Rankine Source Research Articles

A new frequency domain method to solve the potential flow problem

The potential flow theory is a reliable tool widely-used in ocean engineering to solve the fluid hydrodynamic problems such as the wave hydrodynamic problems. The Rankine source and its image about the seabed is often chosen as the Green's function to solve the potential flow problems for its simple form, easy singular-integral treatment and few boundary conditions in the boundary integral equations. However, traditionally, the boundary element method based on it is only executed in the time domain by time stepping technique which requires lots of computation due to the time domain form of the damping zone's boundary condition. To apply this Green's function in frequency domain, this paper derived a parameterized frequency-domain formula for the boundary condition of the damping zone, and successfully applied the Rankine source and its image about the seabed in frequency domain. This paper's work possesses a great potential to promote the efficiency in solving the potential flow problems.

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

Time-domain numerical simulation for multi-ships moving in waves with forward speed

With the development of the shipping industry, the impact of hydrodynamic interaction between two or more ships on navigational safety has received widespread attention. However, limited research has been conducted on multi-ships moving in waves with forward speed due to the increased complexity of hydrodynamic interactions caused by speed. Therefore, this paper establishes a numerical model of multi-ship hydrodynamics under the wave impact based on the time-domain Rankine source panel method. The hydrodynamic performance and motion characteristics of ships advancing in waves are investigated. An artificial damping layer is applied to free surface boundary conditions to absorb reflected waves. The fourth-order Runge-Kutta method is used for time integration to solve the motion of the ship. The hydrodynamic and motion response of the classical Wigley III ship model are calculated at each wavelength, and the simulation results are compared with other calculations to verify the reasonable reliability of the numerical method employed in this study. On this basis, sensitivity analysis of different parameters is conducted to reveal the influence of various parameters such as speed, wave direction, transverse distance, and water depth on the motion response of the Wigley III ship model in waves.

Investigation on wave-body interactions by a coupled High-Order Spectrum method with fully nonlinear Rankine source method

The complex frequency components of ocean waves inherently result in energy transfer during the evolution of wave spectra. This paper introduces a High-order spectrum method coupled with a fully nonlinear hydrodynamic solver (HOS-FNL), for modeling wave-structure interaction within a modulated nonlinear wave train. The approach combines the high-order spectrum method for modeling the incident wave field in a large water domain, with the nonlinear Rankine source method for calculating the disturbed wave field around the floating structure. We validated the HOS code through simulations of free and locked wave cases, and the entire modulation process is analyzed and compared with existing literature. Experimental measurements of the loads acting on a container ship in regular waves are used to validate the core algorithm of the hydrodynamic solver, yielding satisfactory results. Additionally, the behavior of a ship due to finite-amplitude uniform wave train is investigated, visualizing the motion and load response of the ship, wave run-up, diffraction wave patterns, and velocity vectors of the free surface to emphasize the nonlinear characteristics. The HOS-FNL method demonstrates remarkable effectiveness and efficiency, showcasing its potential for a wide range of practical marine applications.

Numerical study of a novel small waterplane area USV advancing in calm water and in waves using the higher-order Rankine source method

The wave loads and motion responses of an Unmanned Surface Vehicle (USV) in water will directly influence the powering requirement, energy supply design and function of the installed sensors. In the present article, to investigate the wave loads and motions response characteristics of a USV advancing on the free surface, a Higher-Order Rankine Source (HORS) method is proposed. During the discretization of the boundary elements, bi-quadratic B-splines are applied to distribute the velocity potentials on the body surface and free surface. Based on the proposed HORS method, steady state simulation and time-domain simulation approaches are both used for predicting the wave loads and motion responses of a novel Small Waterplane Area (SWA) USV in calm water and regular waves, respectively. The predicted wave loads in calm water and the predicted heave and pitch in heading wave are compared with corresponding model test data. The good agreement found indicates the validity of the proposed HORS method. Finally, the heave and pitch motion responses and the wave components of the USV including diffraction and radiation forces and Froude−Krylov forces are investigated when the SWA USV is running against heading wave under different conditions of wavelength, wave steepness and advancing velocity.

Motion characteristics of large arrays of modularized floating bodies with hinge connections

Hinged arrays have garnered increasing interest due to their potential to provide flexible and adaptable solutions for the various challenges faced in ocean development. The effectiveness of these arrays in engineering applications heavily depends on the motion characteristics of each individual module, rather than specific modules, such as the one with the strongest motion. However, the presence of hinge constraints results in coupled motion responses of all modules instead of independent ones. The objective of this study is to investigate the motion behavior of large arrays formed by multiple floaters hinged together, while existing literature mainly focused on two-body hinged systems. Based on the potential flow theory and Rankine source panel method, a numerical program was developed to calculate the hydrodynamic interactions and the coupled motion responses. First, a model test was conducted to validate the developed frequency-domain simulations. A good agreement was achieved. Then, the effects of hinge constraints, the number of modules, and two external constraints on the motion responses of the entire array were discussed. The results indicated that the heave motion of the array subjected to hinge constraints was significantly suppressed, but a strong pitch motion occurred in a larger wavelength range. For hinged arrays, the floaters located at the two ends were most likely to be excited with the strongest motions. Moreover, a shorter hinged array could be used to quantify the trends in the motion of arrays with more floaters. The calculation results also revealed that the motion responses of a hinged array were highly sensitive to the external constraints, e.g., mooring lines.

An eigenvalue method based on the integral equation with a Rankine source for the band structure of water surface waves over periodic scatterers

An eigenvalue model is developed to obtain the band structure of water surface waves in the presence of an infinite array of periodically arranged scatterers, which include both underwater terrains and fixed bodies on the free surface or in the water. Combined with Bloch's theorem, the scattering of waves over arbitrary terrain or bodies is described as a boundary value problem of the Laplace equation, from which the boundary integral equation can be derived and transformed into an eigenvalue problem through the boundary element method to obtain the band structure of water surface waves. Comparisons are made with various existing models and reflection coefficients from experiments, which shows the validity and accuracy of the proposed model.As an application of the present model, the stopbands for convex parabolic terrain, concave parabolic terrain, sinusoidal terrain and cylinders with centres on the still water surface are calculated. The dependence of the stopbands on the relative size of scatterers and relative water depth is investigated and analysed.

Numerical investigations on nonlinear motions of ships sailing in large-amplitude waves based on hybrid HOBEM

This paper presents a nonlinear numerical approach based on hybrid higher-order boundary element method (HOBEM) to investigate the wave-induced motion responses of ships with forward speed. The radiation and diffraction forces in time domain are determined by convolution of the impulsive response functions computed from frequency-domain hydrodynamic data of the hybrid method using a combination of Rankine source and translating-pulsating Green's function. In its numerical implementation, steady flow effects are included in the inner domain encompassing the ship, 9-node bi-quadratic curvilinear elements are employed to discretize the boundaries and analytical quadrature formulas are derived to calculate the influence coefficients related to the Green's function over the control surface so as to improve the accuracy, efficiency and stability. Meanwhile, the nonlinear incident wave force and hydrostatic restoring force are calculated over the instantaneous wetted surface of the ship hull during the motion process to capture the nonlinear properties. Based on the method, a numerical program is developed to predict and analyze the motion response of a modified Wigley hull and a non-wall-sided S175 containership advancing in waves with different amplitudes and heading angles. By comparing the present computed results with experimental data and other numerical solutions, it is found that the nonlinear calculations can produce more accurate motion responses in large-amplitude waves with strong nonlinearity. And the present nonlinear method based on hybrid HOBEM is of higher accuracy and better stability than traditional nonlinear methods based on three-dimensional pulsating source (3DP) and translating-pulsating source (3DTP), especially for S175 with large flare near the waterline.

Numerical investigation of wave interaction with two closely spaced floating boxes using particle method

This paper applies Improved Meshless Local Petrov-Galerkin method based on Rankine source function (IMLPG_R) to simulate two-dimensional wave interaction with two closely spaced floating boxes. The present study considers four different box configurations based on the degree of freedom, namely FF (both boxes fixed), HH (only heave allowed), RR (only roll allowed), and MM (both heave and roll allowed). The influences of the different degrees of freedom of the boxes on the wave elevation in the gap, and forces acting on the boxes, are investigated. Regular waves with a wave period ranging from 0.6s to 1.6s and two focused wave groups based on the JONSWAP spectrum with a peak frequency of 0.869Hz and 1.081Hz, respectively, are employed for the analysis. The piston mode resonance frequencies of different configurations were observed to vary slightly depending on the motion restrained. Compared to the FF configuration, the MM configuration showed a 28.3% increase in wave amplification in the gap. Some unique characteristics observed in the response amplitude operator (RAO) and flow patterns around the boxes are explained using instantaneous velocity and vorticity diagrams. The response of various configurations under a nonlinear transient-focused wave is also examined. The first-order and second-order time series of wave elevation in the gap is obtained by using the four-phase combination method.

Prediction of the Hydrodynamic Forces for a Ship Oscillating in Calm Water by an Improved Higher Order Rankine Panel Method

This paper presents a frequency-domain Rankine source method based on a biquadratic B-spline scheme with an improved radiation mechanism. The improved radiation mechanism, based on the introduction of spatially varying Rayleigh artificial damping in addition to the simplified Seto’s radiation boundary conditions, is considered for modeling radiation of generated waves at various τ conditions, where τ=ωU/g including the undercritical condition (τ<0.25); this condition is present when a ship undergoes slow translation or low oscillatory frequency. In evaluations, the proposed method yields accurate solutions for unsteady flows produced by an oscillating, translationally moving submerged singularity. The radiation problem induced by a RIOS bulker is solved to have the resultant added mass and damping coefficients for further comparisons with the experimental data and the public numerical prediction by a simplified combined method at a wide τ region.

An improved radial basis function for marine vehicle hull form representation and optimization

The marine vehicle is one of the most important platforms in oceanic research and development. During the evolution of marine vehicle design, hull optimization is a key topic consisting of various steps such as hull deformation analysis, optimization algorithm selection and hydrodynamic performance evaluation. For hull deformation analysis, radial based function interpolation technique is usually applied with Wendland’s ψ3,1 function, which requires a great number of fixed points. To improve the optimization effect, this paper proposes a novel radial basis function for better support radius condition. According to the support radius condition, the effect of different deformation functions acting on hull form is studied. To verify the effectiveness of the improved radial basis function in marine vehicle hull optimization, a Series 60 hull is taken as the initial hull, and the Rankine source method is utilized as the solver to calculate the wave resistance coefficient. The Non-dominated Sorting Genetic Algorithm (NSGA) is applied to determine the optimal hull form at given speeds. By comparing the optimization results, the benefits and drawbacks of the improved radial basis function are analyzed. The results show that the number of fixed control points can be reduced by applying the improved radial basis function. When different deformation functions are acting on the marine vehicle hull, the improved radial basis function makes the gradient transition smoother. In the process of hull optimization, the wave resistance coefficient obtained by utilizing the improved function to deform the hull is smaller than that of using Wendland’s ψ3,1 function under different support radii.

Numerical Simulation of the Kelvin Wake Patterns

The ship wave is of great interest for wave drag and coastal erosion. This paper proposes a mechanism of ship wave transformation to explore the effects of ship speed and ship size on the waveform. Firstly, based on the theory of potential flow, the boundary integral equations for the Kelvin ship waves are obtained by deploying the different Kelvin sources or Rankine sources. Then, these integral equations are numerically discretized to a set of nonlinear equations. Finally, the Jacobian−free Newton–Krylov method with a preconditioner is adopted to solve the nonlinear equations. Though imitating plenty of different Kelvin wave patterns, the mechanism of ship wave transformation is proposed to conveniently generate the polymorphic Kelvin wake patterns. The above numerical simulation scheme is verified by comparing simulation results with real ship waves. After that, the wake angle is discussed with the effects of Froude number, source strength and source type by following the mechanism of ship wave transformation. The results show that the wake angle tends to decrease with ship speed but increase with ship size. In addition, for high ship speeds, the effect on the wake angle can be more dramatic.

Study on the Improved Radiation Boundary Conditions Based on the Quadratic B-spline Rankine Panel Method

This study presents a higher-order Rankine source method that is based on a biquadratic B-spline scheme with an improved radiation boundary condition. This method solves flow problems under all conditions, where , including the undercritical condition ( < 0.25); this condition is present when a ship undergoes slow translational motion. The proposed method is thus more flexible than the many Rankine-source models in the literature that are applicable only to the overcritical condition ( > 0.25). Specifically, the improved radiation boundary condition is derived from upstream Seto radiation boundary conditions and by incorporating the Rayleigh damping distribution over a free surface; the improved condition is explicit and efficient for solving flow problems. In evaluations, the proposed method yields accurate solutions for unsteady flows, which are produced by an oscillating, translationally moving submerged singularity. The method is also compared against asymptotic solutions and against analytic solutions obtained using a three-dimensional translating-pulsating-source Green function.

An improved impermeable solid boundary scheme for Meshless Local Petrov–Galerkin method

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson’s Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition.

Ship motion-sloshing interaction with forward speed in oblique waves

Seakeeping properties of large LNG carriers with partially filled tanks are known to be influenced by sloshing inside their tanks, by wave conditions and wave heading, and by their forward speed. An efficient hybrid numerical method was developed to study wave-induced ship response coupled with sloshing-induced compressible fluid response inside the tanks of a 138,000 m3 LNG carrier at zero forward speed and at the constant forward speeds of 12 and 20 kn. Wave conditions considered were oblique regular waves at heading angles of 0, 30, 60, 90, 120, 150, and 180 deg. The numerical approach combined a nonlinear boundary element Rankine source code to solve the weakly nonlinear seakeeping problem with a Reynolds-averaged Navier-Stokes equations solver to compute the compressible sloshing flow inside the tanks. Coupling effects of the six degrees-of-freedom ship motions in beam waves were validated against comparable model test measurements of ship responses and against free surface elevations in tanks. Results demonstrated that sloshing affected the simulated seakeeping behavior of the subject LNG carrier advancing at forward speeds in oblique waves. As sloshing had the greatest influence on the ship's roll motions, a key aspect was the derivation of roll damping from model decay tests.

Research on Design Optimization of High-Speed Ship Bulbous Bow Based on Nelder-Mead Algorithm

In order to reduce the wave-making resistance of ships, this paper proposes an optimization design method for bulbous bow form based on potential flow theory. In this paper, we restrict our interest to parametric deformation of the bow, at the same time constrain the displacement and the length of the ship. Based on Rankine source method, the wave-making resistances of DTMB5512 and ITTC Ship A-2 are calculated. For optimization, Sobol algorithm is selected to create sample space; based on it, we use Nelder–Mead algorithm to obtain the optimal hull form in the space. Additionally, we compare and analyze the hydrodynamic performance of the ship. The results show that the optimization design method reduces the wave-making resistance of DTMB5512 and ITTC Ship A-2, which verifies the feasibility of the method. Introduction Energy Efficiency Design Index (EEDI) guides the future development of the shipbuilding industry toward green ships and accelerates the pace of energy conservation and emission reduction. EEDI is closely related to hull form; therefore, the design optimization of ship hull focus on reducing EEDI becomes the central issue of the current research (Hou et al. 2018).

Numerical investigation on evaluating nonlinear waves due to an air cushion vehicle in steady motion by a higher order desingularized boundary integral equation method

The problem of nonlinear ship waves generated by an air cushion vehicle is studied and modelled in the framework of potential flow theory. A higher order desingularized boundary integral equation method (HODBIEM) using Rankine source is developed. A stationary iterative method is adopted to solve the nonlinear free surface which is discretized into the inner area disturbed by cushion pressure and the outer undisturbed area. Techniques for numerical implementation are described in detail. To save computing time, grid density and domain size are refined.Both straight-ahead and yawed cases are considered for validations. Wave-making resistance versus various speeds in deep and shallow water are calculated for straight-ahead vehicles. The resistance curves obtained by HODBIEM agree well with results of nonlinear CFD simulations. The linear theory predicts higher peak value of resistance than nonlinear methods. The experimental data fit at low Froude number but deviate from numerical predictions when exceeding primary hump speed. Effects of speed and water depth are analyzed by wave patterns and wave profiles. Resistance, side force and yaw moment are calculated at Fn = 0.533 for drift angle from 0° to 90°. Satisfactory results are given by present method. Details about wave patterns and wave angles are discussed.

Scenario-based optimization design of icebreaking bow for polar navigation

Global warming and ice extent reduction in Arctic makes commercial shipping through Arctic waters be promising in the near future. The fuel consumption and emission from ships on the Arctic shipping routes, which mainly depends on the ship resistance, is key factor to determine if it is more cost-effective comparing to traditional shipping route. The shipping route in Arctic water include open water and ice-covered water areas. The polar ships traveling through Arctic water need to have ice-strengthened hull to have ice-breaking capability to proceed safely in ice-covered water. Usually the bow hull form with good ice-breaking capability have high resistance in open water, while the bow form with good open-water efficiency (such as bulbous bow) may not have ice-breaking capability. This paper is to develop a methodology for optimized hull form design so that the minimum overall resistance in open-water and ice-covered water can be achieved, while maintaining desired level of ice-breaking capability for considered shipping route with expected ice conditions. The methodology proposed in this paper is based on the ship's desired route with historic ice data to define its typical operation profile. For the test example of this methodology, based on the desired route, the historic ice data are extracted from public available data source to determine the sections of open-water and ice-covered water, as well as the ice-thickness distribution along the route. The ship's resistance in open-water is calculated based on Rankine source method, while the ice resistance is calculated by Lindqvist formula. The overall resistance is weight-averaged based on the route's sections of open-water and ice-covered water. The ice-breaking capability is measured by the ice loads giving by the IACS Polar UR, which serves as the constraint during the optimization process. The parametric model is established for the optimization process. With an ice-breaking bow base design, the optimized bow hull form is achieved on the specified shipping route with historic ice condition. The result shows that the overall resistance is reduced by 5.42% while maintaining enough ice-breaking capability for the considered sample ship. Discrete Element Method is used to verify the optimized hull form's ice-breaking performance compatible with original hull form. It is shown that the methodology developed in this paper can be generally applied to polar ship hull form design for different shipping route and ice conditions.

Two-Analytical Comparison on Force Measurements and The Wave Pattern

This work presents a comparison of two analytical methods on the components of resistance, transverse and divergent waves, and the profile and contour of the waves. Michell’s thin linearization theory is applied to estimate the wave resistance and analysis of the method compared to the solution of the wave resistance coefficient based on the Rankine source method. The hull of series 60 is utilized as the initial hull, and varying the shape of the bow to the mid-ship with the hull offset variable. Computation of the wave resistance coefficient at low Froude number (Fr<0.3) shows the results of the wave resistance from the thin ship theory are higher, but it gives a graphical agreement with the Rankine source method. The wave profile provides a very similar profile line for both of them. However, the thin ship theory does not establish a significant difference in the wave profile from the initial hull of the improved hull variation. Comparison of contour displays is presented in different versions of the wave contour. Still, it affords a good contour match on the initial hull and the results of hull variations from both methods. At high Froude number (Fr>0.7), the initial hull and variations in the hull have minimal effects with small differences in wave resistance, profile and contour.

Three dimensional numerical modelling for wave radiation problem under arbitrary seabed condition

Three dimensional Rankine source model is utilized to investigate the hydrodynamic responses for wave radiation problems under arbitrary seabed condition in time domain. The free surface, seabed surface and body surface are presented by layer of continuous panels rather than a discretization by isolated points. These panels are positioned exactly on the fluid boundary surfaces and no desingularization technique required. To accommodate the sloping seabed profile, a straightforward source panel distribution on seabed surface is newly developed with two parameters determined by numerical tests. The accuracy and efficiency of the numerical solutions are validated by comparison with published data. The influences of flat seabed, uneven seabed and sloping seabeds are examined. Numerical simulations demonstrate that various seabed profiles have significant and complicated effects on the body hydrodynamic response characteristics.