Possibilities of solving two-dimensional hydrodynamic problems on the basis of the non-divergent form of recording the transport and conservation equations

Abstract

Existing and applied in practice to solve the aerodynamics and hydrodynamics problems finite-difference schemes ensure that the laws of conservation of matter and energy only in limited configurations of velocity fields. In other cases, there are balance errors, which are currently accepted as the norm, and special algorithms of recalculation have been developed to reduce them. Additional calculations are labor-intensive, and when solving small-scale problems, balance errors can completely distort the calculation results. This article attempts to solve two-dimensional hydrodynamic problems using a new finite-difference computational scheme previously developed by the authors, based on the non-divergent form of recording the transfer and conservation equations. Initially, the scheme was developed and tested in one-dimensional space and showed complete conservativity, stability, transportability and adequacy. To solve two-dimensional problems, a transformation of the proposed scheme was performed. The solution of the test problems and comparison with the calculation results of other known schemes showed that in two-dimensional space the proposed scheme surpasses the results obtained by the HEC-RAS and Courant-Isakson-Reese schemes. The proposed scheme makes it possible to use the maximum possible time steps in the calculations, and the resulting scheme viscosity has minimal values. This property of the scheme makes it possible to apply it to solve small-scale aerodynamic and hydrodynamic problems.