Hydroelastic analysis on water entry of a constant-velocity wedge with stiffened panels

Marine Structures 63 (2019) 215–238

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Hydroelastic analysis on water entry of a constant-velocity wedge with stiffened panels

T

Pengyao Yua, Hui Lib, Muk Chen Ongc,∗ a b c

College of Naval Architecture and Ocean Engineering, Dalian Maritime University, Dalian, 116026, China College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China Department of Mechanical and Structural Engineering and Materials Science, University of Stavanger, Stavanger, 4036, Norway

ARTICLE INFO

ABSTRACT

Keywords: Water impact Slamming Elastic wedge Hydroelasticity Mode superposition method

This paper presents a hydroelastic analysis on water entry of a constant-velocity wedge with stiffened panels. Incompressible flow and the potential flow theory are considered in the present study. Through revisiting the pressure distribution around the elastic wedge based on the Wagner theory, a semi-analytical hydrodynamic impact theory is expanded to perform the hydrodynamic analysis of elastic wedges. Mode superposition method is adopted to calculate the structural response. Modal displacements of different two-dimensional (2D) sections corresponding to the cross-sectional fluid domain are obtained from the modal analysis of a three-dimensional (3D) structure. Unlike most hydroelastic studies requiring the analytical mode shapes of structures, the coupling analysis of the 2D section between the discrete mode shapes of the finite element model and the impact hydrodynamic forces is realized; and this makes the present numerical method appropriate for complex 3D structures. By integrating the general force of different sections, the governing equation for the hydroelastic analysis of a complex 3D wedge is established. The numerical scheme of the present method is verified in 2D analysis by comparing with published literature results and the decoupled result of a commercial software. Numerical results show that the present method is more accurate than hydroelastic methods based on the Wagner model, and it can predict the oscillatory response after flow separation, which is usually infeasible in hydroelastic methods based on the Wagner model. By removing the coupled terms in the governing equations, the present method can be used for structural response analysis under the decoupled condition. Then, the numerical scheme is further validated by the comparison with the decoupled result of the commercial software. Through the comparison between coupled and decoupled results of a 3D wedge, it is found that the effect of fluid-structure interaction and the oscillatory response after flow separation are important for predicting the structural responses.

1. Introduction In rough seas, the impact between ship hulls and waves can induce severe local pressure loads, which may cause local or global structural damages [1]. In order to investigate the fundamental mechanisms during slamming, water entry engineering problems of rigid and elastic bodies have been studied extensively over years using analytical or semi-analytical methods [2–18], numerical methods [18–30] and experiments [31–43]. The theory of the rigid body impact is the foundation of the theory of the elastic body impact. Pioneering work on the rigid impact ∗

Corresponding author. E-mail address: [email protected] (M.C. Ong).

https://doi.org/10.1016/j.marstruc.2018.09.007 Received 1 February 2018; Received in revised form 26 July 2018; Accepted 22 September 2018 0951-8339/ © 2018 Elsevier Ltd. All rights reserved.

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had been done by Von Karman [14] and Wagner [15]. With the help of the matched asymptotic expansion method, [2]; Howison et al. [5] and Zhao and Faltinsen [18] corrected the singularity of the contact region edge in the original Wagner model. Logvinovich [44] added some extra terms to the distribution of the velocity potential which made the flow velocity at the edge of the contact region bounded. Korobkin [6] gave a rational derivation of several analytical models, including the Original Logvinovich Model (OLM), the Modified Logvinovich Model (MLM) and the General Wagner Model (GWM). Tassin et al. [12] assessed the accuracy of several analytical models for the prediction of hydrodynamic forces and pressure distributions acting on a body entering initially calm water. During the water entry of a finite wedge, the jet root may detach from the body surface. By introducing a fictitious section, Duan et al. [45] expanded the application of MLM and the vertical entry force of the wedge after the flow separation was calculated. Tassin et al. [13] investigated 2D water entry with separation through different analytical models. This included the separation model suggested by Logvinovich (1972) and the concept of Fictitious Body Continuation (FBC) combined with MLM. By comparing the results of MLM with the numerical result of the commercial explicit FE-code LS-DYNA, Yu et al. [46] proposed a semianalytical model considering the flow separation and negative pressure correction. Hydroelastic method is often adopted to better consider the structural elasticity on water entry problems. Based on analytical hydrodynamic impact theory, many hydroelastic methods have been proposed. Kvalsvold and Faltinsen [36] and Faltinsen et al. [4] studied wetdeck slamming using a hydroelastic beam model. By the direct coupling of the finite element method (FEM) with the Wagner theory, Korobkin et al. [47] illustrated a hydroelastic method to evaluate the structural response of elastic wedge in two dimension [48]. presented a numerical method to study water entry of a 2D elastic wedge, which considered the strong coupling conditions between the hydrodynamic loads and the structural response. Shams and Porfiri [49] proposed a numerical model for the analysis of two-dimensional hydroelastic slamming of flexible wedges, in which the wedge kinematics was modeled by Euler–Bernoulli beam theory and the flow domain was studied using the Wanger theory. This model was validated through comparison with available semi-analytical, computational, and experimental results. Datta and Siddiqui [50] presented a theoretical hydroelastic analysis of an axially loaded uniform Timoshenko beam undergoing hydrodynamic impact-induced bottom slamming. Several fully nonlinear hydrodynamic impact theories are expanded to fully nonlinear methods of elastic body impact. Lu et al. [20] calculated the elastic wedge response by coupling the boundary element method with the finite element method. By using the commercial explicit FE-code LS-DYNA which contains the Arbitrary-Lagrangian Eulerian (ALE) algorithm, Stenius et al. [24, 25] studied the hydroelastic interaction between a 2D elastic wedge and water free surface. A similar analysis has been presented in Das and Batra [51] for sandwich composite hulls. Piro and Maki [23] adopted the open-source computational-fluid dynamics (CFD) library OpenFOAM to solve the fluid domain, and performed the hydroelastic analysis of bodies that enter and exit water. The 3D stiffened plate with several longitudinal stiffeners and transverse frames is a typical structure in ship and ocean engineering. Although water entry of 2D rigid or elastic wedges has been widely studied, dynamic response of the 3D wedge-shaped stiffened plate under the hydrodynamic impact is only marginally studied. Faltinsen [3] performed a hydroelastic analysis of a 3D wedge during water entry. Therein, the cross-fluid domain was solved by the 2D Wagner theory and the analytical modes of the stiffened plate were solved based on the orthogonal plate theory. The applicability of the orthogonal plate theory was verified by comparing with the finite element method in the quasi-steady analysis. Luo et al. [37] performed an decoupled method to study the impact response of a 3D wedge which consists of two stiffened plates. In this method, the hydrodynamic impacting load was solved by the matched asymptotic theory and the structural response was predicted by the FEM code. Luo et al. [52] also performed a coupled study on the dynamic response of a freefall 3D wedge with the commercial code LS-DYNA. The numerical results were in a good agreement with experimental results. Due to the shallow penetration of the wedge, the vibration response after flow separation was not presented. To our knowledge, the limitations of the previous work about the water entry of the 3D wedge include: (a) the vibration response of the stiffened plate after flow separation was usually not considered; (b) the hydroelastic method based on the analytical hydrodynamic was only suitable for simple structures with analytical modes, such as the orthotropic stiffened plate. Compared with fully nonlinear hydrodynamic theories, analytical or semi-analytical models are more efficient in the solution of the fluid domain. Inspired by the coupled idea between the cross-sectional fluid domain and the 3D wedge in Faltinsen [3]; a hydroelastic method for the constant velocity water entry of a 3D wedge is proposed. Compared with the previous studies, the present method contains the following improvements. · A semi-analytical hydrodynamic model is expanded to solve the cross-fluid domain around the elastic wedge. This model is more accurate than the Wanger model for the prediction of impact loads. · The present method can predict the vibration response of the stiffened plate after flow separation, which is usually infeasible for hydroelastic methods based on the analytical hydrodynamic model. · Structure vibration modes in the present method are solved using FEM, which is widely used in the ship engineering. This paper is organized as follows. In section 2, the pressure distribution around a 2D elastic wedge is presented. In Section 3, the expression of general forces in the 2D section by combining hydrodynamic coefficients with principal coordinates is presented. In section 4, a hydroelastic equation of 3D elastic wedges during water entry is established. In Section 5, numerical results of 3D and 2D wedges during water entry are presented and discussed. In Section 6, the main conclusions of the present study are summarized. Furthermore, in Appendix A, the derivation process of the semi-analytical model which considers the correction of the negation pressure and flow separation on the basis of the Modified Logvinovich Model is shown.

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Fig. 1. Impact model in the cross-sectional fluid domain.

2. Pressure distribution around a 2D elastic wedge In this section, the cross-sectional fluid domain considering the effect of structural vibration is analyzed. First, the pressure distribution based on the Wagner model is revisited [3], and the physical meaning of each pressure component is analyzed. Then, by using of the pressure distribution of rigid wedge based on the semi-analytical model [46], the expression of pressure distribution considering the influence of structural vibration velocity is given. 2.1. Pressure distribution based on the Wagner model Fig. 1 shows an impact model of a wedge in the cross-sectional fluid domain. As the wedge discussed in the present study is symmetry, only half of the wedge is used for analysis. As shown in Fig. 1, the global coordinate system o xyz and the local are adopted in this study. The global coordinate system is stationary and its origin is located at the coordinate system o intersection between the symmetry line and the mean free surface. The local coordinate system is fixed to the wedge and its origin is located at the tip of the wedge. The directions of ox , oz , o and o are shown in Fig. 1. The directions of oz , o are determined using the right-hand rule. V is the velocity of the impacting body. The initial state of the water is still. t is the time during the water entry and t=0 means the lowest point of the wedge has just reached the water surface. c (t ) is the half-width of the wetted body surface. s (t ) is the wetted length in the local coordinate system. is the deadrise angle of the wedge. is the location angle of the fictitious section, which is used in the semi-analytical model [46]. b is the half-width of the wedge section; L is the length of the wedge surface. Based on the Wagner model, the velocity potential on the wetted body surface can be approximately described as

(x , t ) =

V c 2 (t )

(1)

x2

where (x , t ) is the velocity potential. c (t ) is determined under the Wagner condition, which can be written as /2

f [c (t )sin ] d = 0

2

Vt

(2)

where f (x ) is the function to describe the section shape. In the case of the wedge with the deadrise angle , Equation (2) can be expressed in the form

c (t ) =

Vt 2 tan

(3)

As shown in Fig. 1, c (t ) = s (t ) cos

( , t) = s (t ) =

V cos

s 2 (t )

and x =

cos , so we can get Equations (4) and (5) in the local coordinate system. (4)

2

Vt 2 sin

(5) 217

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where s (t ) is the wetted length in the local coordinate system. V cos is the component of the impact velocity in the normal direction of the wedge surface. During the impact of an elastic wedge, the structural vibration would change the impact velocity between the fluid and the wedge. In the work of Faltinsen [3] and Datta and Siddiqui [50]; the mean deflection velocity in the normal direction of the wedge surface w¯ is introduced to correct the impact velocity, and the velocity potential and the wetted length can be expressed as follows.

( , t ) = ( V cos (V cos 2 sin

s (t ) =

+ w¯ (s (t )))

s 2 (t )

w¯ (s (t ))) t = cos

(6)

2

w¯ (s (t ))) t

(V cos

(7)

sin 2

In order to consider the change of the structural deflection velocity along the wedge, the mean deflection velocity of the wetted surface w¯ (s (t )) is replaced by the instantaneous deflection velocity of different positions w ( , t ) . Then, the velocity potential can be expressed as:

( , t ) = ( V cos

s 2 (t )

+ w ( , t ))

(8)

2

where ( V cos + w ( , t )) represents the instantaneous impact velocity of different positions considering the vibration of the structure. Based on the linear Bernoulli equation, the distribution of pressure acting on the elastic wedge entering the fluid at a constant velocity can be written in the following form:

P ( , t) =

t

=

w¨ ( , t )

s 2 (t )

2

[ V cos

+ w ( , t )]

s (t )

ds

(9)

2 dt

s 2 (t )

where w¨ is the mean deflection acceleration in the normal direction of the wedge surface. In the Equation (9), it can be seen that the pressure distribution of the Wagner model consists of two terms. The first term 2 w¨ ( , t ) s 2 (t ) is the pressure caused by the acceleration of the wedge and the second term

[ V cos

+ w ( , t )]

s (t )

ds

2 dt

s 2 (t )

is the pressure caused by the velocity of the wedge.

Equation (9) can also be expressed in the following.

P

, t = V cos

s (t ) s 2 (t )

s (t )

2

w¨

s (t )

In the Equation (10), the first term V cos second term

w¨

,t

s 2 (t )

2

+w

s 2 (t )

,t

s 2 (t )

2

s (t )

,t

s 2 (t )

2

2

+w

s (t )

,t

s 2 (t )

2

s (t )

(10)

s (t ) is the impact pressure of the rigid wedge in the Wagner model and the s (t ) represents the effect of structural elasticity on the impact pressure.

2.2. Pressure distribution based on the semi-analytical model According the semi-analytical model in the work of Yu et al. [46]; when considering the correction of the negation pressure and flow separation, the pressure distribution of a rigid wedge can be expressed in Equation (11). The wetted length in the local coordinate system can be expressed in Equations (12a) and (12b).

P ( , t ) = Vs cos

s (t ) s 2 (t )

( )2

1 2 V cos2 2

s 2 (t ) s 2 (t ) ( ) 2

sin2

(11)

ds c V = v before the flow separation dt sin ds = dt sin (1

cv V cos s ) + cos

tan

cos

(12a)

after the flow separation

(12b)

s

The derivation details of Equations (11), (12a) and (12b) are listed in Appendix A. is a parameter introduced to correct the unreasonable negative pressure. s is a parameter that related to the fictitious and it satisfies the equation c (t ) sin s = L cos . c v is the pile-up coefficient of the fluid surface. w ( , t ) and Referring to the effect of the structural vibration in the Wagner model, replace V cos in Equation (11) with V cos add the pressure caused by the acceleration in the Wagner model. Then, the pressure distribution around the 2D elastic wedge based on the semi-analytical model can be expressed as

P ( , t) =

w¨ ( , t )

s 2 (t )

2

s [ V cos

+ w ( , t )]

s (t ) s 2 (t )

(

)2

1 (V cos 2

w ( , t )) 2

s 2 (t ) ( )2

s 2 (t )

tan2 (13)

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The influence of structural deflection on the wetted length is the cumulative influence of the deflection at different positions of the wedge surface. Similar to Equation (7), the wetted length in the local coordinate system can be expressed in Equations (14a) and (14b).

2c (V cos w¯ (s (t ))) ds = v before the flow separation dt sin 2

(14a)

2c v (V cos w¯ (s (t ))) ds = dt sin 2 (1 cos s ) + 2 cos2 tan

(14b)

cos

after the flow separation s

Similar to the pressure distribution based on the Wagner distribution, Equation (13) can also be divided into two parts.

P ( , t ) = Pr ( , t ) + Pe ( , t ) s (t )

Pr ( , t ) = V cos

Pe ( , t ) =

s 2 (t )

w¨ ( , t )

)2

(

s 2 (t )

s (t )

2

s 2 (t ) 1 (V cos )2 2 2 s (t ) ( )2

w ( , t)

s (t ) s 2 (t )

(

)2

s (t )

tan2

1 (w ( , t ) 2 2

2V cos

w ( , t ))

s 2 (t ) ( )2

s 2 (t )

tan2 (15)

where Pr ( , t ) represents the impact pressure of the rigid wedge; Pe ( , t ) represents the effect of structural elasticity on the impact pressure. 3. General force of the 2D section The outer shell deformation of a 3D elastic wedge can be seen as the deformation of a series of 2D wedge sections which are parallel to each other. Using the modal analysis result of the 3D wedge obtained from the finite element method, the modal displacement of the 2D sections can also be achieved. As the impact pressure is acting on the wedge in the normal direction, the displacement mode of the 2D sections in the normal direction is given. ik ( ) is the displacement mode of a 2D section in the normal direction, where i represents the i-th order mode and k represents the k-th section. Combining the displacement mode of the 2D section and the pressure distribution based on the semi-analytical model, the general force of the 2D section can be expressed as s k (t )

fi (s k (t )) =

P ( , t)

k i (

)d

(16)

0

where s k (t ) is the wetted length of the k-th section at the time t in the local coordinate system. The general force of the 2D section can also be divided into two parts.

fi (s k (t )) = fexc, i (s k (t )) + fela, i (s k (t )) fexc, i (s k (t )) = fela, i (s k (t )) =

s k (t )

Pr ( , t )

k i (

)d

Pe ( , t )

k i (

)d

0 s k (t ) 0

(17)

where fexc, i (s k (t )) is the general exciting force and fela, i (s k (t )) is the general hydrodynamic force caused by the vibration of the elastic wedge. s k (t )

fela, i (s k (t )) =

w¨ ( , t )

(s k (t ))2

2

w ( , t)

0

s k (t ) (s k (t )) 2

( )2

s k (t )

1 (w ( , t ) 2 2

2V cos

w ( , t ))

s 2 (t ) s2 (t ) ( ) 2

tan2

k i (

)d

(18) In Equation (18), it can be seen that fela, i is related to the mean deflection velocity w ( , t ) and acceleration w¨ ( , t ) in the normal direction of the wedge surface. If pj (t ) is the j-th order principal coordinate of the 3D elastic wedge, the mean displacement, velocity and acceleration can be achieved.

(s k (t ))

m

w ( , t) = j=1

k j (

) pj (t )

k j (

) pj (t )

(19a)

m

w ( , t) = j=1

(19b) 219

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P. Yu et al. m k j (

w¨ ( , t ) = j=1

m

w¯ (s (t )) =

) p¨j (t )

(19c)

¯ k (s k (t )) pj (t ) j

j =1

(19d)

where m is the number of modes which are used in the calculation; pj (t ) is the first order derivative of pj (t ) ; p¨j (t ) is the second order

derivative of pj (t ) ;

k k j (s (t ))

is the mean deflection of the wetted area.

k s k (t ) j ( ) d 0 s k (t )

¯ k (s k (t )) = j

(20)

Substituting Equations (19b) and (19c) into Equation (18), fela, i m

m

Bijk (s k (t )) pj (t )

j=1

can be expressed as follows.

m

Aijk (s k (t )) p¨j (t )

fela, i (s k (t )) =

(s k (t ))

j=1

j =1

Dijk (s (t )) pj2 (t )

(21a)

s k (t )

Aijk (s k (t )) =

k j (

(s k (t ))2

)

k i (

2

)d

(21b)

0

s k (t )

Bijk (s k (t )) =

k j (

s k (t )

)

(s k (t ))2

0

s k (t )

Dijk (s (t )) = 0

1 2

2 k j (

)2

(

s k (t )

(s k (t ))2

)

(s k (t )) 2

tan2

( )2

k j (

V cos

k i (

)

(s k (t )) 2 (s k (t ))2 ( )2

tan2

where i, j are order numbers of modes; is the general added mass of the k-th section; added damp of the k-th section. By combining Equations (14) and (19d), we can obtain m j =1

2c v V (t )cos

)d

(21c)

)d

Aijk (s k (t ))

ds = dt

k i (

(21d)

Bijk (s k (t ))

and

Dijk (s (t ))

are the general

¯ k (s k (t )) pj (t ) j

before the flow separation

sin 2

m ¯ k (s k (t )) pj (t ) 2c v V (t )cos j=1 j ds = dt sin 2 (1 cos s ) + 2 cos2 tan cos

(22a)

after the flow separation

(22b)

s

4. Hydroelastic equation of the 3D wedge and the decoupled analysis The hydroelastic equation of a 3D wedge can be established based on the general force of 2D section. As the finite element method is used to carry out modal analysis, the analytical expressions for hydrodynamic coefficients are not appropriate, as shown in Equations (21a)–(21d). It is very time-consuming to solve the integral equations of the hydrodynamic coefficient at each time step. Thus, an interpolation calculation method of hydrodynamic coefficients is provided in this section. At last, an decoupled analysis based on the present numerical scheme is illustrated, through eliminating coupled terms of the hydroelastic method. 4.1. Establishment of the hydroelastic equation Integrating the general force of different 2D sections by the trapezoidal method, the general force of the 3D wedge can be expressed in the following. n 1

0.5 × [fi (s k (t ))+fi s k + 1 (t )

Fi (t ) =

dk

(23)

k=1

where Fi (t ) is the i-th order general force of the 3D wedge; n is the number of the 2D sections; sections; dk is the distance between s k (t ) and s k + 1 (t ) . The general force of the 3D wedge is expressed in the following format. m

Fi (t ) =

m

Aij (t ) p¨j (t ) j=1

and

s k + 1 (t )

are two adjacent

m

Bij (t ) pj (t ) j=1

s k (t )

j=1

Dij (t ) pj2 (t ) + Fexc, i (t )

220

(24)

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So the expressions of Aij (t ) , Bij (t ) , Dij (t ) and Fexc, i (t ) can be obtained as follows: n 1

0.5 × [Aijk (s k (t ))+Aijk s k + 1 (t )

Aij (t ) =

dk

(25a)

k=1 n 1

Bij (t ) =

0.5 × [Bijk (s k (t ))+Bijk s k + 1 (t )

dk

0.5 × [Dijk (s k (t ))+Dijk s k + 1 (t )

dk

(25b)

k=1

n 1

Dij (t ) =

(25c)

k=1 n 1

0.5 × [fexc, i (s k (t ))+fexc, i s k + 1 (t )

Fexc, i (t ) =

dk

(25d)

k=1

Based on the above analysis, the general distribution force matrix of the 3D wedge can be written as

{Z } =

[A]{p¨}

A11 (t )

where [A] =

p¨1 (t )

A1m (t )

Am1 (t )

(26)

[D]{p2 } + [Fexc]

[B]{p }

; [B ] =

Amm (t )

B11 (t )

B1m (t )

Bm1 (t )

Bmm (t )

; [D] =

D11 (t ) Dm1 (t )

D1m (t ) Dmm (t )

;

{p } =

p1 (t ) pm (t )

;

p2

1 (t )

{p2 } =

p 2 m (t )

;

Fexc,1 (t )

. Fexc, m (t ) p¨m (t ) When the structural damping is ignored, the vibration equation by the mode superposition method is shown as follows.

{p¨} =

; {Fexc } =

(27)

[a]{p¨} + [c ]{p} = {Z } where [a] is the general structural mass matrix; [c ] is the general structural stiffness matrix. Combining the equations (26) and (27), the hydroelastic equation of the 3D wedge is written as

(28)

{[a] + [A]}{p¨} + [B]{p } + [D]{p2 } + [c ]{p} = {Fexc} Here {q} = {p } is introduced to transform the second order differential equations into the first order differential equations.

{q} {p } = [Fexc] {q}

[D]{q2} [B ]{q} [a] + [A]

[c]{p}

(29)

For the 2D sections in the 3D wedge, Equation (22) needs to be solved. This is necessary to solve (2 m + n) first-order differential equations. Equations (22) and (29) are solved using a Runge-Kutta 4th-order scheme and the matrixes [Fexc ], [A], [B ], [D], ¯j (s k (t )) are updated in each step. The initial conditions are {q} = {q} = {p} = {0} , s k (t ) = s k (t ) = 0 . Then, the stress response of each position can be obtained by the mode superposition method. m

(t ) =

j

pj (t )

(30)

j=1

where (t ) is the stress response history;

j

is the stress corresponding to the jth mode;

j

pj (t ) is the stress component of each mode.

4.2. Numerical details to improve computing speed The updating of the matrixes [Fexc ], [A], [B ] and [D] means that Aijk (s k (t )) , Bijk (s k (t )) and fexc, i (s k (t )) of different sections should also be updated. To improve computational speed, several integral equations are calculated in advance for given snk (n = 1,2,3...) . The discretization of a 2D section and the meaning of snk are shown in Fig. 2.

T1,k ij (snk ) =

snk

k j (

)

k j (

)

k j (

)

(snk )

2

2

k i (

)d

(31a)

0

T2,k ij (snk ) =

snk 0

T3,k ij (snk ) =

snk 0

snk 2 (snk )

(snk )

2 (snk )

( )2

k i (

)d

2

( )2

tan2

(31b) k i (

)d

(31c) 221

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Fig. 2. Schematic diagram of the position Snk .

T4,k ij (snk ) =

snk 0

T5, i (snk ) =

k j (

snk 0

T6, i (snk ) =

2

1 2

snk 2 (snk )

k sn

0

)

(snk )

2 (snk )

( )2

(snk )

2

2 (snk )

k i (

k i (

)d

(31d)

)d

(31e)

2

( )2

tan2

( )2

k i (

tan2

)d

(31f)

When is updated in each step, T1,ijk , T2,ijk , T3,ijk , T4,ijk , T5, i and T6, i can be determined by performing been calculated in advance. Then the instantaneous Aijk (s k (t )) , Bijk (s k (t )), Dijk (s k (t )) and fexc, i (s k (t ))

s k (t )

Aijk (s k (t )) = T1,k ij (s k (t )) Bijk (s k (t )) = T2,k ij (s k (t ))

which have

can be expressed in the following. (32a)

V cos

T3,k ij (s k (t ))

(32b)

Dijk (s k (t )) = T4,k ij (s k (t ))

fexc, i (s k (t )) = V cos

the interpolation of

snk

(32c)

T5, i (s k (t )) s k (t )

1 (V cos ) 2 T6, i (s k (t )) 2

(32d)

It can be seen that and fexc, i can be achieved without the integration in each step, which would improve the computational speed of the hydroelastic equation.

Aijk (s k (t )) ,

Bijk (s k (t )),

Dijk (s k (t ))

(s k (t ))

4.3. Decoupled analysis Through the above hydroelastic analysis, it can be seen that the coupling effect between the fluid and the structure is mainly reflected in two aspects, i.e, (a) the influence of structural deformation on the flow field; (b) the influence of added mass and damp of the flow field on the structural response. Then, the decoupled analysis can be realized in the present numerical scheme by eliminating the coupling effect of these two aspects. The specific operation is setting w¯ (s (t )) in Equations (14a) and (14b) to zero and [A], [B ], [D] in Equation (28) to zero. 5. Results and discussion As there are very few studies on the water entry of a complex 3D wedge, verification data for the impact of a 3D elastic wedge with a constant water-entry velocity has not yet been found. The present method is also applicable to the hydroelastic of a 2D wedge, corresponding to the case that only one 2D section is used in the integral equation of generalized force. In order to validate the numerical scheme of the present method, a hydroelastic analysis of a 2D wedge at a constant water entry velocity is discussed firstly, including a convergence study and comparisons with published literature. Subsequently, the structural response of a 3D complex wedge subjected to water impact is studied. 222

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Fig. 3. The 2D elastic wedge model.

In the subsequent analysis of 2D and 3D wedges, the following parameters remain the same, i.e., the deadrise angle of wedges is 100, the impact velocity is 4 m/s and the location angle α of the fictitious section is set as 40° for 10° deadrise angle, see Yu et al. [46]. 5.1. Results of a 2D wedge This section investigates the hydrodynamic impact of a 2D wedge with a constant water-entry velocity. Firstly, parameters of a 2D elastic wedge model are presented. Secondly, a convergence study of the present method on the mesh density and the number of mode shapes is carried out. Lastly, the coupled and decoupled methods based on the present numerical scheme are illustrated, and they are validated by comparisons with published literature and the result obtained from the commercial software, respectively. 5.1.1. Description of the 2D model A 2D symmetrical elastic wedge is shown in Fig. 3. The length L is 0.5 m. The density of the water is 1000 kg/m3. The material of the structure is linear elastic steel. Its modulus of elasticity is 210 GPa; its mass density is 7850 kg/m3; its Poisson ratio is 0.3. The plate has a thickness h = 18 mm. The boundary conditions of both ends of length L are simply supported. As the wedge is symmetry, only half of the wedge is analyzed. The plate of the 2D wedge is modeled by four-node shell elements. The length of its short side is 0.04 m which contains only one element, as shown in Fig. 4. This 2D model has been used in Piro and Maki [23] and Shams and Porfiri [49] to verify the 2D hydroelastic method during the water entry. Through the analysis of “hydroelasticity factor’’, Piro and Maki [23] pointed out that this 2D model can result in sufficiently large structural deformation such that the hydroelasticity is important. 5.1.2. Convergence study Mesh density and number of mode shapes are discussed in a convergence study. L is the length of the long side of the plate. Mesh density refers to the element number contained in the length L. The symbol “mesh20” denotes the length L is divided into 20 equal elements, and the symbols “mesh40”, “mesh60” and “mesh200” have similar meanings. It is worth noted that the numerical discretization of integral Equation (30) is consistent with the discretization of the finite element model in the length L. The modal analysis of the wedge structure is solved by the commercial finite element software MSC.Nastran. The dry modal frequency, general mass and general stiffness of the plate are listed in Table 1 and the first two displacement modes of the plate are shown Fig. 5. Fig. 6(a) shows the time-history deflection result of the midpoint of the length L with different mesh densities for the entire simulation and the first 10 modes are used in this analysis. It can be seen that the deflection histories of different mesh densities are very close. With the increase of the mesh density, the maximum deflection gradually increases and tends to converge. When the mesh density changes from “mesh60” to “mesh200”, the maximum deflection increases by only 0.039%, see Fig. 6(b). Hence “mesh60” is considered to have sufficient mesh resolution to describe the deflection of the wedge. Fig. 7 shows the deflection of the midpoint of the length L with different numbers of mode shapes and “mesh60” is used in this

Fig. 4. One side of the 2D wedge. 223

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Table 1 Dry modal frequency, general mass and general stiffness. Order

Natural frequency (Hz )

General mass (kg m2 )

General stiffness (N m *105)

1 2 3 4 5 6 7 8 9 10

168.58 671.01 1497.55 2632.74 4056.43 5745.03 7672.80 9813.10 12139.48 14626.51

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

11.22 177.75 885.36 2736.37 6496.02 13029.99 23241.67 38016.54 58178.20 84458.06

Fig. 5. Displacement modes of the 2D wedge.

analysis. It can be seen that the deflection histories of mode numbers are very close to each other. The deviation between the maximum deflection of 5 modes and 10 modes is about 0.002%. It is clear that 5 modes are sufficiently accurate to predict the deflection of the wedge. 5.1.3. Validation of the present method In order to validate hydrodynamic loads of the present method, the pressure distribution of the rigid wedge before the flow 224

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Fig. 6. Time-history deflection of the midpoint of the length L with different mesh densities.

separation is analyzed, as shown in Fig. 8. The similarity solution was reported in the Fig. 6(c) in Zhao and Faltinsen [18]. The results of the Wagner model and the present method are achieved by Equations (8) and (10), respectively. It can be seen that the result of the present method agrees well with the similarity solution, while the pressure distribution of the Wagner model at the root of the jet is infinitely large. This is because the Wagner model only considers the linear term in the Bernoulli equation, which leads to unbounded velocities and unbounded pressure in a small vicinity of the contact area periphery [6]. The semi-analytical model in present method is based on the Modified Logvinovich model [6], which takes account of the nonlinear effects. So, the present method presents a more reasonable pressure distribution than the Wagner model. Fig. 9 shows the comparison of the impact forces in the stages before and after the flow separation, where FZ is the vertical impact force of the half wedge. It can be seen the results of the two method agree well. Fig. 10 shows the comparison of the time-history deflection results at the center of the length L between the present hydroelastic method and the published results reported by Piro and Maki [23] and Shams and Porfiri [49]. It can be seen that the result of Shams and Porfiri [49] presents a higher deflection than the other two results. This is due to that the hydrodynamic analysis of the hydroelasitic method in Shams and Porfiri [49] is based on the Wanger theory, which is known to overpredict the impact force for the wedge with a deadrise angle greater than zero. Since the hydrodynamic model in Shams and Porfiri [49] can only predict the impact force before the flow separation, the oscillatory response of the wedge after flow separation is not given. The maximum deflection of the present method is slightly smaller than that of Piro and Maki [23]. This is due to the hydrodynamic force based on the CFD method is a slightly higher than that of the semi-analytical model, as shown in Fig. 9. Generally speaking, the results of Piro and Maki 225

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Fig. 7. Time-history deflection results with different numbers of mode shapes.

[23] and the present hydroelastic method agree well not only before the flow separation, but also after the flow separation. As mentioned before, the present numerical scheme can be used for decoupled analysis by removing the coupling terms in the algorithm. The decoupled analysis can also be realized by using the commercial finite element software Nastran. In the decoupled analysis of Nastran, the dynamic impact pressure of each element is calculated by using the semi-analytical hydrodynamic model. For the case of the 2D wedge with “mesh60”, pressure histories at the center of shell elements are numbered from “P1” to “P60”, in which “P1” means the lowest one and “P60” denotes the highest one. The pressure histories of different elements are shown in Fig. 11. By applying pressure loads to the finite element model, the dynamic response is achieved by using the direct transient response solver in Nastran. The decoupled results of the present method and the commercial software Nastran are compared in Fig. 12. It can be seen that the two methods are in good agreement, and the decoupled computational ability of the present numerical scheme is verified. 5.2. Results of a 3D wedge In this section the hydroelastic impact of a 3D wedge with a constant water-entry velocity is investigated. Firstly, geometric parameters and a finite element model of the 3D elastic wedge model are introduced. Secondly, the modal convergence study is carried out. Lastly, the dynamic responses of the coupled and decoupled methods based on the present numerical method are compared, and the effect of hydroelasticity is discussed. 226

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Fig. 8. Comparison of the pressure distributions before the flow separation.

Fig. 9. Comparison of the impact forces between the CFD method and the present method.

Fig. 10. Comparison between the present hydroelastic method and the published results.

5.2.1. Description of the model The 2D cross section of the 3D wedge is shown in Fig. 13. The 3D wedge consists of two same stiffened plates. As shown in Fig. 14, the stiffened plate is made up of 7 longitudinal bulb stiffeners and 2 T-shape transverse frames. These stiffeners and frames are equally spaced in the transverse and longitudinal direction of the wedge surface, respectively. The bulb stiffener is converted into an equivalent L-shape stiffener in the finite element analysis. For the L-shape stiffener and the T-shape transverse frame, “tw” is thickness of the web plate, “hw” is height of the web plate, “tf” is thickness of the face plate, “bf” is breadth of the face plate. Relevant structural data in details for the stiffened panel are presented in Table 2. The locations for the discussion of the structural responses 227

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Fig. 11. Pressure histories of different elements.

Fig. 12. Comparison of the decoupled results between the present method and Nastran.

Fig. 13. The 2D section of the 3D wedge.

are presented in Fig. 10. S10-S13 represent the stress responses of the face plate of the longitudinal stiffeners. S4-S6 represent the stress responses of the face plate of the transverse frames. The material of the structure is linear elastic steel. The modulus of elasticity is 210 GPa; the mass density is 7850 kg/m3; the Poisson ratio is 0.3. All components are modeled by four-node shell elements, including longitudinal stiffeners and transverse frames. The finite element mesh of the stiffened panel is determined by referring to the finite element model in Luo et al. [37]. The spacing of longitudinal stiffeners is divided into 10 parts; the spacing of transverse frames is divided into 20 parts; the web plate of the longitudinal stiffener contains 4 elements in its height; the face plate of the longitudinal stiffener contains 1 elements in its width; the web plate of the transverse frame contains 8 elements in its height; the face plate of the transverse frame contains 4 elements in its width. The local coordinate system o which is fixed to the stiffened panel is shown in Fig. 14. o can be determined using the right-hand rule. As the stiffened panel is symmetry about the plane o , only half of the stiffened panel is needed to be analyzed. = U = = 0 is applied to nodes in the symmetry plane o The symmetry boundary and the supported boundary 228

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Fig. 14. The stiffened plate of the 3D wedge. Table 2 Relevant structural data in details for the stiffened panel. Information

value

Unit

Thickness of the plate

12

mm

675 9 200 19.7 39.8 2400 10 400 16 200

mm mm mm mm mm mm mm mm mm mm

Distance between longitudinal stiffeners Dimensions of longitudinal stiffeners

Distance between transverse frames Dimensions of transverse frames

tw hw tf bf tw hw tf bf

Fig. 15. Finite element model and boundary conditions of the stiffened panel. 229

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Table 3 Dry modal frequency, general mass and general stiffness. Order

Natural frequency (Hz )

General mass (kg m2 )

General stiffness (N m *105)

1 2 3 4 5 6 7 8 9 10 11 16 17

52.94 61.18 63.89 65.69 66.32 68.50 69.15 73.16 73.58 77.63 79.56 90.08 91.23

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.11 1.48 1.61 1.70 1.74 1.85 1.89 2.11 2.14 2.38 2.50 3.20 3.29

U = U = U = 0 is applied to nodes in the other three sides, as shown in Fig. 15. 5.2.2. Modal convergence The dry modal frequency, general mass and general stiffness of the stiffened plate are listed in Table 3. Typical displacement modes of the stiffened plate are shown in Fig. 16. In fact, much more modes are used during the calculation, which would be discussed in the following modal convergence. In order to discuss the convergence of the number of modes, different mode numbers are applied to the hydroelastic analysis of this 3D elastic wedge. The stress responses under the different number of modes are shown in Fig. 13. It can be seen that the stress responses of each position tends to converge as the number of the computational modes increases. The stress responses of the first 30 modes are almost the same as that of the first 40 modes, which means that the first 40 modes are enough to capture the structural response of the elastic wedge during water entry. As shown in Equation (30), the stress response of each position is superimposed by the stress responses corresponding to different modes. Important modes can be investigated by comparing the values of the stress components of different modes. For the positions “S5” and “S12”, the times when the absolute peak values appear are 0.10s and 0.15s, respectively. Fig. 18 is the comparison of stress components of different modes at the time of the absolute peak. It can be seen that the first mode makes the largest contribution to

Fig. 16. Typical displacement modes of the stiffened plate.

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Fig. 17. Stress responses of the different number of modes.

the stress response for both the positions “S5” and “S12”. For the position “S12”, the 11th, 16th and 17th modes also make an obvious contribution to the stress response. As shown in Fig. 16, the 1st mode is the overall displacement mode of the stiffened plate. The local deformation of the longitudinal stiffener becomes obvious in the11th, 16th and 17th modes. Therefore, for the position “S12”, the stress response of the first 10 modes is obviously different from that of the first 40 modes, as shown in Fig. 17(b). 5.2.3. Comparison between the coupled and decoupled methods The coupled and decoupled analysis are conducted based on the present model by selectively considering the coupled terms, as described in Section 4.3. Fig. 19 shows the comparison of the structural responses obtained by using the coupled and decoupled methods. Before the time 0.15s, the overall variation trends of the structural responses obtained by using the two methods are similar, even though there are differences in the high frequency oscillation in this stage. Due to the fluid-structure interaction, the absolute peak value of the structural response of the decoupled method is slightly smaller than that of the coupled method for some positions, such as “S11” and “S13”. After the time 0.15s, periodic oscillations become obvious in the structural responses of the two methods, and the oscillation frequency of the coupled method is obviously lower than that of the decoupled method. For positions “S10” and “S12”, the peak value in the stage after the flow separation obtained by using the coupled method is even higher than that in the stage before the flow

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Fig. 18. Stress components of different modes at the time of the absolute peak.

separation, which means the response after the flow separation is also very important to the safety of the structure. This characteristic can also be found in the water-impact hydroelastic analysis of 2D wedges, as shown in Fig. 2 of Khabakhpasheva and Korobkin [48] and in Fig. 10 of Piro and Maki [23]. The difference of structural responses between the two methods can be explained by the physical mechanism during the impact process. Since the decoupled method ignores the influence of added mass and damp of the fluid, the structural vibration characteristics considered in the two methods are different. The fluid begins to separate from the structure at the time 0.15s according to the Equation (12a) in the revised manuscript. At this time, the impact force reaches the maximum as shown in Fig. 9. During the period from 0 to 0.15 s, the impact force increase gradually, and the stiffened plate mainly shows the forced response under the hydrodynamic impact force. Therefore, the overall variation trends of the structural responses of the two methods are close before the time 0.15s. Due to the rapid decrease of the impact force after the flow separation, as shown in Fig. 9, the stiffened plate begins to vibrate after the time 0.15s. Then, the vibration characteristics of the structure itself play an important role on the structural response in this stage. Since the decoupled method ignores the influence of the fluid on the vibration characteristics of the structure, obvious differences can be found in the structural response of the two methods after the time 0.15s. Based on a flat-disc approximation and potential flow theory [37], the 2D added mass can be expressed as ρ(π/2) c2(t). Then, when the wedge is completely submerged, the added mass of the stiffened plate can be calculated, which is 79923.8 kg. The added mass is applied to the finite element model by changing the density of the outer plate. Vibration characteristics of the typical wet modes of the stiffened plate are listed in Table 4. Through the comparisons between Tables 3 and 4, it can be seen that the model frequency of the structure has been reduced due to the effect of the fluid. Fig. 20 shows the Fast Fourier Transform (FFT) on the structural responses of each position obtained by using the coupled and decoupled methods. It can be seen that the FFT results of the

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Fig. 19. Comparison between the coupled and decoupled methods.

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Table 4 Wet modal frequency, general mass and general stiffness. Order

Natural frequency (Hz )

General mass (kg m2 )

General stiffness (N m *103)

1 2 3 4 5 6 7 8 9 10 11 16 17

7.92 9.16 9.56 9.83 9.93 10.25 10.35 10.95 11.01 11.62 11.91 13.48 13.65

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2.48 3.31 3.61 3.82 3.89 4.15 4.23 4.73 4.79 5.33 5.60 7.18 7.36

decoupled method have a peak value near the first dry modal frequency, and the FFT results of the coupled method have a peak value near the first wet modal frequency, which indicates that the first mode plays the most important role in the structural responses. Generally speaking, the effect of fluid-structure interaction on the responses of the stiffened plate during the water entry is mainly reflected in two aspects: (a) the fluid-structure interaction can change the peak value of the structural response in the stage before the flow separation, but it has little effect on the overall trend of the structural response; (b) the fluid-structure interaction can reduce the oscillation frequency of the structure in the stage after the flow separation, and the vibration response of some positions after the flow separation is even higher than the stress before the flow separation. 6. Conclusions This paper presents a hydroelastic analysis on constant-velocity water entry of a 3D wedge. The fluid-structure interaction is analyzed between the cross-fluid domain and the 2D sections of the 3D wedge. The flow domain around the elastic wedge is studied by a semi-analytical hydrodynamic impact theory. The structural response is solved by the modal superposition method. Modal analysis of the structure is realized using the finite element method, which makes the present method appropriate for complex 3D structures. The instantaneous hydrodynamic forces are solved by an interpolation method, avoiding the integrals at each time step. The present numerical results of a 2D wedge show that the present method has good mesh density convergence and modal convergence. Compared with the hydroelasitic analysis based on the Wanger theory, the present method is closer to the hydroelastic analysis based on the CFD method, due to the present model can presents a more reasonable pressure distribution than the Wagner model; and the present model can reasonably give the oscillatory response after the flow separation. The comparison of decoupled results between the present method and the commercial software Nastran shows that the present numerical scheme is appropriate for decoupled analysis. Modal convergence study of the 3D wedge shows that the first 40 modes are required to capture the structural response during the water entry. Important modes can be investigated by comparing the values of the stress components of different mode, which shows that the first mode plays the most important role in the stress response. Comparison between coupled and decoupled methods is performed; and the results shows the overall variation trends of the structural responses of the two methods are very close before the flow separation. After the flow separation, results of the two methods show obvious discrepancy due to the influence of the added mass on the natural frequencies of the structure. For some positions of longitudinal stiffeners and the outer plate, the peak response before flow separation given by the coupled method is even higher than that before the flow separation, which means the response after the flow separation is also very important to the safety of the structure. Generally, the effect of fluid-structure interaction is very important during the water entry of a 3D wedge, especially to the stage after the flow separation. More experimental data is required before a conclusion regarding the validity of the present numerical results can be given. Meanwhile, the present method should be useful as an engineering tool for predicting the hydroelastic response of a 3D wedge with a constant water-entry velocity.

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Fig. 20. Results of the FFT analysis on the structural responses.

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Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 51709030) and the Doctoral Startup Foundation of Liaoning Province (20170520436). Appendix A. The semi-analytical model based on the Modified Logvinovich model The derivation process of the semi-analytical model which considers the correction of the negation pressure and flow separation is shown in the following. This model is based on the Modified Logvinovich model. The definition sketch of an impact model of a wedge is also illustrated in Fig. 1. Note that, the meanings of symbols in this section are the same as that in Section 2. For the Modified Logvinovich model (MLM), when the wedge section enters the fluid at a constant velocity, the distribution of pressure in the contact region is given as follows [6].

P (x , t ) =

1 2 2c (t ) V 2 V

c (t ) c 2 (t )

x2

cos2

c 2 (t ) c 2 (t ) x 2

sin2

(A.1)

c (t ) is determined with the help of the Wagner condition, which can be written as /2

f [c (t )sin ] d = 0

2

Vt

(A.2)

where f (x ) is the function to describe the section shape. /2 on the right side of the equation is the fluid surface rise coefficient and it is denoted as c v in the following expressions. By comparing with results of the CFD method, c v was suggested to be 1.54 [46]. In the case of the wedge, the equation (A.2) can be expressed in the form

c (t ) =

c v Vt tan

(A.3) c2

Influenced by cos2 2 2 in Equation (A.1), the pressure of the contact region edge tends to be negative infinity. Suppose c x P [a (t ), t ] = 0 , then the impact pressure P (x , t ) in the region a (t ) < x < a (t ) is positive. To solve the equation P [a (t ), t ] = 0 , it is convenient to define the ratio = a (t )/ c (t ) and = 1 X 2 . Then, we obtain

X=

sin(2 ) 2c v [1 +

1

cv

2

sin4

(A.4)

]

It can be found that with the increase of the deadrise angle , the value of decreases which means the region of negation pressure increases. During the slamming process of the rigid wedge section entering the fluid at a constant velocity, the pressure of the contact region should always be positive [18]. To correct the unreasonable negative pressure, the pressure distribution of the region a (t ) < x < a (t ) is mapped to the region c (t ) < x < c (t ) by the ratio . So the pressure distribution pressure in the contact region is given as

P (x , t ) =

1 2 2c V 2 V

c c2

(

cos2

x )2

c2

c2 ( x )2

sin2

(A.5)

It can be seen that the pressure is equal to zero when x is equal to c and the unreasonable negative pressure is avoided. As shown in Fig. 1, a fictitious section is introduced to deal with the flow separation. On the assumption that the fluid rises along the fictitious section after flow separation, the section shape function considering the fictitious section is given as

f (x ) =

x tan b tan

+ (x

b) tan

0 x L cos x > L cos

(A.6)

Get the time derivative of both sides of equation (A.2), then we obtain /2

fx [c (t )sin ] c (t ) sin d = c v V

(A.7)

0

where fx (x ) is given as

fx (x ) =

tan tan

0 x L cos x > L cos

Introduce a parameter expression

s

that satisfied the equation c (t ) sin

(A.8) s

= L cos , so Equation (A.7) can be transformed into the following

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s

c (t ) tan

sin d + tan

sin d

0

c (t ) =

tan (1

= cv V

(A.9)

s

cv V cos s ) + tan

cos

(A.10)

s

In general, the expression of pressure distribution before and after flow separation is the same, and the difference is the calculation of c (t ) . Considering the transformation c (t ) = s (t ) cos and x = cos , the pressure distribution of the semi-analytical model can be expressed in the following.

P ( , t ) = Vs cos ds dt

=

ds dt

=

cv V sin

sin (1

s (t ) s 2 (t )

(

)2

1 2 V cos2 2

s 2 (t ) ( )2

s 2 (t )

before the flow separation cv V cos s ) + cos

tan

cos s

sin2

(A.11) (A.12)

after the flow separation

(A.13)

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