Numeircal Simulation On The Second-order Hydrodynamic Problems Based On The Taylor Expansion Boundary Element Method

Due to the work environment and its own characteristics of the floating structures,the second-order wave loads acting on the floating bodies cannot be neglected.It is quite necessary to develop an effective numerical method for the second-order hydrodynamic problems.The solution accuracy of the tangential induced velocity at the unsmooth corners of the floating bodies in the traditional constant panel method(CPM)is too low,which result in the large error in the analysis of wave loads and the divergence of the time-simulation process.Dealing with the singular integration and solving the high-order derivatives at the corners in high-order boundary element method(HOBEM)is very difficult.And some existed methods all avoid the calculation of high-order derivatives in the numerical implementation.Hence,this paper develops a novel boundary element method,named the Taylor expansion boundary element method(TEBEM).The TEBEM method proposed in this paper is based on the framework of the low-order direct method.For three dimension hydrodynamic problems,the complete boundary surface will be dispersed into some small quadrilateral or triangular elements,and take the average values of the nodes as the center in each panel.Applied the Taylor expansion to the dipole strength of boundary integral equation(BIE),and reserve the first or second-order derivatives.Be similar to dipole strength,applied the Taylor expansion to the source strength and only reserve the first-order derivatives,then utilize the corresponding tangential first,second and mixed-order derivatives with respect to the field points on the boundary surface to form the closed equations.Finally,we construct the linear algebraic equations,where the dipole strength and its first,second and mixed-order derivatives and the derivatives of source strength are the unknowns,and the source strength is the known.Many influence coefficients will be involved in the TEBEM method.Some own the integral principal values,such as the tangential first,second and mixed-order derivatives of dipole strength and the first-order derivatives of source strength influence coefficient(normalized to positive/negative 1/2).The accurately solving of second-order derivatives of fluid particle prepares the ground for computing the second-order disturbance potential without forward speed boundary value problems(BVP)and calculating the added resistance for the ship hydrodynamic problems with forward speed.In this paper,the first-order TEBEM in frequency domain is applied to calculate the mean drift force of the floating bodies.The velocity potential at the center of waterline can be obtained,which can be induced form the centeroid of mesh nearby the waterline to the center of waterline by the Taylor expansion.So the accuracy of the integral along the corresponding waterline is quite improved,and the accuracy of the near-field formula for mean drift force is also improved remarkably.When the BVP without forward speed is solved by the Rankine source method in time domain,a hybrid method,which is the composite of the multi-transmitting formula(MTF)and the damping zone(DZ),is applied to suppress the reflection of the disturbance wave on the artificial boundary surface.The numerical results of the mounted and truncated cylinder diffraction and radiation problems show that there are not the phenomenon of wave reflection.In addition,the integral format free surface condition is used to update the velocity potential on the free surface.This integral scheme is more stable compared with general numerical differential method for time marching of the free surface condition.Take the cylinder model for example,the first and second-order wave elevation,first-order wave loads,the second-order double frequency force(involving the first-order quantities and second-order velocity potential contribution),the mean drift force and the induced velocity near the corners between the body and free surfaces.Comparing with the semi-analytical and analytical solutions,a good agreement can be obtained for the numerical results.Under the same amount of calculation,the TEBEM method can get the more accurate results,compared with the CPM.Many papers show that it is necessary to consider the influence of the steady velocity potential for blunt ships in solving the unsteady potential.In this paper,take the mounted cylinder for example to validate the calculation accuracy of mj term by the second-order TEBEM method,which overcomes the shortcoming that the second-order derivative integral in the body surface condition need to be changed into the first-order derivatives integral and integral along the waterline through Stokes formula or solve this BVP in non-inertial coordinate system in HOBEM.With the development of energy saving and emission reduction slogan,wave added resistance of ships has been a focus in the research of ship hydrodynamics.In this paper,a mathematical model of Wigley?,a typical blunt ship KVLCC2 model and 57000 bulk cargo ship are chose to research by the Rankine source method and TEBEM method.The contents mainly include the heave and pitch motion and added resistance calculation under the head wave and the wave-ship length ratio:0.3-2.0.In addition,the convergence study is developed,such as the area of the free surface,time-stepping,damping zone strength,low-passing filter strength and so on.The numerical results show that a good agreement can be obtained,compared with the experimental solution and the other numerical results.