Recovery of flow parameters of viscous heat-conducting fluid by some changes at its surface

A. I. Korotkii ; I. A. Tsepelev

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 1, pp. 27-36

Résumé

Physical characteristics of steady motion of viscous heat-conducting incompressible fluid by changes of temperature and heat flow on its daylight surface are determined in the article. The main desired characteristics are temperature and fluid velocity in the entire simulation area. The problem is formalized as an inverse boundary problem for the flow model of natural thermal convection of highly viscous incompressible fluid. The mathematical flow model of such fluid is described by a stationary Navier-Stokes equations for Newtonian rheology of a medium in Boussinesq approximation in the field of gravity, by the medium incompressibility equation, stationary equation of energy conservation with corresponding boundary conditions. Density and viscosity of the fluid have non-linear dependence on its temperature. The considered inverse problem is incorrect and does not possess the property of stability; small perturbation of initial data on the section of the boundary available for measurement leads to uncontrolled errors in determining the desired values. For numerical solution of unstable problems special methods should be developed. The goal of the article is in developing methods and algorithms of constructive sustainable numerical simulation of the considered inverse problem’s solution. In order to implement this goal, the use of variational method, which is based on reduction of the initial problem to some extremum problem on the minimum of the appropriate objective functional and its sustainable minimization by some appropriate technique, is proposed. Using this strategy, an iteration process of sequential numerical solution of boundary problems of boundary control, which are systems of differential equations with partial derivatives with completely determined boundary conditions, is organized. In order to minimize quality functional, the Polac-Ribiere conjugate gradient method is used. This functional’s gradient and the descent step are determined analytically which allows significantly decreasing the volume of calculations. The method of finite volumes is used for integrating the systems of differential equations with partial derivatives with various types of boundary conditions. The developed algorithms of numerical simulation are implemented in the OpenFOAM calculations package. Calculation of the simulated example is carried out.

Détail

Keywords: thermal convection, inverse boundary problem, variational method, numerical simulation.
Mots-clés : viscous fluid

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