One Boundary Element Algorithm Application to Calculation of Motion near the Solid Boundary

摘要

Motion of small objects near the solid boundary is fundamental problem of applied hydrodynamics. The discrete vortex motion in fluid, small liquid drop motion in gas, small gas bubble motion in liquid, small solid particle motion in liquid and gas can be considered as examples of such problems. The main difficulty in this problem is necessity of very accurate velocity determination near the boundary. Traditional numerical methods (finite difference and finite element method) cannot provide desired accuracy. One of the most sufficient disadvantages of the ordinary boundary element method is fast increasing of error near the boundary. This effect is especially strong on the distances less, than one half of boundary element length from the boundary. The aim of the present work is development of boundary element algorithm, which can overcome mentioned difficulties. As a base of such algorithm the functional equations of V. Kupradze [1] are used. The error sources of boundary element method are considered in paper [2]. Comparing the accuracy of numerical calculations of test problem for different domains it can be concluded, that the approximation of the boundary incomes the most contribution into the total error of method, at least near the boundary. Note that the approximation of the boundary is necessary for singular boundary element method, since otherwise it is difficult to evaluate singular integrals analytically. However for regular method the approximation of the boundary isn't obligatory, since all using integrals are regular. Excluding of the boundary approximation from the regular boundary element method, which is replacing of integration along approximating curve by integration along real boundary, is main idea of proposing algorithm.Several problems with known analytical solutions were used as test problems, for example, external potential ideal fluid flow about the circle without circulation. Considered test problems confirmed high accuracy of the proposed algorithm near the boundary.The proposed approach was applied to the spraying process in protective coating creation and to the problem of discrete vortex motion near the solid boundary in combined boundary element and discrete vortex method proposed in the paper [3].