Numerical analysis of a projection-based stabilization method for the natural convection problems

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2011

Öğrenci: AYTEKİN BAYRAM ÇIBIK

Danışman: SONGÜL KAYA MERDAN

Özet:

In this thesis, we consider a projection-based stabilization method for solving buoyancy driven flows (natural convection problems). The method consists of adding global stabilization for all scales and then anti-diffusing these effects on the large scales defined by projections into appropriate function spaces. In this way, stabilization acts only on the small scales. We consider two different variations of buoyancy driven flows based on the projection-based stabilization. First, we focus on the steady-state natural convection problem of heat transport through combined solid and fluid media in a classical enclosure. We present the mathematical analysis of the projection-based method and prove existence, uniqueness and convergence of the approximate solutions of the velocity, temperature and pressure. We also present some numerical tests to support theoretical findings. Second, we consider a system of combined heat and mass transfer in a porous medium due to the natural convection. For the semi-discrete problem, a stability analysis of the projectionbased method and a priori error estimate are given for the Darcy-Brinkman equations in double-diffusive convection. Then we provide numerical assessments and a comparison with some benchmark data for the Darcy-Brinkman equations. In the last part of the thesis, we present a fully discrete scheme with the linear extrapolation of convecting velocity terms for the Darcy-Brinkman equations.