International Journal of Thermal Sciences 45 (2006) 331–333 www.elsevier.com/locate/ijts CALL FOR CONTRIBUTIONS: Towards numerical benchmark solutions for 3D mixed convection flows in rectangular channels heated from below Marc Medale a,∗ , Xavier Nicolas b,∗ a IUSTI, 5 rue Enrico Fermi, Technopôle de Château-Gombert, 13453 Marseille cedex 13, France b LETEM, Bât. Lavoisier, Université de Marne-La-Vallée, 77454 Marne-La-Vallée cedex 2, France Received 15 December 2005; accepted 15 December 2005 Abstract The main purpose of this paper is to propose two three-dimensional Poiseuille–Rayleigh–Bénard flows (mixed convection flows in horizontal rectangular channels heated from below), covering two different flow ranges, as benchmark problems and to solicit numerical comparisons between various contributors in order to obtain two benchmark solutions for the validation of numerical codes. The second objective is to identify the less perturbing outflow boundary conditions for this flow type. The first test case is a steady longitudinal roll flow in a large aspect ratio channel (A = L/H = 50, B = l/H = 10) at moderate Reynolds number Re = 50, Rayleigh number Ra = 5000 and Prandtl number Pr = 0.7. The second one is a fully-established space and time periodic transversal roll flow in a small aspect ratio channel (A = 25, B = 4) at small Reynolds number Re = 0.1, Ra = 2500 and Pr = 7. The model equations are the incompressible Navier–Stokes equations under the Boussinesq approximation.  2005 Elsevier SAS. All rights reserved. Keywords: Comparison exercise; Numerical benchmark; Poiseuille–Rayleigh–Bénard flow; Mixed convection; Rectangular channel; Outflow boundary conditions 1. Objectives The purpose of this paper is to propose two three-dimensional Poiseuille–Rayleigh–Bénard (PRB) flows as benchmark prob- lems and to solicit interested groups to submit numerical so- lutions for comparison. The main objective is to obtain a nu- merical benchmark solution to validate numerical codes for the computation of thermoconvective instabilities in open channels. The second objective is to evaluate the influence of the outflow boundary conditions on the bulk solutions and to identify the less perturbing outflow boundary conditions for two different flow classes. The third objective is to identify the most efficient numerical methods in terms of CPU time and computational cost to deal with this type of problems. 2. Governing equations The two flows proposed as benchmark cases are PRB flows in horizontal rectangular channels (cf. Fig. 1). A Poiseuille flow * Corresponding authors. E-mail addresses: marc.medale@polytech.univ-mrs.fr (M. Medale), nicolas@univ-mlv.fr (X. Nicolas). Fig. 1. Geometry and top and bottom thermal boundary conditions (the vertical lateral walls are adiabatic). is imposed at the channel entrance and the incoming fluid is cold. After an entrance zone over which a zero heat flux is im- posed on the four walls, the top horizontal wall is maintained at a cold temperature T c and the bottom wall is maintained at a higher temperature T h . The vertical lateral walls are adi- abatic. Let A and B represent the streamwise and spanwise aspect ratios of the computational domain and A e the stream- wise entrance aspect ratio. The working fluid is Newtonian and the flows are governed by the 3D incompressible Navier– Stokes equations under the Boussinesq assumption. Using the channel height H , the mean flow velocity U mean , ρU 2 mean and H/U mean as reference quantities for lengths, velocities, pres- sure and time, respectively, and using the reduced temperature 1290-0729/$ – see front matter  2005 Elsevier SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2005.12.004