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Under the’ assumption of linearization of the free-surface condition, makinguse of Green’s function method and the convolution theorem, analytic solutionsof perturbation velocity potentials which correspond to three dimensional unsteadythickness problem and lifting problem caused respectively by arbitrary motionsof a body and a hydrofoil beneath the water surface can be achieved in the closedform, In general, the whole perturbation velocity potential consists of threeterms, namely φ=φ1+φ2+φ3 , where φ1 denotes the induced velocity potential ofthe surface singularity distribution in an unbounded fluid, φ2 denotes its mirrorimage and φ3 denotes that of wave formation which includes the memory effect ofthe action of the singularity distribution. Utilizing the polynomial expansionof sin[(t-τ)] , the similarity between φ2 and φ3 is discovered and thus asimpler differential relation between them is obtained. Applying this relation,the amount of work in calculation of φ3 which is the most time-consuming one willbe reduced significantly. It is favorable not only for dealing with unsteady wave-making problems but also for solving the steady ones in virtue of evading a majordifficulty which has to be encountered during the evaluation of an improper inte-gral containing a singularity in the Green’s function. The limitation of this new technique turns out to be its slower convergenceas the Froude number is lower.
Under the ’assumption of linearization of the free-surface condition, makinguse of Green’s function method and the convolution theorem, analytic solutionsof perturbation velocity potentials which correspond to three dimensional unsteadythickness problem and lifting problem caused separately by arbitrary motions of a body and a hydrofoil beneath the water surface can be achieved in the closedform, In general, the whole perturbation velocity potential consists of threeterms, ie φ = φ1 + φ2 + φ3, where φ1 denotes the induced velocity potential of the surface singularity distribution in an unbounded fluid, φ2 denotes its mirrorimage and φ3 denotes that the wave formation which includes the memory effect of the action of the singularity distribution. Utilizing the polynomial expansion of sin [(t-τ)], the similarity between φ2 and φ3 is discovered and thus as simpler differential relation between them is obtained. Applying this relation, the amount of work in calculation of φ3 which is t he is time to consume one willbe reduced significantly. It is favorable not only for dealing with unsteady wave-making problems but also for the steady ones in virtue of evading a majordifficulty which has to be encountered during the evaluation of an improper inte-gral containing a singularity in the Green’s function. The limitation of this new technique turns out to be its slower convergenceas the the Froude number is lower.