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     

At-sea towing of a Mega-Float unit

A huge floating offshore platform (359m long, 60m wide, and 3m deep) was towed into the Pacific Ocean for a validation experiment for a floating airport. Full-scale measurements of towline tension and the bending strain on the upper-deck were made during towing. The measured bending moment agreed well with numerical calculation without taking the draught and towing speed into consideration.     

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     

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     

Hydrodynamic loads on ice-class propellers during propeller-ice interaction

Hydrodynamic loads on a propeller blocked with simulated ice were studied using a cavitation tunnel. Comparative predictions were made using a panel method. The propeller was a model of the Canadian Coast Guard's R-class icebreake propeller, and the ice block was simulated using a solid blockage. Experimental results show the open water performance of the propeller, its performance behind a blockage, and the effects of cavitation in these conditions, as well as the loading on the simulated ice block. Panel method predictions were made of the time series propeller performance in the blocked flow. Cavitation during propellerice interaction resulted in a reduction of mean suction load on the ice block. Block load measurements indicated an increase in the oscillation about the mean value of the loads, with a variation in the phase of the loading with respect to blade position as compared with the non-cavitating results. Comparisons of panel method results with the measured block loads support the reliability of the dynamic measurements.List of symbols D propeller diameter - F block drag load - K _T thrust coefficient,T/(n ² D ⁴) - K _B block load coefficient,F/(n ² D ⁴) - K _Q torque coefficient,Q/(n ² D ⁵) - Q propeller torque - T propeller thrust - n propeller rotational speed - J propeller advance coefficientV _A/(nD) - P _A ambient pressure at propeller - P _ATM atmospheric pressure - P _V vapour pressure of water - V _A propeller advance speed - dissolved gas content - _s saturated dissolved gas content at atmospheric pressure - _o open water propeller efficiency - cavitation number, (P _A –P _V )/(0.5(nD)²) - density of water     

Stress intensity factor measurement of cracks using piezoelectric element

A stress intensity factor (SIF) measurement method for cracks using a piezoelectric element and an electrostatic voltmeter is presented. In this method, an isotropic piezoelectric element is first attached near the tip of the crack. Then surface electrodes are attached to three different positions on the piezoelectric element. The electric potentials of the surface electrodes, which are proportional to the sum of the stress ( _x + _y ) on the structural member, are measured by an electrostatic voltmeter during load cycling. The mode I and mode II SIFs of the crack are estimated using the relationship between the SIF and ( _x + _y ). The applicability of the proposed method is examined through experiments and numerical analysis.     

A 2D+T VOF fully coupled formulation for the calculation of breaking free-surface flow

We present the results of simulations obtained with a free-surface flow solver based on the following method. The free surface is simulated by the volume-of-fluid interface capturing method. This code solves the Navier–Stokes equations using a finite-volume method adapted to a structured or unstructured mesh. The system is constructed using a fully coupled approach. This global approach allows the simulation of complex flow as a breaking or merging wave. Moreover, with the use of a 2D+T decomposition, it is possible to simulate three-dimensional steady flow.     

Ductile crack initiation behavior in steels with compressive prestrain

It is well known that compressive prestrain reduces ductility in steels. On the other hand, it has also been found that high stress triaxiality reduces equivalent plastic strain at the onset of ductile fracture. In this research, plate specimens and notched bar specimens, which were prestrained in compression with bending, were used in reversed bending tests, and the effect of compressive prestrain on ductile crack initiation in steels was investigated. It was found that small ductile cracks occurred from the microscopic wrinkles which were formed on the concave surface with compressive prestrain. The critical relationship between stress triaxiality and equivalent plastic strain at ductile crack initiation was investigated by finite-element analysis. It was found that ductile crack initiation in steels with compressive prestrain can be estimated qualitatively by the relationship between stress triaxiality and equivalent plastic strain.     

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     

Numerical study on breaking phenomena of ships’ waves in narrow and shallow waterways

The environmental impact of a ships waves, such as the risk of erosion of coasts and riverbanks, and unacceptable ship movements in a restricted waterway, is now a significant ship design criterion. Therefore, it is necessary to predict ship-wave phenomena accurately in a restricted waterway. In this study, a numerical investigation of the breaking phenomena of a ships waves in restricted waterways was carried out. Incompressible Navier–Stokes and continuity equations were employed. The equations are discretized by a finite-difference method in a curvilinear coordinate system. The interface capturing method was applied to simulation of a ships waves, including wave-breaking. A modification of the level-set method is proposed to find the free surface shape clearly and without difficulty of the implemation of the boundary conditions for the distance function. In order to obtain a high resolution of wave height, a constrained interpolated profile (CIP) algorithm is adopted. In order to check the advantage of the CIP method, computations by two numerical methods, the CIP and the 3rd-order up-wind scheme, were compared. The computations for a Wigley hull in restricted waterways were performed and compared with experiments. The phenomena of ships waves in restricted waterways are discussed in order to understand the mechanism of wave-breaking in relation to the change in water depth along a waterway.     

Hydrodynamics of a body floating in a two-layer fluid of finite depth. Part 1. Radiation problem

A linearized 2-D radiation problem was considered for a general floating body in a two-layer fluid of finite depth. A boundary integral-equation method was developed for directly computing the velocity potential on the wetted surface of a body which is immersed in both the upper and lower layers as a general case. To do this, appropriate time-harmonic Greens functions were derived, and an efficient numerical method of evaluating those functions is proposed. Based on Greens theorem, hydrodynamic relations such as the energy-conservation principle were derived theoretically for a case of finite depth, and we confirm that those relations are satisfied numerically with very good accuracy. Experiments were also carried out using water and isoparaffin oil as the two fluids and a Lewis-form body. Measured results for the added mass, the damping coefficient, and the amplitude of the generated waves are compared with the computed results, and a favorable agreement is found.     

Vibration reduction of main hulls using semiactive absorbers

A semiactive-type absorber for vibration reduction of main hull girders was investigated. The semiactive absorber system includes a moving mass, support springs, dynamic dampers, and a control system. Only a small electrical power supply is needed for control of the damper valve and the operation of the control system. In this paper, the dynamics of the ship's hull and the constraints of the semiactive absorber are described first. Then, a suboptimal operation law is derived based on the properties of the absorber and the theory of optimal vibration reduction. The numerical simulation results show that the semiactive absorber is more efficient in hull vibration reduction than the passive absorber during critical periodical excitation from the propeller. The vibration caused by multifrequency excitation can also be suppressed by the semiactive absorber. In terms of effectiveness, the semiactive absorber is almost as effective as the active absorber. In particular, the performance of the semiactive absorber is excellent in the reduction of high-frequency fluctuations.List of symbols C _h ⁽ⁱ⁾ damping matrices of the segmenti - C _sb structural damping coefficient of bending - C _ss structural damping coefficient of shear - C _v hydrodynamic damping coefficient - EI flexural rigidity - f _a force generated by the absorber - f _ad damper force of the semiactive absorber - f _ext total excitation force - F _ext ⁽ⁱ⁾ generalized load vector in segmenti - teÎ the identity matrix - J performance index - J _r rotatory moment of inertia - k _a stiffness coefficient of the absorber - K _h ⁽ⁱ⁾ stiffnes matrices of the segmenti - K _s A _s G _s shear rigidity - k _v hydrodynamic spring coefficient - l _k length of the segmentk - m _a mass of the absorber - M _ext total exciting moment - M _h ⁽ⁱ⁾ mass matrices of the segmenti - m _v mass moment of inertia - w _h deflection of the center line of the hull - W _h ⁽ⁱ⁾ vertical translation and shear slope of nodes in segmenti - ¯ w _d displacement of the absorber mass relative to the hull - ¯ w _a absolute displacement of the absorber mass - ¯ w (a, t) absolute upward displacement of the hull atx=a - slope deflection due to bending - slope deflection due to shear - Dirac delta function - _k ⁽ⁱ⁾ Kronecker delta function - _k distribution function - shape function vector     

Acoustic excitation of hull surfaces by propeller sources

We present preliminary results from our new ship motion model that includes both strong and weak three-dimensional interactions between environmental surface waves and ship bodies in arbitrary water depth. The linear solutions of steady flow using the new model agree well with those obtained using the Green function methods. When the Froude number Fn is large, the fully nonlinear solutions of our model are significantly different from linear solutions, even in calm water. The interactions between the ship and incident gravity waves are completely different from those in linear solutions even with small Fn and moderate-amplitude surface waves (e.g., Fn = 0.25 and a significant wave height of about 1–3m).     

PIV velocity field measurements of flow around a KRISO 3600TEU container ship model

The main purpose of this investigation was to demonstrate a useful application of the particle image velocimetry (PIV) method to analyze the complex flow characteristics around a ship. For a sample illustration, the KRISO 3600TEU container ship model was chosen. The flow structure in the stern and near-wake region of the model has been investigated experimentally in a circulating water channel. Instantaneous velocity fields measured by the PIV velocity field measurement technique have been ensemble-averaged to give details of flow structures such as the spatial distributions of the local mean velocity, vorticity, and turbulent kinetic energy. The free-stream velocity was fixed at U _o = 0.6m/s, and the corresponding Reynolds number based on the length between perpendiculars was about 9.0 × 10⁵. The container ship model shows a complicated three-dimensional flow structure in the stern and near-wake regions. The PIV results clearly revealed the formation of large-scale bilge vortices in the stern region and their effect on the flow in the near-wake. The results shown here provide valuable information for hull form design and the validation of viscous ship flow codes and of turbulence models.     

A pseudofield model approach to simulate compartment-fire phenomena for marine fire safety design

In order to prevent the spread of marine compartment fires, it is necessary to understand the governing factors or characteristics of fire-spread phenomena. We present a pseudofield model approach to this problem. We first described a field model of turbulent heat convection based on a standard k – turbulence model. Two-dimensional numerical simulations of a two-linked compartment fire were carried out in order to predict the turbulent convection flow induced by the heat released from the fire. Then a more complicated fire-spread problem of multilinked compartment fires was analyzed by means of a zone model, in which the amounts of oxygen consumption and gas generation were solved by a gas-balance equations system. The effect of threshold conditions on fire propagation and the effect of the thickness of the heat insulation were investigated with numerical simulations.     

Synthesis of experimental designs of maneuvering captive-model tests with a large number of factors

An approach to synthesizing D-optimized experimental designs for an arbitrary number of factors was developed and tested on a third-order polynomial regression model with 5–8 factors. Three options were envisaged for the internal optimization procedure: an exhaustive search, a quasirandom search with the help of the Sobol sequences, and a genetic algorithm. The calculations performed have shown the pronounced superiority of the variant involving a genetic algorithm. Captive-model tests with a catamaran model with varying Froude number, drift angle, rate of yaw, sinkage, trim, and heel are presented as an example of the practical synthesis of the experimental design. The linear regression model constructed is a third-order 5-factor polynomial with respect to all factors except the Froude number. The influence of the latter is accounted for by representing the polynomials regression coefficients as functions of the Froude number represented as a truncated Fourier series with a linear term added.     

Numerical modeling of breaking waves generated by a ship’s hull

A new model for the simulation of spilling breaking waves in naval flows is presented. The hydrostatic pressure is used in order to mimic the weight of the breaker on the underlying flow, as in the model of Cointe and Tulin, whereas the algorithm for detecting the breaking inception and the definition of its geometry are completely new and are suitable for the simulation of three–dimensional flows around ships hulls. The model has been implemented in a finite-volume code developed for naval flows, and its performances have been validated against experimental data for a submerged profile, an S60 hull in drift motion, and the US Combatant DTMB 5415 model on a straight course.     

A local model for the simulation of two–dimensional spilling breaking waves

A numerical model for the simulation of two dimensional spilling breaking waves is described. It is derived from a previous model which, in turn, takes its underlying ideas from the Cointe and Tulin theory of steady breakers. With respect to the former model, the present one is local, i.e., the inception, extension, and geometry of the breakers are determined through the local shape of the water surface. The model has been implemented in a RANSE code, which was developed for the simulation of ship flows, through a modification in the boundary conditions. This yields a simple and effective way to reproduce the breakers influence on the underlying flow. The resulting code has been used to simulate the flow past a submerged hydrofoil. The numerical results are compared with those of the previous model and with the experimental data obtained by Duncan.     

Axial vibrations of a propulsion system taking into account the couplings and the boundary conditions

Calculations of the axial vibrations of a marine power transmission system are a very difficult problem owing to the complicated couplings and difficulties in determining the boundary conditions. The torsional–bending–axial coupling action of the system should be accounted for when considering its dynamics. A determination of the mutual interference of system vibrations and their boundary conditions is also necessary. A performance analysis of the main engine bearings, the thrust bearings, and the axial dampers should also be carried out. Thus, the effects of additional bending stresses in the crankshaft and possible vibrations of the ships structure due to the reaction force in the thrust bearings should be considered. I have devised a computer program to analyse the axial vibration problem. The numerical analysis method presented is compared with measurements (performed on real ships) and verified by them.