The Accuracy Study On The Spectral Collocation Method-Artificial Compressibility Method(SCM-ACM) For Solution Of Incompressible Flow

Numerical research is an important approach for the study of the incompressible flow problems,and development of numerical methods with high efficiency and accuracy is an important field for numerical research.The spectral collocation method(SCM)with global approximations,exponential convergence and high accuracy,and the artificial compressibility method which is also simplified,easy to accelerate the convergence and implement are selected to develop the spectral collocation method-artificial compressibility method(SCM-ACM).The SCM-ACM is used for the solution of incompressible steady flow problems.Steps first,the SCM-ACM codes is writed.The artificial compressibility governing equations are given,the spacial partial differential terms of the governing equations are discreted by the SCM,and then,the algebraic equations in matrix form are obtained.The classical case of lid-driven flow in a square cavity is selected to contrast and analyse.The results show that the SCM-ACM inherits the exponential convergence of the spectral method and has the characteristics of simple and easy to implement which derived from the ACM,and it can solve the problem of incompressible flow effectively.Step second,The accuracy of SCM-ACM is analyzed for solving the 2D and 3D incompressible flow.Three groups of two-dimensional and three-dimensional incompressible flows in the square cavity with exact solutions are constructed respectively.The exact solutions of the three groups of flow problems are two order functions,trigonometric functions and exponential functions respectively.The SCM-ACM codes are used to solve three groups of flow problems with exact solutions,and the accuracy of SCM-ACM is studied under different spatial steps,time steps and discrete schemes of time term.The results show that the computational accuracy of SCM-ACM for solving flow problems is affected by the spatial step,time step and discrete scheme of time term,and is related to the boundary conditions of the flow problem.The computational accuracy increases with the increase of the number of grid nodes and the accuracy of time discretization,and tends to be stable when the number of nodes increases to a certain number.The calculation accuracy is affected scarcely by the time step.Step third,SCM-ACM is compared with other computational methods.In order to test the calculation accuracy and calculation efficiency of SCM-ACM,commonly used commercial software FLUENT and FLOTHERM are selected for comparison in this paper.Two dimensional and three-dimensional flow problems in the square cavity are selected as the computational objects,SCM-ACM,FLUENT and FLOTHERM are selected to solve the flow problems with exact solutions respectively.The comparison results show that,under the same computational conditions,SCM-ACM has the highest calculation accuracy and calculation efficiency in solving the two dimensional and three-dimensional flow problems.The SCM-ACM developed in this paper has the characteristics of high precision and exponential convergence,and that simple and easy to operate.The SCM-ACM provides a better choice for solving incompressible flow problems.