Numerical analysis of an adaptive non-linear filter based time regularization model for the incompressible, non-isothermal fluid flows

In this thesis, a time relaxation of the model using a modular, nonlinear filter is studied for incompressible non-isothermal fluids. In obtaining the approximate solutions of the proposed model, the differential Euler method according to time and the finite element method according to space are used. It is shown that the solutions of the proposed numerical algorithm are stable over a finite time interval without any condition on the time step. In addition, it has been shown that the solutions have optimal convergence rates with respect to both time and space. The convergence rates obtained from the mathematical error analysis were verified on a numerical test whose true solutions are known. Finally, the Marsigli experiment was used to demonstrate the effectiveness of the algorithm. In this experiment, a direct numerical simulation of incompressible non-isothermal flow is first obtained on a fine mesh. Then, direct numerical simulations of the algorithm and the working fluid on a thicker mesh were obtained. When these results are compared, it is seen that the algorithm gives a better result than the results of the fluid studied.