Comparison of the boundary element methods for streamline tracking

Abstract

A comparison of the stream function, Euler, and modified Euler methods for tracking streamlines of simple fluid systems is presented. The errors introduced during tracking are evaluated by means of the right-angled wedge problem to examine the accuracy of streamline positioning and the repeated five-spot pattern displacement problem to assess the accuracy of the velocity evaluation. The stream function method, based on the complex variable boundary element method, exhibits the most accurate and stable performance, because the stream function can be directly computed and the tracking points are exactly on the correct streamline regardless of the advance length. Although the accuracy for the velocity evaluation is reduced with larger advance lengths, it is not severe for practical purposes. The Euler method, based on the real variable boundary element method, cannot track streamlines directly and unable of assessing the stream function, because the velocity vector at a certain point is used to find the next point on the streamline, and the stream functions at the two points may not be the same. Consequently, for advance lengths, the Euler method yields inaccurate streamline positioning and velocity evaluation. The modified Euler method is based on the same methodology except for an improved evaluation of the velocity components. Although it requires more computational time, the modified Euler method exhibits the second best performance. The stream function method is recommended to use when the stream-function field can be evaluated. The modified Euler method should be used, if only the potential field is known and an abrupt change in streamline directions is expected.