Convergence Stability of Through-Flow Analyses via the Implementation of the Secant and Bisection Iteration Schemes

Abstract

One of the most frequently encountered problems in engineering is dealing with non-linear equations. For example, the solution of the full Radial Equilibrium Equation (REE) in Streamline Curvature (SLC) through-flow methods is a typical case of a scientific analysis associated with a complex mathematical problem that can not be handled analytically. Various schemes are used routinely in scientific studies for the numerical solution of mathematical problems. In simple cases, these methods can be applied in their original form with success. The Newton-Raphson for example is one such scheme, commonly employed in simple engineering problems that require an iterative solution. Frequently however, the analysis of more complex phenomena may fall beyond the range of applicability of ‘textbook’ numerical methods, and may demand the design of more dedicated algorithms for the mathematical solution of a specific problem. These algorithms can be empirical in nature, developed from scratch, or the combination of previously established techniques. In terms of robustness and efficiency, all these different schemes would have their own merits and shortcomings. The success or failure of the numerical scheme applied depends also on the limitations imposed by the physical characteristics of the computational platform used, as well as by the nature of the problem itself. The effects of these constraints need to be assessed and taken into account, so that they can be anticipated and controlled. This manuscript discusses the development, validation and deployment of a convergence algorithm for the fast, accurate and robust numerical solution of the non-linear equations of motion for two-dimensional flow fields. The algorithm is based on a hybrid scheme, combining the Secant and Bisection iteration methods. Although it was specifically developed to address the computational challenges presented by SLC-type of analyses, it can also be used in other engineering problems. The algorithm was developed to provide a mid-of-the-range option between the very efficient but notoriously unstable Newton-Raphson scheme and other more robust, but less efficient schemes, usually employing some sort of Dynamic Convergence Control (DCC). It was also developed to eliminate the large user intervention, usually required by standard numerical methods. This new numerical scheme was integrated into a compressor SLC software and was tested rigorously, particularly at compressor operating regimes traditionally exhibiting convergence difficulties (i.e. part-speed performance). The analysis showed that the algorithm could successfully reach a converged solution, equally robustly but much more efficiently compared to a hybrid Newton-Raphson scheme employing DCC. The performance of these two schemes, in terms of speed of execution, is presented here. Typical error histories and comparisons of simulated results against experimental are also presented in this manuscript for a particular case-study.