Numerical solution of coupled fluid-thermal problems
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This study develops a numerical steady-state solutions to both buoyancy driven and surface tension driven flow problems in two spatial dimensions. A high-order spectral method is ap- plied for the spatial discretization, while the temporal discretization is done by a backward difference method. By solving the convection-diffusion equation, which governs the tempera- ture distribution, both the spatial and the temporal discretization methods are given. A fast direct solver for the system of algebraic equations is applied, and analytical known solutions verify the expected convergence rates of both the spatial and the temporal discretization. A splitting scheme technique is introduced in solution of the Stokes equation. An extension of this framework is used to solve the incompressible Navier-Stokes equations, which governs the pressure and velocity distribution. Both Rayleigh-B´enard and Marangoni-B´enard convection problems in two-dimensions are studied numerically by solving the Navier-Stokes equations and the convection-diﬀusion equation as a coupled system. Determining critical parameters for the onset of convection rolls, and displaying fluid flow patterns by build in methods. Experimental verification for the buoyancy-driven flow patterns.