Hydrodynamic forces on a semi-displacement ship at high speed

Applied Ocean Research 34 (2012) 68–77

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Hydrodynamic forces on a semi-displacement ship at high speed Hui Sun ∗ , Odd M. Faltinsen Centre for Ships and Ocean Structures, Department of Marine Technology, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

a r t i c l e

i n f o

Article history: Received 3 May 2011 Received in revised form 30 August 2011 Accepted 3 October 2011 Available online 8 November 2011 Keywords: 2D + t theory Semi-displacement ships Seakeeping Round bilge

a b s t r a c t A numerical method based on 2D + t theory (two-dimensional plus time dependent theory) was developed to study the steady and unsteady hydrodynamic problems of a semi-displacement ship with round bilge at high forward speed. The ship was forced to oscillate in heave in the unsteady problem. No incident waves were present. In the 2D + t theory, the original three-dimensional (3D) problem was simplified as fully nonlinear two-dimensional (2D) time-dependent problems in cross planes. A boundary element method was used to solve the 2D problems. The non-viscous flow separation from the round bilge of the ship hull was simulated. The pressure on the hull surface was evaluated and the sectional hydrodynamic vertical forces were obtained. In the steady problems, the sectional vertical forces along the ship were calculated when the ship was displaced to different vertical positions. In the unsteady problems, the sectional added mass and damping coefficients along the ship length were evaluated. The present numerical results were compared with published experimental results and existing numerical results. Good agreement was achieved between the present calculations and the experiments, although some discrepancies near the bow and the stern were observed. The three-dimensional effects at those positions could be the reasons for the discrepancies. In the unsteady problem, the interaction between the local steady flow and unsteady flow were automatically included and the nonlinearities in both steady and unsteady flow were considered. The present method can be generalized to the seakeeping problem in which a semidisplacement ship encounters incident waves. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The conventional strip theory [1] is successfully used in the seakeeping analysis of normal displacement ships. However, its validity can be questioned when it is used for ships with higher maximum operating speeds, such as for semi-displacement vessels (0.4–0.5 < Fn < 1.0–1.2) and planing vessels (Fn > 1.0–1.2). (Those ranges for Froude number are given, e.g. in [2].) Nonlinearities become more significant for higher speed ships, however, the conventional strip theory is a linear theory. Yamamoto et al. [3] developed the so-called nonlinear strip theory by including some nonlinear effects, so that improved results could be obtained for higher speed vessels. However, the free-surface conditions were linearized in calculating the 2D added mass and damping coefficients and the forward speed effect was ignored in the free-surface conditions. Some nonlinear models were developed for planing hulls, e.g. by Zarnick [4], Lin et al. [5] and Lai [6], in which gravity was generally neglected in the local nonlinear water flow due to the very high speed. The gravity effects were simply taken into account by introducing hydrostatic restoring forces. In the present problem

∗ Corresponding author. Tel.: +47 90578796; fax: +47 73595528. E-mail addresses: [email protected], [email protected] (H. Sun), [email protected] (O.M. Faltinsen). 0141-1187/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2011.10.001

for semi-displacement ships, nonlinearities and gravity effects are both important. Strongly nonlinear free-surface flow will appear, such as the splash at the bow and the non-viscous flow separation from the round bilge of the hull. The gravity in the local flow around a semi-displacement ship is more important than that for planing vessels. Although CFD (computational fluid dynamics) solutions have been greatly improved in both speed and accuracy in recent years, it is still very challenging to perform complete nonlinear 3D computations for the present problem. Alternatively, 2D + t theory seems to be a good option. The divergent waves, which are properly accounted for by a 2D + t theory, are more important than the transverse wave system when Fn > 0.5. Further, the 2D + t theory often demands less computational time than fully 3D computations. A 2D + t method has been proven to be efficient in solving strongly nonlinear hydrodynamic problems for high-speed vessels. Both nonlinearities and gravity effects can be easily included in the free-surface conditions. Three-dimensional effects are partly accounted for. The influence from the upstream flow is considered. Fontaine and Tulin [7] gave a good review of the evolution and application of nonlinear 2D + t theory. To name a few, Ogilvie [8] introduced the slender body approximation for slender ships in steady motions. Faltinsen [9] extended the idea to studying the added resistance where unsteady motions in waves were considered. C¸alis¸al and Chan [10] incorporated nonlinear free-surface conditions in a 2D + t method to simulate the bow waves of a wedge

H. Sun, O.M. Faltinsen / Applied Ocean Research 34 (2012) 68–77

model. Using nonlinear 2D + t theory, Maruo and Song [11] and Wu et al. [12] studied the bow waves and the deck wetness of ships in head seas, where they were concerned about the free-surface flow running up in the bow area and the hydrodynamic forces on the ships were not considered. Deck wetness was detected, but the further development of the separated flow from the deck side was not pursued. Kihara et al. [13] presented computations of hydrodynamic forces when the ship oscillates in head seas, although their focus is on the added resistance in waves and they simplified the description of sprays. They also reviewed nicely the literature on the 2D + t method up to 2000s. In the present 2D + t method together with a nonlinear BEM solver, non-viscous flow separation from the knuckle of a ship section or along the round bilge can be simulated. Further, the pressure on the ship section is calculated and therefore the hydrodynamic forces on the ship are obtained. Zhao et al.’s [14] method applied to a planing vessel in steady motion was extended to the steady and unsteady problems of prismatic planing vessels with effects of gravity in Sun and Faltinsen [15–17]. In the present work, the 2D + t theory is generalized to semi-displacement ships. The difficulties for the present problem are mainly related to the more complicated hull shape. First, one must have smooth lines of the hull and the normal vectors must vary smoothly along the ship length. Secondly, the flow separation from the round bilge occurs at varying positions on the hull surface, instead of a fixed point as for a planing hull with hard chine. The separated flow from the round bilge may reattach to the hull surface and introduces more complexities. Therefore, more numerical efforts are needed. The 2D + t theory has been used to solve the seakeeping problem of prismatic planing vessels in [16] where we introduced the effects of incident waves. Similarly, the present method can be extended to study the wave induced motions of the semi-displacement ships. When we solve the steady problem, the present theory is equivalent to the 2.5D theory (another name for 2D + t theory) used by Faltinsen and Zhao [18]. However, for the unsteady problems, the present method is different from their 2.5D theory in two aspects. First, the steady and unsteady flows were separately solved in [18] while in the present method the steady and unsteady flows are simulated as a whole. Secondly, the unsteady flow is assumed linear and solved in the frequency domain in their method. However, in the present approach, time-domain simulations are performed with nonlinear free-surface conditions satisfied. The nonlinear effects in both the steady flow and the unsteady flow are taken into account. We can see that the present results manifest better agreement with experiments especially for lower frequencies, at which interactions between the steady flow and the unsteady flow is stronger. A boundary element method described in [15] was used to solve the 2D water flow in the 2D + t theory. The thin jet running along the body surface and the separated jet were cut away to avoid numerical difficulties. A flow separation model developed in Sun and Faltinsen [19] was applied to simulate the non-viscous flow separation from the round bilge. The calculated sectional forces in the steady motions and the sectional added mass and damping coefficients evaluated in the unsteady problems are compared with Keuning’s [20] experiments and the numerical results by Faltinsen and Zhao [18].

Z

U θ

69

z

Y

y

X0 X

x

Fig. 1. Coordinate systems.

fixed at the centre of gravity (COG) of the ship. The X-axis is pointing to the stern and the Y-axis is towards the starboard. The pitch angle  is defined positive when the bow is going up. The water is assumed inviscid and incomopressible. The water motion is assumed irrotational. A velocity potential ϕ(x, y, z, t) is used which satisfies the three-dimensional Laplace equation. Fully nonlinear free-surface conditions and exact body boundary conditions are satisfied in three dimensions as follows ∂ϕ = n · V = nx (−U + (z − zg )˙ 5 ) + nz (˙ 3 − (x − xg )˙ 5 ) ∂n (1)

on the hull surface 1 ∂ϕ + (ϕx2 + ϕy2 + ϕz2 ) + g = 0 on z = (x, y, t) 2 ∂t

(2)

Dx ∂ϕ = , Dt ∂x

(3)

Dy ∂ϕ = , Dt ∂y

Dz ∂ϕ = Dt ∂z

on z = (x, y, t)

where n = (nx , ny , nz ) is the normal vector expressed in the Earthfixed coordinate system, g is the acceleration of gravity, the free surface is described by z = (x, y, t) and the velocity vector on the hull surface can be written as V = i(−U + (z − zg )˙ 5 ) + k(˙ 3 − (x − xg )˙ 5 ) when U is constant and the unsteady heave 3 and pitch 5 are considered. The unit vectors in positive x- and z-directions are respectively denoted as i and k and the dot above 3 or 5 indicates the first time derivative. The heave and pitch motions are given as 3 = −3a sin(ωt) and 5 = −5a sin(ωt) (t ≥ 0) respectively, where ω is the frequency of oscillations and 3a and 5a are the corresponding amplitudes. Although the results to be presented only involve heave motion, this theory is developed for both heave and pitch. The heave is positive when it is upward and the pitch is positive when the bow is going up. The instantaneous trim angle is (t) =  + 5 (t) where  is the trim angle of the hull in the steady motion. A slenderness ratio is now introduced as ε = d/L, where d is the draft and L is the length of the hull. By using slender body assumption, one has ∂/∂x ∼ O(ε), ∂/∂y ∼ O(1), ∂/∂z ∼ O(1). Further, ∂/∂X ∼ O(ε), ∂/∂Y ∼ O(1), ∂/∂Z ∼ O(1) and it is assumed that the pitch angle is small. The Froude number is assumed as O(1). Neglecting the terms of the order of O(ε2 ) in the governing equation and the boundary conditions given in Eqs. (1)–(3), one can reduce the 3D problem to time-dependent 2D problems in Earth-fixed vertical planes. We call those planes as cross planes. The velocity potential

(y, z, t) in a cross plane satisfies the 2D Laplace equation in the yz coordinates and the following boundary conditions. ∂

= nX (−U cos  + Z ˙ 5 − ˙ 3 sin )+nZ (−U sin  − X ˙ 5 +˙ 3 cos ) ∂N on the hull surface

(4)

2. 2D + t theory

∂

1 + ( y2 + z2 ) + g = 0 on z = (x, y, t) 2 ∂t

(5)

A ship is advancing forward at speed U as shown in Fig. 1. Two coordinate systems are introduced, i.e., an Earth-fixed coordinate system xyz and a hull-fixed coordinate system XYZ. The xy plane is in the calm water surface and the z-axis is pointing upwards. The ship is moving along the negative x-direction. The origin of XYZ is

Dy ∂ϕ = , Dt ∂y

(6)

∂ϕ Dz = Dt ∂z

on z = (x, y, t)

៝ = (ny , nz ) is the projection of the normal vector to the where N vertical 2D plane and the normal vector components nx and nz in

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consists of the body surface, the free surface, the truncation boundary, the bottom surface and the symmetry line surface (Fig. 2). Variables change linearly on each element. Green’s second identity gives

Ship section

z

 

Free surface

2 ϕP =

ϕQ S

y Truncation boundary

∂ϕQ ∂G(P, Q ) − G(P, Q ) ∂nQ ∂nQ



dsQ

(10)

Eq. (1) have been replaced by the normal vector components nX and nZ expressed in the hull fixed coordinate system XYZ by using nx = nX cos  + nZ sin  and nz = nZ cos  − nX sin . When the ship is advancing steadily or with oscillations in heave and/or pitch, the flow field is symmetric about the x–z plane in 3D and symmetric about the z-axis in the cross plane. As shown in Fig. 2, only one half domain is considered. Thus, at y = 0, there is a symmetry line boundary. Zero normal velocity condition is fulfilled on this boundary. In addition, the rigid bottom boundary condition at deep water and the far-field conditions will be satisfied. For a steady problem, the body boundary condition (4) can be written as

where the Green function G(P, Q) = log r(P, Q) and r(P, Q) is the distance from a source point Q on the fluid boundary to the field point P in the fluid domain ˝. By letting the field point P approach the boundary S, an integral equation can be obtained. Now we assume that at a certain time instant the ϕ on the free surface and the ∂ϕ/∂n on the body surface are known. Solving the integral equation, one can obtain the velocity potential ϕ on the body surface and the normal velocity ∂ϕ/∂n on the free surface. The free-surface elevation and the velocity potential on the free surface for the next time instant can be updated by integrating the free-surface conditions with respect to time. In the BEM, the thin jet along the body surface is cut off. If the water flow runs beyond the deck, a flow separation model is applied to simulate the flow separation from the deck side. However, this type of flow separation does not occur in the present numerical study. When the water flow separation occurs along the round bilge, another flow separation model is applied in the BEM solver. The separated water represents a jet flow along the sides of the section that will turn over to hit the water surface underneath. To circumvent the simulation of the water–water impact and thus induced vortices, we cut away the separated thin jet before it falls onto the underlying water surface. Smoothing and regridding procedure described in [21] are applied.

∂

= −nX U cos  − nZ U sin  = −Unx ∂N

3.2. Non-viscous flow separation model

Ω Symmetry line Bottom

Fig. 2. The fluid domain in a 2D vertical plane.

(7)

Now we focus our attention at a cross plane in front of the advancing ship. As the ship is penetrating the plane, we can see in the plane a scenario of a 2D body moving vertically on the free surface. Because the ship hull is not prismatic, both the vertical speed and the shape of the body are varying in the plane. For the steady problem, the initial-boundary-value problem (IBVP) is the same in any cross planes, so we only need to solve an IBVP in one cross plane for a given condition. However, for the unsteady problem, the IBVPs are different at different cross planes, so we need to solve the 2D problems at a number of cross planes. From Bernoulli’s equation, one can express the pressure on the section surface as p − pa = −

1 ∂ϕ − (ϕy2 + ϕz2 ) − gz 2 ∂t

(8)

where pa is the atmospheric pressure and  is the water density. The ∂ϕ/∂t is calculated by ∂ϕ = ∂t

 dϕ  dt

P

− V៝ P · ∇ ϕ

(9)

where dϕ/dt is evaluated by tracking a point P on the section in the plane and V៝ P is the velocity of the point. A finite difference approximation is used to calculate dϕ/dt. 3. Numerical implementation 3.1. Two-dimensional boundary element method The 2D boundary value problem expressed by the Laplace equation and the boundary conditions in Eqs. (4)–(6) is solved by a boundary element method (BEM) described in [15,19,21]. Straight line elements are used to discretize the closed boundary S, which

A non-viscous flow separation models are applied in combination with the BEM solver to simulate the flow separation from the round bilge. This model is briefly described here, but the details that include the local analytical solution of a separated flow, the validation and verification of the flow separation model are given in [21]. The effectiveness of this model has been shown in the application in the symmetric and asymmetric water entry problems by Sun and Faltinsen [19,22]. The physical reason for the flow separation may be the following. As the jet is running fast around the bilge with large curvature, the speed of the running jet can be so high that the pressure on the jet–hull interface relative to the atmospheric pressure outside the jet becomes negative. When the area with pressure lower than atmospheric pressure is large enough, the air can easily ventilate the jet–hull interface and make the jet leave the body surface, which means the flow separation occurs. The numerical scheme in the flow separation model is formulated based on this scenario. Fig. 3 shows the numerical strategy. At each time step, the pressure on a section surface needs to be determined first. Then the size of the section surface where the pressure is lower than pa is checked. If the size is larger than a value specified in advance (more discussions about this parameter are given in [21]), we change this part of water–body interface into a part of the free surface. For example, the segment AB in Fig. 3 is the water–body interface to be separated. The boundary AB is first updated to its new position A1 B1 by using the free-surface conditions, in which the velocities of the points on AB follow from a local analytical solution around the separation position, rather than their velocities on the body surface. Then we connect A1 B1 with the updated free surface B1 C1 to form the new free surface A1 B1 C1 . The point A1 is projected to the body surface to give the new water–hull intersection, i.e., the separation position, which locates near the point where the calculated

H. Sun, O.M. Faltinsen / Applied Ocean Research 34 (2012) 68–77

Section surface

B1 B Free surface

A1 A C

C1

Initially calm free surface

In the 2D cross plane, the problem looks like the water entry of a body into initially calm water, while the body shape is varying as it enters the water. So we need to update both the section profile and the normal vectors on the section at each time step. At the initial time, numerical difficulty is encountered when the bow section in the control plane touches the calm free surface. To avoid this problem, we follow the treatment in [11] by using Mackie’s [23] solution of the water entry of a sharp wedge to give the initial free surface elevation and velocity potential on the free surface. The calculation starts from section 19.5, where a small distance of the section has submerged into the water. The initial conditions are given by

⎧

  1  ˛ 1 ⎨ z¯ (y) ¯ = ln 1 + 2 + 2y¯ tan−1 −2 for y¯ ≤ 5 ⎩ z¯ (y) ¯ =

Fig. 3. Non-viscous flow separation from the round bilge.

pressure on the section changes from larger than pa to lower than pa . Thus the flow separation is numerically simulated. In the unsteady problem studied in this paper, the oscillatory vertical velocity of the ship section adds to complexity in the flow separation especially for the higher frequency cases. The separated flow may reattach to the hull surface. This is simply treated by ignoring the thin air cavity enclosed and setting a new water–body intersection. The reattachment is detected when the section surface intersects the separated free surface with three intersection points. The higher intersection is taken as the new water–hull intersection. This simple treatment may cause jumps of the pressure in its time history. 3.3. Description of the ship hull In order to describe the hull, one needs to know the lines for the hull. In the present work, only the section lines are available for the studied ship hull. First, we read the section lines from the published figure in [20]. Then we find a number of waterlines and buttocks, respectively at different Z and Y positions, by interpolating values from these cross-section lines. Our numerical tests showed that the sectional force distribution along the ship length will display unwanted spatial oscillations if the waterlines and buttocks are not smooth and thus the calculated normal vectors are not smoothly varying along the ship length. Therefore, the section lines, waterlines and buttocks are smoothed by using either a spline approximation, or five-point-third-order smoothing formulae. The offsets at the central plane on each water line are extrapolated. Further, the section line at station 19.5 is obtained by using interpolations. Afterwards, each section line is discretized into N equal elements by N + 1 nodes. In the present calculations N = 80 is used if it is not otherwise explained. The nodes are numbered from the bottom to the deck as 1 to N + 1. The nodes of the same serial number on the 21 stations from station 0 to station 19.5 are connected to give N + 1 curves on the hull surface. Then we can calculate the three components nX , nY , nZ of the normal vector on the hull surface from these N + 1 curves and the 21 section lines. 3.4. Calculation procedure for the steady problem In the steady problem, we only need to solve one initial boundary value problem for a given combination of the forward speed, trim angle and sinkage. From the trim and sinkage, one can find the submergence of any ship section. The section line and normal vectors at any X-position can be interpolated from those values at station 0–19 and station 19.5 by using cubic spline approximations. Thus the body boundary condition is obtained from Eq. (7).

71

¯ = ϕ( ¯ y)



˛ 2



(11)

for y¯ > 5

1

ln 0

y¯

y¯

˛ 3 y¯ 2

y¯ + (¯z + ı)

2

y¯ + (¯z − ı)

2

dı

(12)

¯ = ϕ/(Vs), s is the submergence of the where y¯ = y/s, z¯ = z/s, ϕ wedge, V is the water entry speed of the wedge, ˛ = /(2 − ˇ) is half the apex angle of the thin wedge and ˇ is the deadrise angle. For the present hull shape, V is approximated by V = U tan 45◦ = U, the half angle is approximated as ˛ = 3◦ . It has been tested that the calculations are not sensitive to the approximations of V and ˛. By using the given V and ˛ in Eqs. (11) and (12), one can obtain an approximation of the initial free-surface profile and the velocity potential on it. Then we find the intersection between the free surface and the real ship section in this cross plane and ignore the part of free surface inside the body surface. The velocity potential on the intersection can be interpolated. 3.5. Calculation procedure for the unsteady problems In the computations for unsteady problems, the time-dependent 2D problem is solved in a series of cross planes intersecting the hull surface. The calculation procedure is almost the same as in the calculation of unsteady motions of a prismatic planing hull as in [17]. What is different for the present non-prismatic hull is that we need to find out both the ship section shape and the normal vectors on the section at each time step. It should also be noticed that the way to evaluate the ∂ϕ/∂t term in the pressure is different. The method used to calculate pressure in [17] cannot be applied here because the shape of the section in an Earth-fixed cross plane is varying in time. Now we select 20 cross planes as in Fig. 4 and solve the IBVP problems at those planes. Fewer planes will result in larger numerical errors because of the linear approximation of the sectional force between every two adjacent planes. More cross planes demand more computational time. However, it is believed that the accuracy of the results will not be apparently changed when the number is larger than 20. We can justify this from the sectional force distribution for the steady motion in Fig. 7. From the calculated results in this figure, we can see that the force distribution curve can be well approximated by linearly connecting 20 equally spaced points on it. If we add more planes, i.e., more points in between the existing 20 sectional forces, the approximation of the force distribution does not change much. At t = 0, the hull is travelling at a constant speed with a given trim angle and a given sinkage. The COG at t = 0 locates at the origin of the xyz coordinates. The initial free-surface positions and the velocity potential on the free surface are taken from the calculations of the steady problem. As the hull moves forward with given vertical oscillations, the free surface in each cross plane is updated by integrating the two-dimensional free-surface conditions. At each

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H. Sun, O.M. Faltinsen / Applied Ocean Research 34 (2012) 68–77

U

21 F3

20 …

6

… 2

1

2D

Linear approximation

Fig. 5. The body plan of Keuning’s [20] model.

Obtained by BEM xbow x20 … x6 … x2 x1

x

Fig. 4. Numerical strategies of the 2D + t calculations.

time step, we save the information about the free surface and the displaced position of the ship section at the cross planes. The coordinates and normal vectors on 200 ship cross-sections have been evaluated in advance. From the X-position of a certain section, we can obtain the section line and normal vectors on the section surface by interpolation. If the ship advances forward for a distance equal to the interval of the cross planes, we stop the calculations at the last plane behind the ship and introduce a new plane in front of the ship and initialize the calculation in the new plane. When a new plane is introduced, we use Eqs. (11) and (12) to specify the initial conditions in this plane. The calculated sectional forces at the cross planes are used to approximate the sectional force distribution along the ship (Fig. 4), which is denoted as F32D (x). At the bow, the sectional force is assumed to change linearly from the frontmost hull–water intersection position xbow to the plane 20. The vertical force at a given hull-fixed X-position can then be obtained by interpolation. 4. Keuning’s model Keuning’s [20] model test is studied in this paper. The body plan of the model is reproduced as in Fig. 5, with station 0–8 on the left and station 9–20 on the right. The main particulars are given as follows. The length of the waterline is L = 2 m. The breadth of the waterline is 0.25 m. The draft is 0.0624 m. The displacement is 0.01248 m3 . The block coefficient is 0.396. Two advancing speeds, corresponding to two Froude numbers Fn = U/(gL)1/2 = 0.57 and 1.14, were used in the model tests. Here the case with the higher speed at Fn = 1.14 is studied. The model was divided into seven segments of equal length of 0.285 m. On each segment the dynamic vertical force was measured. When the model was running steadily on the calm water, its trim angle and sinkage were set by using the experimental data from other model tests. The trim angle is  = 1.62◦ at the mean position. The value of sinkage was not directly given

in [20]. However, from the figure of measured wave profile along the model in the report, the sinkage is taken as z0 = −0.004 m at the mean position. In doing so, the centre of rotation in trim is assumed to locate in the midship and at the waterline. The negative value means that the model is moved downwards. Steady running tests were performed by setting constant vertical displacements za = 0.0100, 0.0075, 0.0050, 0.0025, 0.0000, −0.0025, −0.0050, −0.0075, −0.0100 m relative to the mean position to give sectional force distributions for different za . The corresponding sinkage for each case is obtained as z0 + za while the trim angle is the same. In the forced oscillation tests, the model was forced to oscillate in heave from the mean position. The amplitude of the heave motion was 3a = 0.01 m. The various oscillation frequencies were ω = 4, 5, 6, 7, 9, 11, 13, 15 rad/s. Keuning [20] presented the sectional added mass a33 and sectional damping coefficient b33 for the seven segments. The in-phase component and the quadrature component of the dynamic forces were measured. The in-phase component and the measured restoring force coefficients in the static position with trim and sinkage were used to calculate the added mass. The quadrature component was used to calculate the damping coefficient. 5. Numerical results for steady motions In the steady motion, the ship is advancing at a constant speed (Fn = 1.14). The trim angle  = 1.62◦ and sinkage z0 = −0.004 m at the mean position are also constant. Fig. 6 shows the free-surface elevations around the ship sections from station 0 to station 19. The pictures for station 0–8 are shown on the left side, while the rest on the right side. We can see the non-viscous flow separation from the hull surface. The separated water gradually comes down under the effect of gravity and propagates away from the ship. The free-surface elevations shown in Fig. 6 together with the velocity potential on the free surfaces will then used as the initial conditions in the 20 cross planes in the unsteady simulations.The sectional force distribution along the ship is shown in Fig. 7. A hull-fixed X0 coordinate is introduced. It coincides with the X-axis as shown in Fig. 1, but with origin at the aft-perpendicular and its positive direction towards the bow. The buoyancy force FB due to the −gz term in the pressure below z = 0 has been subtracted from the sectional force. The remaining part of force is called dynamic force in the model tests. The experimental results are averaged values on each segment with length 0.285 m. The calculated forces oscillate at the aft ship similarly as in the experimental results, whereas the oscillation amplitudes are larger than those of the experiments. This can be due to the fact that the energy dissipation in the spray in reality is

H. Sun, O.M. Faltinsen / Applied Ocean Research 34 (2012) 68–77

Fig. 6. Free-surface profiles in steady motion at Fn = 1.14,  = 1.62◦ , za = 0.0.

not considered in the numerical simulations. The numerical results by Faltinsen and Zhao [18] are also given in the figure. The difference between their results and the present results are more evident in the region of the aft ship, where the present results show better agreement in the tendency of the variation. The difference can be caused by the different numerical resolutions and treatments of, e.g. the jet and the flow separation, or by the errors in the digitization of the hull shape of the aft ship. The convergence of the present calculation for the steady motion has been verified by changing the size of elements on the body surface and free surface and the time step, as well as the parameter used in the flow separation model. Fig. 8 shows the distribution of the total sectional force and different force components given by the different pressure terms p1 = −gz, p2 = −∂ϕ/∂t, p3 = −0.5|∇ ϕ|2 , respectively. The force due to p1 remains small before the midship and gradually become dominant after the midship. The forces due to p2 and p3 are more important in the bow region. The oscillations of these two force components near the bow are related to the numerical treatment of flow separation. However, the sum of them is not apparently affected by this treatment. The main source of the oscillations of the force distribution at the aft ship is the force component given by the pressure term p2 = −∂ϕ/∂t. In reality, the pressure at the transom equals the atmospheric pressure if the transom plate is dry. Then the total sectional force at the stern should be zero. In a

Fig. 7. Dynamic vertical force distribution along the hull at Fn = 1.14,  = 1.62◦ , za = 0.0. Solid line: experiments by Keuning [20]; dash line: numerical results by Faltinsen and Zhao [18]; dash-dot line: the present numerical results.

73

Fig. 8. Variation of different force components due to different pressure terms. The pressure terms are denoted as p1 = −gz, p2 = −∂ϕ/∂t, p3 = −0.5|∇ ϕ|2 .

2D + t theory, the influence from the downstream flow is neglected, therefore the calculated total force at the stern (X0 = 0) in Fig. 8 does not equal zero. More discussion about the 3D effects neglected in a 2D + t theory can be seen in [15,24]. Fig. 9 shows the measured and calculated sectional forces when the running ship is displaced in the vertical direction relative to the mean position. The displacement is given as za = 0.0100, 0.0075, 0.0050, 0.0025, 0.0000, −0.0025, −0.0050, −0.0075, −0.0100 m. The case with zero displacement, i.e., at the mean position is also included. The hydrostatic force due to the −gz term in the pressure below z = 0 for the mean position has been subtracted for all the cases. The change of the calculated sectional forces with respect to the vertical displacement follows the same tendency as that of the measured results. The change of the sectional force versus za in the fore ship is mainly due to the change of the hydrodynamic force on the bow. However, the change of the sectional force versus za at the aft ship is mainly caused by the difference between the hydrostatic forces at a given vertical displacement and those at the mean position. Near the stern, the calculated dynamic forces are larger than experimental results due to the 3D effects at the stern. We also noticed that the range of change in the dynamic sectional force versus za is larger in the calculation than in the experiments.

Fig. 9. Steady vertical dynamic forces for different vertical displacement za . Stepped lines: experiments from Keuning [20]; continuous lines: the present calculations.

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Fig. 10. Sectional added mass coefficients. Solid lines: experiments by Keuning [20]; lines with symbols: the present numerical results; dashed lines: the numerical results by Faltinsen and Zhao [18].

H. Sun, O.M. Faltinsen / Applied Ocean Research 34 (2012) 68–77

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Fig. 11. Sectional damping coefficients. Solid lines: experiments by Keuning [20]; lines with symbols: the present numerical results; dashed lines: the numerical results by Faltinsen and Zhao [18].

6. Numerical simulation of the unsteady motions Before the ship is forced to oscillate, it is advancing at a constant speed (Fn = 1.14) in its mean position with constant  = 1.62◦ and

z0 = −0.004 m. The free-surface elevations shown in Fig. 6 together with the velocity potential on the free surfaces are used as the initial conditions in the 20 cross planes in the unsteady simulations. There is a transient phase starting from the initial time t = 0 before

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H. Sun, O.M. Faltinsen / Applied Ocean Research 34 (2012) 68–77

a steady state is reached. This transient phase is about 1–2 periods in all the studied cases. The steady-state part of the time history of the sectional forces is then used to calculate the added mass and damping coefficients. In the calculations for the unsteady problem, 20 equally spaced cross planes are used at any time instant. As the ship moves forward, new planes are continuously added and the planes left behind the ship are discarded accordingly. Given a hull-fixed X0 -position, one can interpolate the sectional vertical force at this position. The steady-state part of the time history of the sectional vertical force at a given X0 -position can be expanded as a Fourier series f (t) = C0 +

∞

[Csn sin(nωt) + Ccn cos(nωt)]

(13)

n=1

where C0 , Csn and Ccn (n = 1, 2, 3,. . .) are constants. The sectional added mass coefficient a33 and the sectional damping coefficient b33 can be estimated from the amplitudes of the first harmonic Cs1 and Cc1 as a33 =

−[Cs1 − c33 3a ] ω2 3a

(14)

b33 =

Cc1 ω3a

(15)

where c33 is the sectional restoring force coefficient, which is defined in the same way as in the experiments. The ship model is first settled in its static position with the given trim and sinkage at zero speed, then the sectional restoring force coefficient is calculated by c33 = gswp , where swp is the breadth of the waterline at the section. The sectional hydrodynamic coefficients at 20 hull sections with X0 = 0–1.9 m are calculated. The coefficients at X0 = 2.0 m are simply approximated as zero. The numerical results for the eight different frequencies ω = 4, 5, 6, 7, 9, 11, 13, 15 rad/s are shown in Fig. 10 (added mass coefficients) and Fig. 11 (damping coefficients). The experimental results are the average values on each segment in the model tests. In general, the present numerical results agree well with the experimental results for all frequencies studied here. If we look closely at the bow and the stern, we will see some difference. At the stern, the added mass is underestimated at lower frequencies and overestimated at the higher frequencies by our numerical method. The predicted damping coefficients are higher than the experimental values for all the frequencies at the stern. Due to the flow separation from the transom stern, the pressure at the stern should be atmospheric pressure. However, the 2D + t theory predicts higher pressure on the stern, which will result in hydrodynamic coefficients different from the experiments. At the bow, the averaged added mass is negative on the first two segments starting from the fore perpendicular. The calculated negative added mass values are much lower. The 3D effect at the bow causes the water near the bow to run up along the stem relative to the initial calm water level. This change can affect the hydrodynamic forces near the bow. The difference in a33 near the bow can be caused by this 3D effect. The numerical results in [18] for ω = 5, 11, 15 are also presented in Figs. 10 and 11. The present numerical results show obviously better agreement with experiments than their results for the lower frequency ω = 5 rad/s. The agreement with experiments for ω = 11 rad/s and 15 rad/s looks similar, although the a33 by [18] shows a little better agreement for ω = 15 rad/s. Keuning [20] found through his analysis that the influence of the local steady flow on the added mass is stronger for lower frequencies. The improvements in the present results for the lower frequency imply that the interaction between steady and unsteady flow is better accounted for than those in [18]. They solved the steady and unsteady flow

separately and considered the interactions through some interaction terms; however, the steady and unsteady flow is solved as a whole in the present method. The nonlinearities in the unsteady flow were neglected in their method, but it is not clear whether this effect is important. Nonlinear effects should be the largest around the natural frequency when one considers the responses in regular waves. The natural frequency for heave is around ω = 9 rad/s according to Blok and Beukelman’s [25] results for a similar ship model. However, the results for ω = 9 rad/s were not given in [18], so we could not compare the results for this frequency. It is noticed that the present numerical method encounters numerical difficulties for very high frequencies, such as for ω = 15 rad/s. Frequent detachment and reattachment of the water flow along the hull occur at around station 7 and station 8 and cause many jumps in the time series of the sectional forces at those sections. Smaller time steps can be used to improve the accuracy of the results and then reduce the occurrence and amplitude of the jumps. Because those jumps occur with very short time duration, the time integrations in calculating Cs1 and Cc1 suppress the effects of the jumps. Actually, the 2D + t theory is not expected to be used for very high frequencies since we assume that the incident wavelength should be O(L) or larger when the theory is applied in a seakeeping problem with head seas [16]. 7. Conclusions A 2D + t theory was applied to simulate the steady motion and the forced oscillation in heave of a semi-displacement ship at high forward speed. Strongly nonlinear free-surface flow, such as the jet running up along the hull and the flow separation from the round bilge, were simulated. Hydrodynamic pressure was calculated and integrated to calculate the sectional hydrodynamic forces. The calculated results for the steady and unsteady problems are compared with published model test results. Good agreement is obtained, which validates the present numerical method. In the steady problem, the calculated sectional force distributions show reasonable consistency with the measured results when the running ship is displaced in the vertical direction with different displacements. The spatial oscillation of the sectional force distribution in the aft ship is mainly given by the force component due to the pressure term −∂ϕ/∂t. The 3D effect at the stern causes discrepancies between the calculations and experiments near the stern. The results for the steady motion were used as the initial conditions for the solution of the unsteady problem. In the unsteady problem, the calculated sectional added mass and damping coefficients are compared with the experimental results. Discrepancies were found near the bow and the stern, which could be caused by the three-dimensional effects neglected in the present method at those positions. The interactions between the nonlinear steady and unsteady flow were taken into account automatically because they were solved as a whole. By referring to an earlier work based on a similar 2.5D method, we can see that the coupling between the unsteady flow and the local steady flow is better accounted for in the present method. The present method can be further developed to predict the vertical motions of non-prismatic high-speed vessel in head seas. Because the equations of motions will then need to be solved at each time step, the calculated force and moment on the vessel should be sufficiently accurate so that the solutions can converge. Care must be taken in dealing with the flow separation and attentions should be paid to the 3D effects at the bow and the stern. References [1] Salvesen N, Tuck EO, Faltinsen OM. Ship motions and sea loads. Trans SNAME 1970;78:250–87.

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