Tài liệu Journal of ship research, tập 56, số 03, 2012

Journal of Ship Research, Vol. 56, No. 3, September 2012, pp. 129–145 http://dx.doi.org/10.5957/JOSR.56.3.100031 Journal of Ship Research Validation of Potential-Flow Estimation of Interaction Forces Acting upon Ship Hulls in Parallel Motion Serge Sutulo,* C. Guedes Soares,* and Janne F. Otzen† *Centre for Marine Technology and Engineering, Technical University of Lisbon Instituto Superior Técnico, Lisboa, Portugal { Force Technology, Kongens, Lyngby, Denmark The hydrodynamic interaction problem is of great importance for numerical ship handling simulators and at present, only a relatively simple potential double body panel method can be expected to supply estimates of interaction forces and moments in real time on commonly used hardware, without limitations on the hull shape, and on the mutual position and motion of the interacting bodies. Such a code was developed on the basis of the classic Hess and Smith method and proved to be fast enough to model interaction in real time when a moderate number of panels is used. In the present paper, results obtained with the potential code are validated against experimental data obtained in deep and shallow water towing tanks for the case of a tug operating near a larger vessel. All the tests corresponded to the steady regime and only cases with parallel center planes were considered here. The comparisons carried out for various discretization of the hulls provide useful information about natural limitations in breaking at too close lateral distances and about acceptable trade-off between the computational speed and accuracy. In addition, influence of the nonzero sway and yaw velocities is investigated numerically. Keywords: hydrodynamic interaction; potential flow; panel method; tank experimental data 1. Introduction MARINE VEHICLES of any kind must often maneuver in the presence of various rigid objects such as other vessels, which can be moving, anchored, or moored; water basin boundaries; banks of various shape and length; piers and jetties; and floating production storage, and offloading units. In all those cases, hydrodynamic interaction appears becoming sometimes very significant and jeopardizes navigation safety. Hence, the ability to predict the corresponding hydrodynamic loads is of great practical value and this problem attracted attention of hydrodynamicists for more than 100 years. Results of interaction studies can benefit the seamanship practice in the following ways: Qualitative understanding of the interaction phenomena, working out recommendations for ship operators; Manuscript received at SNAME headquarters May 16, 2010; revised manuscript received October 10, 2011. SEPTEMBER 2012 Estimation of the peak values of the loads, safe distances, and safe velocities in overtaking and encountering situations; Estimation of loads in the mooring lines caused by vessels passing by; Analysis of navigational accidents; Working out maneuvering standards; and Modeling of interaction effects in computerized maneuvering simulators. Specifics of the last and the most recent application include that it requires a combination of reasonable accuracy with high computational efficiency and necessity to predict interaction forces and moments for arbitrary relative position and motion of all interacting bodies. Of course, overtaking and encountering maneuvers continue to be the most critical from the viewpoint of possible accidents, as a too close approach on crossing courses is not likely. At least, this is true for ships of more or less comparable size but, for instance, tugboats can approach the assisted ship in a more or less arbitrary way especially when they work in the pushing mode (van Hilten & Wulder 2006). This paper is mainly aiming to this kind of application. 0022-4502/12/5603-0129$00.00/0 JOURNAL OF SHIP RESEARCH 129 The overall number of publications on the hydrodynamic interactions apparently exceeds 100. As an extensive review of earlier studies starting from first experimental results reported by Taylor in 1909 was presented by Norrbin (1975); the review presented here will mainly focus on more recent contributions. Some of these studies consider specifically the ship-to-ship interactions while some others concentrate on bank-suction problems especially when the ship is sailing in a canal. In many cases, however, both types of problems are solved with similar methods, which can be divided into theoretical, experimental, and numerical. Of course the dividing line between “theoretical” and “numerical” approaches is often fuzzy as every numerical method always has certain theory behind it, whereas any theoretical solution must end in computations. Typically, numerical methods are more straightforward, require better computing power, and are more versatile in applications while theoretical methods mostly are based on certain asymptotic assumptions about the shape of the bodies and their mutual location and motion. A fundamental theoretical study of the ship-to-ship interaction was carried out by Abkowitz et al. (1976). Their formulation was based on the so-called Havelock hypothesis according to which the potential flow model with rigid flat free-surface corresponding to the zero-Froude-number case is sufficient to explain the interaction phenomena. This hypothesis looks quite reasonable at low Froude numbers and, although not quite accurate, was exploited in many studies including the present one. Abkowitz et al. (1976) introduced, however, additional assumptions: the ship hulls were supposed to be slender, they were approximated with bodies of revolution and these were approximated with axial source and dipole distributions. Horizontal dipoles originated from lateral motions of the “own” ship (i.e., the ship of interest) with respect to water and from the induction of the other ship, and vertical dipoles were added in shallow water. The strength of the dipoles was determined as result of a fast iteration process and the forces and moments were computed basing on the Lagally–Cummins formulae. The agreement with experimental data was not very good and to make it acceptable, a certain effective beam of the equivalent body of revolution was to be introduced. It was found that unsteadiness of the flow affects much more the yawing moment, than the sway force. Beck (1977) applied the matched asymptotic expansions method to the problem of ship-to-bank interaction in a shallow channel. In the outer region, the shallow water theory was applied, so the final results depended on the depth Froude number. Besides sway force and yaw moment computed with the Blasius formulae, the sinkage force and trimming moment were estimated from the linearized Bernoulli integral. A conclusion was drawn that the horizontal two-dimensional (2D) model was not satisfactory even at very small bottom clearances. A similar method was much later used by Kijima (1997) who studied interaction between three bodies: two ships running parallel in the proximity of a pier simulated with a circular or oval cylinder. In this study, the hydrodynamic part was combined with dynamics of the ships and time histories for forces and motion were obtained. Interesting theoretical studies were undertaken by Yeung (1978) who applied the slender-body theory and the method of matched asymptotic expansions. The method results in relatively fast algorithms and accounts for the Kutta condition on the aft part of the hull, but it heavily underestimates sway forces and yaw moments in the most interesting case of small separation between the hulls. A similar approach was used by Yeung and Tan (1980) to study interaction with some fixed obstacles like a protruded wall, wedge, or embankment corner. This study was later extended by Hsiung and Gui (1988) to a larger variety of obstacles. Tuck and Newman (1974) performed one of the best asymptotic studies of the ship-to-ship interaction problem. Two situations were investigated, both for the zero Froude number. In the first case, the water depth was assumed to be infinite or finite but relatively large. The so-called first-order theory was constructed, that is, each ship was just sailing in the flow field of its partner. Then, the lateral loads on each of the slender hulls were computed using the generalized Taylor theorem for a body moving in an accelerated uniform flow. The Kutta condition was accounted for on sterns of finite height. The theory resulted in formulae for the sway force and yaw moment having a quadratic form structure. Namely, for a ship in parallel uniform motion with the Nomenclature Ci = G() = H= m= mi = n= N= p= r= r= S= T= u= v= V= X= x, y, z = Y= g rts ; ers = 130 origin of the body axes for the ith body Green’s function depth of the fluid number of interacting bodies components of the generalized normal outer unity normal yaw moment pressure position vector velocity of yaw or distance rigid boundary in the fluid fluid kinetic energy velocity of surge velocity of sway velocity surge force coordinates in the body axes sway force Boltzmann’s symbols SEPTEMBER 2012 x, h, z = mij = pr = r= s= F= f= W= wi = Ñ= coordinates in the global fixed axes added mass coefficients a quasi-coordinate water density source density total velocity potential perturbation potential angular velocity a quasi-velocity the Hamilton operator Subscripts Ci = cur = e= i= I= p= r= related to the origin of the ith body current proper (eigen) number of the body (ship) induced (total) potential force/moment or “pressure” velocity relative JOURNAL OF SHIP RESEARCH SEPTEMBER 2012 + + + + + + + added masses. Ship overtaking maneuvers were simulated and collision-dangerous situations were traced. Finally, a Reynolds-Averaged Navier–Stokes Equation (RANSE) code was applied by Chen et al. (2003) for tackling the problem of ship-ship interaction in shallow water. Good agreement with experimental data was demonstrated except for the case of relatively small distance between the boards when insufficient fineness of the grid was suspected. Pinkster (2004) focused on a rather specific interaction problem related to necessity of estimation of the mooring line loads originated from vessels passing by. Specifics of the situation are that although the passing vessels sail at deeply subcritical Froude numbers even in small water depth and the Havelock hypothesis must hold, these vessels generated low-frequency seiches in the harbor area whose influence dominates the near-field interaction. Hence, in Pinkster’s algorithm the near-field interaction was completely neglected as well as the ship-waves wash. The moving ship was considered as a wave generator and then a diffraction problem was first solved in the frequency domain and further converted to the time domain by means of the fast Fourier transform. An interesting case where the free-surface interaction effects turned out decisive was studied numerically by Söding and Conrad (2005). A real collision case happened on the Elba River and was studied post factum. A larger vessel with dimensions L B T ¼ 264.0 39.9 11.4 m was running at 15 kn and overtaking a smaller ship with particulars L B T ¼ 101.0 18.4 6.3 m cruising at 13.5 kn speed. The water depth was about 14.5 m making the depth-based Froude number equal to 0.65, that is, large enough for most shallow-water wavemaking effects to show up. The distance between the centerplanes of the ships was 150 m, which makes the near-field interaction negligible while the waveorigin interaction was so strong that the smaller ship was captured by depression of the water surface, lost its steering capability, and finally collided with the overtaking vessel. Two different Rankine source nonlinear free surface potential flow codes were used and results were compared with experimental data giving very good agreement for the yaw moment, a satisfactory one for the surge force, and a somewhat worse agreement for the sway force. One of the codes was run with and without vortex distribution aiming at satisfying the Kutta condition. The difference was only significant for the sway force, where presence of vortices improved the agreement. A large group of studies was based on experimental methods. An approximate empirical method for predicting forces in overtaking maneuvers was developed by Brix (1993). Extensive automated tests with scaled models of four different shapes were carried out by Vantorre et al. (2002) covering only parallel motion, that is, overtaking, passing by, and encountering situations in shallow water. A special two-carriage experimental layout was used and time histories of the surge and sway interaction forces and of the yaw moment for various lateral distances and speed values were obtained. On the basis of the huge amount of assembled data a simple approximate empiric method for estimating peak values of the interaction forces was proposed. Later, Vantorre et al. (2003) also studied interaction of a moving ship with banks of various configurations. Results were finally processed to the form of regressions for the sway force and yaw moment, which were found to depend on the under-keel clearance, distance between the ship and the bank, Froude number, propeller slipstream velocity, and a few hull form parameters. + other the formula for the loads had the structure aV1 V2 þ bV22 , where a, b are constant coefficients, V1 is the speed of the “own” vessel upon which the estimated force is acting, and V2 is the speed of another or “target” ship. Here, one of the limitations of the theory is obvious: when the target ship is at rest, no interaction is observed. The second studied case is related to the pair of ships moving with the same speed in extremely shallow water. Here the flow was considered far-field 2D in the horizontal plane and was treated by matched asymptotic expansions. Threedimensional (3D) effects were accounted for by means of the so-called blockage coefficient. The paper also explains why the double-body potential formulation makes sense in the interaction and maneuvering problems: unlike what is happening in seakeeping and wave resistance problems, this approximation still remains nontrivial and captures the main effect. It was also supposed that viscous effects would be more significant than those related to the wavemaking. King (1977) focused on unsteady effects accompanying interaction of two bodies using a 2D formulation, that is, the water depth was assumed to be close to the draft. The wavemaking was completely neglected and the linearized airfoil theory was applied. The lateral distances had to be comparable to the ship lengths. The agreement with experimental data was only judged to be satisfactory for the sway force on the stationary (not moving) body. King and Tuck (1979) investigated a somewhat complementary problem of the steady motion of slender ships parallel to banks. It was shown that in the inner region near a beach with the constant slope at zero Froude number the inner problem becomes isomorphic to the axial flow problem. Gui et al. (1992) discussed application of the Schwarz–Christoffel transformation to the 2D problem of a maneuvering ship interacting with polygonal boundaries. A simplified theoretical study was undertaken by Mastushkin (1977), who studied the case of parallel (in the same or opposite directions) motion of two ships. Within the double-body approach each hull was approximated with an ellipsoid with the axes equal to the length of the ship, its breadth, and doubled draft. The velocity potential was defined approximately through the iteration process. Analytic velocity potentials for isolated ellipsoids were taken as the first approximation. The cited article does not contain any validation against experiments or independent numerical results. That validation was, however, done by the author in his earlier studies related to the case of zero drift angles. The main conclusion drawn from these works is that even a very crude approximation of the ship hull can bring good estimates for the interaction forces at least in some situations. One of the first applications of a numerical method belongs likely to Ivanov (1981) who successfully applied the potential flow algorithm by Hess and Smith to the zero –Froude number interaction problem. However, all the computations were performed offline as the then available computing hardware was too slow. A more advanced but still offline application of another version of the 3D panel potential code was undertaken by Korsmeyer et al. (1993) who studied body-to-body interaction in a rectangular canal. Only parallel motion was studied for two spheroids, two Mariner ships, and two Panamax bulk-carriers. Comparisons with experimental data showed a fairly good agreement. A direct 3D potential-flow formulation was also exploited by Yasukawa (2003) who used an unspecified implementation of the panel method for determining the flow potential and the interaction forces which were expressed through a set of generalized JOURNAL OF SHIP RESEARCH 131 Ch’ng et al. (1993) carried out bank-interaction experiments with MarAd and S-175 models equipped with propellers but without rudders. The varying factors were: the Froude number, relative bottom clearance, lateral distance, bank slope angle, and the thrust. Linear regressions were devised and it was demonstrated that the propeller thrust may influence substantially the interaction sway force and yaw moment. Li et al. (2003) developed another empirical model for predicting interacting forces with banks of three standard shapes: vertical, sloping, and flooded (stepped). The obtained regressions were validated against independent tests and used in simulations. A more specialized experimental study focusing on inland vessels was performed by Gronarz (2006). Interesting details about this study include original measurement records showing uncertainty levels, investigation of the influence of the propeller, and convenient approximations for the forces. Ankudinov et al. (2006) outlined an integrated approach to bank/ obstacle interaction problem. Although no specifics are given, it can be classified as a kind of data assimilation method based, among other things, on certain undisclosed theoretical considerations, although solution of the full hydrodynamic interaction problem was likely avoided. Lee and Kijima (2003) applied some previously developed interaction estimation algorithm to analyze the behavior of interacting ships in restricted waterways and under wind action. Varyani et al. (2003) studied mooring-lines loads originating from the hydrodynamic interaction between the moored vessel and the passing by ship. The zero –Froude number theory was used to estimate the flow field around the moving ship and the pressure integration resulted in estimates for the surge and sway forces and the yaw moment acting on the moored ship. Although the final formulae do not look obvious and the back interaction from the moored ship was completely neglected, comparisons with experimental results showed fair agreement. Later, this and other similar studies resulted in a generic set of approximations of interaction sway forces and yaw moments proposed by Varyani et al. (2004). It seems that lately more focus has been given to experimental studies and attempts to develop some simple, regression-type, algorithms for fast calculation of interaction forces. This is, likely, caused by real-time and accelerated-time simulation requirements. In-loop numerical methods can then seem inconvenient. However, the drawback of all available empirical approximations (and also most analytic solutions) is that they can only cover a limited number of situations. For instance, most of the ship-ship interaction studies were performed for the case of zero yaw velocity and for ships on parallel courses and with parallel center planes. These may be sufficient to predict the most dangerous situations in overtaking maneuvers but are not enough for correct description of the interaction in all possible situations, which is highly desirable for bridge simulators. For example, when simulating maneuvers of a large ship assisted by tugs working also in the pushing mode (van Hilten & Wulder 2006) it would be highly undesirable to impose any restrictions on the mutual motion and position of the involved vessels. On the other hand, the computing power of even average personal computers has grown during late years to such an extent that certain numerical algorithms can be certainly used online without hampering the simulation, although this is still not true for viscous-flow and/or free-surface codes. 132 SEPTEMBER 2012 In general, the hydrodynamic interaction phenomenon can be viewed as being composed of five simpler phenomena: Near-field potential interaction without a free surface; Interaction related to the free-surface effects or wavemaking interaction; Boundary layer and viscous wake interaction; Interaction caused by longitudinal trailing vortices; and Action of propeller slipstreams and of the thrusters jets. It is still difficult to draw definite and precise conclusions about the relative contribution of each component but certain experience and data accumulated so far indicate that the near-field double body potential interaction can be the most important component in many cases, that is, the Havelock hypothesis is pretty much justified. On the other hand, this is one of the few hydrodynamic models which can be treated numerically online in real time at least when the accuracy requirements are eased and the number of approximating panels is reduced. An alternative could be a vortexlattice method, which has the potential advantage of the possibility of accounting for the Kutta condition when it is thought indispensable. The latter is caused by viscosity but once formulated can be applied to nonviscous flows requiring, however, presence of vortices which cannot appear in the perfect fluid. While the Kutta condition can be easily formulated in the case of lifting bodies and surfaces with sharp trailing edges it turned out impossible to create an adequate formulation for most slender ship hulls and it is mostly accounted for through neglecting part of the predicted transverse load on the aft part of the body when it goes about proper forces and moments. It is much less clear how it should be formulated in the three-dimensional interaction case and the literature data are, in general, inconclusive and do not definitely indicate the importance of the corresponding contribution. It is much easier to apply the Kutta condition in the twodimensional models when it is sufficient to impose a circulation (Dand 1976), but such a model is poorly applicable even at extremely low water depth. That is why, sticking with Havelock hypothesis mentioned above, Sutulo and Guedes Soares (2008) have developed a code based on the classic Hess and Smith method suitable for online computations of interaction forces for an arbitrary number of ship hulls with arbitrary relative position and motion, primarily in deep water. This method was later fused with the maneuvering simulation code and an uncontrolled and controlled motion of two vessels in overtaking maneuver was actually simulated (Sutulo & Guedes Soares 2009). After those publications, the code was extended to cover the flat bed shallow water case. All validations of the code were so far either internal (comparisons with analytic solutions, convergence studies) or numerical results were only compared against approximate empirical methods. In the present paper, the code is validated against experimental results obtained in towing tanks for a specific tugboat operating near a typical large vessel. Although the code works at any combination of position and motion of the involved vessels, only results related to parallel motion with equal speeds are validated here. They are also complemented with demonstration of the influence of the sway and yaw motion obtained numerically. The exposure of numerical results is preceded by description of the method and of the experimental setup and followed by conclusions. The general consistency of the approach followed is also confirmed by the fact that a similar online numerical method for estimating JOURNAL OF SHIP RESEARCH the interaction forces was recently and independently applied by Xiang and Faltinsen (2010). 2. Computation of Interaction Forces by means of the Potential-Flow Code 2.1. Problem statement and governing equations Let Oxhz be the Earth-fixed right-handed Cartesian frame of reference with the origin located on the undisturbed free surface of the perfect fluid and with the z-axis directed downwards as illustrated in Figure 1. The fluid is either infinitely deep or limited with a horizontal flat rigid bottom located at z ¼ H. The direction of the x-axis can be chosen arbitrarily, depending on the nature of the specific problem. There are in this fluid some bodies described by their wetted surfaces Si, i ¼ 1, . . . , m, which can be submerged completely, float on the free surface, or be somehow fixed to the bottom. In the general case, the actual surface may depend on the time t at least (for rigid bodies) in terms of their position. These bodies can represent ships, submersibles, pieces of banks, any structures, islands, etc. The number m of these bodies is arbitrary but finite and in fact most situations are covered by m  2. Then, attached to each body is the frame, Ci xiyizi , which can coincide with the Earth axes at some time moment. At any time moment t, the position of each body can be described with the vector ri or with the coordinates: advance xCi, transfer hCi, and submergence (heave) z Ci, as well as with the standard Euler angles: heading y i, pitch qi, and roll wi. The instantaneous motion of each body is described with the vectors of the velocity of the origin VCi and of the angular velocity Wi. Alternatively used are the linear and angular quasi-velocities of surge, sway, heave, yaw, pitch, and roll: ui, vi, wi, pi, qi, ri. Within the formulation followed here, the scalar parameters related to the position and motion in the plane other than horizontal are identically equal to zero. It is supposed that the fluid is perfect and the flow is irrotational, so it is fully described with the potential F(x, h, z, t) at z  0, above the bottom and outside the bodies. Also, it is worthwhile to suppose that a horizontal uniform currently described by the velocities Vxcur and Vhcur or by the vector Vcur is present. The total potential can then be represented as F ¼ Vxcur x þ Vhcur h þ f; ð1Þ where f(x, h, z, t) is the perturbation potential and the perturbation (induction) fluid velocity will then be VI ¼ Ñf. At each time moment, the perturbation potential satisfies the governing Laplace equation Df ¼ 0; ð2Þ and the nonpenetration boundary condition on each body ¶f ¼ Vr  n ð3Þ ¶n where n is the outer unity normal to each body and the relative local velocity is Vr ¼ V  Vcur ; ð4Þ where V is the absolute local velocity of a point on the body surface, depending in the common way on the parameters VCi and Wi. The low –Froude number assumption yields the following condition on the free surface: ¶f ¼0 ¶z ð5Þ and the same condition must be applied on the flat horizontal bottom. In the case of deep fluid, the perturbation potential must vanish at infinity. As the governing equation is elliptic, the time only enters in the formulation as parameter but in the case of moving bodies the boundary is unsteady and the Neumann problem must be resolved at any time moment. 2.2. Solution 2.2.1. General theory. The formulated problem is standard and a common method of solution is to distribute a single layer of sources with density s on the entire wetted surface. Then, the following Fredholm integral equation of the second kind holds: ð 2ps ðMÞ þ s ðPÞ S ¶GðM; PÞ dSðPÞ ¼ f ðMÞ; ¶nM ð6Þ where M(x, y, z) and P(x ¢, y ¢, z ¢) are respectively the field (observation) and the source points belonging here to the surface S which consists of all the wetted surfaces present in the current problem. The Green function G() in the case of deep fluid and rigid free surface takes the form Fig. 1 Frames of reference in the case of two ships SEPTEMBER 2012 1 1 ð7Þ Gðx; y; z; x¢; y¢; z¢Þ ¼ þ ― ; r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ðx  x¢Þ þ ðy  y¢Þ þ ðz  z¢Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ― r ¼ ðx  x¢Þ þ ðy  y¢Þ þ ðz þ z¢Þ. Application of the same mirror-image principle to the finite depth fluid case results in a more complicated formula for the Green function:  1  1 1 Gðx; y; z; x¢; y¢; z¢Þ ¼ ( þ― ; ð8Þ i¼1 ri ri JOURNAL OF SHIP RESEARCH 133 ð VI ðMÞ ¼ s ðPÞÑM GðM; PÞdSðPÞ; ð9Þ S ð fðMÞ ¼ s ðPÞGðM; PÞdSðPÞ: S The pressure distribution is then calculated through the unsteady Bernoulli equation which in moving frames takes the form (Lamb 1968)   ¶f 1  ð10Þ p ¼ r  þ V2r  V2p ; ¶t 2 where r is the fluid density and Vp ¼ VI  Vr : ð11Þ Then, the potential force Fpi and moment Mpi acting on ith body will be: ð ð Fpi ¼  pn dS; Mpi ¼  pr ´ n dS: ð12Þ Si Si The potential hydrodynamic load can be also represented in the standard component form, that is, in the plane motion problem, in terms of the surge force Xp, sway force Yp, and yaw moment Np. The thus defined forces and moments will, however, include the proper inertial hydrodynamic loads which are already accounted for in all consistent maneuvering mathematical models. So, to obtain the pure interaction forces, proper potential loads must be also calculated and subtracted from the loads defined by (12). The pressure Equation (10) includes the local derivative ¶f ¶t , which vanishes for steady flows. When the pressure and loads in the unsteady flow are determined with the local derivative dropped, the corresponding solution is called quasi-steady. 2.2.2. Determination of proper inertial forces and moments. The most convenient way to calculate the proper inertia loads is to apply the added-mass formalism associated with the equations devised by Thomson, Tait, and Kirchhoff (Lamb 1968). The starting point is the following representation of the kinetic energy of the fluid in the presence of a moving rigid body: where wj are the quasi-velocities of the body in concern (i.e., w1 º u—velocity of surge, w 2 º v—velocity of sway,. . ., w6 º r—angular velocity of yaw) and mjk are the added mass coefficients defined for the ith body as ð mjk ¼ r fj mk dS ð14Þ Si where m1, m2, m3 are the projections of the normal n onto the body axes x, y, and z respectively and m4, m5, m6 are similar projections of r n. Each of the potentials fj is obtained as solution to the Equation (6) with the right-hand side f (M) ¼ mj (M). Applying the third Newton law, it is possible to obtain from the Euler–Lagrange equations (Lurie 2002) the following representation for the generalized hydrodynamic reactions Ps, s ¼ 1, . . . , 6:   6 6 d ¶T ¶T ¶T ¶T þ ( g rts wt þ ( esr  ; ð15Þ Ps ¼ r;t¼1 r¼1 dt ¶ws ¶wr ¶p r ¶p s + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ri ¼ ðx  x¢Þ þ ðy  y¢Þ þ ðz  z¢ þ 2iH Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ― that is considered as r i ¼ ðx  x¢Þ þ ðy  y¢Þ þ ðz þ z¢ þ 2iHÞ a series of double hulls with the distance 2H between the waterplanes. In practical computations, the series are truncated as farther mirror images have weak influence on the pressure distribution on the central body. In this context, the point coordinates can be taken in any suitable frame of reference, that is, one can use x instead of x, etc. The right-hand side of the equation is f(M) ¼ Vr (M)n(M) when the boundary condition (3) is in effect. After the Equation (6) is solved, the induction velocity and potential distributions can be expressed through the already known single-layer density s(M): where g rts and esr are the Boltzmann symbols whose definitions and methods of evaluation are given by Lurie (2002) and p s are the quasi-coordinates, that is, p_ s ¼ ws. It can be shown that in the case of a body in the unbounded fluid the fluid kinetic energy does not depend on the generalized coordinates and on the quasi-coordinates. Then, for the horizontal components of the proper hydrodynamic inertial forces Xe, Ye, and Ne, which are most important for the majority of applications, it can be obtained after some evaluations: Xe ¼ m11 u þ m22 vr; ð16Þ Ye ¼ m22 v  m26 r  m11 ur; N ¼ m v  m r þ ð m  m Þuv  m ur: e 26 66 11 22 26 The pure interaction forces will then be XI ¼ Xp  Xe ; YI ¼ Yp  Ye ; NI ¼ Np  Ne : ð17Þ Most of the proper force corrections, except for the Munk moment, are only present in unsteady and/or curvilinear motion. In the most popular case of overtaking in parallel paths all the corrections will be zero if only the motion with zero drift angle is presumed. The elegancy of the added mass approach to evaluating potential forces on an isolated body in unbounded fluid stimulated several attempts to extend it to the case of interacting bodies through introduction of generalized inertial coefficients similar to added masses but accounting for the interaction as done, for instance, by Yasukawa (2003). However, apparently this approach makes little sense as these generalized parameters will inevitably depend on the instantaneous configuration of the hydrodynamic system and, in the general case, must be recomputed at every refreshment step. The main advantage of the added mass formalism consisting in the possibility to describe completely inertial hydrodynamic properties of any rigid body with only 21 constant parameters is therefore lost and the resulting algorithms seem to be even less economical than the direct force calculation. Similar complications occur in the hydrodynamics of deformable bodies. Another approach based on the application of the momentum conservation theorem (Korsmeyer et al. 1993) is more promising but close analysis showed that it does not have serious advantages over the direct pressure integration either. 6 T ¼ 12 (m jk j; k¼1 134 SEPTEMBER 2012 wj wk ; ð13Þ 2.2.3. Numerical algorithm. Most of the developed methods of solution to the Equation (6) and further calculations of the JOURNAL OF SHIP RESEARCH integrals (8) are based on the discretization of the surface S into a n S number of primary panels, that is, S ¼ Sj , further approxima- Table 1 Main particulars of tug and tanker model Tug (Ship 1) j¼1 + tion of each primary panel with a simpler computational panel. Then, certain assumptions about the distribution of the source density are adopted, which make possible the computation of the influence functions from one panel to another analytically or with some efficient integration formulae. The nonpenetration boundary condition is usually satisfied at some collocation point on each panel, which results in a n n linear algebraic set of equations that typically happens if the source density is assumed constant over each panel. If the distribution over each panel is described by more than one parameter, the order of the linear set can be higher than the number of panels in consideration. As the resulting linear set is practically always well-conditioned with a dominant diagonal, iteration methods of solution are normally preferred which gives the computation time approximately proportional to n2. The most popular and, historically, the first potential flow method developed by Hess and Smith (1964, 1967) uses flat quadrilateral panels and constant source density on each panel. A number of improved higher-order methods have appeared later. Examples of these are the method by Kough and Ho (1996) and—one of the latest developments—a method based on the spline approximation of the surface, not using explicit panels (Ni et al. 2005). A thorough inspection of publications on potential flow calculations indicates, however, that while higher-order methods have sensible advantage in accuracy and efficiency for smooth bodies (spheres, ellipsoids etc.), this advantage nearly vanishes for practical ship forms which can include knuckles, regions of rapidly changing curvature, transom, outer keels, etc. Also, in the interaction problems grid refinement at small gaps may be required anyway. Bratu (1975) indicated that inappropriate use of a higherorder method can even decrease the accuracy. A similar opinion was expressed by Bertram (2000, p. 218). Most often, from the viewpoint of overall efficiency the classic Hess and Smith method with flat panels appears to be the best option especially bearing in mind that this method is perfectly and flawlessly described in the easily available publications cited above. That is why in the present study this method was selected and implemented anew, with adaptation to multibody computations. 3. Experimental Determination of Interaction Forces + + + All model tests were conducted in the towing tanks at FORCE Technology. The tests were conducted in both the shallow tank with the horizontal dimensions (25m 8m) and with the depth that can be varied from 0 to 0.25 m, and the deep water towing tank (240m 12m 5.5m). 3.1. Geometry and conditions The hulls of the tug and the assisted ship were considered in model scale 1:25. The assisted ship was a conventional tanker and the tug was a Svitzer M-class ASD tug. Both models were tested unpropelled. The main particulars of the ships are listed in Table 1. The used coordinate system and sign convention of the forces acting on the tug is as illustrated in Figure 1. SEPTEMBER 2012 Main Particulars Unit Full scale Model Length overall, LOA Length between perpendiculars, L Breadth, B Draft, T Displacement, Ñ m m 29.5 25.6 1.18 1.024 m m m3 11.0 4.6 649 0.44 0.184 0.0416 Tanker (Ship 2) Full scale Model 189.5 186.2 7.58 7.44 31.6 10.3 49197 1.26 0.41 3.15 3.2. Model test setup The model of the tanker was attached underneath the carriage, fixed in all degrees of freedom, as seen in Figure 2. The tug model was placed with various heading angles (however, only results corresponding to the zero heading angle are discussed here) and positions around the tanker. The model of the tug was connected to the carriage via two strain gauges positioned fore and aft inside the model, which was free to heave, roll, and pitch. The longitudinal and transverse loads measured by the gauges were transformed in the obvious way into the surge and sway force components and the yaw moment acting on the tug. The tanker model was fully fixed during the tests and no forces acting upon it were measured as being considered not interesting in the case of a much smaller interacting craft. 3.3. Test Program The applied test program included variations of longitudinal and transverse relative positions between tug and tanker for three different water depths and two speeds. One of the depths corresponded to the deep-water tank and could be considered as infinite. The dimensional and dimensionless values of the remaining two depths H1 and H2 are given in Table 2. The tests were performed at two towing speeds corresponding to 4 kn and 6 kn in full scale. More detailed data about these speed regimes are presented in Table 3. All data was acquired as time series using a sampling frequency of 45 Hz. The time series were only recorded after the carriage had reached steady velocity. The force contribution due to the presence of the tanker, that is, interaction forces, was evaluated by subtracting the forces measured in the tests with the tug alone from the forces measured in the tests with both tug and tanker. 3.4. Uncertainty estimation The assumed value of the relative mean square estimate of the error of one measurement based on multiple uncertainty assessment carried out for different conditions and models at the FORCE Technology tanks was estimated to be 5%. Assuming that any possible bias had been eliminated, the observed error was treated as random and Gaussian. As the interaction forces and moment were always obtained as differences between the corresponding components measured on the tug model in presence and absence of the tanker model, the resulting variance was calculated as doubled variance of one measurement. The resulting absolute mean pffiffiffi square was only larger than the original error by the factor 2, although the relative error could be substantially larger when JOURNAL OF SHIP RESEARCH 135 Fig. 2 Model test setup 4. Numerical Results Table 2 Depth conditions 4.1. Body surface discretization Depth Relative to: Absolute Depth in Full Scale Draft of the tug T1 Draft of the tanker T2 2.66 3.38 1.19 1.51 H1 ¼ 12.25m H2 ¼ 15.57m Table 3 Speed regimes data Speed Value in Full Scale Froude Number Based on: Knots m/s Length of the tug L1 Length of the tanker L2 Depth H1 Depth H2 4 6 2.06 3.09 0.13 0.195 0.048 0.072 0.188 0.282 0.167 0.25 the interaction load was obtained as a small difference. On some of the plots in the next section, calculated error bars are shown. As the described estimation procedure presumed access to the loads on the model measured both in free run and in presence of the interacting ship and these data were not always available, the error bars are not shown on all the plots but they should be expected to be of similar length. 136 SEPTEMBER 2012 The two ship forms, a tug (Ship 1) and a tanker (Ship 2), described in the previous section were used for all the calculations presented in this article. The number of offsets in the delivered hull shape description files made possible creation of a very large number of panels for each vessel without interpolation. However, the resulting grid was impractical and a special file management program was developed, which first, transformed the hull offsets into the format required by the panel code, then recreated vertices of the sections in conformity with the given ship draft and, finally, the data were thinned along the sections and along the vertices on each section to obtain rougher but, expectedly, more economical form representations. A parametric cubic spline interpolation was used to redistribute the vertices on a section appropriately. The summary of the number of panels used for computations performed for the present study is given in Table 4 and graphical representation of the ship surfaces with various numbers of panels is shown in Figures 3 to 5. 4.2. Forces and moments: numerical results and comparisons Most of the computations were performed for validating the algorithm against the experimental data and for estimating possible error at various numbers of panels. Hence, reproduced here were situations supported by experimental data. As in course of the experiment both models were always towed with the same JOURNAL OF SHIP RESEARCH Table 4 Number of panels in computations Total Number of Panels 3506 936 570 356 292 Number of Panels on Ship 1 Number of Panels on Ship 2 1392 378 210 128 120 2114 558 360 228 172 velocity V, the calculations were carried out in the quasi-steady mode. The origin of the body axes associated with Ship 2 was placed in the origin of the global frame, so the global ordinate h of Ship 1 also served, with the inversed sign, as the side distance from Ship 2 and the corresponding abscissa x as the longitudinal stagger of Ship 1. The heading angle y was always kept zero for both vessels as well as the drift angle b in case of parallel rectilinear towing. The following nondimensional parameters were used to represent the results: the longitudinal dimensionless stagger x¢s ¼ 2x=L2 , dimensionless lateral offset h¢s ¼ 2h=B2 , ¢ ¼ 2X1;2 =ð rAV 2 Þ, dimensionless dimensionless surge force X1;2 2 ¢ yaw sway forces Y1;2 ¼ 2Y1;2 =ð rAV Þ, and the dimensionless pffiffiffiffiffiffiffiffiffiffiffi ¢ ¼ 2N1;2 = rAL1;2 V 2 , where A ¼ 3 Ñ1 Ñ2 is the moment N1;2 reference area. Results for the deep water, full-scale speed 6 kn and x s¢ ¼ 0:014 that is, with the both ships approximately abreast are shown in Figure 6. It can be seen that the agreement for the surge force is rather poor: the experiment gives definitely negative values or, in other words, an additional interaction resistance while the most precise computation with the finest grid gives this force very close to zero which, however, is expectable from the theoretical viewpoint as the velocity of the perfect fluid in a narrow gap may go to infinity resulting in an unlimited negative pressure. Especially large Fig. 5 Typical computational configuration of two ships with 378 þ 558 ¼ 936 panels discrepancies happen at small distance which can be explained with local free surface deformations resulting in additional increased pressure in the bow region which is not accounted for by the double-body model and which is not counterbalanced by any similar effects in the stern region. Another neglected effect could be the additional friction resistance due to the accelerated flow between the sidewalls, but this rather explains additional resistance observed at larger distances. At very small lateral distances, when the boundary layers on both sidewalls merge together, it results in a kind of additional hydraulic drag increasing the form resistance. As a result of the two described phenomena, the suction first drops and at closer distances even transforms into repulsion. However, as comparative computations showed (Fonfach et al. 2011), in the present case and, likely, for all surface displacement ships, freesurface effects dominate: despite low Froude numbers based on the Fig. 3 Representation of hull for Ship 1: left—1392 panels; center—378 panels; right—120 panels Fig. 4 Representation of hull for Ship 2 (compressed in longitudinal direction by factor 3): left—2114 panels; center—558 panels; right—172 panels SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 137 length for any of interacting ships, the Froude number based on the full-scale 1 meter gap will be close to 2.0 for conditions of the experiment. Hence, it can be assumed that wavemaking effects are responsible for the qualitatively different behavior of predicted and measured values of the interaction sway force at small lateral distances. It is interesting to note that similar disagreement is not observed for the yaw moment which can be explained by symmetric distribution of the small-gap effects with respect to the midship plane of the smaller ship. The fact that the wave making dominates in small-clearance interaction is of some practical importance as wavemaking forces usually are modeled correctly in tank experiments while the viscous interaction is prone to the scale effect even if the laminarturbulent transition lines are reproduced correctly on both models through appropriate stimulation. With a reduced numbers of panels the agreement may seem to become better but obviously this is the effect of occasional mutual cancellation: nonmonotonous convergence of panel methods is not anything exceptional. In general, the surge force on slender bodies is the most difficult for being predicted by panel methods (Söding 1993) and, for instance, Xiang and Faltinsen (2010) even avoided presenting any results for this component. The latter move is partly justified by the surge interaction force being less critical for predicting maneuvering anomalies caused by interaction as any variation of this force can easily be compensated by adjusting the thrust. Results are more encouraging for the sway force: the agreement can be called good at h¢s < 1:4 which, in full scale, corresponds to lateral clearances greater than 1 meter. This is approximately what could have been expected (the comments concerning the surge force apply also here). The plot for the yaw moment presented in Figure 6(c) also shows a fairly good agreement within the same range as for the sway force. But for this component the theory gives surprisingly good predictions even for very small lateral distances. This can be explained by the fact that while the sway force depends greatly on the suction between the sidewalls of the vessels, this suction (or pressure) does not result in a substantial contribution to the yaw moment as it is distributed roughly symmetrically with respect to the midship plane of the ship of interest. Computations for the fore and aft stagger presented in Figures 7 and 8 respectively were only conducted for the number of panels 936 (378 on Ship 1 and 558 on Ship 2). Reasons for such decision and analysis of the influence of the number of panels will be given later, together with the shallow-water case. Comparison of results obtained for different longitudinal positions of Ship 1 present certain difficulties as different trends can be observed for different components. In general, the best agreement is observed for the abreast position but the best agreement for the surge force was obtained at the positive stagger. It must be noted that the experimental signs of this force (negative for the fore stagger and positive for the aft one) are reproduced in the computations and are justified physically: slight longitudinal suction is expected in both situations. The sway force is slightly overestimated in the forward position of the tug while in its aft position it is underestimated especially at larger transfer distances which can be explained by some wavemaking effects from Ship 2. Unlike in the abreast position, absolute values of the yaw moment look relatively large although the sign is captured correctly in the both cases: the yaw moment of the smaller ship is to the target (Ship 2) in aft position and from the target in the fore position. 138 SEPTEMBER 2012 Fig. 6 Interaction forces and moment in deep water at xs¢ ¼ 0.014 (a) surge force, (b) sway force, (c) yaw moment JOURNAL OF SHIP RESEARCH Fig. 7 Interaction forces and moment in deep water at xs¢ ¼ 0.6 (a) surge force, (b) sway force, (c) yaw moment SEPTEMBER 2012 Fig. 8 Interaction forces and moment in deep water at xs¢ ¼ 0.64 (a) surge force, (b) sway force, (c) yaw moment JOURNAL OF SHIP RESEARCH 139 Numerical and experimental results for the longitudinal stagger xs¢ ¼ 0:014, full-scale speed 4 kn and water depth H1 are presented in Figure 9. Here, in addition to the number of panels, influence of one more computational parameter was investigated: the number of reflections n in the truncated version of Equation (8): n Gðx; y; z; x¢; y¢; z¢Þ ¼ ( i¼n   1 1 : þ ri ―r i ð18Þ Concerning the overall agreement and the influence of the hull approximation, the results are qualitatively the same as in deep water although, as could be expected, the quantitative agreement is slightly worse. As to the number of reflected images, even one reflection seemed to work fairly well here. However, for the smaller Ship 1 the relative depth was large enough while a stronger influence could be expected for Ship 2, which is confirmed by results presented in Figure 10. Inspection of these plots shows that n ¼ 8 and even n ¼ 4 will be sufficient in most cases. As to the influence of the number of panels, variants with the coarsest discretization gave naturally the least accurate results especially for the surge force. However, regarding the overall reachable accuracy in maneuvering mathematical models and understanding that accurate prediction of the surge force is less important than for the other two components, for appropriate reproduction of the behavior of a steered ship, a less accurate model with smaller number of panels might be a better practical choice. It must be noted that the apparently less favorable situation with N2¢ in Figure 10c is not characteristic because the absolute value of that moment in this situation (interaction with a small tug near the midship) is too small. This trade off can be necessary anyway because computations with 292 panels are approximately 10 times faster than with 936 panels and 150 times faster than with 3,506 panels. Although the situation is expected to improve in the future, nowadays not more than 300 to 400 panels should be used on an average common PC with a purely sequential code for real-time simulations with a reasonable refreshment frequency of 1 to 3 Hz. Of course, the code parallelization is expected to be efficient for the used algorithm and use of a high-end hardware can also change the situation dramatically. The variant with 936 panels (378 on Ship 1 and 558 on Ship 2) was assessed as optimal for offline computations. That is why it is used in this study as standard in all the cases when the number of panels is not indicated explicitly. The last set of the shallow-water computed data validated against the experiments (Fig. 11) corresponds to Ship 1 always located more than halfway ahead from the midship of Ship 2 at various lateral distances and at a slightly larger fluid depth. However, as the experiment was carried out at a larger speed (6 kn full scale), the depth Froude number was larger than in the first shallow-water case. Although the absolute values of both predicted and measured force/moment coefficients are somewhat different (for instance, a larger turning out moment is observed), the overall agreement remained practically the same. Figure 12 demonstrates, for the deep water case and for the same conditions as in Figure 6, the influence of nonzero instantaneous velocities. The cases “swaying to/from target” in the legends mean the pure sway motion of Ship 1 to/from Ship 2 maintaining the straight heading. The assumed absolute value of the sway velocity 0.54 m/s corresponds to the 10 degrees drift 140 SEPTEMBER 2012 angle. Substantial influence of the sway velocity is evident. These results are, however, somewhat artificial as they were obtained in the quasi-steady mode while influence of the local potential derivative in the pressure equation can be substantial (Sutulo & Guedes Soares 2008). Hence, a strictly steady case of synchronized sway of both vessels was also modeled (marked as “swaying to port/ starboard”). Finally, the case of Ship 1 yawing bow to or from the target with the angular velocity 0.6 deg/s corresponding to the 20 ship lengths steady turn diameter is also presented. These results show that influence of sway velocity can be quite substantial and even change the sign of the sway force and of yaw moment. The yaw, at least with the chosen amplitude of motion (tight turns were thought unlikely in the vicinity of other ships), affects the forces much less. However, the overall conclusion that the interaction simulation must account not only for the instantaneous mutual position but also for the instantaneous motion of involved vessels is obvious. 5. Conclusions A potential-flow code for prediction of interaction between the maneuvering ships has been developed and its output compared with experimental data. The code is based on the double-body concept, that is, it only can be used at low Froude numbers and under condition that the maneuvering motion remains twodimensional (no substantial heel, trim, and sinkage are accompanying the maneuvering). The code can be used in deep water and at constant or slowly varying finite depth. Although the method is applicable to arbitrary motion in the horizontal plane, all comparisons were performed for parallel centerplanes and equal velocities of both vessels, that is, in the cases covered by the experiment. However, additional computations with superimposed sway or yaw motions were also performed. As the computations were carried out with various numbers of panels, it is possible to evaluate its influence on the accuracy and to provide information of an educated trade-off between the accuracy and computational speed. It turned out that from the practical viewpoint, even very coarse and fast grids can be used as the discretization error is usually comparable with the error of the method. The largest substantial discrepancies were discovered for the sway force at a very small horizontal clearance, which in fact corresponds to the fender-limited lateral distance when consideration of the hydrodynamic interaction becomes meaningless. This discrepancy is, however, interesting from the theoretical viewpoint and can be explained by free-surface and viscous effects. These effects could be expected to be more pronounced in nonparallel position of one ship with respect to another. Moreover, in the case of a harbor tug, which in fact was investigated here, the small craft can be affected by longitudinal vortices trailed behind and sidewards of a larger hull, which sometimes can even result in vortex-induced high-frequency motions called bafting (Dand 1975), and by the propeller race. At the same time the propeller works as a sink outside the slipstream which can also alter the interaction scenario at stern shoulders. Unfortunately, not all these effects can be easily accounted for in the double-body potential code and at present more sophisticated codes (RANSE or even perfect fluid free-surface solvers) are too far from being suitable to be incorporated into the online computation loop of a typical bridge simulator. JOURNAL OF SHIP RESEARCH Fig. 9 Interaction forces and moment on Ship 1 in shallow water: depth H1, FnH ¼ 0.188, xs¢ ¼ 0.014 (a) surge force, (b) sway force, (c) yaw moment; left, influence of the number of panels; right, influence of the number of reflections SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 141 Fig. 10 Interaction forces and moment on Ship 2 in shallow water: depth H1, FnH ¼ 0.188, xs¢ ¼ 0.014 (a) surge force, (b) sway force, (c) yaw moment; left, influence of the number of panels; right, influence of the number of reflections 142 SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH Fig. 11 Interaction forces and moment on Ship 1 in shallow water: depth H2, FnH ¼ 0.25, xs¢ ¼ 0.61 (a) surge force, (b) sway force, (c) yaw moment SEPTEMBER 2012 Fig. 12 Influence of elementary maneuvering motions on interaction forces and moment in deep water (a) surge force, (b) sway force, (c) yaw moment JOURNAL OF SHIP RESEARCH 143 + At the same time, it seems to be hopeless to rely on systematic experimental data as, first, they can be only valid for some specific combination of hulls and, second, the number of explaining variables (factors) is too large to create a viable predicting grid function. Application of optimized experimental designs (Sutulo & Guedes Soares 2002) is also problematic in this case as no established structures of regression models are known and working out such structures requires multiple complicated nonsteady experiments and, apparently, creation of specialized facilities. Hence, a double-body potential flow method, despite all its limitations, seems to be currently the most suitable solution for online prediction of interaction forces in computerized maneuvering simulators. In the coarse case of 292 panels, when, however, quite reasonable estimates were obtained in most cases, it was possible to update instantaneous interaction loads approximately twice per second using a rather modest 1.9 GHz processor AMD Turion 64 2 TL-58 without any parallelization and any other special efforts to optimize the code. It must be kept in mind that any category of the maneuvering force is typically estimated with the average error of 15 to 20% and adjustment of simulation codes is performed anyway (Hensen 1999) and the presented method can be recommended for practical use for any number of ships moving arbitrarily though remaining approximately parallel to each other. Of course, certain improvements can, probably, be made besides the obvious development of parallel codes. First, it was noticed that the worst agreement happens with the surge force. In fact, it is a known difficulty with the panel method and probably certain improvements in the spirit of the patch method by Söding (1993) could be successful. As the number of panels is the main parameter affecting the computational speed, certain reserves are hidden in the optimized or adaptive distribution size and shape of the panels although this would complicate the code greatly. Acknowledgments The study was supported by the project “Effective Operations in Ports” (EFFORTS) funded by the European Commission, contract No. FP6-031486. The experimental research of tug-ship interaction forces was sponsored by Den Danske Maritime Fond. References ABKOWITZ, M. A., ASHE, G. M., AND FORTSON, R. M. 1977 Interaction effects of ships operating in proximity in deep and shallow water, Proceedings, 11th Symposium on Naval Hydrodynamics, March 28–April 2, 1977, London, UK, 671– 691. ANKUDINOV, V. K., FILIPPOV, I. I., AND SOBOLEV, P. K. 2006 Modelling of ship motions in restricted channels on marine simulators, Proceedings, International Conference on Marine Simulation and Ship Manoeuvrability, June 25–30, Terschelling, The Netherlands, M-17-1–M-17-10. BECK, R. F. 1977 Forces and moments on a ship moving in a shallow channel, J. Ship Res., 21, 2, 107–119. BERTRAM, V. 2000 Practical Ship Hydrodynamics, Butterworth Heinemann, Oxford. 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JOURNAL OF SHIP RESEARCH 145 Copyright of Journal of Ship Research is the property of Society of Naval Architects & Marine Engineers and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. Journal of Ship Research, Vol. 56, No. 3, September 2012, pp. 183–196 http://dx.doi.org/10.5957/JOSR.56.3.110018 Near-Field Hydrodynamics of a Surface-Effect Ship Lawrence J. Doctors The University of New South Wales, Australia A consistent near-field linearized potential-flow analysis of a surface-effect ship traveling at a steady speed in laterally restricted water of finite depth is described. Additionally, the influence of seals, in terms of their drag and lift on the vessel, is included in the theory. The predicted results include the sinkage and trim, the resistance components, and the profile of the free surface of the water along the sidehulls. These predictions are compared with the corresponding results measured on a towing-tank model. It is demonstrated that the inclusion of the influence of the seals substantially improves the numerical predictions for the sinkage and trim. Good agreement is achieved for the total resistance provided a frictional form factor of 1.306 is used. The correlation between theory and experiment for the profile of the free surface is high, except in the vicinity close to the bow, where the nonlinear effects in that region are strong. Keywords: hydrodynamics; surface-effect ships 1. Introduction 1.1. Background THE SURFACE-EFFECT ship (SES) can be described as a marine aircushion vehicle (ACV), which is characterized by also possessing a pair of sidehulls. These sidehulls contain the air cushion on the sides. This reduces the lift-fan flow requirements in comparison with an ACV, but there is a penalty of increased frictional drag. The presence of the sidehulls provides the vessel with better control and permits one to use waterjet propulsion rather than air propulsion. The air cushion is typically contained at the bow by a finger seal and at the stern by means of a laterally two-dimensional multilobe bag seal pressured at a level marginally above the cushion pressure. Historical descriptions of the SES and its early development have been provided by Ford (1964), Morris (1964), and Butler (1985), and (somewhat later) by Lavis and Spaulding (1991). Broadly speaking, the wave resistance of an SES is not considerably less than that of an equivalent surface (displacement) vessel, because in both cases, the total weight is borne by the water. Thus, any difference in wave resistance can be traced back to relatively small differences in the way that its weight is distributed, essentially by choosing the length-to-beam ratio. This point was studManuscript received at SNAME headquarters May 9, 2011; revised manuscript received April 17, 2012. SEPTEMBER 2012 ied in depth by Ford, Bush, Wares, and Chorney (1978). They demonstrated that the choice of a high length-to-beam ratio would lead to a lower wave resistance over much of the speed range of interest; however, this advantage is lost at the higher speeds of operation. The key gain in efficiency of an SES over a traditional surface vessel is related to its much reduced wetted surface. This efficiency is traditionally defined, as for any transport craft, through the transport factor. This is the product of the weight and speed of the vessel divided by the installed power. From a practical or economic viewpoint, the weight should be that of the payload rather than the total weight. Thus, the naval architect must aim for improving both the hydrodynamic and the structural characteristics. The concept of the transport factor was first developed by Gabrielli and von Kármán (1950) and further considered by Jewell (1980). In more recent years, considerable additional insight into this important idea has been provided by Templeman and Kennell (1999), Broadbent and Kennell (2001), and Kennell (2001). In this project, we are principally concerned with evaluating the resistance of an SES as well as its sinkage and trim. To this end, we wish to effect both theoretical calculations and to execute corresponding model tests. Tests on SES models have been widely reported. We quote here the example of Wilson, Wells, and Heber (1979), who discussed the principal resistance components. Of great interest here is the research of Ozawa, Yamashita, and Tanaka (1979) and Tanaka, Ozawa, and Yamashita (1980). They presented comparisons of the predicted wave resistance with 0022-4502/12/5603-0183$00.00/0 JOURNAL OF SHIP RESEARCH 183