Benchmark solution for a three-dimensional mixed-convection flow, part 2: analysis of Richardson extrapolation in the presence of a singularity

Xavier Nicolas (1) , S. Gounand (2) , M. Médale (3) , S. Glockner (4)

1 MSME - Laboratoire de Modélisation et Simulation Multi Echelle
2 STMF - Service de Thermo-hydraulique et de Mécanique des Fluides
3 IUSTI - Institut universitaire des systèmes thermiques industriels
4 I2M - Institut de Mécanique et d'Ingénierie de Bordeaux

A reference solution to a benchmark problem for a 3D mixed convection flow in a horizontal rectangular channel differentially heated (Poiseuille-Rayleigh-Bénard flow) has been proposed in "Part 1: reference solution" of the present paper [Num. Heat Trans. A, vol. 60, pp.325-345 (2011)]. Since mixed Dirichlet and Neumann thermal boundary conditions are used on the horizontal walls of the channel, a temperature gradient discontinuity is generated. The aim of this paper is to analyze the consequences of this singularity on Richardson extrapolation (RE) of the numerical solutions. The convergence orders of the used numerical methods (finite difference, finite volume, finite element), observed from RE of local and integral quantities are discussed with an emphasis on singularity influence. With the grids used, it is shown that RE can increase the accuracy of the discrete solutions, preferentially with the discretization methods of low space accuracy order, but only in some part of the channel and for a restricted range of the extrapolation coefficient. A correction to the Taylor expansion involved in the RE formalism is proposed to take into account the singularity and to explain the majority of the RE behaviors observed.

Mots clés: Richardson extrapolation; singularity; boundary conditions; convergence order; Benchmark; mixed convection; Poiseuille-Rayleigh-Bénard; reference solution

Domaines: Analyse numérique [cs.NA]; Modélisation et simulation; Mécanique des fluides [physics.class-ph]; Thermique [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Thermique [physics.class-ph]

Xavier Nicolas, S. Gounand, M. Médale, S. Glockner. Benchmark solution for a three-dimensional mixed-convection flow, part 2: analysis of Richardson extrapolation in the presence of a singularity. Numerical Heat Transfer, Part B Fundamentals, 2011, 60 (5), pp.346-369. ⟨10.1080/10407790.2011.616761⟩. ⟨hal-00692094⟩