INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2011) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2674 An algorithm for the simulation of thermally coupled low speed flow problems Pavel Ryzhakov * ,† , Riccardo Rossi and Eugenio Oñate International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia, Barcelona, Spain SUMMARY In this paper, we propose a computational algorithm for the solution of thermally coupled flows in subsonic regime. The formulation is based upon the compressible Navier–Stokes equations, written in nonconserva- tion form. An efficient modular implementation is obtained by solving the energy equation separately and then using the computed temperature as a known value in the momentum-continuity system. If an explicit single-step time integration scheme for the energy equation is used, the decoupling results to be natural. Integration of the momentum-continuity system is carried out using a semi-explicit method, combin- ing Runge–Kutta and Backward Euler schemes for the momentum and continuity equations, respectively. Implicit treatment of pressure leads to favorable time step estimates even in the low Mach number (Ma 1) regimes. The numerical dissipation introduced by the Backward Euler scheme ensures absence of the spurious high frequencies in the numerical solution. The key point of the method is the assumption of linear variation of the temperature within a time step. Combined with a fractional splitting of the momentum-continuity system, it allows to solve the continuity only once per time step. Omitting the necessity of solving for the pressure at every intermediate step of the Runge–Kutta scheme minimizes the computational cost associated to the implicit step and leads to an efficiency close to that of a purely explicit scheme. The method is tested using two benchmark examples. Copyright © 2011 John Wiley & Sons, Ltd. Received 3 May 2011; Revised 6 July 2011; Accepted 16 July 2011 KEY WORDS: subsonic flows; semi-explicit methods; natural convection; computational fluid dynamics; 8:1 cavity 1. INTRODUCTION Motivation Many problems of practical importance deal with thermally coupled flows at low speeds. These range from heating of liquid semiconductors to ignition of domestic objects under fire situations. In such problems, the flow’s velocity is low (typically the subsonic range), but the encountered temperature gradients can be very large. These in turn induce density alterations, thus precluding the use of incompressible models (often used for isothermal low-speed flows). When the temperature gradients and therefore density changes are small, the Boussinesq hypothesis is often employed [1, 2]. This hypothesis is based on assuming the buoyant term of the momentum equation to be temperature dependent, while keeping the density in the rest of the terms in the gov- erning equations constant. Boussinesq solvers require the subsequent solution of the energy and momentum-continuity system. The popularity of the Boussinesq hypothesis is especially indebted to the simplicity of its implementation, a reason why in many situations, the analysts discard the use of compressible models. *Correspondence to: Pavel Ryzhakov, Technical University of Catalonia, CIMNE, Barcelona, Spain. † E-mail: pryzhakov@cimne.upc.edu Copyright © 2011 John Wiley & Sons, Ltd.