Steady wave drift force on basic objects of symmetry

Transcription

1 Fluid Structure Iteractio ad Movig Boudary Problems IV 3 Steady wave drift force o basic objects of symmetry S. Chakrabarti & A. Gupta Uiversity of Illiois at Chicago, Chicago, IL, USA Abstract The steady wave drift force o a submerged body is a secod-order quatity. With a potetial flow assumptio, the force arises from the diffractio ad radiatio of the waves from the iteractio with the body. For a fixed body i waves the steady force is cotributed from the wave diffractio effect aloe. Numerical solutios are geerally eeded for the computatio of the steady drift force o submerged structures. I this paper the steady wave drift forces o several fixed bodies of basic shapes are derived i closed form. The paper addresses the steady drift forces o the followig basic structures: a vertical circular cylider, a submerged horizotal cylider, a bottom-seated horizotal half cylider, ad a bottom-seated hemisphere. The results developed demostrate the importace of various idepedet o-dimesioal parameters. A umerical program based o liear diffractio/radiatio theory is used to validate the closed form solutio. Keywords: basic objects of symmetry, closed form solutio, desig curves, steady drift force, wave structure iteractio. Itroductio The liear wave forces o a large structure are computed from the liear diffractio theory based o the (Beroulli s) liear pressure term i its equilibrium positio up to the still water level. However, the structure motio, wave free surface, ad the (Beroulli s) oliear pressure terms itroduce oliear forces o the structure ot predictable by the liear theory. The steady wave drift force has bee show to derive from the first order wave potetial. The 3-D diffractio/radiatio theory is well established to compute these forces. WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) doi:.495/fsi7 7 WIT Press

2 4 Fluid Structure Iteractio ad Movig Boudary Problems IV For floatig offshore structures, the steady drift force ca be a importat desig cotributio, especially for soft moored systems. The purpose of this paper is to derive the steady drift forces o a few fixed bodies of basic shapes i closed form ad discuss the cotributio of various parameters ad differet terms i the total drift forces o these fixed bodies. The results preseted here for these basic shapes should be useful to desigers of offshore structures, which is geerally composed of these basic compoets, i order to evaluate the magitudes of these forces ad to determie the importace of the steady drift force for these compoets. Literature review Much work has already bee doe with the umerical computatio of steady drift forces. These works have bee carried out assumig the structure to be rigid ad either freely floatig or fixed. They are too may to cite here. Oe oteworthy oe is by Pikster [5] who studied secod order steady drift forces ad low frequecy forces usig 3-D diffractio/radiatio theory. Ogilvie [6] summarized both mometum ad pressure itegratio approach for drift force calculatios. Skourup et al. [7] ivestigated o-liear loads o fixed body due to waves ad curret usig potetial theory ad 3-D Boudary Elemet Method ad computed secod order steady ad oscillatory forces o a vertical circular cylider. 3 Theoretical backgroud The liear diffractio theory is based o a velocity potetial fuctio Φ which is composed of a icidet wave potetial, ad a scattered potetial from the fixed body. Oce the velocity potetial is kow the steady drift forces o the fixed body are computed from two sources. If the body exteds above the mea water level, the steady drift force due to free surface effect is F i = ρg ηr idl () WL where ρ = water mass desity, g = gravitatioal acceleratio, η r = wave elevatio at the body surface icludig diffractio effect, i = directio of the drift force compoet, i = surface directio ormal compoet alog i, ad WL = water lie of the body i its equilibrium positio. The secod compoet is the steady drift force due to Beroulli s velocity-squared term 3 Fi = ρ u j ids () = S where u,j represets the water particle velocity compoets alog the three body coordiates based o the first-order velocity potetial ad ds is the elemetal area o the submerged part of the body surface up to the still water level, S. j WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

3 4 Closed form formulas The steady drift force due to the icidet wave o ay symmetric basic shape (with respect to the wave directio) ca be show to be idetically zero due the symmetry of the pressure distributio. Thus the cotributio to the secod-order drift force comes from the diffractio potetial. 4. Vertical cylider Fluid Structure Iteractio ad Movig Boudary Problems IV 5 Let us cosider a vertical cylider restig o the ocea floor ad extedig above the free surface. Vertical cylider is the most researched shape for wave forces. It has led to may studies for decades (aalytical, umerical ad experimetal), ad liear results, secod-order results ad fully o-liear results for the wave forces ad the wave ruup o the cylider ca be foud i the literature. For a bottom-seated cylider exteded above the free surface, MacCamy ad Fuchs [4] developed the forces o a fixed vertical cylider i a closed form usig the Bessel fuctio series. The closed form expressio for the first-order velocity potetial was used by several researchers to derive the secod order compoets of the forces (see, e.g.. Chakrabarti []). While it is oe of the simplest shapes, it has several offshore applicatios icludig the deep draft floatig SPAR i deep water i waves whose pressure decays to ear-zero at the bottom of the SPAR. The total liear velocity potetial icludig diffractio at the surface of a bottom-mouted cylider, r = a (e.g., Chakrabarti [3]) is writte as Hg Φ( a, θ; t) = ω = δ [ a cosωt + b si ωt] cos θ (3) where a = cylider radius, H = wave height, ω = wave frequecy, k = wave umber, t = time, (r,θ) = cylidrical polar coordiates, ad the coefficiets a J ( = ( ) ; + Y + ( a+ = ( ) ; Y ( b = ( ) ; A ( A + ( A ( J + ( b+ = ( ) ; ad A = [ J ( ] + [ Y ( ] A ( + ad the fuctios J ( ad Y ( are the derivatives of the Bessel fuctio of the first ad secod kid. The first-order wave profile at the surface of the cylider, r = a, becomes: H η( a, θ ; t) = = δ [ a si ωt + b cosωt] cos θ (4) WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

4 6 Fluid Structure Iteractio ad Movig Boudary Problems IV Upo itegratio of Eq. () aroud the cylider the o-dimesioal compoet of the horizotal steady drift force due to the free surface at the vertical cylider reduces to Fx 4 = ( ) aa+ + bb + (5) g( H / ) a ( ρ = Substitutig a a the above expressio reduces to ρg ( + ) ( A A + + bb + = F x ( ) = 3 ( H / ) a ( = ( A A + (6) (7) which is a fuctio of oly ad does ot deped o the water depth. The odimesioal mea velocity-squared force has the form: F x ρg( H / ) 4 = a ( 3 kd sih kd = kd + sih kd ( ( + ) + ( A A + ( + ) + ( ) = ( A A + (8) I deep water combiig the two expressios, the total drift force o the vertical cylider is ρg F 4 ( ) x + = 3 + ( H / ) a ( = ( A A + (9) The o-dimesioal steady forces (Eqs. (7)-(8)) o a bottom-seated vertical cylider for various values of d/a are give i Fig. as fuctios of. The free surface cotributio is idepedet of the water depth ad is positive. The velocity-squared term is egative ad is oly a weak fuctio of water depth at the low values of (log periods). The et steady drift force o the cylider is positive ad icreases with the higher values (short periods). It is also relatively idepedet of the water depth. The closed form solutio is validated i Fig. by comparig the results with the umerical solutio based o the well-established boudary elemet liear diffractio theory. The cylider i the umerical solutio was represeted by paels. Note that the results from Skourup et al. [7] also match well (Fig. ). WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

5 Fluid Structure Iteractio ad Movig Boudary Problems IV 7.5 Nodim Forces FS VEL. ^ d/a = VEL. ^ d/a = VEL. ^ d/a = 4 TOTAL d/a = TOTAL d/a = Figure : Horizotal steady drift force o a vertical cylider. Nodimesioal Forces FS Closed Form VEL. SQ. Closed Form FS VEL. SQ Nodim Forces TOTAL d/a = Skuorup, et al Figure : Validatio of steady drift force o a vertical cylider. 4. Bottom-mouted horizotal half-cylider Let us ow cosider a horizotal half-cylider seated o the ocea bottom (Chakrabarti []). A possible applicatio of this geometry is the buried ocea floor pipelies used for the trasportatio of crude oil from the productio platform to shuttle takers or shore. A appropriate cylidrical coordiate system (r,θ) is chose to describe the velocity potetial. Based o the liear diffractio theory for a bottom-seated halfcylider the velocity potetial at the surface of the cylider, r = a, ca be expressed i a closed form as log as the half cylider is removed from the free surface so that the effects of the free surface o the velocity potetial is cosidered small ad ca be igored. I this case the velocity potetial becomes cosh( kr siθ ) exp( ikr cosθ ) + gh Φ( r, θ; t) = i a a exp( iωt) ω cosh kd cosh k siθ exp ik cosθ r r () WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

6 8 Fluid Structure Iteractio ad Movig Boudary Problems IV For the half-cylider o the ocea bottom it has bee show that a accurate first order solutio ca be obtaied as log as d/a is equal to or greater. The steady horizotal drift force for a submerged half-cylider due to the velocity-squared pressures is idetically zero. The vertical compoet of the drift force is obtaied from the itegral F y θ = L p siθ ( ad ) () where the dyamic pressure is derived from the velocity-squared term. Sice the ormal velocity o the surface of the cylider is zero, the oly cotributio to the velocity-squared term o the -D half-cylider is Φ p = ρu θ = ρ () r θ The F y F y = = [ sih ( siθ ) cos θ ] siθdθ (3) ρgl( H / ) sih kd which reduces to = I( + ; I( = cosh( si θ ) si θdθ sih kd 3 F y (4) The values of the itegral I ( versus are give i Table. d s y θ r x Steady Vert. Drift Force D/A =.5 D/A =.75 D/A = D/A =.5 D/A = Figure 3: Vertical steady drift force o a horizotal half-cylider. WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

7 Fluid Structure Iteractio ad Movig Boudary Problems IV 9 The results o the steady drift force versus are give i Fig. 3 for various depths of submergece. For brevity the results are show for d/a value dow to.5. It ca be see that the forces icrease i value as the value decreases. As expected, the magitude of the o-dimesioal force decreases rapidly with icrease i the water depth. 4.3 Deep-submerged horizotal cylider For a deeply submerged horizotal cylider away from the free surface the effect of the free surface may be igored. It is also assumed that the cylider is somewhat removed from the ocea floor. I this case, a closed form expressio of the velocity potetial has the form quite similar to the velocity potetial for the half-cylider, Eq. (), by itroducig the depth of submergece ks i the cosh terms. By the same reasoig the validity of this potetial fuctio for the first-order force ca be show to be applicable for a d/a value of at least equal to. A possible applicatio of this geometry is a submerged compoet of a offshore structure. The time idepedet velocity-squared pressure is obtaied from the velocity potetial as ρ g H cos θ sih ( si θ + ks ) p = (5) a sih kd + si θ cosh ( si θ + ks ) The horizotal compoet of the force is idetically zero. The vertical compoet is give by F y θ = L p siθ ( ad ) (6) The ρgl ( H ) cos θ sih ( si θ + ks ) si θdθ + si θ cosh ( si θ + ks ) F y = / sih kd (7) which reduces to Fy = I(, s / a) + ; I(, s / a) = cosh(( (si θ + s / a)))siθdθ sih kd 3 (8) Table lists the values of I (, s /a) for differet values of s /a. The results o the steady drift force versus are give i Fig. 4 for various depths of submergece s /a. It ca be see that the forces peak at itermediate values. As expected, the magitude of the o-dimesioal force decreases rapidly with the depth of submergece as well as values. WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

8 Fluid Structure Iteractio ad Movig Boudary Problems IV Table : Values of itegrals I ( ad I (, s /a). I ( I (, s /a) I (, s /a) I (, s /a) I (, s /a) s/a= s/a=.5 s/a= s/a= e e e e e e e e e e e-6.997e e-7.98e e e e e e e e-6 5.5e e e-6 3.5e WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

9 Fluid Structure Iteractio ad Movig Boudary Problems IV d y θ r so x No Dimesioalized Force s/a= No Dimesioalized Force s/a= s/a=.5 s/a= No Dimesioalized Force s/a=3 s/a=.5 s/a= s/a=.5 s/a= Figure 4: Vertical steady drift force o a horizotal cylider. 4.4 Bottom-mouted hemisphere Let us cosider a hemisphere seated o the ocea bottom (Chakrabarti ad Naftzger []). The spherical coordiate system is adopted i terms of θ ad µ. Based o the liear diffractio theory for a bottom-seated hemisphere the velocity potetial at the surface of the cylider, r = a, ca be expressed i a closed form as log as the hemisphere is removed from the free surface with a d/a ratio of at least.5, so that its effects o the velocity potetial ca be igored. I this case the velocity potetial becomes where r r a ix a a cosh( Y)exp( i X) + + cosh( )exp( ) Y i X gh a a r R r r Φ (, r θ;) t = i exp( iωt) ω coshkd Y a a ix sih( Y)exp( i X) R r r R X = cosθ si µ Y = siθ si µ R = si µ λ = gh /( ω cosh kd) The the θ- ad µ-compoets of the particle velocity respectively are (9) Φ λ X uθ ( r, θ; t) = = T sihy cos( X ωt) + T coshy si( X ωt) siωt asi µ θ asi µ R () ad cot µ Φ λ cot µ X uµ ( r, θ; t) = = T cosh Y si( X ωt) + T sih Y cos( X ωt) si ωt a θ a R () WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

10 Fluid Structure Iteractio ad Movig Boudary Problems IV where T = X R+ X / R ad T = Y Y / R. For a bottom-seated hemisphere the steady horizotal surge drift force due to velocity-squared pressures is writte as F x = ρ a ( uθ uµ ) cosθdθ si µ dµ + () Substitutig the value of λ the ormalized force reduces to F x = ρg( H / ) a 4 sih kd Y T sih Y + T cosh Y + R + Y Y T Y X T Y X sih si + cosh cos R R X T cosh Y + T sih Y + R cos X X T Y X T Y X cosh cos + sih si R R dµ θdθ cos µ (3) The vertical compoet of steady drift force is computed from F y = ρ a ( uθ uµ ) siθdθ si µ dµ + (4) so that the oly chage i Eq. 3 is to replace cosθ with siθ. The results o the o-dimesioal steady horizotal ad vertical drift forces versus are give i Fig. 5. It ca be see that the forces peak at a itermediate value. As expected, the magitude of the o-dimesioal force decreases rapidly with the depth f submergece. Steady Horiz. Drift Force D/A =.5 D/A =.6 D/A =.75 D/A = D/A = Steady Vert. Drift Forc D/A =.5 D/A =.6 D/A =.75 D/A = D/A = Figure 5: Horizotal ad vertical steady drift forces o a bottom-seated hemisphere. 5 Coclusios The secod-order steady wave drift forces o a few fixed basic objects of symmetry have bee computed based o the total diffractio potetial. The WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press

11 Fluid Structure Iteractio ad Movig Boudary Problems IV 3 objects chose are the vertical cylider, submerged horizotal cylider, bottomseated horizotal half-cylider, ad bottom-seated hemisphere. For the vertical cylider extedig above the still water level, the horizotal drift forces receive cotributios from the free surface as well as the Beroulli s velocity-squared terms. For the other fixed bodies the cotributio comes from the velocitysquared term oly, as the objects do ot peetrate the mea water level. The expressios for the steady forces are obtaied i closed forms. It is show that the forces deped o the o-dimesioal diffractio parameter ad the depth to radius ratio d/a. For the submerged horizotal cylider, the forces are also a fuctio of the o-dimesioal submergece depth, s /a. Comprehesive results are preseted for these shapes for differet values of these o-dimesioal parameters so that they ca be used directly to estimate the drift force i a desig of these compoets. While the results are limited to bodies of specific basic shapes, they costitute members of most offshore structures, icludig submerged pipelies, caissos, TLP, Spar, ad semisubmersible. It is expected that the formulas ad umerical results preseted here will have applicatios i the desig of such structures. Refereces [] Chakrabarti, S.K., Noliear Wave Forces o Vertical Cylider, Joural of Hydraulics Divisio, ASCE, Vol. 98, November 97. [] Chakrabarti, S.K. ad Naftzger, R.A., Noliear Wave Forces o Half- Cylider ad Hemisphere, Joural of Waterways, Harbors ad Coastal Egieerig Divisio, ASCE, Vol., WW3, August 974. [3] Chakrabarti, S.K., Hydrodyamics of Offshore Structures, WIT Press, Ashurst, Southampto, 99. [4] MacCamy, R. C., ad Fuchs, R. A., Wave Forces o a Pile: A Diffractio Theory, Techical Memoradum No.69, Beach Erosio Board, 7 pp, 954. [5] Pikster, J.A., Mea ad Low Frequecy Wave Driftig Forces o Floatig Structures, Ocea Egieerig, Vol. 6, pp , 979. [6] Ogilvie, T. F., Secod-Order Hydrodyamics Effects o Ocea Platforms, Proceedigs Iteratioal Workshop o Ship ad Platform Motios, Berkeley, Uiversity of Califoria, pp. 5-65, 983. [7] Jesper Skourup, Cheug Kwok Fai, Harry B. Bigham ad Bjare Buchma, Loads o a 3D body due to Secod Order Waves ad a Curret, Ocea Egieerig, Vol. 7 () WIT Trasactios o The Built Eviromet, Vol 9, ISSN (o-lie) 7 WIT Press