A new finite difference method to solve the velocity gradient equation in streamline curvature method

Abstract

For the inverse proposition design of a three-dimensional centrifugal impeller using the streamline curvature method, the cubic spline curve fitting method is extensively used to solve the velocity gradient equation. Given the deficiency in stability with the cubic spline curve fitting method, a new finite difference method is proposed to solve the velocity gradient equation on the S2m stream surface. In the finite difference scheme, the relative velocity derivative along the streamline direction is decomposed into two terms. One term uses forward difference, and the other uses backward difference. The difference schemes of all other parameter derivatives in the velocity gradient equation adopt forward difference. The method can guarantee the matrix main diagonal elements of the dominant, indicating stable convergence in solving the velocity field. Robustness analysis is performed for both methods, and the new finite difference method shows excellent superiority in stability. Finally, the finite difference method is applied to redesign the Krain impeller. Through computational fluid dynamics, the efficiency of the redesigned impeller at the design operating point is increased by approximately 0.3% and the pressure ratio by approximately 5%. These results show that the difference method is feasible to solve the S2m stream surface velocity gradient equation.