Comparative measurements on the flow structure of a marine propeller wake between an open free surface and closed surface flows

In order to investigate the effects of a free surface on the wake behind a rotating propeller, experiments were carried out in a circulating water channel for two cases: one with an open free surface and one with a closed free surface covered by a rigid plate. Four hundred instantaneous velocity fields were measured using a two-frame particle image velocimetry (PIV) technique at four different blade phases. These were ensemble-averaged to investigate the time-averaged flow structure in the near-wake region. For a surface ship, the flow behind the propeller is influenced by the hull wake and the free surface. The phase-averaged mean velocity fields show the potential wake and the viscous wake formed by the boundary layers developed on the blade surfaces. The interaction between the bilge vortices and the incoming flow along the ship’s hull deforms the wake structure. Tip vortices are generated periodically, and the slipstream contraction occurs in the near-wake region. The free surface was found to affect the axial velocity component and vortex structure behind the propeller. As the flow goes downstream, the tip and trailing vortices dissipate due to turbulent diffusion and active mixing with adjacent vortices.     

船舶尾流作用对抛石基床冲刷稳定性的影响研究

文章提出了抛石基床块石理想排布状态的二维模型,并通过有限元软件对抛石基床的冲刷问题进行了数值模拟。采用ADINA软件建立二维数值模型,针对船舶尾流的分布特点,计算分析抛石基床在船舶尾流作用下的起动流速,得出抛石基床在尾流作用下的冲刷机理,并拟合得到抛石基床块石的起动流速经验公式。     

Surf-riding and oscillations of a ship in quartering waves

The behavior of a ship encountering large regular waves from astern at low frequency is the object of investigation, with a parallel study of surf-riding and periodic motion paterns. First, the theoretical analysis of surf-riding is extended from purely following to quartering seas. Steady-state continuation is used to identify all possible surf-riding states for one wavelength. Examination of stability indicates the existence of stable and unstable states and predicts a new type of oscillatory surf-riding. Global analysis is also applied to determine the areas of state space which lead to surf-riding for a given ship and wave conditions. In the case of overtaking waves, the large rudder-yaw-surge oscillations of the vessel are examined, showing the mechanism and conditions responsible for loss of controllability at certain vessel headings.List of symbols c wave celerity (m/s) - C(p) roll damping moment (Ntm) - g acceleration of gravity (m/s²) - GM metacentric height (m) - H wave height (m) - I _x ,I _z roll and yaw ship moments of inertia (kg m²) - k wave number (m^–1) - K _H ,K _W ,K _R hull reaction, wave, rudder, and propeller - K _p forces in the roll direction (Ntm) - m ship mass (kg) - n propeller rate of rotation (rpm) - N _H ,N _W ,N _R hull reaction, wave, rudder, and propeller - N _P moments in the yaw direction (Ntm) - p roll angular velocity (rad/s) - r rate-of-turn (rad/s) - R(,x) restoring moment (Ntm) - Res(u) ship resistance (Nt) - t time (s) - u surge velocity (m/s) - U vessel speed (m/s) - v sway velocity (m/s) - W ship weight (Nt) - x longitudinal position of the ship measured from the wave system (m) - x _G ,z _G longitudinal and vertical center of gravity (m) - x _S longitudinal position of a ship section (S), in the ship-fixed system (m) - X _H ,X _W ,X _R hull reaction, wave, rudder, and propeller - X _P forces in the surge direction (Nt) - y transverse position of the ship, measured from the wave system (m) - Y _H ,Y _W ,Y _R hull reaction, wave, rudder, and propeller - Y _p forces in the sway direction (Nt) - z _Y vertical position of the point of action of the lateral reaction force during turn (m) - z _W vertical position of the point of action of the lateral wave force (m) Greek symbols angle of drift (rad) - rudder angle (rad) - wavelength (m) - position of the ship in the earth-fixed system (m) - water density (kg/m³) - angle of heel (rad) - heading angle (rad) - _e frequency of encounter (rad/s) Hydrodynamic coefficients K roll added mass - N _v ,N _r yaw acceleration coefficients - N _v N _r N _rr N _rrv ,N _vvr yaw velocity coefficients K. Spyrou: Ship behavior in quartering waves - X _u surge acceleration coefficient - X _u X _vr surge velocity coefficients - Y _v ,Y _r sway acceleration coefficients - Y _v ,Y _r ,Y _vv ,Y _rr ,Y _vr sway velocity coefficients European Union-nominated Fellow of the Science and Technology Agency of Japan, Visiting Researcher, National Research Institute of Fisheries Engineering of Japan     

舰船气泡尾流数值模拟(英文)

Based on a volume of fluid two-phase model imbedded in the general computational fluid dynamics code FLUENT6.3.26, the viscous flow with free surface around a model-scaled KRISO container ship (KCS) was first numerically simulated. Then with a rigid-lid-free-surface method, the underwater flow field was computed based on the mixture multiphase model to simulate the bubbly wake around the KCS hull. The realizable k-ε two-equation turbulence model and Reynolds stress model were used to analyze the effects of turbulence model on the ship bubbly wake. The air entrainment model, which is relative to the normal velocity gradient of the free surface, and the solving method were verified by the qualitatively reasonable computed results.     

Simple Lagrangian formulation of bubbly flow in a turbulent boundary layer (bubbly boundary layer flow)

For the theoretical consideration of a system for reducing skin friction, a mathematical model was derived to represent, in a two-phase field, the effect on skin friction of the injection of micro air bubbles into the turbulent boundary layer of a liquid stream. Based on the Lagrangian method, the equation of motion governing a single bubble was derived. The random motion of bubbles in a field initially devoid of bubbles was then traced in three dimensions to estimate void fraction distributions across sections of the flow channel, and to determine local bubble behavior. The liquid phase was modeled on the principle of mixing length. Assuming that the force exerted on the liquid phase was equal to the fluid drag generated by bubble slip, an equation was derived to express the reduction in turbulent shear stress. Corroborating experimental data were obtained from tests using a cavitation tunnel equipped with a slit in the ceiling from which bubbly water was injected. The measurement data provided qualitative substantiation of the trend shown by the calculated results with regard to the skin friction ratio between cases with and without bubble injection as function of the distance downstream from the point of bubble injection.List of symbols B law of wall constant - C _f local coefficient of skin friction - C _f0 local coefficient of skin friction in the absence of bubbles - d _b bubble diameter [m] - g acceleration of gravity [m/s²] - k ₁ k₄ proportional coefficient - k _L turbulent energy of the liquid phase [m²/s²] - L representative length [m] - l _b mean free path of a bubble [m] - m _A added mass of a single bubble [kg] - m _b mass of a single bubble [kg] - N _x ,N _y ,N _z force perpendicular to the wall or ceiling exerted on a bubble adhering to that wall or ceiling [N] - P absolute pressure [Pa] - Q _G rate of air supply [/min] - q _L ⁽ⁱ⁾ turbulent velocity at the ith time increment [m/s] - R> _ex Reynolds number defined by Eq. 32 - T _*L integral time scale of the liquid phase [s] - U velocity of the main stream [m/s] - ,¯v,¯w time-averaged velocity components [m/s] - u,v,w turbulent velocity components [m/s] - û _L ,v_L root mean square values of liquid phase turbulence components in thex- and y-directions [m/s] - V volume of a single bubble [m³] - X,Y,Z components of bubble displacement [m] - x _s ,y _s ,z _s coordinate of a random point on a sphere of unit diameter centered at the coordinate origin - root mean square of bubble displacement in they-direction in reference to the turbulent liquid phase velocity [m] - local void fraction - _m mean void fraction in a turbulent region - regular random number - R _v increment of the horizontal component of the force acting on a single bubble, defined by Eq. 22 [N] - t time increment [s] - ₁ reduction of turbulent stress [N/m²] - _L rate of liquid energy dissipation [m²/s³] - _m coefficient defined by Eq. 30 - law of wall constant in the turbulent region in absence of bubbles - ₁ law of wall constant in the turbulent region in presence of bubbles     

Effect of a drag-reducing polymer solution ejection on tip vortex cavitation

Experiments regarding the modification of the foil geometry and/or active or passive mass injection in the vortex core have been performed to investigate the possibility of inhibiting tip vortex cavitation. The ejection at very low flow rates of drag-reducing polymer solutions at the tip of hydrofoils and propeller blades has demonstrated effectiveness as a tip vortex cavitation inhibitor. This paper reports the results obtained with an elliptical hydrofoil, of 8cm maximum chord and 12cm haif-span, operating at Reynolds numbers, of =10⁶, much larger than those previously reported in the literature. Lift coefficients and critical cavitation numbers were determined for a variety of flow and polymer solution ejection conditions. Tangential and axial components of the mean velocity as well as velocity fluctuations along the vortex path were also measured. At 12.5 m/s free stream velocity and a variety of angles of attack, the ejection of a 500 ppm aqueous solution of a drag-reducing polymer at a flow rate of about 5 cm³/s leads to a decrease of up to 30% in the cavitation number. This occurs without modification of the lift coefficient and, hence, of the midspan bound circulation of the foil. Moreover, water injection does not cause any appreciable change in the cavitation numbers. The tangential velocity profiles along the vortex path during polymer ejection indicate that the potential region remains the same, while the viscous core dimension increases, and the maximum tangential velocity decreases substantially as compared to the no ejection or water ejection experiments. Thus, the pressure coefficients at the vortex axis are smaller than for the no ejection or water ejection cases and cause the reduction of the critical cavitation numbers. It is speculated that this inhibition effect is due only to swelling of the polymer solution when exiting the ejection orifice.List of symbols a core radius (distance to the vortex axis for maximum tangential velocity) - C ₁ lift coefficient - c _max maximum chord - Cp pressure coefficient at the vortex axis - Cp _min minimum pressure coefficient at the vortex axis - d _e diameter of the ejection port - m ejection flow rate - P reference pressure - P _v vapor pressure - V free stream velocity - V _a axial velocity - V _t tangential velocity - v _r radial component of the velocity resulting from jet swelling - x downstream distance from the tip of the foil - y, r distance to the vortex axis - angle of attack - r difference between the swollen jet and the ejection port radii - boundary layer thickness - tip vortex intensity - _d ( _de ) desinent cavitation number (with ejection) - _i ( _ie ) inception cavitation number (with ejection) - _ii normal stresses - viscosity - v kinematic viscosity - p specific mass     

Distribution of void fraction in bubbly flow through a horizontal channel: Bubbly boundary layer flow, 2nd report

A method of enveloping the hull with a sheet of microbubbles is discussed. It forms part of a study on means of reducing the skin friction acting on a ship's hull. In this report, a bubble traveling through a horizontal channel is regarded as a diffusive particle. Based on this assumption, an equation based on flow flux balance is derived for determining the void fraction in approximation. The equation thus derived is used for calculation, and the calculation results are compared with reported experimental data. The equation is further manipulated to make it compatible with a mixing length model that takes into account the presence of bubbles in the liquid stream. Among the factors contained in the equation thus derived, those affected by the presence of bubbles are the change of mixing length and the difference in the ratio of skin friction between cases with and without bubbles. These factors can be calculated using the mean void fraction in the boundary layer determined by the rate of air supply into the flow field. It is suggested that the ratio between boundary layer thickness and bubble diameter could constitute a significant parameter to replace the scale effect in estimating values applicable to actual ships from corresponding data obtained in model experiments.List of symbols a ₁ proportionality constant indicating directionality of turbulence - B law-of-the-wall constant - C _f local skin-friction coefficient in the presence of bubbles - C _f0 local skin-friction coefficient in the absence of bubbles - d _b bubble diameter (m) - g acceleration of gravity (m/s²) - j _g flow flux of gas phase accountable to buoyancy (m/s) - j _t flow flux of gas phase accountable to turbulence (m/s) - k ₄ constant relating reduction of liquid shear stress by bubble presence to decrease of force imparted to bubble by its displacement due to turbulence - l _b mixing length of gas phase (m) - l _m mixing length of liquid phase (m) - l _mb diminution of liquid phase mixing length by bubble presence (m) - Q _G rate of air supply to liquid stream (l/min) - q _/g velocity of bubble rise (m/s) - 2R height of horizontal channel (m) - T _* integral time scale (s) - U _m mean stream velocity in channel (m/s) - U friction velocity in channel (m/s) - V volume of a bubble (m³) - u, ¯ v time-averaged stream velocities inx- andy-directions, respectively (m/s) - u, v turbulent velocity components inx- andy-directions, respectively (m/s) - v root mean square of turbulence component in they-direction (m/s) - root mean square of bubble displacement iny-direction with reference to turbulent liquid phase velocity (m) - y displacement from ceiling (m) - local void fraction - _m mean void fraction in boundary layer - _m constant relating local void fraction to law-of-the-wall constant - _t reduction of turbulent stress (N/m²) - law-of-the-wall constant in turbulent liquid region in absence of bubbles - ₁ law-of-the-wall constant in turbulent liquid region in presence of bubbles - ₂ law-of-the-wall constant in gas phase - _m constant indicating representative turbulence scale (m) - viscosity (Pa × s) - v kinematic viscosity (m²/s) - density (kg/m³) Suffixes G gas - L liquid - 0 absence of bubbles     

螺旋桨射流对抛石基床冲刷的数值计算

抛石基床遭螺旋桨射流冲刷破坏而危及码头使用安全,是近几年码头检测评估工作中的新问题,国内海港工程设计规范中未考虑船舶尾流作用对抛石基床的冲刷影响。采用计算流体力学软件Fluent建立螺旋桨射流流场三维数值模型,并利用螺旋桨射流流速的理论研究成果对数值模型进行验证。通过数值模拟,研究螺旋桨射流在码头岸壁和基床底面影响下的变化规律,建立了船舶尾流在重力式码头前沿的最大冲刷流速及其位置的经验公式。     

大型集装箱船尾轴分段建造工艺优化

以23 000 TEU大型集装箱船为例，进行尾轴分段建造工艺策划，论述尾轴分段施工计划管理，对尾轴分段建造工艺的关键路径进行优化，提升大型集装箱船的建造效率。     

Linear seakeeping and added resistance analysis by means of body-fixed coordinate system

This paper presents an alternative formulation of the boundary value problem for linear seakeeping and added resistance analysis based on a body-fixed coordinate system. The formulation does not involve higher-order derivatives of the steady velocity potential on the right-hand side of the body-boundary condition, i.e., the so-called m _j -terms in the traditional formulation when an inertial coordinate system is applied. Numerical studies are made for a modified Wigley I hull, a Series 60 ship with block coefficient 0.7, and the S175 container ship for moderate forward speeds where it is thought appropriate to use the double-body flow as basis flow. The presented results for the forced heave and pitch oscillations, motion responses, and added resistance in head-sea waves show good agreement with experiments and some other numerical studies. A Neumann–Kelvin formulation is shown to give less satisfactory results, in particular for coupled heave and pitch added mass and damping coefficients.     

Estimation of CO2 reduction for Japanese domestic container transportation based on mathematical models

This research discusses domestic feeder container transportation connected with international trades in Japan. Optimal round trip courses of container ship fleet from the perspective of CO₂ emission reduction are calculated and analyzed to obtain basic knowledge about CO₂ emission reduction in the container feeder transportation system. Specifically, based on the weekly origin–destination (OD) data at a hub port (Kobe) and other related transportation data, the ship routes are designed by employing a mathematical modeling approach. First, a mixed integer programming model is formulated and solved by using an optimization software that employs branch and bound algorithm. The objective function of the model is to minimize the CO₂ emission subject to necessary (and partially simplified) constraints. The model is then tested on various types of ships with different speed and capacity. Moreover, it is also tested on various waiting times at hub port to investigate the effect in CO₂ emission of the designated fleet. Both the assessment method of container feeder transportation and the transportation’s basic insights in view of CO₂ emission are shown through the analysis.     

超大型集装箱船艏艉砰击强度直接计算方法

超大型集装箱船的船艏显著外飘、船艉宽平外悬,使其在恶劣海况下航行时容易发生严重的砰击。为确保船体艏艉部结构在砰击中不发生损坏,需要研究作用到艏艉外板上的砰击压力,并以此为设计载荷来校核外板和相连结构的强度。目前对集装箱船砰击局部强度的校核要求仍以经验公式为主,但是为提高对超大尺度船舶强度校核的可靠性,近年来推出了砰击的直接分析方法。本文初步分析了砰击直接分析方法的基本原理,并运用该方法对20,000 TEU集装箱船的艏、艉部砰击压力以及最小板厚要求进行了研讨,其结果可为超大型集装箱船的结构设计提供重要的参考。     

限制性航道船周回流速度与船体下沉研究

通过船舶在限制性航道(宽32 m,水深2.5 m)的航行试验,研究了500 t和300 t船队航行时的船周回流速度与船体下沉。试验表明,船周回流和船体下沉与航速、航行方式和断面系数等因素有关,其中交错航行时的船尾下沉量是渠道水深设计的控制条件。     

船舶尾流冲刷作用下抛石基床块石稳定性计算方法研究

船舶尾流是导致重力式码头抛石基床冲刷破坏的重要因素之一,国内港口、海岸工程设计规范中未考虑船舶尾流作用对抛石基床块石稳定性的影响。文章通过总结、对比、分析国外船用螺旋桨射流流速分布研究成果,得到了螺旋桨射流流速分布规律,提出了船舶尾流冲刷作用下抛石基床块石稳定性计算方法,并利用该方法计算并分析了某重力式码头抛石基床块石的冲刷情况。计算结果表明:螺旋桨射流起始段和主体段的流速均可能对抛石基床块石产生冲刷影响,且主体段水流对块石稳定性影响更大,船用螺旋桨桨轴线距基床顶面高度2倍桨直径内的区域可作为船舶尾流冲刷影响区。     

基于 LES 和 BPBE 的舰船气泡尾流数值分析方法

提出一种基于大涡模拟（LES）和气泡群平衡方程(BPBE)的舰船气泡尾流特征参数研究的数值分析法：首先建立舰船尾流场 LES 控制方程,结合实船初边始条件,对流向的速度进行数值模拟;然后再将 LES 计算得到的速度代入 BPBE 方程进行尾流中气泡数密度分布（BND）的求解。计算结果表明：BND 沿尾流流向的分布基本符合指数分布,在500 m 以内的区域衰减速度很快,500 m 以外的区域衰减速度缓慢;在离船尾越近的地方,小气泡的相对 BND 越大;在3000 m 距离处 BND 最大的气泡尺寸约为70滋m。该方法计算效率较高、占用空间较少,可望为舰船气泡尾流特征的研究提供新的有效手段。     

The interaction of tidal advection, diffusion and mussel filtration in a tidal channel

Time series measurements of flow and pigment concentrations (Chl) in the Menai Strait have revealed that the strong residual flow in a tidal channel ( 500 m³ s^− 1) transports phytoplankton from the open sea into the channel where much of it is consumed by suspension feeders, mainly in commercial beds of Mytilus edulis. The progressive depletion of phytoplankton along the channel results in a strong horizontal gradient of plankton and hence Chl. Tidal displacement of this gradient causes large (± 50% of mean) oscillations of Chl in the vicinity of the mussel beds. Vertical mixing by the strong tidal flows is sufficiently vigorous for most of the tidal cycle to ensure that downward diffusion can resupply the near-bed layer although there are indications of some transient depletion around slack water.This paradigm of the interaction of advection, diffusion and filtration determining the distribution of plankton and its supply to mussels has been encapsulated in a series of simple models forced only by boundary values. In the first, a 1-D model of tidal flow in the channel reproduces the principal features of the observed currents including the unusually large spatial change in phase of the currents and the variation of the residual transport with tidal range. The flow field from this physical model is used to drive a second model based on the advection diffusion equation for Chl with a source at the Irish Sea boundary and a sink over the mussel bed. This model illustrates the formation of a strong Chl gradient along the channel and simulates the amplitude and phase of the M₂ oscillations of Chl and the development of the M₄ variation apparent in the observations. This second model has been extended to 2-D over the mussel beds to allow investigation of the effects of water column mixing. The model indicates that only for a short period ( 30 min), close to slack water, is mixing sufficiently reduced to permit the development of a depletion boundary layer and then only within  1 m from the bottom, a result which is consistent with the observations.     

A simple formulation for predicting the ultimate strength of ships

The aim of this study is to derive a simple analytical formula for predicting the ultimate collapse strength of a single- and double-hull ship under a vertical bending moment, and also to characterize the accuracy and applicability for earlier approximate formulations. It is known that a ship hull will reach the overall collapse state if both collapse of the compression flange and yielding of the tension flange occur. Side shells in the vicinity of the compression and the tension flanges will often fail also, but the material around the final neutral axis will remain in the elastic state. Based on this observation, a credible distribution of longitudinal stresses around the hull section at the overall collapse state is assumed, and an explicit analytical equation for calculating the hull ultimate strength is obtained. A comparison between the derived formula and existing expressions is made for largescale box girder models, a one-third-scale frigate hull model, and full-scale ship hulls.List of symbols A _B total sectional area of outer bottom - A _B total sectional area of inner bottom - A _D total sectional area of deck - A _S half-sectional area of all sides (including longitudinal bulkheads and inner sides) - a _s sectional area of a longitudinal stiffener with effective plating - b breadth of plate between longitudinal stiffeners - D hull depth - D _B height of double bottom - E Young's modulus - g neutral axis position above the base line in the sagging condition or below the deck in the hogging condition - H depth of hull section in linear elastic state - I _s moment of inertia of a longitudinal stiffener with effective plating - l length of a longitudinal stiffener between transverse beams - M _E elastic bending moment - M _p fully plastic bending moment of hull section - M _u ultimate bending moment capacity of hull section - M _uh ,M _us ultimate bending moment in hogging or sagging conditions - r radius of gyration of a longitudinal stiffener with effective plating [=(I _s /a _s )^1/2] - t plate thickness - Z elastic section modulus at the compression flange - Z _B ,Z _D elastic section modulus at bottom or deck - slenderness ratio of plate between stiffeners [= (b/t)(_y/E)^1/2] - slenderness ratio of a longitudinal stiffener with effective plating [=(l/r)(_y/E)^1/2] - _y yield strength of the material - _yB , _yB , _yD yield strength of outer bottom, inner bottom - _yS deck, or side - _u ultimate buckling strength of the compression flange - _uB , _uB , _uD ultimate buckling strength of outer bottom - _uS inner bottom, deck, or side     

数值模拟现代水面舰船带自由面的湍流绕流场

为了研究带自由面的船舶湍流绕流场,选择了一艘ITTC推荐的公开船模作为模拟对象,数值求解RANS方程。此模型为带声呐导流罩和方形尾封板的复杂水面舰船。采用Ogrid块拓扑算法生成质量较高的纯六面体多块结构化网格,计算中采用RNG k-ε湍流模式和标准壁面函数,并使用VOF算法来捕捉自由面。空间离散采用QUICK格式,压力-速度解耦采用PISO算法。将阻力和波形的数值结果与实验数据相比较,总阻力系数误差约2%,船侧波形、船首自由面吻合良好,显示了CFD方法在船舶水动力学中预测带自由面湍流绕流场的有效性。     

Self-propulsion computations using a speed controller and a discretized propeller with dynamic overset grids

A method that can be used to perform self-propulsion computations of surface ships is presented. The propeller is gridded as an overset object with a rotational velocity that is imposed by a speed controller, which finds the self-propulsion point when the ship reaches the target Froude number in a single transient computation. Dynamic overset grids are used to allow different dynamic groups to move independently, including the hull and appendages, the propeller, and the background (where the far-field boundary conditions are imposed). Predicted integral quantities include propeller rotational speed, propeller forces, and ship’s attitude, along with the complete flow field. The fluid flow is solved by employing a single-phase level set approach to model the free surface, along with a blended k−ω/k−ɛ based DES model for turbulence. Three ship hulls are evaluated: the single-propeller KVLCC1 tanker appended with a rudder, the twin propeller fully appended surface combatant model DTMB 5613, and the KCS container ship without a rudder, and the results are compared with experimental data obtained at the model scale. In the case of KCS, a more complete comparison with propulsion data is performed. It is shown that direct computation of self-propelled ships is feasible, and though very resource intensive, it provides a tool for obtaining vast flow detail.     

Hydrodynamic optimization of a catamaran hull with large bow and stern bulbs installed on the center plane of the catamaran

Here, a numerical optimization procedure is proposed for a fundamental study of a fast catamaran, and we compare the wave-making characteristics of a catamaran hull form with and without large bow and stern airship-type bulbs installed on the center plane of a catamaran operating at high speed. The method involves coupled ideas from two distinct research fields: numerical ship hydrodynamics and a nonlinear programming technique. The wave-making characteristics of catamaran hulls with and without bulbs were investigated using the panel method applied to free surface flow (PAFS), in which Morinos method for lifting bodies is extended to analyze the problem of free surface flow, and PAFS is linked to the optimization procedure of the sequential quadratic programming (SQP) technique. An optimum hull form for a catamaran can be obtained through a series of iterative computations, subject to some design constraints. Here, only the hull shape of a catamaran is optimized with and without center-plane bow and stern bulbs. The optimization is carried out at two Froude numbers, 0.45 and 0.5, which are around the last hump of the wave-making resistance curve. The numerical results show that a reduction in wave-making resistance is achieved around the design speed.