This article is cited in 11 papers

NUMERICAL METHODS AND THE BASIS FOR THEIR APPLICATION

Difference scheme for solving problems of hydrodynamics for large grid Peclet numbers

A. I. Sukhinov^a, A. E. Chistyakov^a, E. A. Protsenko<sup>b

a</sup> Don State Technical University, 1 Gagarin Square, Rostov-on-Don, 344000, Russia
^b Taganrog Institute, named after A. P. Chekhov — branch of Rostov State University of Economics, 48 Initiative st., Taganrog, 347936, Russia

Abstract: The paper discusses the development and application of the accounting rectangular cell fullness method with material substance, in particular, a liquid, to increase the smoothness and accuracy of a finite-difference solution of hydrodynamic problems with a complex shape of the boundary surface. Two problems of computational hydrodynamics are considered to study the possibilities of the proposed difference schemes: the spatial-two-dimensional flow of a viscous fluid between two coaxial semi-cylinders and the transfer of substances between coaxial semi-cylinders. Discretization of diffusion and convection operators was performed on the basis of the integro-interpolation method, taking into account the fullness of cells and without it. It is proposed to use a difference scheme, for solving the problem of diffusion–convection at large grid Peclet numbers, that takes into account the cell population function, and a scheme on the basis of linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error at small Courant numbers. As a reference, an analytical solution describing the Couette–Taylor flow is used to estimate the accuracy of the numerical solution. The relative error of calculations reaches 70 % in the case of the direct use of rectangular grids (stepwise approximation of the boundaries), under the same conditions using the proposed method allows to reduce the error to 6 %. It is shown that the fragmentation of a rectangular grid by 2–8 times in each of the spatial directions does not lead to the same increase in the accuracy that numerical solutions have, obtained taking into account the fullness of the cells. The proposed difference schemes on the basis of linear combination of the Upwind and Standard Leapfrog difference schemes with weighting factors of 2/3 and 1/3, respectively, obtained by minimizing the order of approximation error, for the diffusion–convection problem have a lower grid viscosity and, as a corollary, more precisely, describe the behavior of the solution in the case of large grid Peclet numbers.

Keywords: accounting of cell fullness, Upwind Leapfrog difference scheme, grid Peclet numbers.

UDC: 519.8

Received: 14.07.2019
Revised: 11.08.2019
Accepted: 26.08.2019

DOI: 10.20537/2076-7633-2019-11-5-833-848