A Determinant Aided Fixed Point Method for Nonlinear Coupled Constitutive Relation of Rarefied Nonequilibrium Flows

Abstract

The Nonlinear Coupled Constitutive Relation (NCCR) model is derived from the generalized hydrodynamic equations of Eu and has the capability to describe some significant characteristics of rarefied flows. However, the NCCR model is a complicated nonlinear system, and previous iterative methods for solving the NCCR equations have been observed to be associated with unphysical solutions and instability in some unfavourable conditions. In this study, a new numerical method for solving NCCR equations is proposed to enhance the reliability of the NCCR model. An objective function for a single variable is employed within a fixed point perspective to determine the solution, and the NCCR equation system is reorganized into a smaller linear matrix system for iterative processes. The determinant of the matrix system is used to search the valid solution region, ensuring the method’s robustness. Three typical flow problems in transition regimes are conducted to validate the numerical performance of the proposed method. Results show that the computational time of the proposed method is only approximately 2 to 6 times that of the NS solution, representing efficiency at the same magnitude order of NS solvers and enabling broader engineering applications.