Formulations of Hydrodynamic Force in the Transition Stage of the Water Entry of Linear Wedges with Constant and Varying Speeds

There is an increasing need to develop a two-dimensional (2D) water entry model including the slamming and transition stages for the 2.5-dimensional (2.5D) method being used on the take-off and water landing of seaplanes, and for the strip theory or 2D+t theory being used on the hull slamming. Motivated by that, this paper numerically studies the transition stage of the water entry of a linear wedge with constant and varying speeds, with assumptions that the fluid is incompressible, inviscid and with negligible effects of gravity and surface tension, and the flow is irrotational. For the constant speed impact, the similitude of the declining forces of different deadrise angles in the transition stage are found by scaling the difference between the maximum values in the slamming stage and the results of steady supercavitating flow. The formulation of the hydrodynamic force is conducted based on the similitude of the declining forces in the transition stage together with the linear increasing results in the slamming stage. For the varying speed impact, the hydrodynamic force caused by the acceleration effect in the transition stage is formulated by an added mass coefficient with an averaged increase of 27.13% compared with that of slamming stage. Finally, a general expression of the hydrodynamic forces in both the slamming and transition stages is thus proposed and has good predictions in the ranges of deadrise angles from 5 deg to 70 deg for both the constant and varying speed impacts.

slamming and transition stages is thus proposed and has good predictions in the ranges of deadrise angles from 5 • to 70 • for both the constant and varying speed impacts.
Keywords: water entry, wedge, hydrodynamic force, transition stage

Introduction
The take-off and water-landing of seaplanes had been studied since 1920s and the procedures to assess the structural crashworthiness of airframe is mainly based on a 2.5-dimensional (2.5D) method [1,2,3], which is similar to the formulations of slamming forces of the strip theory for the hull slamming. Every individual cross-section normal to the longitudinal direction of hull independently experiences a single process of water entry (see Fig. 1), where the 2D water entry model is formulated by the added mass method [4,1,2] or the Wagner theory [5,3]. The traditional procedures had been identified to be less accurate as the numerical methods have fast developed in recent decades, and make it possible to conduct numerical simulations of water-landing of seaplanes [6] and transport airplanes [7,8]. Although a whole-time history of water entry of aircraft can be reproduced, massive computational resources are required and several days of calculations are taken. This could not be treated as a practical method for the engineers to complete the initial designs of seaplanes or high-speed planning hulls [9,10]. The alternative method is to improve the 2D model of the 2.5D methods because the errors of early 2.5D methods are mainly resulted from the transverse pressure distribution, especially for that with chine immersion. For the cross-section C Fig. 1 without chine immersion, they used the Wagner theory; For the cross-sections A and B in Fig. 1 with chine immersion, they adopted the pressure distribution of a steady supercavitating flow [11,12] to formulate the transverse flow. However, the pressure distribution of Wagner theory is quite different from that of steady supercavitating flow. How the pressure distribution continuously changes from that of Wager theory to that of steady supercavitating flow is missing in the early 2.5D methods.
From the perspective of a 2D transverse flow in a constant speed, the crosssection without chine immersion is corresponding to a slamming stage in which the hydrodynamic force increases linearly with the increasing wetted length, and the cross-section with chine immersion is corresponding to a transition stage in which the body experiences a fast drop of hydrodynamic force when the spray root leaves the chine. The pressure distribution and the hydrodynamic force gradually decline and finally approach those of steady supercavitating flow, as the experimental results of Zhao et al. [13] indicated. The poor accuracy of original 2.5D methods is due to the weakness of their 2D model for the transverse flow and lack of involvement of the hydrodynamic forces acting on afterbody of hull (see Fig. 2). Since there is no effective theoretical method to formulate the transition stage, Sun and Faltinsen [14,15]  method called 2D+t theory for the high speed hulls. Their BEM [16,13] can address both the slamming and transition stages with increasing accuracy of predictions, but at the same time greatly increases the computational cost, which undermines the efficiency of the 2D+t theory. Therefore, there is an increasing need to improve the efficiency of the 2.5D methods and the 2D+t theory by proposing an analytical solution as accurate as the numerical methods to predict the hydrodynamic forces in both slamming and transition stages. For the slamming stage, many researchers had contributed to the formulations of hydrodynamic force by added mass methods [4,5,17], asymptotic theories [18,19,20] and approximate solution [21]. Among them, the approximate solution of Wen et al. [21] provided the most accurate model for the constant and varying speed cases by a similarity solution of Dobrovol1'skaya [22]. For the transition stage, due to the different hydrodynamic characteristics, the formulation of hydrodynamic force is more difficult and is yet to be fully addressed. In this paper, by following the formulation work of our previous research [21], the transition stage of the water entry of linear wedges with constant and varying speeds, as shown in Fig. 3, is formulated to complete the 2D water entry model.
There are three different formulations of the hydrodynamic forces in the transition stage in literature. The first one was proposed by Logvinovich [18] based on the boundary condition at the separation (point C in Fig. 3). The pressure at C is required to be same with the atmospheric pressure, which is denoted as zero pressure condition. Logvinovich derived an ordinary differential equation (ODE) to obtain a virtual wetted length but integrated the pressure to obtain the hydrodynamic force on the real wetted surface AC. Tassin et al. [23] improved the Logvinovich's model by introducing the correction of 1+tan 2 β (β being the deadrise angle) to the pressure expression on the wall surface and re-derived the ODE of virtual wetted length. The predictions are in better agreement with BEM results of Iafrati and Battistin [24] than the original one. The second way is called fictitious body continuation (FBC) and was also developed by Tassin et al. [23] based on a modified Logvinovich's model (MLM) of Korobkin [20] who addressed the formulation of the slamming stage. The FBC is a virtual wall surface extended from the separation C with an angle of β L with respect to the horizonal line (see Fig. 3). The wetted length is solved from the combination of the real wall surface AC and FBC, while the hydrodynamic force is only integrated on AC as it does in the Logvinovich's model of the zero pressure condition. Tassin et al. [23] had to compare with the numerical results to identify the parameter β L . Although the agreement with the numerical results is better than the Logvinovich's model with correction 1+tan 2 β, there is still some discrepancy and the FBC needs more improvement. Wen et al. [25] proposed the curved FBC to improve the accuracy of linear FBC of Tassin et al. [23] based on a modified Wagner's model (MWM). They introduced another parameter k and provided explicit equations to determine β L and k. Their predictions were in better agreement with the numerical results than the linear FBC of Tassin et al. [23]. Since these methods are all based on the Wagner theory [5], their predictions will become less accurate when the deadrise angle is larger than 30 • . The last and the most sophisticated formulation was conducted by Semenov and Wu [26] by extending their integral hodograph method (IHM) [27,28] from the slamming stage to transition stage. Their results show a larger declining force than the BEM simulations of Iafrati and Battistin [24] [16] and their previous method [29] did, they combined the use of a BEM solver in the bulk of fluid and a simplified finite element method (FEM) in the thin jet developing along the body contour. The hybrid method was denoted as a hybrid BEM-FEM (HBF) approach and can provide a detailed description of the flow and free surface dynamics, together with an improved prediction of the separation, while keeping the computational effort still reasonable. The HBF method was recently extended by Del Buono et al. [30] to deal with the water entry with varying speed and the water exit problems. Considering the good application of the BCs switching, Wang and Faltinsen [31] also develop their BEM of jet cutting replacing the lowest-order expansion of the Kutta condition with the BCs switching for the water entry problems. In contrast to the BCs switching and FEM solver, Bao et al. [32] used the least orders of equations to update the normal velocity of the free surface AC and a shallow water assumption for the jet region in the transition stage. The abovementioned BEM methods have different strategies for the jet region and new free surface AC, but are basically consistent with each other. They can reduce the computational cost and provide the velocity potential distribution on the wedge surface, which is important for the varying speed impact. Different from the complexity of the free surface modelling of BEM, the CFD methods are more flexible to deal with the modelling of free surface. The most widely used method for the free surface problems, volume of fraction (VOF), was first proposed by Hirt and Nichols [33] based on the framework of finite volume method (FVM). A modified high-resolution interface capturing scheme (Modified HRIC) was developed by Muzaferija et al. [34] to solve volume fraction equations and has good applications to the water entry problems [8,35,36]. Compared with the BEM approaches, the CFD methods provide more information about the distributions of pressure and velocities in the whole region which intuitively picture the rapid changes of flow field in the transition stage of the constant speed impact. The FVM with VOF has been the most successful method to deal with the water entry problems and will be adopted in this paper to produce the numerical results of constant speed impact. In order to study the distribution of velocity potential on the wall surface for the varying speed impact, the HBF of Iafrati and Battistin [24] will also be used to provide the required data.
In this paper, the slamming and transition stages of water entry of linear wedges with constant and varying speeds are numerically studied by the FVM with VOF [36] and the HBF of Iafrati and Battistin [24]. To propose a semianalytical solution of a combination of numerical results and theoretical results to address the high speed impact problems, the fluid is considered to be incom- The general formulations of hydrodynamic force of both the slamming and transition stages will be provided and can address the constant speed impact and the acceleration effect.

Computational Approaches
A CFD method of FVM with VOF technique is adopted to provide the detailed results of flow field of water entry in a constant speed and the HBF [24,37] is used to calculate the velocity potential (the CFD method can't provide the velocity potential) and the pressure distributions on the wedge surface during the water entry in a varying speed.

Flow solver
The unsteady incompressible Euler equations ignoring the surface tension force are solved using ANSYS FLUENT as follows where V is the velocity of fluid, ρ is the density, p is the pressure and g = (0, −g) representing the gravity of fluid (the gravity of fluid can be neglected for the high speed impact). The semi-implicit method for pressure linked equation consistent algorithm (SIMPLEC) is used to deal with the pressure-velocity coupling. The unsteady terms are discretized by first order implicit scheme, the convention terms are discretized by second order upwind scheme, and the pressure term is discretized by body force weighted scheme.

VOF method
The VOF method was firstly proposed by Hirt and Nichols [33], which can capture the free interfaces between two or more immiscible fluids by introducing a variable, called volume fraction, for each phase. If the volume fraction of the q th fluid in a certain cell is denoted as c q , c q = 0 represents the cell is empty of the q th fluid; c q = 1 represents the cell is full of the q th fluid; and 0 < c q < 1 represents the cell contains the interface between the q th fluid and other fluids.
The sum of the volume fractions of all phases must be 1 in each cell. The volume fraction equation of the q th fluid is written as follows: where V q is the velocity of q fluid. The first term in the left hand is discretized by one order implicit scheme, and the second term is discretized by modified high resolution interface capturing (Modified HRIC) scheme [34].

GMM method and VOF boundary conditions
In this paper, the global moving mesh method (GMM) [38] is used to deal with the motion of wedge. The whole computational domain (including the cells and boundaries) moves together with the wedge like a rigid body. The volume fraction boundary conditions can ensure that the free water surface keeps a given level when the computational domain moves. This condition is set according to the cell coordinates of the boundaries in the earth fixed coordinate system, i.e., the volume fraction of water c q = 0.5 for the cell located on the interface between air and water; c q = 0 for the cells located above the interface; c q = 1 for the cells located below the interface.

HBF approach
In this section, the fully nonlinear potential flow model based the hybrid BEM-FEM approach is presented. The method is mainly based on the studies of Refs. [24,37], where full details are provided.

Governing Equations
The water entry problem of rigid bodies is faced under the hypotheses of an inviscid and incompressible fluid. The flow is assumed irrotational and the problem is formulated in terms of the velocity potential φ. Surface tension effects are neglected. The flow is therefore governed by the following initial-boundary value problem: where n is the unit vector of normal to the wall surface, n y is the projection of n in the vertical direction y, W is the entering speed of the body, and x is the position of the particle lying on the free surface. At each time step, the solution of the boundary-value problem for the velocity potential is solved in the form of the boundary integral representation provided by the second Green's identity where G (P, Q) = 1 2π log (|P − Q|). According to Eqs. (4)-(5), the velocity potential is known on the free surface while its normal derivative is assigned on the body contour, which belongs to a boundary integral equation of mixed first and second kind. Once Eq. (6) is solved, the velocity potential and its normal derivative are known on the body contour and the free surface. The solution of the boundary integral equation Eq. (6), providing the normal derivative of the velocity potential on the free surface, allows the determination of the velocity field on the free surface, which is integrated in time through a two-step Runge-Kutta scheme to update the position of particle of the free surface.

Pressure distribution
The pressure distribution along the body contour is obtained through the Bernoulli's equation: and the total hydrodynamic load is obtained by integration of the pressure field along the wetted part of the body φ has to be provided before the pressure distribution on the wall surface is given. The calculation ofφ is similar to φ. On the free surface, theφ is known − gy according to Eq. (5). On the wall surface, the normal derivative ofφ is known and calculated as where a is the body acceleration, w s and w n are the tangential and normal

Jet model
In the simplified FEM solver used for the description of the thin jet, a part of the jet region is divided in control volumes in which the vertices corresponding to the panel centroids (P i−1 ,P i ,P i−1 ,P i ), as shown in Fig. 4. In each control volume, the velocity potential is written in the form of a harmonic polynomial expansion, up to second order. Details about the approach in the slamming stage can be found in Battistin and Iafrati [24] and Del Buono et al. [30]. For the transition stage, the harmonic polynomial expansion, φ J i is reduced to first orders and reads The corresponding normal derivative is where (x * i , y * j ) is the centroid of the fluid control volume P * i , n x,i and n y,i are the unit vector of the i th panel which are directed along the x-axis and y-axis, α i , χ i and µ i are new unknown variables and can be determined by enforcing the free surface condition and by enforcing the continuity of the φ n at adjacent elements

Grid independence
where U w is the entering speed of the body. Three grids of cell numbers of 50,000, 100,000 and 200,000 are adopted to simulate the impact flow with adaptive time steps. The courant number is set to be 0.95 during the adaptive time steps.
The C s of the whole time-history between the three grids match well. The C p distribution and free surface of h/h 0 = 0.8 (h 0 = l tan β) are also consistent between the three grids except for some discrepancies at the tip of jet. The grid independence of CFD method is successfully validated. Therefore, the CFD grid with cell number of 100,000 is adopted for further simulations by balancing both the accuracy of spatial resolution and the computational cost.  grid is adopted for most of the HBF calculations in present study.

Comparisons between different methods
For the slamming stage of constant speed impact in which the self-similar flow is satisfied, the present CFD and HBF methods are compared with the similarity solution of Dobrovol'skaya [22,36].  It can be concluded that the CFD method and HBF method are consistent with each other and can deal with the slamming and transition stages of the constant and varying speed impacts. In this paper, the CFD method is mainly used to produce the numerical results of constant speed impact, while the HBF method is to used produce those of varying speed impact. because the water under the wedge has to accumulate in the spray root and turns into a jet with high speed, as the wetted area of wedge keeps increasing.
In the transition stage, the wetted length of the wedge stops increasing and the wall surface of the spray root no longer exists, and thus the high pressure region disappears. Fig. 12 shows the pressure distributions on the wedge surface.
With the increasing penetration depth, the pressure on the whole wedge surface decreases and finally approaches the distribution of steady supercavitating flow.
It is difficult to figure out the way how the pressure drops from the original distribution of slamming stage. In this paper, we focus on the formulation of the force and expect to find a general expression of slamming and transition stages . are shown in Fig. 32 in Appendix 7.1, which is calculated by a potential theory [12]. In order to formulate the slamming coefficient C s0 in the transition stage, new variables are adopted:

Hydrodynamic force caused by the constant speed impact
where h 2 is the penetration depth corresponding to the maximum slamming coefficient C smax . The new parameter λ is determined by a gradient algorithm enforcing on the following function where σ i (λ) is the standard deviation of C * s0 between different deadrise angles for the i th node in the range of h * ∈ [0, 7] and thereby becomes a function of λ. An uniform distribution with N =7001 nodes in the range of h * ∈ [0, 7] is adopted, and the initial values are chosen as λ = 1.4 and 1.35. After 35 steps, the gradient algorithm is convergent with λ = 1.3075 (∆λ <1e-6). The C * s0 of different deadrise angles with λ = 1.3075 are also shown in Fig. 13 (b), and it can be seen that the C * s0 of different deadrise angles in the transition stage coincide well with each other.
Due to the fitting errors, C * s0 in Eq. (19) is larger than 1 when h * ∈ [0, 0.067]. In order to deal with the continuity between Eq. (19) and the linear increasing C * s0 of slamming stage, C * s0 of is approximately as 1 in h * ∈ [0, 0.067]. The linear increasing C * s0 appears the following form By taking the derivative of C * s0 with respect to h * , the slope of the linear expression of Eq. (20) where k 1 = h 2 /h 0 , k 2 = dCs0 d(h/h0) . C s max = k 1 k 2 according to the definations of k 1 and k 2 . Fig. 15 shows the k 1 and k 2 of various deadrise angles from 15 • to 45 • , which are calculated by the present CFD method. In the Wagner theory, k 1 = 2/π. k 1 from CFD results match 2/π when β ≤ 30 • . Thus, k 1 = 2/π can be adopted for small deadrise angle. k 2 appears to be equal to B 2 tan β of the similarity solution [22,39,21]. The CFD results of k 2 are in good agreement with the similarity solution. In this paper, the results of given by a least square fitting method (LSF) of a quadratic function k 1 = 0.2027β 2 − 0.1295β + 0.655 and a quadratic function multiplying cot β, e.g., k 2 = 1.585β 2 − 6.856β + 7.764 cot β from the CFD results, and the maximum fitting errors are 0.62% and 1.91% respectively. Therefore By combining the Eqs. (16) - (17) and (22), it is possible to find the C s0 value and then the hydrodynamic force acting on the wedge surface (half model) for the cases in a constant speed is given

Comparisons with CFD results
In Sec. 4.2, the formulation of slamming stage in the slamming and transition stages is set up by Eqs. (16)- (17) and (22)

Varying speed impact
In this section, the varying speed impact of a linear wedge is numerically studied by the HBF method described in Sec.   From the studies of slamming stage [36,21], the pressure changing caused by the acceleration of wedge can be quantified by the dimensionless velocity potential −2Φ. Fig. 20 shows the dimensionless velocity potential −2φ/ (U w c) Similarity solution Tassin et al. [23] Hybrid added mass coefficient The other distributions show large differences with −2Φ of the similarity solution and the ∆p/ (0.5ρa w c) distributions in Fig. 19 (b). It can be concluded that the pressure changing caused by the acceleration of wedge can not be quantified by the velocity potential in the transition stage as it does in the slamming stage. Although the ∆p/ (0.5ρa w c) distributions have a similar distribution and small increase as the penetration depth increases, there is still some challenging problems to formulate the pressure changing caused by the acceleration effect.
In this paper, we focus on the formulation of the hydrodynamic force caused by the acceleration effect and hope to find an effective method to formulate the hydrodynamic force caused by the acceleration effect.

Hydrodynamic force caused by the acceleration effect
Similar to the pressure distribution on the wedge surface, the force can also be divided into two part: (1) the force caused by the constant speed impact in the slamming stage [21]. In the slamming stage, the acceleration effect on the hydrodynamic force is given as ∆F = ρA 2 a w h 2 from the similarity solution [36], indicating an added mass coefficient C a0 = A 2 (k 1 tan β) 2 . The C a in the slamming stage can be given as C a0 = 1.0459, and in the transition stage, C a has an averaged result of 1.3297 in h/h 0 ∈ [0.639 1.0], which is ξ = 27.13% larger than the result of similarity solution C a0 . In this paper, a hybrid added mass coefficient is adopted to formulate the acceleration effect corresponding to Eq. (22) (1 + ξh * /0.067)C a0 , The results of h * ∈ [0, 0.067] are given by a linear distribution from C a0 to (1 + and (22), and the final expression of the force (half model) has the following form: where c = min {h cot β/k 1 , l}.
In the linear FBC of Tassin et al. [23] based on MLM, the acceleration effect was addressed by the following form of C a of in the slamming and transition As can be seen in Fig. 21, the present model has a different C a from that of Tassin et al. [23]. Eq. (23) Eq. (25) slamming transition The case in Fig. 10 is also used for a validation for the present model. Fig. 25 shows A comparison of the forces between an experimental test of Zhao et al. [13] and the present model is shown in Fig. 26. The case is a three-dimensional water Wagner's original model [5] π 2 π 3 4 π 2

Validations by numerical and experiment results
Wagner's new model [5] π 2 Faltinsen [17] π 2 Tassin et al. [23] π 2 Tab. 4: The correction factor γ of the wetted length, the dimensionless coefficient of constant speed impact Cconst and the added mass coefficient Ca of slamming stage for different theories. and varying speeds.

Comparisons with other theories
For the slamming stage, the traditional added mass methods have the forms of added mass and the hydrodynamic force (half model) where c = γh cot β for the traditional methods. Tab. 4 shows the correction factor γ of the wetted length, the dimensionless coefficient of constant speed impact C const and the added mass coefficient C a of slamming stage for different theories [4,5,17,19,23]. The Wagner's new model [5] was proposed to improve the predictions of forces of constant speed impact for different deadrise angles, but the acceleration term remained unchanged. γ Mei can be found in Refs. [19,21]. B 2Mei and A 2Mei are calculated by direct pressure and velocity potential integrations of Mei et al.'s model [19]. C Korobkin is from the theoretical results of Korobkin [20]. B 2 and A 2 of the present model were calculated from the similarity solution [22], and the results are shown in Ref. [21]. Wagnner [2] Faltinsen [35] Mei et al. [36] Tassin et al. [14] C a o (a) Present model Von Karman [1] Wagnner [2] Faltinsen [35] Mei et al. [36] Tassin et al. [16] Wagner's new model [2] C const o In early researches of Refs. [4,5,17,19,20] for the slamming stage, the hydrodynamic force of constant speed impact received the most attention and the researchers proposed their models by comparing the results of similarity solution [22,16] to correct the underestimated results of Von Karman [4] and overestimated results of Wagner's original model [5], as shown in Fig. 27 (a).
The models of Mei et al. [19], Tassin et al. [23] and Wagner's new model [5] have been approximately close to the results of similarity solution, but these models actually violate the framework of added mass methods. Their C const are no longer 2C a γ 2 , which are different from the models of Von Karman [4], Wagner [5] (original model) and Faltinsen [17]. Although these models [19,23,5] brought good predictions for the constant speed impact, they still can not properly model the acceleration effect, as shown in Fig. 27 Fig. 27 (b). In contrast to an increase of C a in the transition stage, the C a of Tassin et al.'s model declines and finally reaches a constant value lower than C a0 , which is inconsistent with the HBF results in Fig. 27 (b).
In general, we rewrite the formula of hydrodynamic force and extend it to the transition stage for the constant and varying speed impacts. Although the present model is more sophisticated, it works better than the traditional added mass methods and the asympotic theories and can work in a larger range of deadrise angle.

Conclusions
In this paper, the transition stage of the water entry of a linear wedge with constant and varying speeds is numerically studied by FVM with VOF and HBF.
The hydrodynamic force acting on the wedge is formulated by a combination of the numerical results and theoretical results of the steady supercavitating flow. The fluid is assumed to be incompressible, inviscid, weightless and with negligible surface tension, and the flow is irrotational for the high speed impact. The water entry in the infinite penetration depth becomes a steady supercavitating flow with an incoming currency with velocity v 0 . The theory can be extended from the water entry on a plate by the theory of Gurevich [12]. The free surface CD and C D of impact flow is shown in Fig. 28 after the solution is solved. The direction of incoming flow is upward and the stream-function on the streamline passing A is 0. The velocity potential at A is chosen as 0. A parameter plane τ mapping to the geometry in Fig. 28 is shown in Fig. 29 and thus the complex velocity potential of impact flow can be calculated as where φ 0 is the velocity potential at C and C . Another parameter plane ω = v 0 dz dw = v 0 e iθ /v (i being the imaginary unit) mapping to the velocity is shown in Fig. 30 and can be formulated by the Schwarz-Christoffel formula as The complex coordinate can be derived By considering the distance from A to C, the velocity potential at C and C can be determined 2φ where Therefore, the x on AC is given as x and the velocity on AC is The pressure coefficient C p has the form The C p distributions of different deadrise angles β are shown in Fig. 31. The y (τ ) = l tan β + l A cos β Im The force acting on the wedges is calculated by integrating the pressure with the C p expression Eq. (36). The slamming coefficient C s∞ of different deadrise angles is shown in Fig. 32. The result of β = 0 is equal to 0.88, which is consistent with that of water entry on a plate derived by Gurevich [12]. The C s∞ decreases with the increasing deadrise angle and reaches 0 at β = 90 • .
Korvin-Kroukovsky and Chabrow [11] in 1948 had already presented the calculation of pressure distribution on the wedge surface. The present theory is consistent with their model.