Numerical simulation of viscous fluid flow based on thermal measurements at its surface

The viscous heat-conducting fluid flow characteristics are determined based on temperature measurements and heat flow on its daylight surface. The desired characteristics are temperature and fluid velocity in the whole model area. The problem is considered in a stationary setting and formalized as an inverse boundary problem for the model of high-viscosity incompressible fluid. A mathematical model of this fluid flow is described with the help of the Navier-Stokes equations for a Newtonian fluid in the Boussinesq approximation in a gravity field, incompressible fluid equation, and equation of the energy conservation with the appropriate boundary conditions. The fluid density and viscosity depend on the temperature. The considered problem is incorrect and does not possess the property of stability. Therefore, a small perturbation of the initial data on the accessible part of the border leads to uncontrolled errors in the determination of the unknown quantities in the model area. Conventional classical numerical methods are not suitable for solving the problem, which is why a variation method is used for its numerical solution, which reduces the solution of the original inverse problem to a series of solutions for stable problems. The Polak-Ribiere conjugate gradient method is used to minimize a merit functional in a variation method. This method steadily solves a corresponding extremal problem. The gradient of merit functional is defined analytically as a sequential solution of the direct and conjugate boundary problems. Direct and conjugate problems are numerically solved by the classical method of finite volumes. Constructed algorithms of numerical simulation are implemented in OpenFOAM software. The calculations of model problems are done.