Study on fast integration method for Bessho form translating–pulsating source Green’s function distributing on a panel

Abstract

The singularities, oscillatory performance of the panel source functions, which are derived by using a single integral expression of Bessho form translating–pulsating source Green’s function and represent the velocity potentials due to the translating–pulsating sources distributing on a panel, are analyzed. Relative fast numerical integration methods, such as the “LOBATTO Rule”, the “Variable Substitution Method (VSM)” and the “Steepest Descent Integration Method (SDIM)”, are used to evaluate this type of Green’s function. For SDIM, complex domain is restricted only on the θ-plane; meanwhile, the integral along the real axis is computed by use of the VSM to avoid the complication of numerical search of the steepest descent line. Auxiliary techniques, such as local refinement of integral steps in narrow zone near inflection points when using VSM, are applied so as to improve accuracy. Contour integral technique is adopted, and a simple numerical method for justifying the relative positions of singular points and integral contour is established based on Green’s theorem. The numerical method is validated by comparison to other existing results, and is shown to be efficient and reliable.