用保角转绘坐标解二维和轴对称可压流Euler方程

本文推导出用保角转绘正交曲线坐标表示的二维与轴对称Euler方程,方程中的未知量以曲线坐标方向的速度分量表示,并有不含未知量导数的源项。方程组的形式与直角坐标系中的接近,因而简化了计算。本文将该方程组用于计算二维任意翼型和旋成体轴对称绕流,采用分辨率较高的Eberle特征平均显式有限体积差分算法求解,在不同区域中用不同的显式推进步数加速收敛。本方法已分别对翼型和旋成体轴对称绕流编制了计算程序,可用于自由流为亚声速、跨声速和超声速的绕流计算。大量计算结果表明,本方法精度好,激波分辨率高,编程序简单。

SOLUTION OF EULER EQUATIONS TO 2-D AND AXISYMMETRIC COMPRESSIBLE FLOWS USING CONFORMAL MAPPING COORDINATES

The common used Euler equations of conservative form in curvilinear coordinates contain contravariant velocity components as well as Cartesian velocity ones, which leads to complexities of the formulation, especially the boundary conditions. It is well known that 2-D Laplace equation is invariant under comformal mapping, and that, for compressible flow problems, the conservative full-potential equation is kept the same form as that in Cartesian system under the transformation by conformal mapping, it means that the use of the orthogonal coordinates formed by conformal mapping greatly simplifies the govering equation for problems with boundary of complex shapes. In the present paper, conformal mapping is employed to transform two-dimensional and axisymmetric Euler equations. It is found that the Euler equations can be cast in quasi-conservative form with source terms containing no unknown derivatives, and that there are, as the unknown velocity components, only those in aligning with the directions of curvilinear coordinates in transformed equtaions. The equations derived are almost as simple as those in Cartesian system. These equations are applied to computation of two-dimensional and axisymmetric flows past respec- tively arbitrary airfoils and bodies of revolution, wherein the explicit finite difference formulation of characteristic averaging, proposed by Eberle, is used. To accelerate convergence to steady state, we take different number of time marching steps in different regions. With the proposed algorithm, the Fortran programs are developed for computation of compressible flows past airfoils or bodies of revolution with subsonic, transonic or supersonic free stream. As examples of computation by the computer code developed, the pressure distributions and isomach contours for the flows past RAE 2822, NACA 0012 airfoils and hemisphere-cylinder are given by computation. The results show good accuracy and shock resolution for the present method.