Generalized two-dimensional Lagally theorem with free vortices and the method of fundamental solutions to the application of fluid-body interaction problems

本論文發展出一個可用於研究二維完全耦合的流固互制問題的數值模式。 Lagally定理使用那些可用以表示流場的奇異點的強度及位置來描述作用於物體上的力。因此本模式利用Lagally定理來求得作用於物體的力並進一步得到物體運動的控制方程式。但是目前為止，Lagally定理只考慮流場中存在單源或偶源。故本文首先將Lagally定理推廣至渦源存在中的流場。本模式首先驗證三個在文獻中已被研究過的問題:(1)一個自由圓柱與渦源之交互作用，(2)移動的Foppl 問題 ，(3)一個等速的圓柱接近另一相同圓柱。接著，我們研究當一個強制移動的Foppl 系統接近另外一個目標圓柱的問題。並且考慮目標圓柱是固定的或是可以自由移動的。我們考慮五種不同的渦偶對的初始結構，並研究兩個圓柱的受力情形。為了將本模式推廣到包含不規則幾何形狀的物體的問題，我們進一步使用基本解法來求得可用於描述流場的奇異點的強度及位置，並且利用這些資訊結合Lagally定理來得到作用於物體上的力。為了驗證，我們利用這個新的模式來重新計算前述的單個圓柱以及渦源的交互作用。為了展現本模式的效果。最後，我們考慮一個渦偶對正向接近一個橢圓形圓柱之運動。A potential flow based numerical model which can be utilized to study the fully coupled fluid-structure interaction problems in two dimensions is developed in this work. The Lagally theorem is adopted in the current model to obtain the equation of motion for object. This theorem describes the unsteady hydrodynamic force on a body exhibiting arbitrary motion in an ideal fluid by the properties of the singularities employed to generate the flow. So far, only sources and dipoles have been considered and the present work extends the theorem to include free vortices in a two-dimensional flow. The developed model is validated by reproducing the system dynamics or force evolution of three existing problems reported in the literature: (1) a free cylinder interacting with a free vortex, (2) the moving Foppl problem, and (3) a cylinder in constant normal approach to a fixed identical cylinder. The present theorem is then employed to study the problem when the cylinder in the moving F‥oppl system is forced to move at constant velocity normally towards an identical target cylinder that is fixed or free to move. The force developed on each cylinder is investigated with five different initially tail vortices configurations for each example. To extend the current model to further consider the problems involving irregular shaped objects, the method of fundamental solutions is applied to numerically solve the strength as well as the position of the internal image singularities employed to represent the flow. The new model is validated by re-examining the single vortex-object interaction. Then the inline approach of a vortex pair towards an ellipse is considered to demonstrate the capability of the proposed model