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Simulation of hydrodynamic interaction forces acting on a ship sailing across a submerged bank or an approach channel
Ocean Engineering 103 (2015) 103–113
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Simulation of hydrodynamic interaction forces acting on a ship sailing across a submerged bank or an approach channel Xueqian Zhou, Serge Sutulo, C. Guedes Soares n Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
art ic l e i nf o
a b s t r a c t
Article history: Received 7 March 2014 Accepted 28 April 2015 Available online 27 May 2015
A solution is proposed to handle complex water boundaries based on the panelled moving patch method, in which the moving patch on the seabed is dynamically meshed by an optimized version of the paving algorithm and smoothed by a hybrid smoother. Using this approach, two typical hydrodynamic interaction scenarios with the non-even seabed are studied, and numerical results are obtained for a variety of values of the relative ship course as well as of characteristics of the seabed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Hydrodynamic interaction Potential flow Restricted waters Panelled moving patch Dynamic mesh generation
1. Introduction Experimental study of the hydrodynamic interaction forces on a ship in confined waters dates back to experimental investigation of the ship-bank interaction done in the 1970s by Norrbin, where a scaled model was towed in the tank along vertical sidewalls of a dredged channel and the sway force and yawing moment were measured. Ch’ng (1991) carried out bank-interaction experiments with MarAd and S-175 models. Based on these experimental data, a set of regressions approximating the bank-induced sway force and yaw moment were obtained (Ch’ng et al., 1993). In the experimental results presented by Li et al. (2001), model tests in extremely shallow water and near the bank were carried out. In particular, it was discovered that the bank suction changed into repulsion when the ratio of water depth to draft decreased to approximately 1.10. A more recent experimental study on ship-to-bank interaction was conducted by Lataire et al. (2007), who investigated bank effects induced by sloped surface-piercing as well as submerged banks. Based on the formula proposed by Norrbin (1974), a new formula of the relationship between the height of the sidewall and the magnitude of the sway force and yaw moment was derived. Although experimental methods can usually provide realistic and reliable estimates, they are of little help in the context of irregular water boundaries for at least two reasons. First, water
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Corresponding author. Tel.: þ 351 218 417607. E-mail address: [email protected] (C. Guedes Soares).
http://dx.doi.org/10.1016/j.oceaneng.2015.04.067 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
boundaries of the real world are usually different from one to another, and in practice only 2 types of them can be relatively easily reproduced in the laboratory: (1) shallow water with either a horizontal or sloped flat bottom, (2) a channel with water surface piercing or completely submerged sidewalls of vertical, inclined or sloped profiles. Second, motions of the interacting vessels are typically limited to moving on courses parallel to the bank. Numerical methods have no restrictions on the geometry of water boundaries or on the motions of vessels. For instance, Skejic et al. (2012) performed a numerical simulation of turning manoeuvres of a ship in calm water with randomly generated seabed profiles and discussed its influence on the ship’s trajectories. This particular advantage of numerical methods makes them the most feasible solution to the problem of interaction loads involving complex water boundaries. Among all the numerical methods for studying the hydrodynamic interaction problem, real fluid computational methods (Chen et al., 2003) and perfect fluid freesurface methods (Söding and Conrad, 2005) can produce rather accurate estimates compared to experimental results, but neither of them is suitable for online simulation. On the other hand, as compared to the two numerical approaches mentioned above, the double-body potential-flow methods are less accurate but are fast enough to be run online and can still capture the main effect of ship hydrodynamic interaction (Tuck and Newman, 1974). An implementation based on the panel method (Hess and Smith, 1964) for real-time simulation was first accomplished and applied to the interaction loads on interacting ships in deep water (Sutulo and Guedes Soares, 2008), and which was later extended to the case of shallow water with a flat horizontal bottom by using a truncated series of mirrored images. Validation of the interaction
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code against experimental data for the case of two ships in parallel motion (Sutulo et al., 2012) showed that the prediction accuracy for the sway force and yaw moment varies from satisfactory to acceptable, and the agreement only breaks at very small lateral clearance between sidewalls. Comparative computations with various models carried out by Fonfach et al. (2011) showed that the local wavemaking effects rather pronounced in this case are responsible for the discrepancy. In order to eliminate the limitation of horizontal flat bottom required by the mirrored image technique and gain the ability to deal with complex water boundaries, a solution using the panelled moving patch technique was proposed (Zhou et al., 2014). The approach consisted in distributing a layer of sources not only on the wetted surface but also on the seabed beneath the moving vessel. Although the obtained good agreement between the moving patch method and the mirror image method in the flat bottom case encouraged belief that the new method will also handle appropriately the arbitrary bathymetry case, the actual capability of dealing with complex water boundaries was still limited by restrictions on the geometry and on the motion of the vessel (Zhou et al., 2012), and the scenario had to be one of the following: 1. the ship sailing over a flat and sloped bottom, 2. the ship on a course parallel to an either piercing or submerged bank with various forms of profile: vertical, inclined, or stepped etc. One can see that in either case there is no difficulty in meshing the moving patches as they can be easily discretized with rectangular elements. However this option is no longer available when it goes about any situation different from those mentioned above. For instance, the patch under a vessel in a canal on a course non-parallel to the bank is a non-rectangular polygon requiring an advanced meshing technique. Among all available algorithms suitable for meshing the moving patch on the seabed, the advancing front meshing algorithm— the paving (Blacker and Stephenson, 1991) has two features most desired by a real-time simulation: a high computational efficiency and automatic all-quadrilateral meshing. Although the paving algorithm has been improved by later contributions (White and Kinney, 1995; Kinney, 1997) and successfully applied in various engineering and scientific areas including, for instance, electromagnetics (Moreno et al., 2011), it still sometimes generates meshes containing poor quality elements, and even invalid elements that cannot be improved by its quality improving procedure. This issue is particular critical for online simulations that require dynamical meshing, and must be solved prudently as inferior or invalid elements can cause a meshing failure and thus halt the simulation. Several of such circumstances are discussed in this paper, focusing on the causes of the irregularities, their detection, and possible solutions. The smoothing procedure, as a technique independent of mesh generation, plays a very important role in the process of producing meshes of good quality. The Laplacian smoothing algorithm that accompanied the paving algorithm is computationally efficient and easy to implement, but in many cases it can only worsen the quality of the mesh. On the other hand, the optimization-based smoothing methods typically result in a very good mesh quality (Canann et al., 1993; Batdorf et al., 1997), but their computational cost is unacceptably high. As real-time simulations of hydrodynamic interaction loads have two very strict requirements: computational efficiency and acceptable mesh quality, neither of the Laplacian smoothing or the optimization-based smoothing is a unique suitable solution. In this paper, the panelled moving patch method is used to deal with hydrodynamic interaction problems involving arbitrary geometry
of the seabed, while the mesh for the moving patch is dynamically generated by the paving algorithm and smoothed by a hybrid smoother combining the Laplacian algorithm and an angle-based algorithm (Xu and Newman, 2006). Finally, this approach is applied for studying two cases of hydrodynamic interaction involving complex water boundaries: (1) the ship sailing across a submerged bank from deep to shallow water that can be interpreted as a ship entering a continental shelf and the hydrodynamic interaction loads are computed for a variety of approach angles, and (2) the ship sailing obliquely across an approach channel, various approach angles and at various values of the channel width.
2. General theory of hydrodynamic interaction loads Consider the global coordinate system Oξηζ, with the ξ-axis laid on the undisturbed water surface, the ζ-axis directed vertically downwards, and the η axis defined by right-handed convention. The instantaneous advance ξC , transfer ηC , and the heading angle ψ of the ship are defined with the help of the body-fixed frame Cxyz attached to the vessel in question, with its origin located at the midship, the x-axis pointing to the bow, y-axis to the starboard, and the z-axis downwards. The geometry of any complex flow boundaries, like a canal bottom, a bank, or a jetty, is expressed and loaded in the global coordinate system at the beginning of the simulation, and converted at each computational instant to the body-fixed frame whose instantaneous location and orientation in the global system can be calculated from the velocity of the origin C–VC and the angular velocity ΩC . A layer of sources is distributed on the wetted ship surface and also on a moving patch of some sufficient size placed beneath the ship in consideration to express the local geometry of the seabed. Under the assumption of a sufficiently low Froude number, the mirror-image principle is applied to the undisturbed free surface, so that the fluid domain is defined by the doubled hull and the panelled moving patch is considered. The total velocity potential Φ takes the form: Φ ¼ V ξcur ξ þ V ηcur η þ ϕ;
ð1Þ
where V ξcur and V ηcur are the components of the velocity of the current (if any) in the global axes, and ϕ ¼ ϕðξ; η; ζ; tÞ is the perturbation potential. Then the induced velocity is: VI ¼ ∇ϕ:
ð2Þ
At any time moment, the perturbation potential should satisfy the Laplace equation Δϕ ¼ 0;
ð3Þ
and on the ship wetted surface and on the moving patch, the following non-penetration boundary condition is applied: ∂ϕ ¼ Vr Un; ∂n
ð4Þ
where n is the outward unity normal to the local geometry, Vr is the relative local velocity: Vr ¼ V Vcur ;
ð5Þ
where V is the absolute local velocity on the body surface (is zero on the moving patch). The ship’s wetted surface and the moving patch on the water flow boundaries chosen in the vicinity of the ship constitute the total surface S in the following equations: Z ∂GðM; PÞ 2πσðMÞ þ σðPÞ dSðPÞ ¼ f ðMÞ; ð6Þ ∂nM S where σ is the source density, Mðx; y; zÞ is the field point, Pðx0 ; y0 ; z0 Þ is the source point belonging to the surface S, and Gð Þ is the Green
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
105
function:
Predefined boundary nodes
1 1 Gðx; y; z; x0 ; y0 ; z0 Þ ¼ þ ; r r
Interior geometry boundary
w Ne
Once Eq. (6) is solved, the induced velocity and induced potential at a point can be obtained by summing up the contribution of each panel. In turn, the pressure at this point can be calculated using the unsteady Bernoulli equation: ∂ϕ 1 p ¼ ρ þ ðV2r V2p Þ ; ð9Þ ∂t 2
ts
ð8Þ
n me el e
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz þ z0 Þ2 :
New elements
ð7Þ
Exterior geometry boundary
where V p ¼ VI Vr ;
ð10Þ
where VI is the induced velocity The total hydrodynamic inertial force Fpi and moment Mpi can be calculated by: Z Z ð11Þ Fpi ¼ pndS; Mpi ¼ pr ndS: Si
Si
and their components in the body axes are: the surge force X P , sway force Y P , and yaw moment N P , respectively. The proper hydrodynamic inertial forces and moments are X e ¼ μ11 u_ þ μ22 νr Y e ¼ μ22 ν_ μ26 r_ μ11 ur N e ¼ μ26 ν_ μ66 r_ þ ðμ11 μ22 Þuv μ26 ur;
ð12Þ
where μij are the added mass coefficients. Finally the pure interaction loads are: X I ¼ X p X e ;
Y I ¼ Y p Y e;
NI ¼ Np Ne :
ð13Þ
3. Geometry discretization and mesh generation In the present implementation according to the Hess and Smith method, distributing sources requires meshing the underwater portion of the ship hull and the moving patch on the seabed. The former geometry is pre-discretized and loaded before the simulation starts; while the latter is dynamically meshed using a modified version of the paving algorithm with improved robustness. 3.1. Review of the paving algorithm The paving algorithm starts meshing from the nodes predefined on the geometry boundaries. In this context a boundary, unlike its definition in describing a fluid domain where it refers to a surface that encloses a volume in space, is a circuit of line segments by which a non-closed surface is bounded. New elements are created according to the boundary nodal configurations and are properly placed starting from the boundary toward the interior. The paving boundary or the advancing front is being updated as the new elements are inserted. The paving boundaries are advanced alternatively if there is more than one boundary. When intersections of the paving boundaries (including the selfintersection) are detected, an intersection handling module is invoked to connect the fronts and, depending on the situation, to merge two paving boundaries or split one boundary into two. This process continues until all the paving boundaries are closed. Fig. 1 shows an example of meshing a rectangle plate with a hole in it, where the arrows show the directions in which the elements
Fig. 1. Illustration of the paving algorithm.
are installed along the exterior and interior geometry boundaries, respectively. The cracks, characterized by the small angles between two adjacent edges of elements and often emerging during a paving process, are closed by the procedure called seaming which can be viewed as a “tailor” who stitches up the cracks between the elements of a mesh. The boundary is checked for the necessity of seaming after any action that might form cracks, i.e. after placing new elements, boundary node smoothing, row adjustment, connecting intersections, etc. As new elements are being added to the existing mesh, the unmeshed region shrinks and tends to close. A paving region that is about to close is detected by monitoring the number of nodes remaining in the boundary. When this number reduces to a certain value (usually 4 or 6), the boundary is closed by filling one or more elements into the remaining unpaved region in a predefined pattern. However, as will be discussed later in this paper, in some situations the standard closing procedure may cause a meshing failure. After the entire geometry is meshed, a “clean up” procedure is applied to evaluate the sizes and interior angles of all the generated elements, and to improve the quality by means of merging neighbouring elements or by a localized re-meshing when necessary. 3.2. Robustness improvements The original paver may fail to generate a mesh in some situations. The failure that happens most frequently is that the paver is unable to close a relatively small paving region due to a harsh transition of element size, which, unfortunately, is quite the case in this real-time hydrodynamic interaction simulation when the moving patch is divided into several paving regions. The paving algorithm closes a paving region when there are six or less nodes remaining in the paving boundary. For a paving boundary containing only two nodes, a seaming process is applied to close the regain. When there are four nodes “hanging”, the boundary is closed simply by inserting an additional element. In the case of six nodes, two, three, or four quadrilateral elements are inserted depending on the configurations of those nodes, which is achieved by identifying the configuration of the nodes and then filling the paving region with elements in a pattern that is predefined accordingly. However in the simpler case of four nodes remaining, as shown in Fig. 2, insertion of the element not only increases the valency of the node N 4 from five to six (this is an unfavourable circumstance
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per se as high-valency nodes result in elements with small interior angles), but also produces the unacceptable bivalent Node N 3 . In another case shown in Fig. 3, no bivalent nodes are produced during the closure process, but the quality of the new element and its neighbouring elements is not guaranteed. These elements have either invalid (very large, very small, or even negative) interior angles or very large element side ratios, which will challenge the smoother invoked afterwards to improve the mesh quality. In fact, these situations mostly arise from a transitional failure of the seaming process where bad elements are produced if the seaming procedure is executed unconditionally. Note also that both described cases are accompanied by an abrupt change of the element size. Although these unfavourable situations do not usually happen, they present a critical robustness issue as the real-time simulation will immediately break down due to the meshing failure. The solution is to select a number of installed elements and remove them so that the paver can close the boundary successfully. To accomplish this, an installed node on the paving boundary with the lowest valency is chosen, and then all its attached elements are removed. In the first case, see Fig. 2, the paving region is closed by removing the element attached to the bivalent node N4 and then merging the nodes N 1 and N 3 . Fig. 4 shows the closed region after the smoothing was applied. When there are no bivalent nodes in the paving boundary, the decision on which elements are to be removed must be carefully made depending on the configurations of the interior angles, node valences and involved side length ratios. As shown in Fig. 3, where the nodes N3 and N4 are candidates for removal, comparison of their configurations shows that removing the elements attached to the node N 4 and merging N 3 with N 4 is a better choice. Then, the new paving boundary contains four nodes, and can be closed by inserting an additional element (formed by the nodes N 1 or N 3 , N6 , N7 and N 8 ) in the new paving boundary. Fig. 5 shows the closed paving region after the applied smoothing.
surrounding nodes. The nodal location updating process deals with one node at a time. After all the nodes in the given set are updated once, the convergence is considered attained if the changes of the location of the nodes are smaller than some predefined tolerance, otherwise the procedure continues to iterate until such a tolerance is met. Due to the intrinsic mechanism of the algorithm described above, the Laplacian smoothing fails often in many situations for at least two reasons. The optimality of the nodal location proposed by the algorithm relies highly on the distribution of the surrounding nodes, and there are situations that are not favourable for reaching the optimality. Fig. 6 shows the case where the node to be smoothed is located near a permanent boundary corner and the Laplacian smoother tends to move the node N6 to a worse location. The second reason is that its smoothing performance also depends on the order in which the nodes of an element are optimized. Fig. 7 shows the case where the smoothing procedure starting with the node N2 results in a sliver-like element, but when it starts with the node N4 the desirable element is obtained. Aiming at overcoming the shortcomings of the Laplacian smoothing with the minimum computational cost, the anglebased mesh smoothing algorithm is chosen to handle the elements untreatable by the Laplacian smoothing. The idea of the angle-based smoothing is to mount a system of virtual torsion springs between the nodes, and to achieve the mesh smoothing goal by minimizing the system potential energy. As shown in Fig. 8, let Vji be the vector from the surrounding node Ni to the node Nj , then the vector V0ji from the node N i to the
N3
N2
3.3. Hybrid mesh smoother The Laplacian smoothing algorithm accompanying the paving algorithm optimizes the mesh by updating the nodal locations with optimal ones calculated from the coordinates of the
N5
N6
Fig. 4. Remedy of a closing paving boundary with a bivalent node.
N4 N3
N2
N2
N1 N5
N6
N1
N8
N6
N5
N7
Fig. 2. A closing boundary with a bivalent node that requires special treatment, the hatching denotes the unpaved region. Fig. 5. Remedy of a closing paving region containing no bivalent nodes.
N3
N2
N1
N1
N4
N2 N5
N6
N7
N6
N8
Fig. 3. A closing boundary without bivalent nodes requiring special treatment, the hatching denotes the unpaved region.
N3
N4
N7 N5
Fig. 6. A case that the Laplacian smoothing algorithm can hardly improve.
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
107
optimal location calculated based on the location of Ni , V0ji , can be expressed as ½V0ji T ¼ RðθÞVji
ð14Þ
where RðθÞ ¼
cos θ
sin θ
sin θ
cos θ
;
θ¼
αβ 2
ð15Þ
Then the change in location of the node N j due to N i is ΔVji ¼ V0ji Vji ;
ð16Þ
Fig. 9. A breakwater meshed.
Repeating the process for all the surrounding nodes, the contribution of each node to the change in the location of the node N j is obtained, and then the average of the contributions of all the surrounding nodes is taken as the final modification ΔVj . It must be noted that the smoothing solution employed here combines the Laplacian smoothing and the angle-based smoothing, the former being the primary smoother while the latter is only used when the Laplacian smoothing fails, i.e. when the convergence of the Laplacian smoothing is still not reached after a certain number of iterations. Fig. 9 shows a breakwater meshed together with the adjacent bank and with a square of sea bottom around it. This mesh contains 9989 quadrilateral elements and none of them is misshapen or sliver-shaped. Fig. 10. Discretization of the ship hull and the moving patch on the seabed.
N3
N 3'
N2
N 2'
N 4'
4. Examples of interaction force computation
N4 N 1'
N1
Fig. 7. The solid lines represent the original element, while the dashed lines denote the resulting elements with each node being the starting node, respectively.
N i+1
4.1. Ship sailing across a submerged bank
Ni i i
V j i'
V ji
Incorporation of the paving algorithm described above boosts the effectiveness of hydrodynamic interaction simulation originally only applicable to the case of simplified courses. Two scenarios requiring handling of complex water boundaries were modelled and the interaction forces with difference configurations were computed. The first scenario implies a ship sailing across a submerged bank from the deep to the shallow water, which may represent a ship entering the continental shelf. The second scenario simulates oblique sailing across a dredged channel. The still water assumption is made although steady current could be present in either scenario. The ship form used for the calculation and analysis throughout this work has main dimensions L B T of 189.6 m 31.6 m 10.3 m, and its block coefficient is 0.815. The hull was approximated with 542 panels.
N i-1
N j i' ΔV ji Nj
Fig. 8. Node N j , with 4 elements attached to, is to be smoothed. The dashed lines denote the diagonals of each element passing Node N j .
In this simulation, the ship is advancing with the speed of 3 m/s across a vertical bank from the deep (22 m, d=T ¼ 2:136) to shallow (12 m, d=T ¼ 1:158) water. A rectangular patch with dimensions of 400 m 200 m is placed beneath the ship, which is dynamically discretized at each time into quadrilateral elements with side lengths approximately equal to 8 m, as shown in Fig. 10. Of course, the moving patch method requires an increased number of panels. The total number of panels contained in the resultant mesh varied from 1250 (before the moving patch reached the bank) to 1392 (when the ship was passing exactly over the bank). The computation time required to solve the flow state for one ship position was about 1 to 3 s, and this speed was achieved without any implementation of parallelism and executed on a rather outdated computer equipped with Intel Pentiums E5200 CPU and 2 Gigabytes of memory. It can be expected that, with processors powered by up-to-date technology and/or parallel
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X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
computing, the method is able to simulate this and even more complicated scenarios in real-time. In order to investigate the behaviour of the hydrodynamic interaction forces on a ship entering the shelf at different angles, as shown in Fig. 11, the simulation was repeated for θ ¼15, 30, 45, 60, and 75 degrees. The numerical results for the surge, sway forces and the yaw moment are nondimensionalized by Fig. 12: X0 ¼
2X ρLTV
; 2
Y0 ¼
2Y ρLTV
; 2
N0 ¼
2N
0.03
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.02 0.01
ð17Þ
ρL2 TV 2
where ρ is the water density, L the ship length, T the ship draught, and V the ship speed. These results are presented in Figs. 13–15 as a 0 function of the relative distance d ¼ d=L, where d is the distance between the origin of the frame fixed to the ship hull and the bank in the ξη-plane of the global frame. The simulation shows that a ship sailing across the submerged bank from deep to shallow water experiences a negative surge force which can be treated as drag. This surge force has two negative peaks in the time history. None of them appears when the midship is passing exactly over the submerged bank as the absolute value of the surge force reaches a minimum at this position. During the process, a positive sway force is present pushing the ship from deep to shallow water. This force increases rapidly with the angle θ, with the peaks appearing after the midship passes the bank. The time history of the yaw moment is of particular interest, the ship experiences a negative yaw moment until the midship passes the submerged bank where it reaches a negative peak, and then changes its sign and reaches its positive peak. 4.2. Ship sailing obliquely across the approach channel
-0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 13. Sway force acting on a ship sailing across a submerged bank.
0.004
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.002 0 -0.002 -0.004
-1
0 d'
1
ban
k
In this scenario, Fig. 15, the ship sails across a dredged channel with the speed of 3 m/s at various angles between the ship
0
Fig. 14. Yaw moment acting on a ship sailing across a submerged bank.
s ub
me rg
ed
deep water
θ
shallow water Fig. 11. A ship sailing across a submerged bank at different angles.
0 -0.002 -0.004 Fig. 15. A ship sailing obliquely across a dredged channel.
-0.006 -0.008 15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
-0.01 -0.012 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 12. Surge force acting on a ship sailing across a submerged bank.
centreplane and the central line of the channel, i.e. θ ¼15, 30, 45, 60, and 75 degrees. The channel width varies from 20 to 200 m, while the water depth is 22 m inside the channel, and 12 m outside it. The moving patch beneath the ship had the same size as used in the previous scenario. The numerical results for the forces and moments presented in Figs. 16–30 are nondimensionalised in the same way as before. However, there is a slight difference in the definition of the 0 abscissa d here, which is in this simulation the nondimensional
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
distance between the origin of the body frame and the central line of the channel. The channel width is also nondimensionalized by the ship length. Throughout all the simulated cases with various channel widths and course angles, a two-staged behaviour of the surge force over the time history is observed. The first stage begins when the ship starts to “feel” the approach channel and ends when the midship section reaches the channel centre, and the second stage follows. It is observed that an ever increasing surge force is acting on the ship from the beginning of the first stage until it reaches the peak, which is followed by a trough for small channel widths and small course angles, or by a second peak for large channel widths and large course angles (Fig. 16). However, this second peak or trough shrinks and merges with the first peak as the relative heading angle increases (see Figs. 19, 22, 25, and 28). In the second stage, a similar but reversed pattern is observed, with slight differences in the absolute values of the peaks and troughs. Compared to the surge force, the pattern of the time history of the sway force is rather simpler. A negative sway force dominates through the first stage with the maximum value in the middle, while it is the opposite for the second stage. The effect of the channel width is negligible except for the two smallest channel widths where the cancellation effect of the pair of sidewalls takes on. On the other hand, the maximum values of the sway force increases rapidly with the course angle. See Figs. 20, 23, 26 and 29. Independent of the channel width and course angle, the time history of the yaw moment starts and ends with a positive peak,
109
0.01
0.105L 0.264L 0.422L 0.738L 1.055L
0.005 0 -0.005 -0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 18. Yaw moment on the ship with a course angle of 15 degrees.
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.005 0 -0.005
0.005 -0.01 -1.5
0
-1
-0.5
0 d'
0.5
1
1.5
Fig. 19. Surge force on the ship with a course angle of 30 degrees.
-0.005
0.03 0.105L 0.264L 0.422L 0.738L 1.055L
-0.01 -0.015 -1.5
-1
-0.5
0.105L 0.264L 0.422L 0.738L 1.055L
0.02 0.01 0 d'
0.5
1
1.5
Fig. 16. Surge force on the ship with a course angle of 15 degrees.
0 -0.01 -0.02
0.06 0.105L 0.264L 0.422L 0.738L 1.055L
0.04
-0.03 -1.5
0.02
-1
-0.5
0 d'
0.5
1
1.5
Fig. 20. Sway force on the ship with a course angle of 30 degrees.
0 -0.02 -0.04 -0.06 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 17. Sway force on the ship with a course angle of 15 degrees.
and in-between it is dominated by a negative moment which, for the cases of large channel widths and large crossing angles, may fluctuate several times before reaching the negative peak (Figs. 21, 24, 27, and 30). The results obtained for the 60 m (0:316L) and 200 m (1:055L) channel width are represented in the different form in Figs. 31–36 clearly demonstrating influence of the crossing angle. The cancellation effect of the channel walls diminishes as the width increases, and the dredged channel case degrades to two separate submerged banks. When the width exceeds some value,
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X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.004
0.004
0.105L 0.264L 0.422L 0.738L 1.055L
0.002
0.002
0
0
-0.002
-0.002
-0.004
-0.004
-0.006 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 21. Yaw moment on the ship with a course angle of 30 degrees.
-0.006 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 24. Yaw moment on the ship with a course angle of 45 degrees.
0.01
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.005
0.105L 0.264L 0.422L 0.738L 1.055L
0.005 0
0
-0.005
-0.005 -0.01 -1.5
0.105L 0.264L 0.422L 0.738L 1.055L
-1
-0.5
0 d'
0.5
1
1.5
-0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 25. Surge force on the ship with a course angle of 60 degrees. Fig. 22. Surge force on the ship with a course angle of 45 degrees.
0.015
0.03
0.105L 0.264L 0.422L 0.738L 1.055L
0.02
0.005
0.01
0
0
-0.005
-0.01
-0.01
-0.02 -0.03 -1.5
0.105L 0.264L 0.422L 0.738L 1.055L
0.01
-1
-0.5
0 d'
0.5
1
1.5
-0.015 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 26. Sway force on the ship with a course angle of 60 degrees.
Fig. 23. Sway force on the ship with a course angle of 45 degrees.
one of these two submerged banks represents the first scenario considered before—a ship sailing across a submerged bank from deep to shallow water. In order to find the similarities between the two cases, the interaction forces are compared for 45 degrees of course angle in Figs. 37–39, where the abscissa is the nondimensionalized distance between the ship and the corresponding bank. It can be seen for each force component that the two cases have very different patterns in the first half of the interaction process due to the presence
of the other wall of the channel; however, as the channel width increases, both cases converge to each other.
5. Conclusions Based on the panelled moving patch method and the paving algorithm, a solver to hydrodynamic interaction is proposed and
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.001
0.003 0.105L 0.264L 0.422L 0.738L 1.055L
0.002
0
0
-0.0005
-0.001
-0.001
-0.002 -1
-0.5
0 d'
0.5
1
1.5
Fig. 27. Yaw moment on the ship with a course angle of 60 degrees.
0.008
-0.0015 -2
-1
0 d'
0
-0.004
-0.005
0 d'
1
2
Fig. 28. Surge force on the ship with a course angle of 75 degrees.
2
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.005
0
-1
1
Fig. 30. Yaw moment on the ship with a course angle of 75 degrees.
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.004
-0.008 -2
0.105L 0.264L 0.422L 0.738L 1.055L
0.0005
0.001
-0.003 -1.5
111
-0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 31. Surge force for 60 m wide channel and various crossing angles.
0.06
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.005
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.04 0.02 0
0
-0.02 -0.04
-0.005 -0.01 -2
-0.06 -1.5 -1
0 d'
1
2
-1
-0.5
0 d'
0.5
1
1.5
Fig. 32. Comparison of sway force between various angles at 60 m of channel width.
Fig. 29. Sway force on the ship with a course angle of 75 degrees.
applied to the study of the hydrodynamic interaction loads on a ship crossing submerged banks and dredged channels. The simulated cases demonstrate its ability to deal with those situations involving complex flow boundaries where experimental methods are problematic. Previous studies showed that even in the case of ships sailing in shallow water with a horizontal flat bottom, more accurate
numerical results can be obtained by increasing the panel density in the areas on the moving patch close to the ship hulls. Such a technique, which can be implemented through integrating an automatic mesh adaptation algorithm, can certainly improve the accuracy of the numerical results for the cases with complex flow boundaries, while, however, introducing additional complications in the geometric processing of the patch.
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X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.004
0.005
0.002 0
0
-0.002 -0.004
-0.005
-0.006
-0.01 -1.5
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
-0.008 -1
-0.5
0 d'
0.5
1
1.5
Fig. 33. Comparison of yaw moment between various angles at 60 m of channel width.
0.01
-1
0 d'
1
2
Fig. 36. Comparison of yaw moment between various angles at 200 m of channel width.
0.01
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.005
-0.01 -2
Sub. Bank 0.422L 0.633L 0.844L 1.055L
0.005
0
0
-0.005 -0.005
-0.01 -0.015
-2
-1
0 d'
1
2
-0.01 -3
-1
0
1
2
d'
Fig. 34. Comparison of surge force between various angles at 200 m of channel width.
Fig. 37. Surge force for 45 degrees crossing angle and various channel width.
0.03
0.06 15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.04
0.01
0
0
-0.02
-0.01
-0.04
-0.02 -1
0 d'
Sub. Bank 0.422L 0.633L 0.844L 1.055L
0.02
0.02
-0.06 -2
-2
1
2
-0.03 -3
-2
-1
0
1
2
d'
Fig. 35. Comparison of sway force between various angles at 200 m of channel width.
Fig. 38. Sway force for 45 degrees crossing angle and various channel width.
The hydrodynamic interaction computation method had been validated for the particular case of the flat seabed, and there are reasons to believe that it remains consistent in more complex cases although direct experimental confirmation is desirable and should be attempted in the future.
The squat effect has not been accounted for in the present study focused mainly on the improvement on the moving patch method but it is known that the squat in shallow water can significantly influence the interaction forces and accounting for it is also envisaged for future research work.
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.004 Sub. Bank 0.422L 0.633L 0.844L 1.055L
0.002 0 -0.002 -0.004 -3
-2
-1
0
1
2
d' Fig. 39. Yaw moment for 45 degrees crossing angle and various channel width.
Acknowledgement The study was carried out within the framework of the project Energy Efficient Safe SHip OPERAtion (SHOPERA) financed by the EU under contract number 605221 of the 7th Framework Programme. The first author was supported by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e Tecnologia), under contract no. SFRH/BD 75354/2010. References Batdorf M., Freitag L.A., Ollivier-Gooch C. (1997) Computational study of the effect of unstructured mesh quality on solution efficiency. In: Proceedings of the 13th Annual Computational Fluid Dynamics Meeting. Snowmass Village, CO. Blacker, T.D., Stephenson, M.B., 1991. Paving: a new approach to automated quadrilateral mesh generation. Int. J. Numer. Methods Eng. 32, 811–847. Canann, S.A., Stephenson, M.B., Blacker, T.D., 1993. Optismoothing: an optimization-driven approach to mesh smoothing. Finite Elem. Anal. Des. 13, 185–190. Chen H.-Ch., Lin W.-M., Liut D.A., Hwang, W.-Y. (2003) An advanced viscous flow computation method for shi-ship dynamic interactions in shallow and restricted waterway. In: Proceedings International Conference on Marine Simulation and Ship Maneuverability (MARSIM’03), 25–28 Aug, Kanazawa, Japan, vol. 3, pp. RC-35-1–10. Ch’ng, P.W., 1991. An Investigation into the Influence of Bank Effect on Ship Manoeuvring and its Mathematics Modeling for a Ship-Handling Simulator ME.
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Thesis. School of Mechanical and Manufacturing Engineering, The University of New South Wales. Ch’ng, P.W., Doctors, L.J., Renilson, M.R., 1993. A method of calculating the shipbank interaction forces and moments in restricted water. Int. Shipbuild. Prog. 421, 7–23. Fonfach J.M.A., Sutulo S., Guedes Soares C. (2011) Numerical study of ship-to-ship interaction forces on the basis of various flow models. In: Proceedings of the Second International Conference on Ship Manoeuvring in Shallow and Confined Water: Ship to Ship Interaction. May 18–20, Trondheim, Norway, pp. 137–146. Hess, J.L., Smith, A.M.O., 1964. Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. J. Ship Res. 8, 22–44. Kinney P. (1997) CleanUp: improving quadrilateral finite element meshes. In: Sixth International Meshing Roundtable, pp. 437–447. Lataire E., Vantorre M., Laforce E., Eloot K., Delefortrie, G. (2007) Navigation in confined waters: influence of bank characteristics on ship-bank interaction. In: Proceedings of the Second International Conference on Marine Research and Transportation, Ischia, Naples, Italy, 28–30 June, pp. 1–9. Li, D.-Q., Leer-Andersen, M., Ottosson, P., Trägårdh, P., 2001. Experimental Investigation of Bank Effects under Extreme Conditions. PRADS’, Shanghai, ChinaPRADS’, Shanghai, China. Moreno, J., Algar, M.J., Gonzalez, I., Catedra, F., 2011. Redesign and optimization of the paving algorithm applied to electromagnetic tools. Prog. Electromagnet. Res. B 29, 409–429. Norrbin N.H. (1974) Bank effects on a ship moving through a short dredged channel. In: Proceedings of the Tenth Symposium on Naval Hydrodynamics, Cambridge. Skejic R.; Kirimoto K.; Berg T.E.; Pedersen E. (2012) Maneuvering performance of ships in calm water with variable finite water depths. In: International Conference Marine Simulation and Ship Manoeuvrability. , pp. 1–9. Söding, H., Conrad, F., 2005. Analysis of overtaking manoeuvres in a narrow waterway. Ship Technol. Res. 52, 189–193. Sutulo S., Guedes Soares C. (2008) Simulation of the hydrodynamic interaction forces in close-proximity manoeuvring. In: Proceedings of the 27th Annual International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2008), Estoril, Portugal, 15–19 June, Paper OMAE2008-57938, pp. 1–10. Sutulo, S., Guedes Soares, C., Otzen, J.F., 2012. Validation of potential-flow estimation of interaction forces acting upon ship hulls in side-to-side motion. J. Ship Res. 56 (3), 129–145. Tuck E.O., Newman J.N. (1974) Hydrodynamic interactions between ships. In: Proc. Tenth Symposium on Naval Hydrodynamics, Cambridge, Mass., USA, pp. 35–69. White D.R. Kinney P. (1995) Redesign of the paving algorithm—robustness enhancements through element by element meshing. In: Proceedings of the Sixth International Meshing Roundtable, pp. 291–304. Xu, H., Newman, T.S., 2006. An angle-based optimization approach for 2D finite element mesh smoothing. Finite Elem. Anal. Des. 42 (13), 1150–1164. Zhou, X.-Q., Sutulo, S., Guedes Soares, C., 2012. Ship hydrodynamic interaction forces in restricted waters. In: Guedes Soares, C., Garbatov, Y., Sutulo, S., Santos, T.A. (Eds.), Maritime Engineering and Technology. Taylor & Francis Group, UK, pp. 241–249. Zhou, X.-Q., Sutulo, S., Guedes Soares, C., 2014. Computation of ship-to-ship interaction forces by a 3D potential flow panel method in finite water depth. J. Offshore Mech. Arct. Eng. 136 041301-1 a 041301-8.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Simulation of hydrodynamic interaction forces acting on a ship sailing across a submerged bank or an approach channel Xueqian Zhou, Serge Sutulo, C. Guedes Soares n Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
art ic l e i nf o
a b s t r a c t
Article history: Received 7 March 2014 Accepted 28 April 2015 Available online 27 May 2015
A solution is proposed to handle complex water boundaries based on the panelled moving patch method, in which the moving patch on the seabed is dynamically meshed by an optimized version of the paving algorithm and smoothed by a hybrid smoother. Using this approach, two typical hydrodynamic interaction scenarios with the non-even seabed are studied, and numerical results are obtained for a variety of values of the relative ship course as well as of characteristics of the seabed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Hydrodynamic interaction Potential flow Restricted waters Panelled moving patch Dynamic mesh generation
1. Introduction Experimental study of the hydrodynamic interaction forces on a ship in confined waters dates back to experimental investigation of the ship-bank interaction done in the 1970s by Norrbin, where a scaled model was towed in the tank along vertical sidewalls of a dredged channel and the sway force and yawing moment were measured. Ch’ng (1991) carried out bank-interaction experiments with MarAd and S-175 models. Based on these experimental data, a set of regressions approximating the bank-induced sway force and yaw moment were obtained (Ch’ng et al., 1993). In the experimental results presented by Li et al. (2001), model tests in extremely shallow water and near the bank were carried out. In particular, it was discovered that the bank suction changed into repulsion when the ratio of water depth to draft decreased to approximately 1.10. A more recent experimental study on ship-to-bank interaction was conducted by Lataire et al. (2007), who investigated bank effects induced by sloped surface-piercing as well as submerged banks. Based on the formula proposed by Norrbin (1974), a new formula of the relationship between the height of the sidewall and the magnitude of the sway force and yaw moment was derived. Although experimental methods can usually provide realistic and reliable estimates, they are of little help in the context of irregular water boundaries for at least two reasons. First, water
n
Corresponding author. Tel.: þ 351 218 417607. E-mail address: [email protected] (C. Guedes Soares).
http://dx.doi.org/10.1016/j.oceaneng.2015.04.067 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
boundaries of the real world are usually different from one to another, and in practice only 2 types of them can be relatively easily reproduced in the laboratory: (1) shallow water with either a horizontal or sloped flat bottom, (2) a channel with water surface piercing or completely submerged sidewalls of vertical, inclined or sloped profiles. Second, motions of the interacting vessels are typically limited to moving on courses parallel to the bank. Numerical methods have no restrictions on the geometry of water boundaries or on the motions of vessels. For instance, Skejic et al. (2012) performed a numerical simulation of turning manoeuvres of a ship in calm water with randomly generated seabed profiles and discussed its influence on the ship’s trajectories. This particular advantage of numerical methods makes them the most feasible solution to the problem of interaction loads involving complex water boundaries. Among all the numerical methods for studying the hydrodynamic interaction problem, real fluid computational methods (Chen et al., 2003) and perfect fluid freesurface methods (Söding and Conrad, 2005) can produce rather accurate estimates compared to experimental results, but neither of them is suitable for online simulation. On the other hand, as compared to the two numerical approaches mentioned above, the double-body potential-flow methods are less accurate but are fast enough to be run online and can still capture the main effect of ship hydrodynamic interaction (Tuck and Newman, 1974). An implementation based on the panel method (Hess and Smith, 1964) for real-time simulation was first accomplished and applied to the interaction loads on interacting ships in deep water (Sutulo and Guedes Soares, 2008), and which was later extended to the case of shallow water with a flat horizontal bottom by using a truncated series of mirrored images. Validation of the interaction
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code against experimental data for the case of two ships in parallel motion (Sutulo et al., 2012) showed that the prediction accuracy for the sway force and yaw moment varies from satisfactory to acceptable, and the agreement only breaks at very small lateral clearance between sidewalls. Comparative computations with various models carried out by Fonfach et al. (2011) showed that the local wavemaking effects rather pronounced in this case are responsible for the discrepancy. In order to eliminate the limitation of horizontal flat bottom required by the mirrored image technique and gain the ability to deal with complex water boundaries, a solution using the panelled moving patch technique was proposed (Zhou et al., 2014). The approach consisted in distributing a layer of sources not only on the wetted surface but also on the seabed beneath the moving vessel. Although the obtained good agreement between the moving patch method and the mirror image method in the flat bottom case encouraged belief that the new method will also handle appropriately the arbitrary bathymetry case, the actual capability of dealing with complex water boundaries was still limited by restrictions on the geometry and on the motion of the vessel (Zhou et al., 2012), and the scenario had to be one of the following: 1. the ship sailing over a flat and sloped bottom, 2. the ship on a course parallel to an either piercing or submerged bank with various forms of profile: vertical, inclined, or stepped etc. One can see that in either case there is no difficulty in meshing the moving patches as they can be easily discretized with rectangular elements. However this option is no longer available when it goes about any situation different from those mentioned above. For instance, the patch under a vessel in a canal on a course non-parallel to the bank is a non-rectangular polygon requiring an advanced meshing technique. Among all available algorithms suitable for meshing the moving patch on the seabed, the advancing front meshing algorithm— the paving (Blacker and Stephenson, 1991) has two features most desired by a real-time simulation: a high computational efficiency and automatic all-quadrilateral meshing. Although the paving algorithm has been improved by later contributions (White and Kinney, 1995; Kinney, 1997) and successfully applied in various engineering and scientific areas including, for instance, electromagnetics (Moreno et al., 2011), it still sometimes generates meshes containing poor quality elements, and even invalid elements that cannot be improved by its quality improving procedure. This issue is particular critical for online simulations that require dynamical meshing, and must be solved prudently as inferior or invalid elements can cause a meshing failure and thus halt the simulation. Several of such circumstances are discussed in this paper, focusing on the causes of the irregularities, their detection, and possible solutions. The smoothing procedure, as a technique independent of mesh generation, plays a very important role in the process of producing meshes of good quality. The Laplacian smoothing algorithm that accompanied the paving algorithm is computationally efficient and easy to implement, but in many cases it can only worsen the quality of the mesh. On the other hand, the optimization-based smoothing methods typically result in a very good mesh quality (Canann et al., 1993; Batdorf et al., 1997), but their computational cost is unacceptably high. As real-time simulations of hydrodynamic interaction loads have two very strict requirements: computational efficiency and acceptable mesh quality, neither of the Laplacian smoothing or the optimization-based smoothing is a unique suitable solution. In this paper, the panelled moving patch method is used to deal with hydrodynamic interaction problems involving arbitrary geometry
of the seabed, while the mesh for the moving patch is dynamically generated by the paving algorithm and smoothed by a hybrid smoother combining the Laplacian algorithm and an angle-based algorithm (Xu and Newman, 2006). Finally, this approach is applied for studying two cases of hydrodynamic interaction involving complex water boundaries: (1) the ship sailing across a submerged bank from deep to shallow water that can be interpreted as a ship entering a continental shelf and the hydrodynamic interaction loads are computed for a variety of approach angles, and (2) the ship sailing obliquely across an approach channel, various approach angles and at various values of the channel width.
2. General theory of hydrodynamic interaction loads Consider the global coordinate system Oξηζ, with the ξ-axis laid on the undisturbed water surface, the ζ-axis directed vertically downwards, and the η axis defined by right-handed convention. The instantaneous advance ξC , transfer ηC , and the heading angle ψ of the ship are defined with the help of the body-fixed frame Cxyz attached to the vessel in question, with its origin located at the midship, the x-axis pointing to the bow, y-axis to the starboard, and the z-axis downwards. The geometry of any complex flow boundaries, like a canal bottom, a bank, or a jetty, is expressed and loaded in the global coordinate system at the beginning of the simulation, and converted at each computational instant to the body-fixed frame whose instantaneous location and orientation in the global system can be calculated from the velocity of the origin C–VC and the angular velocity ΩC . A layer of sources is distributed on the wetted ship surface and also on a moving patch of some sufficient size placed beneath the ship in consideration to express the local geometry of the seabed. Under the assumption of a sufficiently low Froude number, the mirror-image principle is applied to the undisturbed free surface, so that the fluid domain is defined by the doubled hull and the panelled moving patch is considered. The total velocity potential Φ takes the form: Φ ¼ V ξcur ξ þ V ηcur η þ ϕ;
ð1Þ
where V ξcur and V ηcur are the components of the velocity of the current (if any) in the global axes, and ϕ ¼ ϕðξ; η; ζ; tÞ is the perturbation potential. Then the induced velocity is: VI ¼ ∇ϕ:
ð2Þ
At any time moment, the perturbation potential should satisfy the Laplace equation Δϕ ¼ 0;
ð3Þ
and on the ship wetted surface and on the moving patch, the following non-penetration boundary condition is applied: ∂ϕ ¼ Vr Un; ∂n
ð4Þ
where n is the outward unity normal to the local geometry, Vr is the relative local velocity: Vr ¼ V Vcur ;
ð5Þ
where V is the absolute local velocity on the body surface (is zero on the moving patch). The ship’s wetted surface and the moving patch on the water flow boundaries chosen in the vicinity of the ship constitute the total surface S in the following equations: Z ∂GðM; PÞ 2πσðMÞ þ σðPÞ dSðPÞ ¼ f ðMÞ; ð6Þ ∂nM S where σ is the source density, Mðx; y; zÞ is the field point, Pðx0 ; y0 ; z0 Þ is the source point belonging to the surface S, and Gð Þ is the Green
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
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function:
Predefined boundary nodes
1 1 Gðx; y; z; x0 ; y0 ; z0 Þ ¼ þ ; r r
Interior geometry boundary
w Ne
Once Eq. (6) is solved, the induced velocity and induced potential at a point can be obtained by summing up the contribution of each panel. In turn, the pressure at this point can be calculated using the unsteady Bernoulli equation: ∂ϕ 1 p ¼ ρ þ ðV2r V2p Þ ; ð9Þ ∂t 2
ts
ð8Þ
n me el e
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz þ z0 Þ2 :
New elements
ð7Þ
Exterior geometry boundary
where V p ¼ VI Vr ;
ð10Þ
where VI is the induced velocity The total hydrodynamic inertial force Fpi and moment Mpi can be calculated by: Z Z ð11Þ Fpi ¼ pndS; Mpi ¼ pr ndS: Si
Si
and their components in the body axes are: the surge force X P , sway force Y P , and yaw moment N P , respectively. The proper hydrodynamic inertial forces and moments are X e ¼ μ11 u_ þ μ22 νr Y e ¼ μ22 ν_ μ26 r_ μ11 ur N e ¼ μ26 ν_ μ66 r_ þ ðμ11 μ22 Þuv μ26 ur;
ð12Þ
where μij are the added mass coefficients. Finally the pure interaction loads are: X I ¼ X p X e ;
Y I ¼ Y p Y e;
NI ¼ Np Ne :
ð13Þ
3. Geometry discretization and mesh generation In the present implementation according to the Hess and Smith method, distributing sources requires meshing the underwater portion of the ship hull and the moving patch on the seabed. The former geometry is pre-discretized and loaded before the simulation starts; while the latter is dynamically meshed using a modified version of the paving algorithm with improved robustness. 3.1. Review of the paving algorithm The paving algorithm starts meshing from the nodes predefined on the geometry boundaries. In this context a boundary, unlike its definition in describing a fluid domain where it refers to a surface that encloses a volume in space, is a circuit of line segments by which a non-closed surface is bounded. New elements are created according to the boundary nodal configurations and are properly placed starting from the boundary toward the interior. The paving boundary or the advancing front is being updated as the new elements are inserted. The paving boundaries are advanced alternatively if there is more than one boundary. When intersections of the paving boundaries (including the selfintersection) are detected, an intersection handling module is invoked to connect the fronts and, depending on the situation, to merge two paving boundaries or split one boundary into two. This process continues until all the paving boundaries are closed. Fig. 1 shows an example of meshing a rectangle plate with a hole in it, where the arrows show the directions in which the elements
Fig. 1. Illustration of the paving algorithm.
are installed along the exterior and interior geometry boundaries, respectively. The cracks, characterized by the small angles between two adjacent edges of elements and often emerging during a paving process, are closed by the procedure called seaming which can be viewed as a “tailor” who stitches up the cracks between the elements of a mesh. The boundary is checked for the necessity of seaming after any action that might form cracks, i.e. after placing new elements, boundary node smoothing, row adjustment, connecting intersections, etc. As new elements are being added to the existing mesh, the unmeshed region shrinks and tends to close. A paving region that is about to close is detected by monitoring the number of nodes remaining in the boundary. When this number reduces to a certain value (usually 4 or 6), the boundary is closed by filling one or more elements into the remaining unpaved region in a predefined pattern. However, as will be discussed later in this paper, in some situations the standard closing procedure may cause a meshing failure. After the entire geometry is meshed, a “clean up” procedure is applied to evaluate the sizes and interior angles of all the generated elements, and to improve the quality by means of merging neighbouring elements or by a localized re-meshing when necessary. 3.2. Robustness improvements The original paver may fail to generate a mesh in some situations. The failure that happens most frequently is that the paver is unable to close a relatively small paving region due to a harsh transition of element size, which, unfortunately, is quite the case in this real-time hydrodynamic interaction simulation when the moving patch is divided into several paving regions. The paving algorithm closes a paving region when there are six or less nodes remaining in the paving boundary. For a paving boundary containing only two nodes, a seaming process is applied to close the regain. When there are four nodes “hanging”, the boundary is closed simply by inserting an additional element. In the case of six nodes, two, three, or four quadrilateral elements are inserted depending on the configurations of those nodes, which is achieved by identifying the configuration of the nodes and then filling the paving region with elements in a pattern that is predefined accordingly. However in the simpler case of four nodes remaining, as shown in Fig. 2, insertion of the element not only increases the valency of the node N 4 from five to six (this is an unfavourable circumstance
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per se as high-valency nodes result in elements with small interior angles), but also produces the unacceptable bivalent Node N 3 . In another case shown in Fig. 3, no bivalent nodes are produced during the closure process, but the quality of the new element and its neighbouring elements is not guaranteed. These elements have either invalid (very large, very small, or even negative) interior angles or very large element side ratios, which will challenge the smoother invoked afterwards to improve the mesh quality. In fact, these situations mostly arise from a transitional failure of the seaming process where bad elements are produced if the seaming procedure is executed unconditionally. Note also that both described cases are accompanied by an abrupt change of the element size. Although these unfavourable situations do not usually happen, they present a critical robustness issue as the real-time simulation will immediately break down due to the meshing failure. The solution is to select a number of installed elements and remove them so that the paver can close the boundary successfully. To accomplish this, an installed node on the paving boundary with the lowest valency is chosen, and then all its attached elements are removed. In the first case, see Fig. 2, the paving region is closed by removing the element attached to the bivalent node N4 and then merging the nodes N 1 and N 3 . Fig. 4 shows the closed region after the smoothing was applied. When there are no bivalent nodes in the paving boundary, the decision on which elements are to be removed must be carefully made depending on the configurations of the interior angles, node valences and involved side length ratios. As shown in Fig. 3, where the nodes N3 and N4 are candidates for removal, comparison of their configurations shows that removing the elements attached to the node N 4 and merging N 3 with N 4 is a better choice. Then, the new paving boundary contains four nodes, and can be closed by inserting an additional element (formed by the nodes N 1 or N 3 , N6 , N7 and N 8 ) in the new paving boundary. Fig. 5 shows the closed paving region after the applied smoothing.
surrounding nodes. The nodal location updating process deals with one node at a time. After all the nodes in the given set are updated once, the convergence is considered attained if the changes of the location of the nodes are smaller than some predefined tolerance, otherwise the procedure continues to iterate until such a tolerance is met. Due to the intrinsic mechanism of the algorithm described above, the Laplacian smoothing fails often in many situations for at least two reasons. The optimality of the nodal location proposed by the algorithm relies highly on the distribution of the surrounding nodes, and there are situations that are not favourable for reaching the optimality. Fig. 6 shows the case where the node to be smoothed is located near a permanent boundary corner and the Laplacian smoother tends to move the node N6 to a worse location. The second reason is that its smoothing performance also depends on the order in which the nodes of an element are optimized. Fig. 7 shows the case where the smoothing procedure starting with the node N2 results in a sliver-like element, but when it starts with the node N4 the desirable element is obtained. Aiming at overcoming the shortcomings of the Laplacian smoothing with the minimum computational cost, the anglebased mesh smoothing algorithm is chosen to handle the elements untreatable by the Laplacian smoothing. The idea of the angle-based smoothing is to mount a system of virtual torsion springs between the nodes, and to achieve the mesh smoothing goal by minimizing the system potential energy. As shown in Fig. 8, let Vji be the vector from the surrounding node Ni to the node Nj , then the vector V0ji from the node N i to the
N3
N2
3.3. Hybrid mesh smoother The Laplacian smoothing algorithm accompanying the paving algorithm optimizes the mesh by updating the nodal locations with optimal ones calculated from the coordinates of the
N5
N6
Fig. 4. Remedy of a closing paving boundary with a bivalent node.
N4 N3
N2
N2
N1 N5
N6
N1
N8
N6
N5
N7
Fig. 2. A closing boundary with a bivalent node that requires special treatment, the hatching denotes the unpaved region. Fig. 5. Remedy of a closing paving region containing no bivalent nodes.
N3
N2
N1
N1
N4
N2 N5
N6
N7
N6
N8
Fig. 3. A closing boundary without bivalent nodes requiring special treatment, the hatching denotes the unpaved region.
N3
N4
N7 N5
Fig. 6. A case that the Laplacian smoothing algorithm can hardly improve.
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
107
optimal location calculated based on the location of Ni , V0ji , can be expressed as ½V0ji T ¼ RðθÞVji
ð14Þ
where RðθÞ ¼
cos θ
sin θ
sin θ
cos θ
;
θ¼
αβ 2
ð15Þ
Then the change in location of the node N j due to N i is ΔVji ¼ V0ji Vji ;
ð16Þ
Fig. 9. A breakwater meshed.
Repeating the process for all the surrounding nodes, the contribution of each node to the change in the location of the node N j is obtained, and then the average of the contributions of all the surrounding nodes is taken as the final modification ΔVj . It must be noted that the smoothing solution employed here combines the Laplacian smoothing and the angle-based smoothing, the former being the primary smoother while the latter is only used when the Laplacian smoothing fails, i.e. when the convergence of the Laplacian smoothing is still not reached after a certain number of iterations. Fig. 9 shows a breakwater meshed together with the adjacent bank and with a square of sea bottom around it. This mesh contains 9989 quadrilateral elements and none of them is misshapen or sliver-shaped. Fig. 10. Discretization of the ship hull and the moving patch on the seabed.
N3
N 3'
N2
N 2'
N 4'
4. Examples of interaction force computation
N4 N 1'
N1
Fig. 7. The solid lines represent the original element, while the dashed lines denote the resulting elements with each node being the starting node, respectively.
N i+1
4.1. Ship sailing across a submerged bank
Ni i i
V j i'
V ji
Incorporation of the paving algorithm described above boosts the effectiveness of hydrodynamic interaction simulation originally only applicable to the case of simplified courses. Two scenarios requiring handling of complex water boundaries were modelled and the interaction forces with difference configurations were computed. The first scenario implies a ship sailing across a submerged bank from the deep to the shallow water, which may represent a ship entering the continental shelf. The second scenario simulates oblique sailing across a dredged channel. The still water assumption is made although steady current could be present in either scenario. The ship form used for the calculation and analysis throughout this work has main dimensions L B T of 189.6 m 31.6 m 10.3 m, and its block coefficient is 0.815. The hull was approximated with 542 panels.
N i-1
N j i' ΔV ji Nj
Fig. 8. Node N j , with 4 elements attached to, is to be smoothed. The dashed lines denote the diagonals of each element passing Node N j .
In this simulation, the ship is advancing with the speed of 3 m/s across a vertical bank from the deep (22 m, d=T ¼ 2:136) to shallow (12 m, d=T ¼ 1:158) water. A rectangular patch with dimensions of 400 m 200 m is placed beneath the ship, which is dynamically discretized at each time into quadrilateral elements with side lengths approximately equal to 8 m, as shown in Fig. 10. Of course, the moving patch method requires an increased number of panels. The total number of panels contained in the resultant mesh varied from 1250 (before the moving patch reached the bank) to 1392 (when the ship was passing exactly over the bank). The computation time required to solve the flow state for one ship position was about 1 to 3 s, and this speed was achieved without any implementation of parallelism and executed on a rather outdated computer equipped with Intel Pentiums E5200 CPU and 2 Gigabytes of memory. It can be expected that, with processors powered by up-to-date technology and/or parallel
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X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
computing, the method is able to simulate this and even more complicated scenarios in real-time. In order to investigate the behaviour of the hydrodynamic interaction forces on a ship entering the shelf at different angles, as shown in Fig. 11, the simulation was repeated for θ ¼15, 30, 45, 60, and 75 degrees. The numerical results for the surge, sway forces and the yaw moment are nondimensionalized by Fig. 12: X0 ¼
2X ρLTV
; 2
Y0 ¼
2Y ρLTV
; 2
N0 ¼
2N
0.03
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.02 0.01
ð17Þ
ρL2 TV 2
where ρ is the water density, L the ship length, T the ship draught, and V the ship speed. These results are presented in Figs. 13–15 as a 0 function of the relative distance d ¼ d=L, where d is the distance between the origin of the frame fixed to the ship hull and the bank in the ξη-plane of the global frame. The simulation shows that a ship sailing across the submerged bank from deep to shallow water experiences a negative surge force which can be treated as drag. This surge force has two negative peaks in the time history. None of them appears when the midship is passing exactly over the submerged bank as the absolute value of the surge force reaches a minimum at this position. During the process, a positive sway force is present pushing the ship from deep to shallow water. This force increases rapidly with the angle θ, with the peaks appearing after the midship passes the bank. The time history of the yaw moment is of particular interest, the ship experiences a negative yaw moment until the midship passes the submerged bank where it reaches a negative peak, and then changes its sign and reaches its positive peak. 4.2. Ship sailing obliquely across the approach channel
-0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 13. Sway force acting on a ship sailing across a submerged bank.
0.004
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.002 0 -0.002 -0.004
-1
0 d'
1
ban
k
In this scenario, Fig. 15, the ship sails across a dredged channel with the speed of 3 m/s at various angles between the ship
0
Fig. 14. Yaw moment acting on a ship sailing across a submerged bank.
s ub
me rg
ed
deep water
θ
shallow water Fig. 11. A ship sailing across a submerged bank at different angles.
0 -0.002 -0.004 Fig. 15. A ship sailing obliquely across a dredged channel.
-0.006 -0.008 15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
-0.01 -0.012 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 12. Surge force acting on a ship sailing across a submerged bank.
centreplane and the central line of the channel, i.e. θ ¼15, 30, 45, 60, and 75 degrees. The channel width varies from 20 to 200 m, while the water depth is 22 m inside the channel, and 12 m outside it. The moving patch beneath the ship had the same size as used in the previous scenario. The numerical results for the forces and moments presented in Figs. 16–30 are nondimensionalised in the same way as before. However, there is a slight difference in the definition of the 0 abscissa d here, which is in this simulation the nondimensional
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
distance between the origin of the body frame and the central line of the channel. The channel width is also nondimensionalized by the ship length. Throughout all the simulated cases with various channel widths and course angles, a two-staged behaviour of the surge force over the time history is observed. The first stage begins when the ship starts to “feel” the approach channel and ends when the midship section reaches the channel centre, and the second stage follows. It is observed that an ever increasing surge force is acting on the ship from the beginning of the first stage until it reaches the peak, which is followed by a trough for small channel widths and small course angles, or by a second peak for large channel widths and large course angles (Fig. 16). However, this second peak or trough shrinks and merges with the first peak as the relative heading angle increases (see Figs. 19, 22, 25, and 28). In the second stage, a similar but reversed pattern is observed, with slight differences in the absolute values of the peaks and troughs. Compared to the surge force, the pattern of the time history of the sway force is rather simpler. A negative sway force dominates through the first stage with the maximum value in the middle, while it is the opposite for the second stage. The effect of the channel width is negligible except for the two smallest channel widths where the cancellation effect of the pair of sidewalls takes on. On the other hand, the maximum values of the sway force increases rapidly with the course angle. See Figs. 20, 23, 26 and 29. Independent of the channel width and course angle, the time history of the yaw moment starts and ends with a positive peak,
109
0.01
0.105L 0.264L 0.422L 0.738L 1.055L
0.005 0 -0.005 -0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 18. Yaw moment on the ship with a course angle of 15 degrees.
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.005 0 -0.005
0.005 -0.01 -1.5
0
-1
-0.5
0 d'
0.5
1
1.5
Fig. 19. Surge force on the ship with a course angle of 30 degrees.
-0.005
0.03 0.105L 0.264L 0.422L 0.738L 1.055L
-0.01 -0.015 -1.5
-1
-0.5
0.105L 0.264L 0.422L 0.738L 1.055L
0.02 0.01 0 d'
0.5
1
1.5
Fig. 16. Surge force on the ship with a course angle of 15 degrees.
0 -0.01 -0.02
0.06 0.105L 0.264L 0.422L 0.738L 1.055L
0.04
-0.03 -1.5
0.02
-1
-0.5
0 d'
0.5
1
1.5
Fig. 20. Sway force on the ship with a course angle of 30 degrees.
0 -0.02 -0.04 -0.06 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 17. Sway force on the ship with a course angle of 15 degrees.
and in-between it is dominated by a negative moment which, for the cases of large channel widths and large crossing angles, may fluctuate several times before reaching the negative peak (Figs. 21, 24, 27, and 30). The results obtained for the 60 m (0:316L) and 200 m (1:055L) channel width are represented in the different form in Figs. 31–36 clearly demonstrating influence of the crossing angle. The cancellation effect of the channel walls diminishes as the width increases, and the dredged channel case degrades to two separate submerged banks. When the width exceeds some value,
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X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.004
0.004
0.105L 0.264L 0.422L 0.738L 1.055L
0.002
0.002
0
0
-0.002
-0.002
-0.004
-0.004
-0.006 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 21. Yaw moment on the ship with a course angle of 30 degrees.
-0.006 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 24. Yaw moment on the ship with a course angle of 45 degrees.
0.01
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.005
0.105L 0.264L 0.422L 0.738L 1.055L
0.005 0
0
-0.005
-0.005 -0.01 -1.5
0.105L 0.264L 0.422L 0.738L 1.055L
-1
-0.5
0 d'
0.5
1
1.5
-0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 25. Surge force on the ship with a course angle of 60 degrees. Fig. 22. Surge force on the ship with a course angle of 45 degrees.
0.015
0.03
0.105L 0.264L 0.422L 0.738L 1.055L
0.02
0.005
0.01
0
0
-0.005
-0.01
-0.01
-0.02 -0.03 -1.5
0.105L 0.264L 0.422L 0.738L 1.055L
0.01
-1
-0.5
0 d'
0.5
1
1.5
-0.015 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 26. Sway force on the ship with a course angle of 60 degrees.
Fig. 23. Sway force on the ship with a course angle of 45 degrees.
one of these two submerged banks represents the first scenario considered before—a ship sailing across a submerged bank from deep to shallow water. In order to find the similarities between the two cases, the interaction forces are compared for 45 degrees of course angle in Figs. 37–39, where the abscissa is the nondimensionalized distance between the ship and the corresponding bank. It can be seen for each force component that the two cases have very different patterns in the first half of the interaction process due to the presence
of the other wall of the channel; however, as the channel width increases, both cases converge to each other.
5. Conclusions Based on the panelled moving patch method and the paving algorithm, a solver to hydrodynamic interaction is proposed and
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.001
0.003 0.105L 0.264L 0.422L 0.738L 1.055L
0.002
0
0
-0.0005
-0.001
-0.001
-0.002 -1
-0.5
0 d'
0.5
1
1.5
Fig. 27. Yaw moment on the ship with a course angle of 60 degrees.
0.008
-0.0015 -2
-1
0 d'
0
-0.004
-0.005
0 d'
1
2
Fig. 28. Surge force on the ship with a course angle of 75 degrees.
2
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.005
0
-1
1
Fig. 30. Yaw moment on the ship with a course angle of 75 degrees.
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.004
-0.008 -2
0.105L 0.264L 0.422L 0.738L 1.055L
0.0005
0.001
-0.003 -1.5
111
-0.01 -1.5
-1
-0.5
0 d'
0.5
1
1.5
Fig. 31. Surge force for 60 m wide channel and various crossing angles.
0.06
0.01 0.105L 0.264L 0.422L 0.738L 1.055L
0.005
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.04 0.02 0
0
-0.02 -0.04
-0.005 -0.01 -2
-0.06 -1.5 -1
0 d'
1
2
-1
-0.5
0 d'
0.5
1
1.5
Fig. 32. Comparison of sway force between various angles at 60 m of channel width.
Fig. 29. Sway force on the ship with a course angle of 75 degrees.
applied to the study of the hydrodynamic interaction loads on a ship crossing submerged banks and dredged channels. The simulated cases demonstrate its ability to deal with those situations involving complex flow boundaries where experimental methods are problematic. Previous studies showed that even in the case of ships sailing in shallow water with a horizontal flat bottom, more accurate
numerical results can be obtained by increasing the panel density in the areas on the moving patch close to the ship hulls. Such a technique, which can be implemented through integrating an automatic mesh adaptation algorithm, can certainly improve the accuracy of the numerical results for the cases with complex flow boundaries, while, however, introducing additional complications in the geometric processing of the patch.
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X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.004
0.005
0.002 0
0
-0.002 -0.004
-0.005
-0.006
-0.01 -1.5
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
-0.008 -1
-0.5
0 d'
0.5
1
1.5
Fig. 33. Comparison of yaw moment between various angles at 60 m of channel width.
0.01
-1
0 d'
1
2
Fig. 36. Comparison of yaw moment between various angles at 200 m of channel width.
0.01
15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.005
-0.01 -2
Sub. Bank 0.422L 0.633L 0.844L 1.055L
0.005
0
0
-0.005 -0.005
-0.01 -0.015
-2
-1
0 d'
1
2
-0.01 -3
-1
0
1
2
d'
Fig. 34. Comparison of surge force between various angles at 200 m of channel width.
Fig. 37. Surge force for 45 degrees crossing angle and various channel width.
0.03
0.06 15 degrees 30 degrees 45 degrees 60 degrees 75 degrees
0.04
0.01
0
0
-0.02
-0.01
-0.04
-0.02 -1
0 d'
Sub. Bank 0.422L 0.633L 0.844L 1.055L
0.02
0.02
-0.06 -2
-2
1
2
-0.03 -3
-2
-1
0
1
2
d'
Fig. 35. Comparison of sway force between various angles at 200 m of channel width.
Fig. 38. Sway force for 45 degrees crossing angle and various channel width.
The hydrodynamic interaction computation method had been validated for the particular case of the flat seabed, and there are reasons to believe that it remains consistent in more complex cases although direct experimental confirmation is desirable and should be attempted in the future.
The squat effect has not been accounted for in the present study focused mainly on the improvement on the moving patch method but it is known that the squat in shallow water can significantly influence the interaction forces and accounting for it is also envisaged for future research work.
X. Zhou et al. / Ocean Engineering 103 (2015) 103–113
0.004 Sub. Bank 0.422L 0.633L 0.844L 1.055L
0.002 0 -0.002 -0.004 -3
-2
-1
0
1
2
d' Fig. 39. Yaw moment for 45 degrees crossing angle and various channel width.
Acknowledgement The study was carried out within the framework of the project Energy Efficient Safe SHip OPERAtion (SHOPERA) financed by the EU under contract number 605221 of the 7th Framework Programme. The first author was supported by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e Tecnologia), under contract no. SFRH/BD 75354/2010. References Batdorf M., Freitag L.A., Ollivier-Gooch C. (1997) Computational study of the effect of unstructured mesh quality on solution efficiency. In: Proceedings of the 13th Annual Computational Fluid Dynamics Meeting. Snowmass Village, CO. Blacker, T.D., Stephenson, M.B., 1991. Paving: a new approach to automated quadrilateral mesh generation. Int. J. Numer. Methods Eng. 32, 811–847. Canann, S.A., Stephenson, M.B., Blacker, T.D., 1993. Optismoothing: an optimization-driven approach to mesh smoothing. Finite Elem. Anal. Des. 13, 185–190. Chen H.-Ch., Lin W.-M., Liut D.A., Hwang, W.-Y. (2003) An advanced viscous flow computation method for shi-ship dynamic interactions in shallow and restricted waterway. In: Proceedings International Conference on Marine Simulation and Ship Maneuverability (MARSIM’03), 25–28 Aug, Kanazawa, Japan, vol. 3, pp. RC-35-1–10. Ch’ng, P.W., 1991. An Investigation into the Influence of Bank Effect on Ship Manoeuvring and its Mathematics Modeling for a Ship-Handling Simulator ME.
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