The Accuracy Study On The Spectral Collocation Method-artificial Compressibility Method (SCM-ACM) For Solution Of Incompressible Flow In Cylindrical Coordinates

The incompressible flow problems in a circular tube is very common in the engineering field.As one of the methods to study and analyze the incompressible flow problems,numerical simulation method has been paid more and more attention by researchers with the rapid development of computer technology in recent years.In this paper,the spectral collocation method(SCM)is selected to discretize partial differential equations,which is favored by researchers because of its high accuracy.The artificial compressibility method(ACM)which is easy to converge is used to solve the discrete unsteady governing equations,and the two methods are combined in cylindrical coordinates,the numerical method SCM-ACM is developed for the solution of incompressible flow problems in the circular tube.Firstly,the validity of SCM-ACM program to solve the incompressible flow problems in cylindrical coordinates is verified.According to the requirements of artificial compressibility method,the control equation of the artificial compressibility scheme is established,and then the spectral collocation method is used to discretize the control equation into an algebraic equation in matrix form.In this paper,the Taylor-Couette flow between concentric cylinders is selected as the calculation object to verify the validity of the SCM-ACM program.The data results show that the SCM-ACM developed in this paper can effectively solve the incompressible flow problems in cylindrical coordinates,it not only retains the characteristics of high accuracy and exponential convergence of the spectral method,but also has the characteristics of simple form and easy convergence of the artificial compressibility method.Secondly,the accuracy of SCM-ACM in two-dimensional and three-dimensional cylindrical coordinates is tested.Three sets of incompressible flow problems with exact solutions are constructed in two-dimensional and three-dimensional cylindrical coordinates respectively,three sets of incompressible flow problems are different forms of function equations,which are used to test the calculation accuracy of SCM-ACM in solving incompressible flow problems.The effects of different spatial step sizes,time steps and discrete schemes of time term on the calculation accuracy are studied.The results show that the calculation accuracy of SCM-ACM increases exponentially with the number of nodes before reaching the grid independence.After reaching the grid independence,the number of nodes continues to increase,and the calculation accuracy of SCM-ACM decreases slightly;the time step has almost no influence on the calculation accuracy;the discrete schemes of time term has a great influence on the calculation accuracy of SCM-ACM,and the high-order discrete schemes of time term has higher calculation accuracy;The form of function distribution of the flow problem has a significant influence on the accuracy of the calculation.Finally,the SCM-ACM developed in this paper is compared with other algorithms.In this paper,SCM-ACM is compared with the commonly used simulation software FLUENT and COMSOL in the calculation accuracy and calculation efficiency.The values of the three groups of incompressible flow problems are constructed in two-dimensional and three-dimensional cylindrical coordinates.Applied to SCM-ACM,FLUENT and COMSOL,the results show that under the same calculation parameters,SCM-ACM takes less time to achieve higher calculation accuracy.In summary,the SCM-ACM developed in this paper can solve the incompressible flow problems in two-dimensional and three-dimensional cylindrical coordinates,and retain the advantages of both.Compared with other algorithms,it can achieve higher calculation accuracy in less time.It provides a new numerical method for researchers to solve the incompressible flow problems in the circular tube,and also expands the applicability of SCM-ACM to solve the incompressible flow problems.