**Purpose** – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number models with particular emphasis on the numerical implementation of the algorithm developed.

**Design/methodology/approach** – The formulation, that allows us to consider convection
dominated problems using equal order interpolation for all the valuables of the problem, is based on the subgrid scale concept. The full Newton linearization strategy gives rise to monolithic treatment of the coupling of variables whereas some fixed point schemes permit the segregated treatment of velocity-pressure and temperature. A relaxation scheme based on the Armijo rule has also been developed.

**Findings** – A full Newtown linearization turns out to be very efficient for steady-state problems and very robust when it is combined with a line search strategy. A segregated treatment of velocity-pressure and temperature happens to be more appropriate for transient problems.

**Research limitations/implications** – A fractional step scheme, splitting also momentum and continuity equations, could be further analysed.

**Practical implications** – The results presented in the paper are useful to decide the solution strategy for a given problem.

**Originality/value** – The numerical implementation of a stabilized finite element approximation of thermally coupled flows is described. The implementation algorithm is developed considering several possibilities for the solution of the discrete nonlinear problem.

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Published on 03/09/19

DOI: 10.1108/09615530810898980

Licence: CC BY-NC-SA license

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