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.
Highlights
The streamline curvature method plays an important role in the preliminary design phase of radial[1,2,3,4] and axial turbomachinery applications[5] because of its simple and compact equations, definite physical meanings, and straightforward solving algorithm
The velocity profile on the stream surface is specified by solving the S2m stream surface velocity gradient equation
The fluctuation algorithm involves the convergence analysis of the nonlinear calculation. This is why the finite difference method is superior to the cubic spline fitting method in solving the velocity gradient equation
Summary
The streamline curvature method plays an important role in the preliminary design phase of radial[1,2,3,4] and axial turbomachinery applications[5] because of its simple and compact equations, definite physical meanings, and straightforward solving algorithm. Most through flow codes use the streamline curvature method and derive from those of Hamrick et al.,[6] Smith,[7] and Novak,[8] based on the general theory of Wu.[9] The cubic spline function and quasi-orthogonal grids were applied to the streamline curvature method to enhance its accuracy and efficiency in calculation.[10] With the method, the blade shape is generated by specifying the distributions of relative velocity on both pressure and suction surfaces along the streamline direction.[11] Considering that secondary flow and Department of Fluid Machinery and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, China. For the inverse proposition design of a full threedimensional (3D) centrifugal impeller with streamline curvature method, this article proposes a new finite difference method to solve the velocity gradient equation. For the design of a full (3D) centrifugal impeller, the velocity gradient equation on the S2m stream surface is solved. The velocity gradient equation can be derived from a combination of the first law of thermodynamics, the Euler equation of turbomachinery, and the inviscid momentum equation in a cylindrical coordinate system for the flow on the mean stream surface, such as the following form dWm dq
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