Analysis of solution algorithms for the three-dimensional Navier-Stokes equations in the natural variables

Abstract

Numerical modeling methods have long been widely used in convection analysis. Problems arising in science and technology are a continuing source of new challenges. Their solution requires developing and using special algorithms that take into account the specific features of the processes to be modeled. Production technology of advanced materials, especially bulk monocrystals and semiconductor structures, is one of the fields in which, owing to the development of efficient numerical methods, mathematical modeling has become the most important tool of analysis [1, 2]. The choice of an algorithm for solving the Navier–Stokes equations for an incompressible fluid plays a very important role in the numerical analysis of convective heat and mass transport processes. Although a large body of experience has been accumulated in this field, process modeling on the basis of these equations, especially in the three-dimensional setting, remains a laborious task, and the properties of the corresponding algorithms are still not clearly understood. The choice of a computational algorithm is especially critical if one deals with the modeling of long-term nonstationary processes, where the solution method should provide physically correct results on coarse grids and large time intervals. The difficulties arising here are mainly due to the fact that the continuity equation for an incompressible fluid is not an evolution equation; it contains only the space derivatives of the velocity vector components, which formally corresponds to the infinite perturbation propagation velocity. At the same time, there is no equation where pressure is the main variable. Determination of pressure on the domain boundary also has some specific features; in particular, the mathematical statement of the problem does not require pressure to be specified on rigid walls [3, p. 372]. In these circumstances, algorithms that simultaneously solve the equations of motion and the continuity equation for the vector (V, p), where V and p are the velocity and pressure, respectively, should be preferred [4, 5]. This approach permits one to construct a numerical procedure in the framework of the original physical-mathematical statement of the problem without resorting to additional equations and boundary conditions for pressure. This provides high accuracy and stability of algorithms of this class. However, the use of simultaneous algorithms is largely restricted by difficulties encountered when solving the corresponding system of grid equations. The most widespread algorithms for solving the three-dimensional Navier–Stokes equations in the natural variables are based on the momentum conservation equation and the Poisson equation for pressure [6, p. 134; 7, p. 390 of the Russian translation]. Essentially, the continuity equation is not used directly. On each time layer, the velocity vector components and pressure are computed consecutively, and the incompressibility condition is ensured by a predictor–corrector procedure. Here one needs to pose additional boundary conditions for pressure. The previous experience in numerical analysis of dynamic problems for a viscous incompressible fluid show that the accuracy and stability of sequential algorithms essentially depend on the choice of these boundary conditions and the method used for solving the pressure equation [7, p. 398 of the Russian translation]. The present paper deals with the construction and analysis of numerical methods for solving the three-dimensional nonstationary Navier–Stokes equations in the natural variables. We consider