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HYDRODYNAMICS AND ACOUSTICS
2021 ◊ Volume 2 (92) ◊ Issue 2 ◊ p. 126-148
A. O. Borysyuk *
* Institute of Hydromechanics of NAS of Ukraine, Kyiv, Ukraine
Studying the flow in a straight flat rigid channel with two axisymmetric rigid-wall narrowings of a rectangular shape. Part 1. Theory
Gidrodin. akust. 2021, 2(2):126-148
TEXT LANGUAGE: Ukrainian
ABSTRACT
The paper deals with developing a numerical method for solving the problem of fluid movement in a straight flat rigid channel with two axisymmetric rigid-wall narrowings of a rectangular shape. The method has the second order of accuracy in terms of spatial coordinates and time. At the same time, the Navier-Stokes and continuity equations are solved in the velocity-pressure variables using integration over elementary volumes into which the computational domain is divided. The resulting integral equations are subject to spatiotemporal discretization, which leads to the need to solve nonlinear algebraic equations. The base of the temporal discretization part is an implicit three-point asymmetric scheme with backward differences, and the base of the spatial one is the TVD scheme with the appropriate discretization scheme of spatial derivatives. The solving of specified algebraic equations includes three stages. First, rewrite the discrete momentum equation as an equation for velocity. Then, based on the discrete continuity equation, derive the pressure equation. After that, apply the procedure of finding and matching successive approximations of these quantities to the resulting coupled nonlinear algebraic equations for velocity and pressure. When calculating the first approximation in the velocity equation, substitute the unknown pressure and velocity (in the expression for the flow) by their values found at the previous time moment. When finding the following approximations, replace these values with already-known previous approximations. The given solution accuracy determines the approximation number. It makes it possible to move from the solution of coupled systems of nonlinear algebraic equations for velocity and pressure to the corresponding independent linear equations. To solve systems of linear algebraic equations, apply an iterative algorithm using methods of delayed correction and conjugate gradients, as well as ICCG (for symmetric matrices) and Bi-CGSTAB (for asymmetric matrices) solvers.
KEY WORDS
flow, channel, narrowing, the Navier-Stokes equations, iterative method