NUMERICAL SIMULATION OF WATER ENTRY OF WEDGES BASED ON THE CIP METHOD

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1 Journal of Marne Scence and Technology, Vol., No., pp. -5 (5) DOI:.69/JMST--6- NUMERICAL SIMULATION OF WATER ENTRY OF WEDGES BASED ON THE METHOD Zhao-Yu We,, He-ye Xao, Zh-dong Wang, and Xu-Hua Sh Key words: water entry mpact, wedge, constraned nterpolaton profle, THINC scheme. ABSTRACT The hydrodynamc problems for water entry of wedges are studed by a constraned nterpolaton profle () based on Cartesan grd method. The scheme was appled to the Cartesan grd based on flow solver, whch was frst descrbed. The tangent of hyperbola for nterface capturng (THINC) scheme was appled to capture the free surface. The method was extended to capture the dynamcs n the water entry approach. The smulatons were performed n three aspects. Frst, pressure dstrbutons for the wedges enterng nto water wth constant speed were predcted, and the results were compared wth the smlarty solutons by the boundary element method (BEM). Then smulatons were conducted for the free-fallng wedges enterng nto water wth one degree of freedom. The results were compared wth the publshed expermental data. Fnally, the mpact of a wedge on the water surface n a free fall moton wth an ntal heel angle was studed. The moton of the wedge was subjected to three degrees of freedom, and the results were also compared wth those of experments. I. INTRODUCTION Water entry mpact problem has wde applcatons n ocean engneerng. More than eghty years ago, Von Karman (99) provded the frst approxmate soluton for the load predcton as the V-shaped body mpacts the water surface, thus ntated the feld of flud-structure mpact. Hs work was further developed by Wagner (9), who accounted the local up rse of water. Later, Dobrovolskaya (969) obtaned the frst complete soluton for a two-dmensonal wedge enterng nto the Paper submtted 9/6/; revsed /7/; accepted /6/. Author for correspondence: Zhao-Yu We (e-mal: wemuru@6.com). School of Naval Archtecture and Ocean Engneerng, Jangsu Unversty of Scence and Technology, Zhenjang, Jangsu, P. R. Chna. No. research nsttute of chna ordnance ndustres, x'an, P. R. Chna. School of Marne Engneerng, Northwestern Polytechncal Unversty, X'an, Shaanx, P. R. Chna. water at constant velocty. Hs work was based on the velocty potental theory, where the gravty of flud flow was gnored. Snce there was no length scale, the flud was self-smlar. Zhao and Faltnsen (99) reconsdered the problem and solved t usng a boundary element method n the tme doman. The thn jettng was cut to avod numercal errors. Khara () also studed the water entry problem of wedges at constant speed. In hs work, the boundary element method (BEM) and a mxed Euler and Lagrangan method (MEL) were employed, ntroducng an analytcal descrpton of the jettng flow. Other works on water entry problems of wedges nclude those by Semenov and Iafrat (6) and Xu et al. (8). For the free surface water entry, an mportant work was one by Wu et al. (). By usng BEM wth Taylor expanson for the jet, Wu et al. obtaned the tme marchng soluton for the water entry of a wedge n free fall moton. The soluton was further expanded to three degrees of freedom by Xu et al. (). Besdes these potental-flow based methods, the recently developed computatonal flud dynamcs (CFD) methods have made t possble to solve the water entry problems and also treat hghly dstorted or breakng free surface breakng by solvng the governng equatons. Oger et al. (6) solved the water entry of wedges usng smoothed partcles hydrodynamcs (SPH) method. Other examples nclude the use of volume of flud (VOF) for water entry problems by Kleefsman et al. (5), and the level set as descrbed by Yang and Stern (9). In addton, the constraned nterpolaton profle () method was also appled to nvestgate the water entry problem. Zhu et al. (7) appled the method to study the water entry and ext of a two-dmensonal horzontal crcular cylnder. Yang and Qu () also used the same method to solve the water mpact of two- and three-dmensonal bodes. In ths work, the constraned nterpolaton profle () based Cartesan grd method s presented and appled to the water mpact problems of wedges. The Cartesan grd method used for the numercal soluton does not depend on the locatons of the body boundary and the free surface, whch makes the computaton of strongly nonlnear problems wth complcated free-surface deformaton more effcent and convenent. In Sect. we frst gve a bref descrpton of the -based flow solver. The THINC scheme (tangent of hyperbola for

2 Z.-Y. We et al.: Numercal Smulaton of Water Entry of Wedges Based on the Method nterface capturng) for the nterface capturng s also ntroduced and explaned. To valdate the method and the THINC scheme for free surface capturng, the water entry of two-dmensonal wedges were nvestgated. In Sect.., the water entry of wedges wth constant speed s numercally smulated and compared wth the smlarty soluton by the BEM. Then n Sect.. we present the numercal results of symmetry wedges enterng water n free fall moton and the results are compared wth that from experments. In Sect.., the water mpact of a wedge n free fall moton wth an ntal heel angle s studed and the results are also compared wth expermental data. Ar Free surface Lqud Sold Vrtual partcles at body surface Fg.. Concept for the Cartesan grd approach durng water entry of wedges. II. NUMERICAL METHODS. Governng Equatons Assumng that there s no temperature varaton n the problem, the flow s consdered as unsteady, vscous and ncompressble. The governng equatons can therefore be wrtten as and u x () u u p uj ( Sj) f, () t x x x j j where Sj ( u / xj uj / x )/, f stands for the gravty. Eq. () was solved by a fractonal step method, n whch the equaton was dvded nto three calculaton steps: one advecton step and two non-advecton steps (Hu and Kashwag, ). The scheme was appled to calculate the advecton step whle the two non-advecton parts were calculated by the Euler explct scheme and the mplct scheme ntegraton, respectvely. The Posson equaton was used to obtan the pressure dstrbuton, whch was treated n a non-advecton calculaton step and can be wrtten as n p u ( ) x x t x **, () here the superscrpt n+ stands for the quanttes of the next tme step, and ** denotes nterm values obtaned from the frst non-advecton phase. After Eq. () was numercally solved, we were able to obtan the pressure dstrbuton over the whole computaton doman, where Eq. () was assumed vald. More detals about the calculaton of the Posson equaton and fractonal step treatment used n the flow solver can be found n a prevous study by Hu and Kashwag ().. The Inner Interface Treatment Method The free surface and wedge boundary n the present numercal method were treated as mmersed nterfaces, as shown n Fg.. A densty functon m was defned to recognze the dfferent phases, n whch m =,, denote the lqud, gas and sold phases, respectvely. Each computatonal cell satsfes the densty functon Σ m =.. The nterfaces n the multphase problem need to be determned n the computaton, whch can be performed by solvng the followng equaton for the densty functon m m u t x. () Only two densty functons need to be solved, e.g., for the lqud and for the sold; and the thrd can be obtaned by. (5) As all of the densty functons are determned, any physcal property, such as densty and vscosty, can be obtaned for each computaton cell as. (6) m m m In order to mantan numercal stablty, the densty and vscosty of the sold wedge are set to be the same as that of the lqud. Two knds of nterface should be consdered; gas-lqud nterface (free surface) and sold-flud nterface (wedge boundary). The THINC (tangent of hyperbola for nterface capturng) scheme s appled to capture the free surface whle the vrtual partcle method was appled to track the wedge surface (Hu and Kashwag, 9). The THINC scheme has the features needed for our computatons: mass conservaton and effectvty n elmnatng numercal dffuson and oscllaton (Hu and Kashwag, 9; Xao et al., 5, Zhao and Hu, ). Eq. () can be ntegrated by an upwnd scheme, such as the scheme (Hu and Kashwag, ). The nterface s then

3 Journal of Marne Scence and Technology, Vol., No. (5) φ g + / F ( x) F ( ) x + u + /.5 φ φ φ + φ + x x / x + / x + / Δ up = u Δt + / Fg.. Concept of THINC scheme. Y (m) determned by the so-surface of =.5. Smlar to scheme, the profle of nsde a computaton cell s approxmated by an nterpolaton functon and the tme ntegraton of s performed by a sem-lagrangan scheme. Instead of usng a polynomal functon as adopted by the scheme, the THINC scheme uses a hyperbolc tangent functon to approxmate the profle (Xao et al., 5). To explan the THINC scheme, a one-dmensonal case s consdered. The mult-dmensonal calculaton can be done usng a drectonal splttng technque (Godunov, 959). The advecton equaton on (x, t) for THINC scheme s wrtten n a conservaton form as ( u). (7) t x Integratng Eq. (7) over [t, t + t] and [x -/, x +/ ] gves n n g / g / / x, (8) x / where (/ x) ( xtdx, ) s the cell averaged densty x/ tt functon, g / ( u) /dt s the flux at the cell boundt ary. The flux s computed by a sem-lagrangan scheme as shown n Fg.. The shaded area denotes the flux at x = x +/ for the case of u /. Snce the varaton of ( ) across the free surface s step-lke, we can approxmate the profle nsde the computatonal cell [x -/, x +/ ] by a pecewse modfed hyperbolc tangent functon as F( x) tanh[ (( xx ) / x )] /, (9) / where x = x +/ x /, and,,, are the parameters to be specfed. Parameter controls the steepness of the profle, and a value of =.5 was used n the present computatons. Parameter represents the mddle pont of the hyperbolc tangent functon and was determned by the followng relaton X (m) Fg.. The confguraton for the grd cells. x/ n F( x) dx x. () x/ After F (x) was determned from Eq. (9), the flux across the cell boundares was then calculated and the cell ntegrated value at a new tme step n was computed. As ths cell- ntegrated value was used to determne the free surface, the mass conservaton of lqud s automatcally satsfed. III. NUMERICAL RESULTS AND DISCUSSIONS Convergence study was frst undertaken. The schematc llustraton of the basc grd confguraton for all the computatons performed n ths study s shown n Fg.. The grd spacng was vared, wth the mnmum but equally spaced cells at the regon near the free surface n the y-drecton and around the wedge n the x-drecton. The cell sze ncreased gradually away from the mpact regon and the free surface to reduce the CPU requrement and memory. The mnmum grd spacng was. m, whch was found to gve ndstngushable results. Care was provded to ensure that there were at least fve vrtual partcles wthn each grd cell. As a result, the sze of the largest cell furthest away from the wedge was about. m. In the present model, calculatons started for the wedges wth a very small dstance above the quescent water surface. The wedge fell freely n at frst, before t mpacted the free surface of water.. Test Cases of Water Impact for Symmetrc Wedges wth Constant Speed To valdate the method, we frst appled t to the cases of symmetrc wedges mpactng the free surface wth constant velocty. The results of our smulaton were compared wth the smlarty solutons of BEM by Wu et al. (). The wedges used n the method had the same sde length of.5 m. Four deadrse angles were consdered, correspondng

4 Z.-Y. We et al.: Numercal Smulaton of Water Entry of Wedges Based on the Method BEM 5 5 x/vt (a) BEM 6 8 x/vt (b) BEM 6 x/vt (c) BEM x/vt (d) Fg.. The comparson of pressure dstrbuton for the wedge enterng water wth constant speed (: method, BEM: smlarty soluton by the BEM). (a) β =, (b) β =, (c) β =, (d) β = 5. to =,, and 5. In the BEM, the gravty of flud was gnored and the mpact speed was constant. As there was no flow separaton, t led to a self-smlar soluton. Furthermore, the mass of the structure was also gnored. Fg. gves the pressure dstrbuton over the wedges surface before the flows separated from the sde face of the wedges. In Fg., the pressure dstrbuton has been non-dmensonalzed by V, where s the densty of water, V =. m/s s the vertcal velocty for all wedges. Good agreement wth BEM method was evdent. In partcular, as the lnear jet approxmaton was used n the BEM (Wu et al., ) smulaton, the pressure n the long and thn jet was zero, resultng n the absence of spke and oscllaton. However, the smulaton presents a multphase problem. Each grd cell was flled wth ar, flud and sold phase. When solvng the Posson equaton, the pressure was calculated over the whole computatonal doman, thus the pressure value exsts n each cell. Therefore the pressure dstrbuton n the jet for the method was not zero. On the other hand, as the SOR (successve over relaxaton) teraton method was used to solve the pressure, the dvergence caused by the teraton number and the accuracy resulted n the saw teeth behavor of these pressure curves, as shown n Fg.. Fg. 5 gves the wave elevaton for four dfferent deadrse angles by method. It was found that as the wedge wth small deadrse angle mpacts the water surface; the jettng was very thn and moves wth very hgh speed. There were some spkes and oscllatons formed along the free surface. Ths was because the surface tenson was not consdered n the smulaton, thus the free surface easly broke.. Test Cases of Water Entry of Wedges n Free Fall wth One Degree of Freedom The cases of free-fallng wedges mpactng the water surface wth one degree of freedom were also nvestgated and compared wth the results from two other experments. The frst experment was carred out by Wu et al. (). In the experment, both the acceleraton and the stran were recorded. Here, only the former was used to valdate the numercal smulaton. The V-shaped model s L = 6 cm long and B = cm wde, as shown n Fg. 6. The model was set up along a steel made gldng platform. When released, the wedge would glde nto a tank wth the dmensons of.65 m.8 m.5 m. In ther experment, frcton between the gldng platform and the wedge frame were consdered. The acceleraton g e of wedge was not equal to the gravtatonal acceleraton, even before the mpact occurred, and hence was taken nto account n the analyss. The second experment was carred out by Xu et al. (998). Both the symmetrc and asymmetrc drop tests were nvestgated, wth the wedges released freely nto the water. In ths secton only the symmetrc cases were analyzed. The man nterest n such a physcal problem came from the need to understand the dynamc response of hgh-speed shps durng water mpacts so as to reduce the hull damage resultng from boat slammng (Okada and Sum, ).

5 6 Journal of Marne Scence and Technology, Vol., No. (5).6 Table. Parameters of the confguratons studed Name Sym-lght Sym-lght Asym-lght H (m) M s (kg) CG (m).6 I G (kg m ) (a) t =.5 s..6 L.6. B. -. Fg. 6. Model of a V-shaped body (b) t =.8 s.5. m.6 m (a) x(m). (c) t =. s. CG Free surface C G H (b) α Ar Water Fg. 7. Expermental confguraton of drop tests (a) Geometry of the prsmatc hull, (b) Expermental parameters x(m). (d) t =.5 s..6 Fg. 5. The free surface deformaton for the wedge enterng water wth constant speed (a) β =, (b) β =, (c) β =, (d) β = 5. Images represent t =.5 s,.8 s,. s and.5 s after mpact, respectvely. The physcal model was a hgh-aspect-rato prsmatc wedge wth deadrse angle of =, as llustrated n Fg. 7(a). Dfferent expermental confguratons of drop heght H, wedge mass M S, and heel angle α (Fg. 7(b)) were consdered and s summarzed n Table. For the asymmetrc tests, the poston of the center of gravty (G) and the nerta were also mportant parameters, hence the roll and vertcal acceleraton tmehstory were recorded by nstruments wth accelerometers. Fg. 8 shows the curves of acceleraton (m/s ) aganst tme t(s), the deadrse angles of the wedges are = 5 and, respectvely. The acceleraton of the wedges had been non-dmensonalzed by gravtatonal acceleraton, g. For the cases wth deadrse angle of = (Fg. 8(c, d)), the smulaton curves and the experment curves are n excellent agreement, whle there are small dscrepances at the later stage for the cases of = 5 (Fg. 8(a, b)). Ths could be

6 Z.-Y. We et al.: Numercal Smulaton of Water Entry of Wedges Based on the Method 7 a/g a/g a/g a/g Experment (a) Experment -.. (b) 8 6 Experment (c) Experment (d) Fg. 8. Acceleraton at β = 5 : (a) g e = 8.5 m/s, M b =.5 kg, V s =.5797 m/s, (b) g e = m/s, M b =.88 kg, V s =.6967 m/s, and β = : (c) g e = 9.8 m/s, M b =. kg, V s =.577 m/s, (d) g e = 9.8 m/s, M b =. kg, V s =.89 m/s. dsplacement (m) angular dsplacement (degree) horzontal x vertcal y.5..5 (a) 5 Experment (b) Fg. 9. Dsplacement of the wedge wth the deadrse angle of β =, heel angle of α = 5, and the released heght s H =.6 m. (a) Horzontal of x and vertcal dsplacement of y, (b) Rotatonal dsplacement. explaned by the wedge beng pulled to a stop towards the end of the mpact (Wu et al., ), n whch the pullng force was not ncluded n the smulaton. Another mportant factor affectng the comparson s the end effect. In the smulaton, t was a completely D problem. However n the experment, as the wedge enters the water, only the mddle of the wedge s a D problem, but not at the two ends. Smlar to the pressure curves, the acceleraton curves also showed obvous spke and saw tooth behavor due to dvergence of SOR teraton method and accuracy.. Test Case of Water Entry of a Wedge n Free Fall Moton wth Three Degrees of Freedom In ths secton, the wedge n Fg. 7(b) was released freely nto the water wth complete three degrees of freedom. The whole dynamcs of the wedge was governed by both the nteractons wth water and gravty. The man dffculty that was encountered here s the couplng appearng between lnear and angular acceleratons, because the flows around the wedge are asymmetrc. The deadrse angle of the wedge was =, whle the heel angle was = 5, as shown n Fg. 7(b). The wedge was released from a heght of H =.6 m, thus the ntal vertcal velocty was V =. m/s and the ntal horzontal velocty was U =. m/s. To lmt the CPU processng tme,

7 8 Journal of Marne Scence and Technology, Vol., No. (5) t =. t =.5 t = (a) t = s x/h Fg.. Pressure dstrbuton evoluton of the wedge at the same tme correspondng to the case n Fg., at the tme of t = s, the wedge touches the water surface (b) t =.88 s -.. (c) t =.87 s -.. (d) t =.75 s Fg.. Free surface profle of the wedge wth the deadrse angle of β =, heel angle of α = 5, and the released heght s H =.6 m. At the tme of t = s, the wedge touches the water surface.... the free fall moton was not computed. The wedge was placed at a small heght just above the free surface at the begnnng of the smulaton, and the ntal knematc condtons for the wedge were corresponded as much as possble to those n the experment. Fg. 9(a) gves the tme hstory curves of horzontal and vertcal dsplacement for the wedge. As the ntal heel angle caused the wedge to rotate, the horzontal dsplacement also ncreased. The effect of the ntal heel angle on the vertcal dsplacement was much smaller. In Fg. 9(b), the rotatonal angle dsplacement was compared wth that of the experment. Although there were some dscrepances, the overall trend agreed well. All these show that the nteractons between the three degrees of freedom are mportant durng the free fall moton. Fg. shows the free surface deformatons at t =. s,.88 s,.87 s and.75 s, respectvely. It provdes nformaton on the successve shapes of the two non-symmetrcal jettng generated. The presence of fragmentaton at the tp of these jettng were also notced. Fg. presents the correspondng pressure dstrbuton for the free surface deformatons n Fg.. The ntal heel angle and clockwse rotaton ncreased the peak value of pressure on the rght-hand sde of the wedge. However, the pressure on the left sde was reduced. Over tme, the flows moved wth hgh speed and clmbed up the sde face of wedge, thus the wdth of the pressure dstrbuton ncreased whle the peak value decreased. Fg. (a) shows the tme hstory of the horzontal acceleraton. Fg. (b, c) shows the comparson of tme hstory curves of vertcal and angular acceleraton between the and expermental results. It should be noted that because of the errors made n estmatng the ntal condtons for computaton, the results were adjusted to make the curves match. Ths was to ensure better comparsons between the two results. Here, the wedge was relatvely flat. As there was an ntal heel angle of 5, the true deadrse angle on the rght sde was 5. As a result, some non-neglgble ar effects seemed to occur at the ntal stages of mpact. Indeed, the progressve

8 Z.-Y. We et al.: Numercal Smulaton of Water Entry of Wedges Based on the Method 9 du/dt/g U (m/s) (a) (b) (c) dv/dt/g V (m/s) Experment Experment (d) (e) (f) Fg.. Moton of the wedge wth the deadrse angle of β =, heel angle of α = 5, and the released heght s H =.6 m. dω/dt ω (degree/s) 5-5 Experment evoluton of the vertcal acceleraton due to ar cushon effects at the ntal stage could not be accurately captured by the model. Ths resulted n a stffer vertcal acceleraton slope than that n the experment. Nonetheless, the ampltude of maxmum loads was n good agreement wth the experment. Concernng the tme evoluton of angular acceleraton, some oscllatons were notced on the expermental data. These oscllatons were produced by structure vbratons whch cannot be captured n the context of rgd body smulatons (Xu et al., 998). These expermental vbratons thus make the comparson wth the angular acceleraton dffcult. Fg. (e) shows the tme hstory of vertcal velocty, whch showed good agreements between the smulaton and expermental results. Fg. (f) shows the tme hstory of the angular velocty. As the wedge touches the water surface, the heel angle and the ntal mpact causes the wedge to rotate clockwse. Then as the pressure on the rght half sde s much larger than that on the left sde, the wedge begns to rotate antclockwse. Ths trend also corresponds to the tme hstory curve of horzontal velocty (Fg. (d)), where the center of wedge frst moves slghtly towards the rght, and then towards the left, and fnally towards the rght agan. IV. CONCLUSIONS The water entry problems of wedges have been solved by a constraned nterpolaton profle () based on Cartesan grd method. The smulatons nclude three parts: n secton., the pressure dstrbuton on the wedges as they enter nto the water wth constant speed are predcted and compared wth the smlarty solutons by BEM; n secton., the cases of symmetrc water entry of wedges n a free fall moton were calculated and the vertcal acceleratons were also compared wth publshed expermental results; n secton., the water entry of a wedge wth an ntal heel angle n a free fall moton was calculated, wth detaled smulatons showng that there are some strong couplngs between the three degrees of freedom. For all these three parts, good agreements were found. These results have some mportance n applcatons for practcal problems such as shp slammng, whch wll be further nvestgated n future works. ACKNOWLEDGMENTS The authors would lke to thank the support of the Project Funded by the Prorty Academc Program Development of Jangsu Hgher Educaton Insttutons (PAPD). REFERENCES Dobrovolskaya, Z. N. (969). Some problems of smlarty flow of flud wth a free surface. Journal of Flud Mechancs 6, Godunov, S. K. (959). Fnte dfference methods for numercal computaton of dscontnuous solutons of the equatons of flud dynamcs. Matematchesk Sbornk 7, 7-95.

9 5 Journal of Marne Scence and Technology, Vol., No. (5) Hu, C. H. and M. Kashwag (). A -based method for numercal smulatons of volent free surface flows. Journal of Marne Scence and Technology 9, -57. Hu, C. H. and M. Kashwag (9). Two-dmensonal numercal smulaton and experment on strongly nonlnear wave-body nteractons. Journal of Marne Scence and Technology, -. Khara, H. (). Numercal modelng of flow n water entry of a wedge. Proceedng of 9th Internatonal Workshop on Water Waves and Floatng Bodes, Cortona, Italy, 8- March. Kleefsman, K. M. T., G. Fekken, A. E. P. Veldman, B. Iwanowsk and B. Buchner (5). A volume-of-flud based smulaton method for wave mpact problems. Journal of Computatonal Physcs 6, 6-9. Oger, G., M. Dorng, B. Alessandrn and P. Ferrant (6). Two-dmensonal SPH smulatons of wedge water entres. Journal of Computatonal Physcs, 8-8. Okada, S. and Y. Sum (). On the water mpact and elastc response of a flat plate at small mpact angles. Journal of Marne Scence and Technology 5, -. Semenov, Y. A. and A. Iafrat (6). On the nonlnear water entry problem wedges. Journal of Flud Mechancs 57, -56. Von Karman, T. (99). The mpact of seaplane floats durng landng. NACA TN, October, Washngton. Wagner, H. (9). Uber Stoss-und Gletvorgange an der Oberfluche von Flussgketen. Zetschrft für Angewandte Mathematk und Mechank, 9-5. Wu, G. X., H. Sun and Y. S. He (). Numercal smulaton and expermental study of water entry of a wedge n free fall moton. Journal of Fluds and Structures 9, Xao, F., Y. Honma and T. Kono (5). A Smple Algebrac Interface Capturng Scheme Usng Hyperbolc Tangent Functon. Internatonal Journal for Numercal Methods n Fluds 8, -. Xu, G. D., W. Y. Duan and G. X. Wu (8). Numercal smulaton of oblque of an asymmetrcal wedge. Ocean Engneerng 5, Xu, G. D., W. Y. Duan and G. X. Wu (). Smulaton of water entry of a wedge through free fall n three degrees of freedom. Proceedngs of the Royal Socety A 66, 9-9. Xu, L. X., A. W. Troesch and R. Petterson (998). Asymmetrc hydrodynamc mpact and dynamc response of vessel. Proceedngs of 7th Internatonal Conference on Offshore Mechancs and Arctc Engneerng 98-. Yang, J. M. and F. Stern (9). Sharp nterface mmersed- boundary/level-set method for wave-body nteractons. Journal of Computatonal Physcs 8, Yang, Q. Y. and W. Qu (). Numercal smulaton of water mpact for D and D bodes. Ocean Engneerng, Zhao, R. and O. Faltnsen (99). Water-entry of two-dmensonal bodes. Journal of Flud Mechancs 6, Zhao, X. Z. and C. H. Hu (). Numercal and expermental study on a -D floatng body under extreme wave Condtons. Appled Ocean Research 5, -. Zhu, X. Y., O. M. Faltnsen and C. H. Hu (7). Water entry and ext of a horzontal crcular cylnder. Journal of Offshore Mechancs and Arctc Engneerng 9, 5-6.

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