Parametric investigation of the effects of deadrise angle and demi-hull separation on impact forces and spray characteristics of catamaran water entry

J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 DOI 10.1007/s40430-016-0679-3 TECHNICAL PAPER Parametric investigation of the effects of deadrise angle and demi‑hull separation on impact forces and spray characteristics of catamaran water entry Roya Shademani1 · Parviz Ghadimi1 Received: 8 July 2015 / Accepted: 11 November 2016 / Published online: 23 November 2016 © The Brazilian Society of Mechanical Sciences and Engineering 2016 Abstract The growing application of catamaran planing hulls implies the significance of comprehensive studies on different aspects of these types of vessels. In this context, slamming is an important phenomenon which is generally simplified to the wedge water entry of the planing hulls. In the present paper, the constant speed water entry of twin wedges, as a simplification of a planing catamaran hull, is numerically modeled at three deadrise angles and five demi-hull separations. Control volume-based finite element has been used to solve two-dimensional Navier–Stokes equations coupled with volume of fluid method for twophase flow modeling. The results of numerical method have been validated against experimental data. It has been illustrated that low demi-hull separation causes an increase in the impact forces by up to 19%, an effect which decreases rapidly with an increase in demi-hull separation. The obtained results also indicate that secondary impact forces between and outside the hulls are asynchronous due to the reciprocal effects of demi-hulls. Moreover, spray characteristics have been defined and it has been demonstrated that the height of spray between hulls decreases by a decrease in the demi-hull separation. Keywords Twin wedge · Water entry · Demi-hull separation · Deadrise angle · FEM-FVM method · VOF · Impact force · Spray parameters · Secondary impact Technical Editor: Marcio S. Carvalho. * Parviz Ghadimi pghadimi@aut.ac.ir 1 Department of Marine Technology, Amirkabir University of Technology, Hafez Ave, No 424, P.O. Box 15875‑4413, Tehran, Iran 1 Introduction The water entry of wedges has long been used as a simplification of planing hull slamming. Also, the solution of this problem is used in 2D + t method where the planing hull is simplified to several prismatic wedge sections and the lift force of the hull is calculated using an integration of wedge impact forces along the hull length. The growing application of planing catamarans and twin hull planing vessels has emphasized the importance of investigating different hydrodynamic and structural aspects of these vessels; slamming as an important phenomenon among others. As mentioned before, this can be done by simplifying the problem to wedge water entry or more precisely twin wedge water entry. The stated applications of this problem among others have lead many researchers to focus on this apparently simple, but highly complex phenomenon. Recently, the efforts of many researchers in the field have vastly been directed towards the hydro-elastic study of the wedge water entry. For instance, Panciroli [1, 2] analyzed the water entry of flexible wedges, while Piro et al. [3] addressed the water entry and exit of flexible bodies. Other researchers who have studied this issue are Khabakhpasheva et al. [4], Yamada et al. [5], Alaoui et al. [6], Luo et al. [7] [8], Mo et al. [9], and Mutsuda et al. [10]. However, there still remain some unknown corners in the pure hydrodynamics of the water entry problem which require astute attention of the scientific communities. This is indeed why the researchers are still working on this issue. Water entry of a wedge of finite deadrise angle was analyzed by matched asymptotic expansions by Faltinsen [11].Yang et al. [12] numerically investigated the vertical and oblique water entry of 3D bodies using CIP-based finite difference method in 2012. Wu [13] investigated the 13 1990 J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 water entry problem with varying speed. The free surface shape, jet and pressure were recently simulated by Wang et al. [14] using the boundary element method. Luo et al. [15] applied an explicit finite element code to calculate the water entry loads on wedges. Velocity potential theory was used by Gao et al. [16] to simulate this problem. Ghadimi et al. [17] introduced an analytical solution for the water entry of wedges using the Schwartz-Christoffel conformal mapping. Sun et al. [18] by the use of boundary element method calculated the forces acting on a planning hull through the integration of water entry force on the wedges. Korobkin and Scolan [19] considered perturbation about the normal symmetric impact to approximate the vertical entry of an inclined cone. Molin and korobkin [20] also investigated a perforated wedge when entering the water. List of other researchers who have worked on this problem in the last five years include Qian et al. [21], Li et al. [22], Wu et al. [23], Xu et al. [24], Miloh [25], Malenica [26], Feizi Chekab et al. [27], Farsi and Ghadimi [28–30], Ghadimi et al. [31–34], Tavakoli et al. [35], Shademani and Ghadimi [36], and many others. It is noteworthy that investigations of this problem has been pioneered by Von Karman [37] and Wagner [38] in late 20s, followed later by Zhao et al. [39–41] and Faltinsen [42]. Many recent works are still based on these studies. As observed in the literature, most studies have been focused on single wedges and very limited number of works has been devoted to the study of catamaran slamming problems. Wu [43] and Yousefnezhad et al. [44] used boundary element method to model the constant velocity water entry of twin wedges neglecting gravitational effects. The aim of the present paper is to numerically investigate the water entry of twin wedges at different deadrise angles. This is conducted for different hull separations. Accordingly, the effect of hull separation on impact forces and spray formation, which has been investigated by He et al. [45] for a specific catamaran, is now extensively studied for different deadrise angles and hull separations. In the following sections, the governing equations and discretization are presented. 2 Governing equations The transient, incompressible Navier–Stokes equations with negligible dissipation effects are solved for two-phase flow using Control volume-based finite element (CVFEM) method coupled with the VOF method. The two-dimensional incompressible form of the conservation of mass is as follows: ∂u ∂v + =0 ∂x ∂y 13 (1) where u and v are the flow velocity components in the x and y directions, respectively. Also, the two-dimensional incompressible conservation of momentum may be written as  2  ∂u ∂u ∂ u ∂ 2u ∂P ∂u +u +v =ν + 2 − 2 ∂t ∂x ∂y ∂x ∂y ∂x  2  ∂v ∂v ∂v ∂P ∂ v ∂ 2v + 2 − +u +v =ν + By , ∂t ∂x ∂y ∂x 2 ∂y ∂y (2) ∂P where ν is the viscosity, ∂P ∂x and ∂y are the pressure gradients in x and y directions, respectively, and By is the gravitational body force. Also, the applied volume of fluid (VOF) method is based on the conservation of volume fraction (α) with respect to time and space, expressed as follows: ∂(uα) ∂(vα) ∂α + + =0 ∂t ∂x ∂y (3) In the VOF method, the volume fraction (α) is a scalar parameter which represents the fraction of a cell filled with one fluid, while assuming (1 − α) represents the volume fraction of the second fluid. When α = 1, the cell is filled by the first fluid and the density/viscosity of the first fluid is used. On the other hand, when α = 0, the cell is filled by the second fluid and the properties of the second fluid should be implemented. When 0 < α < 1, the cell contains a portion of both fluids and equivalent properties should be used. To consider a general formulation of these three cases using the volume fraction approach, the equivalent density and viscosity of the cells are calculated as follows: ρeq = αρ1 + (1 − α)ρ2 µeq = αµ1 + (1 − α)µ2 , (4) where α is the volume fraction, while ρ1 or ρ2 and µ1 or µ2 are the density and viscosity of the fluids, respectively. As evident in these relations, Eq. 4 covers all three options. When using VOF method, Eq. 4 is implemented in place of fluid properties of the Navier–Stokes equation. This way, although the Navier–Stokes equations are solved simultaneously for both fluids as one, the free surface can be extracted using the volume fraction. In the next section, the applied numerical method is briefly explained. 3 Numerical method In the present work, the momentum equations are implemented in a control volume resulting in four terms presented as follows: J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 1991 ∂ ∂t  ◦ ρϕd∀ ∼ =ρ  ∀i i ∀ ∂ϕi ϕi − ϕ i = ρ∀i ∂t �t (7) The discretization of the advection term on the control line j,k is shown in Eq. 8, using the upwind difference scheme (UDS).   − − → →  ρ V ϕds = ρ V ds ϕj,k (8) j,k Sj,k Fig. 1  Sub-control volumes of elements (∀i ), control surfaces (sm |j,k), nodes of elements (i) and integration points on the control lines (P) in a triangular element  ∂ρϕ → −→ − ∀i + ρuϕdsx |j,k + ρvϕdsy |j,k = Γ ∇ϕ · ds |j,k + ∫ bd∀, ∂t i       ∀i       Advection Diffusion  Transient Also, to eliminate the false diffusion problem, the method proposed by Karimian and Schneider [36] has been utilized which is shown in Eq. 9.    − �¯ →  ρ V ds ϕj,k = ρ V̂ ds ϕj,k , (9) j,k � where V̂¯ is the velocity vector defined using Karimian and Schnider [46] scheme to prevent the checkerboard problem. The components of the vector are calculated using Eqs. 10 and 11. ûj = UL−S Source (5) where ϕ. represents the flow velocity. A rigid body of constant impact speed is considered in the current study. In this method, each triangular element is divided into three partial control volumes as shown in Fig. 1. In Eq. 5, as shown in Fig. 5, i is the node of the triangular element and j and k are integration points on control lines of the partial control volume. Shape functions are defined at each node of the elements  as ϕ = Node i=1 Ni ϕi, where Ni is the shape function at node i while ϕi is the quantity of ϕ at that node. The shape functions for triangular elements are as follows: j,k �SU − ρ V̄    �SU − ρ V̄    ∂u ∂v − v −ρ u ∂y ∂y ∂P ∂u + − µ∇ 2 u + ρ ∂t ∂x  (10) ûj = VL−S ∂v ∂u − v −ρ u ∂x ∂x ∂P ∂v + − µ∇ 2 v + ρ ∂t ∂x  (11) Here, P is the pressure, SU is the distance of the integration point to the upwind point as illustrated in Fig. 1, and UL−S and VL−S are calculated using a bilinear interpolation      x3 − x2 x2 y3 − x3 y2 y2 − y1 x+ y+ DET DET DET       y3 − y1 x1 − x3 x3 y1 − x1 y3 N2 = x+ y+ (6) DET DET DET       x2 − x1 x1 y2 − x2 y1 y1 − y2 x+ y+ , N3 = DET DET DET N1 =  where DET is the determinant of the coordinates matrix which is given by DET = (x1 y2 + x2 y3 + x3 y1 − y1 x2 − y2 x3 − y3 x1 ) The terms of Eq. 6 should be discretized on the control lines between each two nodes. Transient terms are expanded on partial volume (∀i ) as in Fig. 2  The numerical Algorithm 13 1992 J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 Fig. 6  Schematics of the problem setup Fig. 3  Parameters definition on the velocity. Also, V̄ is the velocity at the previous time step. The bar sign on any variable indicates the previous time step. The final form of the continuity equation is presented in Eq. 12. (ρ ûdsx )j + (ρ v̂dsy )j + (ρ ûdsx )k + (ρ v̂dsy )k = 0 (12) Also, in Eq. 9, ϕj is calculated using Eq. 13 ∂ϕ �S, ∂s (13) Fig. 4  Impact force (F) vs. normalized depth (Z/d): numerical results (−) versus experimental data (.) [47] for a 10° deadrise and b 15° deadrise Fig. 7  Simplified problem setup 1600 1000 1400 900 800 1200 700 F (N) 1000 600 F (N) ϕj = ϕupj + 800 600 500 400 300 400 200 200 100 0 0 0 0.5 0 1 0.5 (a) (b) 0.4 0.4 0.35 0.3 0.3 Y (m) 0.25 Y (m) Fig. 5  Free Surface profiles from the numerical method (−) versus experimental data (.) [37] for a 10° at z/d = 3.03 and b 15° deadrise angle at z/d = 2.65 1 Z/d Z/d 0.2 0.2 0.15 0.1 0.1 0.05 0 0 0.3 13 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 x (m) x (m) (a) (b) 0.7 0.8 J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 1993 where ϕupj is the upwind velocity as shown in Fig. 1, and ∂ϕ ∂s �S is a linear estimation of the variation of ϕ from the upwind point to the integration point. The diffusion and pressure terms are modeled as in Eqs. 14 and 15, respectively. To discretize the pressure terms, the Bilinear Interpolation is used:   ∂p − dv = −Pdsx ∂x v s   (14) ∂p − dv = −Pdsy , ∂y v s    ∂ϕ ∂ϕ → −→ − dsx + Γ dsy  , −Γ ∇ϕ · ds |j,k = − Γ ∂x ∂y j,k (15) ∂ϕ where ∂ϕ ∂x and ∂y are calculated using the shape functions and dsx and dsy are the projection of ds is x and y direction. The overall algorithm of the code is illustrated in Fig. 2. As observed in Fig. 2, the Navier–Stokes equations are solved, first. Subsequently, the VOF equations are solved using the velocities resulting from the Navier–Stokes equations. Afterward, using the obtained volume fraction, new density and viscosity in each element are calculated and fed into the Navier–Stokes equations for the next time step. Fig. 8  Comparison of the results based on the full and simplified setups Fig. 9  Definition of spray parameters Fig. 10  Impact force vs. Time: 15° deadrise with different demi-hull separations 10 9 F/(1/2)(ρV²A) 8 7 6 S/W= 0.25 5 S/W= 0.5 4 S/W= 0.75 3 S/W= 1 2 S/W= 1.5 1 0 0 5 10 15 τ =vt/d 13 1994 J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 Fig. 11  Impact force vs. time: 30° deadrise with different demi-hull separations 10 9 8 F(1/2)(ρV²A) 7 6 5 S/W=0.25 4 S/W=0.5 3 S/W=0.75 S/W=1 2 S/W=1.5 1 0 -1 Fig. 12  Impact forces vs. time: 50° deadrise with different demi-hull separations 0 1 2 3 τ =vt/d 4 5 6 7 4 3.5 F/(1/2)(ρV²A) 3 2.5 S/W=0.25 2 S/W=0.5 S/W=0.75 1.5 S/W=1 1 S/W=1.5 0.5 0 0 0.5 1 1.5 2 τ =vt/d 4 Validation The applied numerical method is validated using the results provided by Tveitnes et al. [47]. To this end, wedges of 10 and 15 degrees deadrise angles and 0.6 m width are modeled at constant entry speed of 0.94 m/s as shown in Fig. 3. Figure 4 illustrates the calculated forces against the experimental results for two different deadrise angles. Also, the free surface profiles provided by Tveitnes et al. [47] have been digitized and overlaid on the numerically obtained free surface in Fig. 5. As observed in Figs. 4 and 5, reliable results have been obtained for impact force and free surface compared with 13 10 9 8 F/(1/2)(ρV²A) This algorithm is used to develop a computer code for solving the two-phase flow problem of wedge water entry. In the next section, the code is validated using previously published water entry data. 7 6 5 15 deg 4 30 deg 3 50 deg 2 1 0 0.25 0.5 0.75 1 1.5 S/W Fig. 13  Peak impact forces vs. demi-hull separations for different deadrises experiments with an impact force error of 5.36 and 7.48% for 10° and 15° deadrise angles, respectively. Also good agreement is observed in free surface evaluation, despite the J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 5.5 Original 2nd impact 5 F/(1/2)(ρV²A) 1995 4.5 S/W= 0.25 4 S/W= 0.5 3.5 S/W= 0.75 Retarted 2nd impact 3 2.5 S/W= 1 S/W= 1.5 Precipitated 2nd impact 2 5 6 τ =vt/d 7 8 Fig. 14  A closer look at the secondary impact forces slight difference in the breaking of spray which may be due to the nebular nature of the spray in that region. Next section addresses the simulation of twin wedge water entry for different deadrises and demi-hull separations. 5 Results and discussion The definition of the problem is illustrated in Fig. 6. In Fig. 6, 2S is the demi-hull separation, W is the width of the demi-hull, V is the constant entry velocity, d is the Fig. 15  Sequential illustration of water entry for deadrise 15 and S/W = 0.25 and 0.5 chine height from the wedge apex, and z is the entry depth at a specific simulation time. Three deadrise angles of 15, 30, and 50 degrees have been selected and the problem has been solved for 5 demi-hull separations (S/W = 0.25, 0.5, 0.75, 1 and 1.5). In all the considered cases, wedge is 1 m wide and V = 2 m/s. Before proceeding to the solution, a simplification can be applied due to the symmetry of the problem. The simplified setup is illustrated in Fig. 7. To ensure that this simplification is valid, a particular case has been modeled with both setups (full or symmetry consideration), and compared in Fig. 8. As observed in Fig. 8, similar results have been displayed for both setups. Therefore, the simplified setup is considered for the rest of the study. As pointed out earlier, the aim of the current study is to analyze the effect of deadrise and hull separation on the impact force and spray of the wedges. To this end, free surface parameters are defined in Fig. 9 for a better comparison of the results. In Fig. 9, Hs and Ws are the height and width of the spray, while the spray characteristics for the spray inside and outside the hulls are defined with additional subscripts i (inner) and o (outer). In this paper, the results are compared in the form of impact forces and spray parameters. S/W = 0. 25 S/W = 0.5 Precipitated inner 2nd impact Retarded inner 2nd impact 13 1996 15 deg. 30 deg. 50 deg. 0.25 Demi - Hull Fig. 16  Profile of spray for the inner (left side of each wedge) and outer (right side of each wedge) sides of demi-hulls with different deadrises and at different demi-hull separations J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 0.5 0.75 1.0 1.5 5.1 Impact forces The results of impact forces for different deadrise angles and hull separations are presented in Figs. 10, 11, and 12. These quantities are displayed in dimensionless form. A is the effective area of wedge and d is the height of the wedge, as pointed out before. As observed in Figs. 10, 11, and 12, the impact force for low demi-hull separation (S/W = 0.25) is slightly different than the other cases. The first observation is the peak force which is higher compared to other separations. To quantitatively assess this observation, the peak forces are illustrated in Fig. 13 for different cases. As observed in Fig. 13, the increase in peak impact force is in the range of 1 to 19% for S/W = 0.25, while the peak decreases dramatically with an increase in the separation. However, more deadrises should be analyzed to make a reliable deduction on the effect of deadrise on the separation effect. Another observation in Fig. 10 is the secondary impact forces. A closer look at the secondary impact forces of 15° deadrise is illustrated in Fig. 14. As observed in Fig. 14, two secondary impacts occur for S/W = 0.25 and 0.5, while there is only one secondary impact for other cases of S/W which possess higher impact force than those of S/W = 0.25 and 0.5. This phenomenon 13 is attributed to the effect of the separation on the inner spray (spray between the demi-hulls). This can be better understood after a sequential observation of the water entry in Fig. 15. As evidenced in these profiles, the separation causes the secondary impact to occur asynchronously for the inner and outer part of the hulls. It seems that for S/W greater than 0.5, the secondary impact is synchronized and summed to make one single secondary impact force as observed in Fig. 14. It should be emphasized that small separation can cause the precipitation or delay of the inner secondary impact, a fact that should be more investigated by analyzing more cases in this range of S/W. 5.2 Spray characteristics To study the spray characteristics of the wedges, the spray parameters should be extracted for each deadrise at different demi-hull separations. The free surface profiles for different deadrises and separations are presented in Fig. 16. As observed in Fig. 16, the geometry of the inner spray for S/W = 0.25 is very different from the outer spray. As should be expected, the symmetric shape of the spray for a demi-hull increases with respect to the growth of demihull separation. This may be better observed in the Fig. 17 J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 1997 0.13 0.125 0.12 0.115 0.11 0.105 0.1 Wsi Wso 0.25 0.5 0.75 1 Hsi/W and Hso/W 15 deg. Wsi/W and Wso/W Wsi and Wso 1.5 0.165 0.16 0.155 0.15 0.145 0.14 0.135 0.13 0.125 0.3 0.25 0.2 0.15 0.1 0.05 0 Wsi Wso 0.25 0.5 0.75 1 1.5 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Hso 0.25 0.5 0.75 1 1.5 Hsi Hso 0.25 0.5 0.75 1 1.5 S/W 0.39 Wsi Wso Hsi/W and Hso/W 50 deg. Wsi/W and Wso/W S/W 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Hsi S/W Hsi/W and Hso/W 30 deg. Wsi/W and Wso/W S/W Hsi and Hso 0.38 0.37 0.36 Hsi 0.35 Hso 0.34 0.25 0.5 0.75 1 1.5 0.25 0.5 0.75 1 1.5 S/W S/W Fig. 17  Spray characteristics for different deadrises at different demi-hull separations where the characteristics of the spray are illustrated with respect to the demi-hull separation and dearise. It is quite clear in Fig. 17 that spray is highly affected by the demi-hull separation, especially at low separations. However, to find a particular trend for the variation of spray characteristic is very difficult. However, one may observe that the spray height is mainly decreased in the inner part of the demihull. In the meantime, the spray characteristics converge for the inner and outer parts as the separation increases. 6 Conclusions In the present paper, the water entry of twin wedges has been addressed for different deadrise angles and mid-hull separations at constant impact speeds. To numerically analyze the problem, the 2D Navier–Stokes equations coupled with the Volume of Fluid method have been solved using the control volume-based finite element (CVFEM) numerical scheme. The results of the developed computer code have been validated against the available experiment and it has been demonstrated that due to the symmetry of the problem, solving half of the domain does not affect the results. Therefore, the problem has been simplified to a half domain and solved for deadrises 15, 30 and 50, with five demi-hull separations. The computed impact forces have been presented versus time and it has been shown that the demi-hull separation can affect the peak impact force with up to 19% error which is increased by a decrease in the separation. It has also been observed that secondary impact force is highly 13 1998 affected by the separation in a way that by decreasing the separation, the secondary impact between the hulls is precipitated or delayed regarding the demi-hull separation. In other words, the secondary impact of the wedge is asynchronous inside and outside the hull. After analyzing the impact force, spray characteristics have been defined to describe and compare the spray generated by different twin wedges. It has been shown that no unique trend may be deduced from the obtained results for the spray variations. However, the height of the spray generated between the demi-hulls is generally decreased with respect to the demi-hull separation decrease, and as the separation increases, the inside spray converges to the outer spray. References 1. Panciroli R (2013) Water entry of flexible wedges: some issues on the FSI phenomena. Appl Ocean Res 39:72–74 2. Panciroli R, Abrate S, Minak G, Zucchelli A (2012) Hydroelasticity in water-entry problems: comparison between experimental and SPH results. Compos Struct 94(2):532–539 3. Piro DJ, Maki KJ (2013) Hydroelastic analysis of bodies that enter and exit water. J Fluids Struct 37:134–150 4. Khabakhpasheva TI, Korobkin A (2012) Elastic wedge impact onto a liquid surface: wagner’s solution and approximate models. J Fluids Struct 36:32–49 5. Yamada Y, Takami T, Oka M (2012) Numerical study on the slamming impact of wedge shaped obstacles considering fluidstructure interaction (FSI). Proceedings of the International Offshore and Polar Engineering Conference 6. Alaoui AEM, Neme A (2012) Slamming load during vertical water entry at constant velocity. Proceedings of the International Offshore and Polar Engineering Conference 7. Luo H, Wang H, Soares CG (2012) Numerical and experimental study of hydrodynamic impact and elastic response of one freedrop wedge with stiffened panels. Ocean Eng 40:1–14 8. Luo H, Wang H, Soares CG (2011) Comparative study of hydroelastic impact for one free-drop wedge with stiffened panels by experimental and explicit finite element methods. Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering—OMAE 9. Mo LX, Wang H, Jiang CX, Xu C (2011) Study on dropping test of wedge grillages with various types of stiffeness. J Ship Mech 4:394–401 10. Mutsuda H, Doi Y (2009) Numerical simulation of dynamic response of structure caused by wave impact pressure using an Eulerian scheme with Lagrangian particles. Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering—OMAE 11. Faltinsen OM (2002) Water Entry of a wedge with finite deadrise angle. J Ship Res 46(1):39–51 12. Yang Q, Qiu W (2012) Numerical solution of 3-D water entry problems with a constrained interpolation profile method. J Offshore Mech Arct Eng 134(4):041101 13. Wu G (2012) Numerical simulation for water entry of a wedge at varying speed by a high order boundary element method. J Mar Sci Appl 11(2):143–149 14. Wang YH, Wei ZY (2012) Numerical analysis for water entry of wedges based on a complex variable boundary element method. Explosion Shock Waves 32(1):55–60 13 J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 15. Luo H, Wang H, Soares CG (2011) Numerical prediction of slamming loads on a rigid wedge subjected to water entry using an explicit finite element method. Advances in Marine Structures—Proceedings of the 3rd International Conference on Marine Structures 16. Gao J, Wang Y, Chen K (2011) Numerical simulation of the water entry of a wedge based on the complex variable boundary element method. Appl Mech Mater 90–93:2507–2510 17. Ghadimi P, Saadatkhah A, Dashtimanesh A (2011) Analytical solution of wedge water entry by using Schwartz-Christoffel conformal mapping. Int J Model Simul Sci Comput 2(3):337–354 18. Sun H, Zou J, Zhuang J, Wang Q (2011) The computation of water entry problem of prismatic planning vessels. 3rd International Workshop on Intelligent Systems and Applications—Proceedings 19. Korobkin AA, Scolan YM (2006) Three-dimensional theory of water impact. Part 2. Linearized wagner problem. J Fluid Mech 549:343–374 20. Molin B, Korobkin AA (2001) Water entry of a perforated wedge. Proc. 16th Int. Workshop on Water Waves and Floating Bodies, Japan, pp 121–124 21. Qian L, Causon D, Mingham C (2012) Comments on an improved free surface capturing method based on Cartesian cut cell mesh for water-entry and exit problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 22. Li Y, Li Y, Hu S (2011) Numerical simulation of water entry of two-dimensional body. Journal of Huazhong University of Science and Technology (Natural Science Edition) 23. Wu GX, Xu GD, Duan WY (2010) A summary of water entry problem of a wedge based on the fully nonlinear velocity potential theory. J Hydrodyn 22(5):859–864 24. Xu GD, Duan WY, Wu GX (2008) Numerical simulation of oblique water entry of an asymmetrical wedge. Ocean Eng 35:1597–1603 25. Miloh T (1991) On the oblique water-entry problem of a rigid sphere. J Eng Math 25(1):77–92 26. Malenica S, Korobkin AA (2007) Some aspects of slamming calculations in seakeeping, Proceedings of 9th International Conference on Numerical Ship Hydrodynamics, Michigan, 5-8 August 27. Feizi Chekab MA, Ghadimi P, Farsi M (2015) Investigation of three-dimensionality effects on aspect ratio on water impact of 3D objects using smoothed particle hydrodynamics method. J Brazilian Soc Mech Sci Eng. Published. doi:10.1007/ s40430-015-0367-8 28. Farsi M, Ghadimi P (2014) Finding the best combination of numerical schemes for 2D SPH simulation of wedge water entry for a wide range of deadrise angles. Int J Naval Archit Ocean Eng. 6:638–651 29. Farsi M, Ghadimi P (2016) Effect of flat deck on catamaran water entry through smoothed particle hydrodynamics. Inst Mech Eng Part M: J Eng Maritime Environ 230(2):267–280 30. Farsi M, Ghadimi P (2015) Simulation of 2D symmetry and asymmetry wedge water entry by smoothed particle hydrodynamics method. J Brazil Soc Mech Sci Eng 37(3):821–835 31. Ghadimi P, Dashtimanesh A, Djeddi SR (2012) Study of water entry of circular cylinder by using analytical and numerical solutions. J Brazil Soc Mech Sci Eng 37(3):821–835 32. Ghadimi P, Feizi Chekab MA, Dashtimanesh A (2014) Numerical simulation of water entry of different arbitrary bow sections. J Naval Archit Marine Eng 11(2):117–129 33. Ghadimi P, Tavakoli S, Dashtimanesh A (2015) An analytical procedure for time domain simulation of roll motion of the warped planing hulls. Institution of Mechanical J Braz. Soc. Mech. Sci. Eng. (2017) 39:1989–1999 34. 35. 36. 37. 38. 39. 40. Engineering Part M: J Engineering for the Maritime Environment. doi:10.1177/1475090215613536 Ghadimi P, Tavakoli S, Dashtimanesh A, Zamanian R (2016) Steady performance prediction of heeled planing boat in calm water using asymmetric 2D + T model. Inst Mech Eng Part M J Eng Maritime Environ. doi:10.1177/1475090216638680 Tavakoli S, Ghadimi P, Dashtimanesh A, Sahoo P (2015) Determination of hydrodynamic coefficients related to roll motion of high-speed planing hulls. In: Proceedings of the 13th International Conference on Fast Sea Transportation, FAST 2015 (2015), Washington DC, USA Shademani R, Ghadimi P (2016) Estimation of water entry forces, spray parameters and secondary impact of fixed width wedges at extreme angles using finite element based finite volume and volume of fluid methods. Brodogradnja 67(2):101–124 Von Karman T (1929) The impact of seaplane floats during landing. NACA TN 321, Washington DC, USA Wagner H (1932) The phenomena of impact and planning on water. National Advisory Committee for Aeronautics, Translation, 1366, Washington, DC ZAMM. J Appl Math Mech 12(4):193–215 Zhao R, Faltinsen OM (1993) Water entry of two-dimensional bodies. J Fluid Mech 246:593–612 Zhao R, Faltinsen OM, Aarsnes J (1996) Water entry of arbitrary two dimensional sections with and without flow separation, 1999 41. 42. 43. 44. 45. 46. 47. 1st Symposium on Naval Hydrodynamics, Tronheim, Norway, National Academy Press, Washington, DC Zhao R, Faltinsen OM (1998) Water entry of arbitrary axisymmetric bodies with and without flow separation. In: 22nd Symposium on Naval Hydrodynamics, Washington, DC Faltinsen OM, Landrini M, Greco M (2004) Slamming in marine applications. J Eng Math 48(3–4):187–217 Wu GX (2006) Numerical simulation of water entry of twin wedges. J Fluids Struct 22:99–108 Yousefnezhad R, Zeraatgar H. A parametric study on water-entry of a twin wedge by boundary element method. J Mar Sci Technol (2024), 19:314–326 He W, Castiglione T, Kandasamy M, Stern F (2011) URANS Simulation of Catamaran Interference, 11th International Conference on Fast Sea Transportation, FAST 2011, Honolulu, Hawaii, USA Karimian SMH, Schneider GE (1994) Pressure-based computational method for compressible and incompressible flows. J Thermo Phys Heat Transfer 8(2):267–274 Tveitnes T, Fairlie-Clarke AC, Varyani K (2008) An experimental investigation into the constant velocity water entry of wedgeshaped sections. Ocean Eng 35(14–15):1463–1478 13