DOI: https://doi.org/10.15588/1607-3274-2018-1-6

### METHOD OF NUMERICAL ANALYSIS OF THE PROBLEM OF STATIONARY FLOW PAST BODIES OF REVOLUTION BY VISCOUS FLUID

S. N. Lamtyugova, M. V. Sidorov, I. V. Sytnykova

#### Abstract

Context. The nonlinear stationary problem of flow past a body of revolution by a viscous incompressible fluid is examined in this article.
Objective. The purpose of this work is to develop a new method of numerical analysis of stationary problems of flow around bodies of
revolution by viscous incompressible fluid.
Method. The mathematical model of the process under consideration is a nonlinear boundary value problem for the stream function
obtained by the transition from the system of Navier-Stokes equations to one nonlinear equation of the fourth order. A special feature of the
formulation the task of the flow past body is that the boundary value problem is considered in an infinite region and both boundary conditions
on the boundary of the streamlined body and the condition at infinity are imposed for the stream function. Using the structural method (the
R-functions method), the task solution structure, that exactly satisfies all the boundary conditions of the task, and also guarantees the
necessary behavior of the stream function at infinity, is constructed. Two approaches are proposed to approximate the uncertain components
of the structure. The first approach is based on the use of the successive approximations method, which makes it possible to reduce the solution of the initial nonlinear task to the solution of a sequence of linear boundary value problems. These linear tasks are solved by the Bubnov-Galerkin method at each step of the iteration process. The second approach for approximating the uncertain components of the structure is based on the usage of the nonlinear Galerkin method and it is proposed to use it in the case of divergence of successive approximations. In this case, the solution of the initial nonlinear task reduces to solving a system of nonlinear algebraic equations.
Results. A computational experiment was carried out for the task of flow past a sphere, an ellipsoid of rotation and two articulated
ellipsoids for various Reynolds numbers.
Conclusions. The conducted experiments have confirmed the efficiency of the proposed method of numerical analysis of stationary
problems of flow around bodies of revolution by viscous incompressible fluid. The prospects for further research may consist in using the
method developed for the implementation of semi-discrete and projection methods for solving non-stationary problems.

#### Keywords

flow past bodies; stationary flow; viscous fluid; stream function; R-functions method; successive approximations method; Galerkin method.

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