SIMULATION OF EXTREME WAVE INTERACTION WITH MONOPILE MOUNTS FOR OFFSHORE WIND TURBINES

SIMULATION OF EXTREME WAVE INTERACTION WITH MONOPILE MOUNTS FOR OFFSHORE WIND TURBINES Feng Gao, Clve Mngha and Derek Causon Extree wave run-up and pacts on onople foundatons ay cause unexpected daage to offshore wnd far facltes and platfors. To assess the forces due to wave run-up, the dstrbuton of run-up around the ple and the axu wave run-up heght need to be known. Ths paper descrbes a nuercal odel AMAZON-3D study of wave run-up and wave forces on offshore wnd turbne onople foundatons, ncludng both regular and rregular waves. Nuercal results of wave force for regular waves are n good agreeent wth experental easureent and theoretcal results, whle the axu run-up heght are lttle hgher than predcted by lnear theory and soe eprcal forula. Soe results for rregular wave sulaton are also presented. Keywords: extree wave; onople; nuercal sulaton; wave run-up; wave force INTRODUCTION Durng the last decade, a large nuber of offshore wnd fars were bult. Observatons on exstng wnd fars have clearly shown that wave nteracton wth onople foundatons can be qute sgnfcant (De Vos et al. 007). Wave run-up and wave pacts ay cause unexpected daage to wnd far facltes and platfors. To assess the forces due to wave run-up, the dstrbuton of run-up around the ple and the axu run-up heght need to be known. Prevously, run-up on crcular cylnders has been studed experentally and atheatcally. By usng lnear wave dffracton theory, Sarpkaya and Isaacson (98) obtaned results for the wave elevaton around a crcular cylnder surface. Nedzweck and Duggal (99) perfored a sall-scale experental study to nvestgate wave run-up on rgd full length and truncated crcular cylnders under regular and rando wave condtons. They found that lnear dffracton theory underestates the wave run-up for all but very low wave steepness and eployed a se-eprcal varaton of the forula to predct the wave run up. Chan et al. (995) studed the run-up and especally the forces on a crcular cylnder under the nfluence of a plungng breaker. Fro that study t becae clear that the breakng process has a great pact on the axu horzontal forces and also nfluences the run-up. Krebel (998) focused on the run-up for perodc waves on a plane bed. The results were copared wth st and nd order analytcal wave dffracton theores and ndcated that the non-lnearty of the waves has a large effect on the total run-up. Büchann et al. (998) used a second order boundary ntegral ethod to study run-up on a structure wth and wthout an abent current. Mase et al. (00) set up analytcal equatons for the run-up on sall daeter foundatons wthout explctly ncludng the daeter of the cylnder, even though basc dffracton theores appear to show that the daeter has a clear effect on the run-up. Martn et al. (00) nvestgated run-up on coluns caused by steep regular waves n deep water. They copared ther experental results wth varous theores and conclude that ost theores underestate the run-up values and the se-eprcal ethod suggested by Nedzweck and Huston (99) overestated the run up n ost of the test cases consdered. Recently, De Vos et al. (007) suggested a new forula for predcton of the axu wave run-up on onople foundaton, whch s based on a sall-scale experental study that exanes both regular and rregular wave run-up on cylndrcal ple foundatons. In ths paper, soe results of nuercal sulatons nvolvng regular and extree wave pact and run-up on onople foundatons for offshore wnd turbnes wll be presented. Sulatons were carred out by usng the AMAZON-3D code, whch solves the ncopressble Naver-Stokes equatons n both ar and water regons sultaneously wth the free surface captured autoatcally as a contact surface n the densty feld. A te-accurate artfcal copressblty ethod and hgh Godunov-type schee was adopted to replace the pressure correcton solver used n other ethods (Qan et al. 006). The Cartesan cut cell technque was used to generate a boundary ftted esh. The advantages of ths approach were outlned by Causon et al. (00) ncludng ts flexblty for dealng wth arbtrarly coplex geoetres and ovng bodes. Centre for Matheatcal Modellng and Flow Analyss, School of Coputng, Matheatcs & Dgtal Technology, Manchester Metropoltan Unversty, Chester Street, Manchester, M 5GD, Unted Kngdo

COASTAL ENGINEERING 0 NUMERICAL METHOD Governng equatons and nuercal soluton For ncopressble, unsteady, vscous flows, the Naver-Stokes equatons wth a varable densty feld can be odfed usng the artfcal copressblty ethod and wrtten n the ntegral for: t Q F F v S S c B () n whch Q represents the vector of flow varables, F c and F v are the convectve and vscous flux ters, and B stands for the source ters. They are defned as follows: u Q v, w p U uu nx p Fc vu ny p, wu nz p U nx xx Fv nx yx nx zx 0 n y n y n y 0 xy yy zy n z xz nz yz, nz zz 0 0 B g 0 0 where ρ s the densty, p s the pressure, β s the copressblty coeffcent, g s the gravtatonal acceleraton. U u n vn wn s the contra-varant velocty and n = (n x, n y, n z ) s the outward x pontng unt noral vector at a esh cell face. The vscous stress tensor s defned as S rate of the stran tensor defned as S y u x j z u x j (), n whch µ s the dynac vscosty and S s the The flow equatons () are dscretsed by the cell-centred fnte volue ethod over each cell of the flow doan, whch gves Q t F F ds B RQ (3) c v (4) where Ω ndcates the grd cell ndexed by the subscrpt and R s the resdual of the flow equatons. Supposng the grd cell Ω has faces then ntegraton of the fluxes across the faces wll result n Fc Fv k lk Fc Fv k k The convectve flux across a grd cell face s coputed by Roe s approxate Reann solver k where Q and atrx whch can be expressed as k l Q F Q AQ Q Fc, k Fc k c k k k (6) k Qk are the reconstructed data values on the rght and left of face k, A s the Jacoban The egenvalues of the atrx are 0 nx ny nz 0 uu unx U uny unz nx A vu vn x vn y U vn z ny (7) wu wnx wny wnz U nz U nx ny nz 0 U, U c, c U 4 (8),,3 4,5 (5)

COASTAL ENGINEERING 0 3 More detals of the fnte volue soluton ethod for ncopressble two-flud flows on a Cartesan cut cell esh can be found n the work of Qan et al. (006) and Gao et al. (007). By dscretzng equaton (4) n te and ottng the subscrpts for splcty, the followng frstorder Euler plct dfference schee can be eployed n n Q Q n RQ t To acheve a te-accurate soluton at each physcal te step n unsteady flow probles, equaton (9) ust be further odfed to obtan a dvergence free velocty feld. Ths s accoplshed by ntroducng a pseudo te dervatve nto the syste of equatons, as n, n, n, n, Q Q Q Q n, I RQ ta t where τ s the pseudo te and I ta = dag[,,,, 0]. The rght hand sde of equaton (0) can be lnearzed usng Newton's ethod at the + pseudo-te level to yeld I R Q Q n, n, n, n, n, Q Q n, Q Q I RQ where I = dag[/δτ+/δt, /Δτ+/Δt, /Δτ+/Δt, /Δτ+/Δt, /Δτ]. When Δ(Q n+ ) =Q n+,+ - Q n+, s terated to zero at each te step, the densty and oentu equatons are satsfed dentcally and the dvergence of the velocty at te level n + s zero. The syste of equatons can be wrtten n atrx for as D L U Q s RHS ta () where D s a block dagonal atrx, L s block lower trangular atrx, and U s a block upper trangular atrx. Each of the eleents n these atrces s a 5 5 atrx. An approxate LU factorzaton (ALU) schee can be adopted to obtan the nverse of equaton () n the for t (9) (0) () D LD D UQ s RHS (3) Wthn each te step of the plct ntegraton process, the sub-teratons are ternated when the L nor of the change n successve sub-teratons L N s s Q Q N s less than a specfed lt ɛ. In the present study ths value s set ɛ = 0-4. Wave generaton In the nuercal wave tank, the three coponents of velocty are specfed at the nflow boundary to generate the requred wave wth the pressure and densty extrapolated fro the nteror of the coputatonal doan by assung zero spatal gradents. Ths defnton allows the desred waves to propagate nto the coputatonal doan through ths boundary. For regular wave generaton, lnear wave theory s used to calculate the nput velocty profle and wave elevaton; however, for an extree wave, the exact velocty profle for a true physcally realsable nonlnear wave under the gven condtons s not known a pror. Thus, a vable approach s to nput reasonable approxate wave condtons along the nflow boundary to sulate the real phenoenon. Ths leads to the noton of the extree wave forulaton as a focused wave group n whch any wave coponents n a spectru are focussed sultaneously at a partcular poston n space n order to odel the average shape of an extree wave profle consstent wth the rando process wthn a specfed wave energy spectru. The dervaton here refers to the works of Dalzell (999) and Nng et al. (009) n whch a frst or second-order Stokes focused wave can be posed n such a anner. Assung waves are focused at a specfed pont x f at te t f, lnear wave theory defnes the wave elevaton at an arbtrary pont as (4)

4 COASTAL ENGINEERING 0 N x, t a cosk x x f f t t f (5) where N s the total nuber of wave coponents, and a, k, f represent the wave apltude, the wave nuber and the wave frequency of the th wave coponent, respectvely. The dsperson relaton establshes the relaton between space and te,.e. between k and f. Wth a chosen wave energy frequency spectru and by settng the phases of all the wave coponents as zero at the focal pont, the apltude a of each wave coponent can be calculated fro a A F S ( f ) f N S f f where S ( f ) s the desred frequency spectru, f s the ncreent n frequency dependng on the nuber of wave coponents and the frequency band wdth and AF s the total nput wave apltude of the focused wave. In ths study, a JONSWAP wave spectru forulated as (7) s selected to generate the extree wave. Where H s the sgnfcant wave heght; T 3 p and f p are the peak wave perod and frequency respectvely. The peak enhanceent factor γ a s chosen as 3.3. 4 5 f H T f exp.5t f J 3 p 4 exp T p f p a (6) S (7) a 0.0638.094 0.095ln J (8) 0.30 0.0336 0.85.9 a 0.07 f f p (9) 0.09 f f p Force calculaton The analytcal soluton of the lnearzed dffracton proble for a crcular cylnder at arbtrary water depths was gven by MacCay and Fuchs (954). Accordngly, the frst-order non-densonal axu horzontal force F ax F ax s a kr Y' kr Fax J' (0) ghhr tanh kh kh kr where ρ s the water densty, h s the water depth, k s the wave nuber, H s the ncdent wave heght, r s the cylnder radus, and J (kr) and Y (kr) are dervatves of the Bessel functons of the frst and second knd of order one respectvely. For a vertcal crcular cylnder n fnte water depth, Krebel (990) presented a coplete closedfor soluton for the velocty potental resultng fro the nteracton of second-order plane waves; Rahan et al. (999) also presented an analytcal soluton for the second order wave force. In ths nuercal sulaton, the pressure p can be obtaned fro the derved p / by solvng the governng equatons (). The total force s obtaned by ntegraton of the pressure feld around the cylnder contour where S b F pn ds () S b s the cylnder surface as defned approxately by the boundary ftted cut cell surface.

COASTAL ENGINEERING 0 5 Wave run-up Usng dfferent approaches Krebel (99) and Martn et al. (00) have carred out an extenson of dffracton theory to the second order. They found that there s a large nfluence of usng second order theory to calculate run-up and t s not suffcent to attept an extrapolaton based on lnear dffracton theory. However, they have gven the followng approxate result for run-up on the up-wave sde of a crcular cylnder R D u ax L where R u s the predcted wave run-up, D s the daeter of cylnder, L s the wave length and η ax s the wave crest. The threshold of lnear dffracton s wdely regarded as D/L<0.. In ths range, lnear dffracton theory suggests that the scattered wave energy s neglgbly sall. However, ths s not the case for steep waves. There are sgnfcant nonlnear contrbutons n the case of steep waves; thus, fully nonlnear odelng s advsable for a steep wave run-up calculaton. Recently, De Vos et al. (007) suggested a new forula to predct the axu wave run-up on a cylndrcal foundaton based on a sall-scale experental study as follows () u R u ax. ax.7 (3) g where η ax s the axu wave elevaton and u s the horzontal partcle velocty at the wave crest. H H H cosh kd ax k coshkd (4) 3 8snh kd H gk cosh k ax d u cosh kd 3 H k 4 4 cosh k ax 4 snh kd where H s the ncdent wave heght, g s the gravtatonal acceleraton, d s the stll water depth, k s the wave nuber and ω s the wave frequency. NUMERICAL RESULTS The present nuercal sulatons were frst carred out wth regular waves and then wth extree waves. The nuercal wave tank has densons 8 3.6 0.9, wth a stll water depth of 0.45 (Fg. ). A crcular cylnder of daeter 0.35 s postoned wth ts center at the ddle of the tank. Dfferent waves are generated at the left nflow boundary wth propagaton to the rght. The rght sde boundary s set as open boundary whch allows fluds to freely enter or leave the coputatonal doan accordng to the local flow velocty and drecton. The front and back sde wall boundares are set as sold walls. Several wave gauges are set along the center lne of the tank to record wave elevatons. d (5) Fgure. Nuercal wave tank set up A non-unfor block structure esh was used n the background of the coputaton doan; a relatvely fne esh was used n the area near the cylnder and around the water free surface. Around the structure of cylnder, the Cartesan cut cell technque was used to generate a fully boundary-ftted esh. Detals of the Cartesan cut cell technque can be found n the works of Causon et al. (00) and Ingra et al. (003). Part of the 3D coputatonal esh around the vertcal crcular cylnder s shown n Fg.. Totally there are 58 39 6 cells n the doan wth the sze of sallest cell of denson 0.0.

6 COASTAL ENGINEERING 0 Fgure. 3D coputatonal esh around the vertcal cylnder (blue parts are n ar and red parts are n water) Regular wave sulaton Four sulatons are carred out for regular waves and copared wth theoretcal and experental data found n Krebel(998). The paraeters of these test cases are shown n Table, where A s wave apltude, T s wave perod and kr s the scatterng paraeter correspondng to wave nuber k and cylnder radus r. Table. paraeters of test case for coparsons wth experents reported n Krebel (998). Case Case Case3 Case4 A () 0.0535 0.048 0.06 0.074 T (s).95.75.50.5 kr 0.7 0.308 0.374 0.48 (a) case: kr = 0.7, kh = 0.78 (b) case: kr = 0.308, kh = 0.8 (c) case3: kr = 0.374, kh = 0.86 (d) case4: kr = 0.48, kh = 0.438 Fgure 3. Coparson of wave force te seres for varous cobnatons of kr and kh

COASTAL ENGINEERING 0 7 Saple te seres of horzontal force on the cylnder copared to easureents fro Krebel (998) and both lnear and second order analytcal predctons are presented n Fg 3. Each fgure shows the results over one wave perod n whch the wave crest phase s centered n the fgure. It can be seen that nuercal results are n generally good agreeent wth the experental and theoretcal results. The nuercal results for axu wave run up copared wth the approxaton forulae are shown n Fg. 4. Although the present nuercal results are a lttle hgher than the eprcal Eq. (3) predctons, they are acceptable, whle Eq. () whch s based on lner dffracton theory s uch less accurate as Martn et al. (00) has entoned. Fg.5 shows the te hstory of wave elevaton at dfferent wave gauges for frst two test cases. The locatons of the wave gauges are along the center lne of the wave tank; gauge s located just n front of the cylnder and gauge s just behnd the cylnder. The wave run up stuaton can be seen clearly. Fgure 4. Coparson of axu wave run-up (a) case (b) case Fgure 5. Te hstory of wave elevaton at two dfferent wave gauges for varous cases Extree wave sulaton The calculaton doan for the extree wave sulatons are alost the sae as for the regular wave sulatons, the only dfference beng that the center of the cylnder s oved to x = 3.78. The focus pont for the extree wave s set just n front of the cylnder at x = 3.6 and focus te s about 5.s. Followng the work of Nng et al. (009), two test cases are chosen fro ther dfferent experental cases for these nuercal sulatons. The nput characterstcs of the relevant wave groups are lsted n Table. The JONSWAP energy spectra wth the sae peak frequency fp = 0.83Hz s used. Table. Input characterstcs of wave groups for extree wave Case 3 Frequency band f (Hz) 0.6-.3 0.6-.4 Input apltude AF () 0.063 0.0875 Wave perod T (s).0.5 Wave length λ ().00.8

8 COASTAL ENGINEERING 0 The te hstory of the horzontal forces actng on the cylnder for the two test cases s presented n Fg. 6. It can be seen that the axu force appears around the focus te and larger apltude waves produce larger pact forces. The axu force s about 8.6N for case and 00.N for case3, whch are qute slar to the experent results. (a) case (b) case 3 Fgure 6. Te hstory of horzontal force actng on cylnder for varous cases Fg. 7 shows the te hstory of wave elevaton at dfferent wave gauges for these test cases. The locaton of the wave gauges s along the center lne of wave tank wth the one n front of the cylnder set at x = 3.6 and the other just after the cylnder at x = 3.95. The wave run-up can be seen clearly, and the axu run-up appears close to the focus te. The value of axu run-up heght s about 0.0 for case and 0.5 for case3, whch are a lttle hgher than the predctons by eprcal forula (3) as n the regular wave sulatons. (a) case (b) case3 Fgure 7. Te hstory of wave elevaton at dfferent wave gauges for varous cases CONCLUSIONS The characterstcs of wave run-up and horzontal wave force on the onople foundaton of an offshore wnd turbne have been nvestgated nuercally usng a Naver-Stokes solver. The nuercal results for wave force have been shown to be n good agreeent wth experent easureents and results usng second order theory. The axu run-up heght s only a lttle hgher than the theoretcal and eprcal forulae predctons. It can be concluded that the present flow code AMAZON-3D has the potental to be a usable tool for the detaled nvestgaton of wave nteractons wth structures of ths type. As the code can sulate breakng waves, further sulatons ncludng pacts fro extree waves breakng on offshore onople ounts could be perfored. In addton wave nteractons wth ultple structures wll also be perfored n the future.

COASTAL ENGINEERING 0 9 ACKNOWLEDGMENTS Ths work was fnancally supported by the EPSRC (UK), Supergen Wnd Energy Technologes Core, Towards the Offshore Wnd Power Staton, under grant Ref: EP/H0866/. REFERENCES Büchann, B., Skourup, J. and Cheung, K.F. 998. Run-up on a structure due to second order waves and a current n a nuercal wave tank, Appled Ocean Research, 0, 97-308. Causon D.M., Ingra D.M. and Mngha C.G. 00. A Cartesan cut cell ethod for shallow water flows wth ovng boundares. Advances n Water resources, 4, 899-9. Chan, E.S., Cheong, H.F. and Tan, B.C. 995. Laboratory study of plungng wave pacts on vertcal cylnders, Coastal Engneerng, 5, 87-07. Dalzell J.F. 999. A note on fnte depth second-order wave-wave nteractons. Appled Ocean Research,, 05. De Vos L., Frgaard P and De Rouck J. 007. Wave run-up on cylndrcal and cone shaped foundatons for offshore wnd turbnes. Coastal Engneerng. 54, 7 9. Gao F., Ingra D.M., Causon D.M. and Mngha C.G. 007. The developent of a Cartesan cut cell ethod for ncopressble vscous flows, Internatonal Journal for Nuercal Methods n Fluds, 54, 033-053. Ingra D.M., Causon D.M. and Mngha C.G. 003. Developents n Cartesan cut cell ethods. Matheatcs and Coputers n Sulaton, 6, 56 57. Krebel, D.L. 990. Nonlnear wave dffracton by vertcal crcular cylnder. Part I: dffracton theory. Ocean Engneerng, 7, 345 377. Krebel, D.L. 99. Nonlnear wave nteracton wth a vertcal crcular cylnder. Part II: Wave run-up, Ocean Engneerng, 9, 75-99. Krebel, D.L. 998. Nonlnear wave nteracton wth a vertcal crcular cylnder: Wave Forces, Ocean Engneerng, 5, 597-605. MacCay, R.C. and Fuchs, R.A. 954. Wave forces on ples: A dffracton theory. Techncal eorandu no. 69. Beach Eroson Board Offce of the Chef Engneers. Departent of the Ary. 7. Martn, A.J., Easson, W.J. and Bruce, T. 00. Run-up on coluns n steep, deep water regular waves. Journal of Waterway, Port, Coastal, and Ocean Engneerng 7, 6 3. Mase, H., Kosho, K. and Nagahash, S. 00. Wave run-up of rando waves on a sall crcular per on slopng seabed. Journal of Waterway, Port, Coastal and Ocean Engneerng. 7, 9 99. Nedzweck, J.M. and Duggal, S.D. 99. Wave run-up and forces on cylnders n regular and rando waves. Journal of Waterway, Port, Coastal, and Ocean Engneerng. 8, 65 634. Nedzweck, J.M. and Huston, J.R. 99. Wave nteracton wth tenson leg platfors. Ocean Engneerng. 9, 37. Nng D.Z., Zang J., Lu S.X., Eatock Taylor R. Teng B. and Taylor P.H., 009. Free-surface and wave kneatcs for nonlnear focused wave groups, Ocean Engneerng. 36, 6-43. Qan L., Causon D.M., Mngha C.G. and Ingra D.M. 006. A free-surface capturng ethod for two flud flows wth ovng bodes, Proceedngs of the Royal Socety London A. 46, -4. Rahan, M., BORA, S.N. and Satsh, M.G. 999. A Note on Second-Order Wave Forces on a Crcular Cylnder n Fnte Water Depth. Appled Matheatcs Letters,, 63-70 Sarpkaya, T., Isaacson, M. 98. Mechancs of Wave Forces on Offshore Structures. Van Nostrand Renhold Co., New York.