| At present,in the study of wave-body interactions on the free surface within the framework of the potential flow theory,especially considering wavy properties of the free surface,the frequency domain method is mature.However,frequency domain approaches are not suitable to deal with transient problems.On the other hand,due to the singularity and high oscillations of free surface Green functions,it is difficult to get convergent results by using free surface Green functions.For these reasons,a theory based on time-domain elementary solutions is proposed in this thesis,and it is applied to study the cylinder radiation and wave diffraction around a vertical circular cylinder by transient waves.In this thesis,the hybrid Green function method composed of the Rankine source method and the free surface Green function method is taken as the starting point.Based on analyses of roles of the control surface dividing the fluid domain in the hybrid Green function method,the concept of time-domain elementary solutions is proposed.And the boundary integral equation is given based on time-domain elementary solutions in the multi-domain method.Secondly,an infinite vertical circular cylinder is selected as the analytical control surface in present thesis,and the velocity potential and its normal derivative on the control surface are expanded into Fourier-Laguerre series.Expressions of time-domain elementary solutions based on the Fourier-Laguerre expansion are obtained.Different from traditional boundary element methods in which the control surface needs to be panelized,the control surface in present work does not need to be panelized.The boundary integral equation,which is constructed on the analytical control surface in the sense of Galerkin collocation,represents the relationship between expansion coefficients of the velocity potential and its normal derivative,not the relationship between the velocity potential and its normal derivative on discrete panels on the control surface.By using the orthogonality of basis functions,elementary solutions and related coefficients in the boundary integral equation could be analytically simplified from multi-folds integrals to single variable integrals with respect to wavenumber.In this thesis,a numerical algorithm based on the numerical steepest descent method is also developed to analyze and calculate oscillatory and highly oscillatory integrals related to time-domain elementary solutions.The comparison with results obtained by LevinRule in the software Mathematica shows that the algorithm proposed in this study is effective.The time-domain free surface Green function(point solution)is compared with the time-domain elementary solution based on Fourier-Laguerre series expansion and numerical results show that the elementary solution does not inherit the singularity and high oscillation of the point solution.As applications of time-domain elementary solutions,waves generation by radiation of a cylinder in deep water is discussed in this thesis.In order to better verify the boundary element method based on time-domain elementary solutions and proposed numerical algorithms,the semi-analytical solution of cylinder radiation in the time domain is derived which is based on velocity potential decompositions.Relationships between the instantaneous term and the memory term are also analyzed.Numerical results from two methods verify each other.On the one hand,it shows the effectiveness of time-domain elementary solutions proposed in this thesis;on the other hand,it shows the correctness of the time-domain semi-analytical solution and corresponding asymptotic analyses derived in this thesis.The semi-analytical solution enriches examples with time-domain analytical solutions in hydrodynamic researches and it also provides a new benchmark for numerical verifications of transient problems by time-domain analysis methods.In order to make the time-domain analysis method give full play to its advantages in dealing with transient issues,an in-depth analysis of plane progressive waves with wavefront,which have a remarkable transient feature,in deep water is made.And they are used as incident waves in the study of wave diffraction.By using the algorithm for highly oscillatory integrals based on the numerical steepest descent method and presented in this thesis,the transient wave elevation on the free surface is decomposed into a steady-state term,an initial term,a wavefront term and a local term.After variable substitution,the wavefront term can be expressed as analytical forms of a special function associated with the error function.By using this special function,the wavefront term decreases according to(x/xF)5/2 and(x/xF)-3/2 on two sides of the wavefront demarcation point,respectively.The maximum of the transient wave elevation on the free surface is obtained analytically,which is about 17%higher than the steady-state amplitude.It also reveals the propagation mechanism that the wavefront demarcation point moves forward with the group velocity.Finally,time-domain elementary solutions are used to study the wave diffraction around a circular cylinder,and transient solutions based on the Fourier transform are also given.From numerical results,it can be found that,similar to incoming waves,there exists a maximum of the wave force which is greater than the steady state amplitude(about 3.5%higher in some cases)and it is also found that values of wave forces at the initial time are not equal to zero.These initial values are asymptotically analyzed and a physical explanation is given.Numerical results and discussions show that time-domain elementary solutions proposed in this thesis are effective,and evaluations of the point solution are sometimes unnecessary in solving wave-body interactions on the free surface in the framework of the potential flow theory.It can also conclude that considering interactions between transient waves and structures shall be safer for the design of ships and offshore structures. |
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