensoness) Page 6 of 7

7 Gaussan swe The Gaussan swe spectrum s defned as: Where: σ =.8 s a standard vaue for sgma. T 1 = T = T p Page 7 of 7