ORIGINAL Natural convection in a vertical plane channel: DNS results for high Grashof numbers P. Kis ˇ • H. Herwig Received: 19 February 2013 / Accepted: 29 January 2014 Ó Springer-Verlag Berlin Heidelberg 2014 Abstract The turbulent natural convection of a gas (Pr = 0.71) between two vertical infinite walls at different but constant temperatures is investigated by means of direct numerical simulation for a wide range of Grashof numbers (6.0 9 10 6 [ Gr [ 1.0 9 10 3 ). The maximum Grashof number is almost one order of magnitude higher than those of computations reported in the literature so far. Results for the turbulent transport equations are presented and compared to previous studies with special attention to the study of Verteegh and Nieuwstadt (Int J Heat Fluid Flow 19:135–149, 1998). All turbulence statistics are available on the TUHH homepage (http://www.tu-harburg. de/tt/dnsdatabase/dbindex.en.html). Accuracy consider- ations are based on the time averaged balance equations for kinetic and thermal energy. With the second law of ther- modynamics Nusselt numbers can be determined by eval- uating time averaged wall temperature gradients as well as by a volumetric time averaged integration. Comparing the results of both approaches leads to a direct measure of the physical consistency. 1 Introduction The plane channel flow with two infinite walls is one of the most frequently used benchmark scenarios for direct numerical simulations (DNS). Turbulent time averaged quantities such as Reynolds stresses or turbulent heat fluxes are subject to one dimensional variations along the wall normal direction only. Therefore plane channel flow also is a valuable benchmark case for studying turbulent transport phenomena and their modelling in RANS computations. Whereas numerous studies of the plane channel flow focus on forced convection (Poiseuille flow) or natural convection in a horizontal channel (Rayleigh-Benard con- vection) only little attention is paid to the vertical channel. While the gravity vector and the mean temperature gradient are parallel for Rayleigh-Benard convection, they are per- pendicular for the vertical channel. As a result Rayleigh- Benard convection and the vertical channel represent the two extremes of the broad spectrum of natural convection confined between two infinite plane walls. It is the chal- lenge for turbulence modeling to account for the physics of these two extremes of natural convection properly in order to give reasonable results for any other angle between the gravity vector and the mean temperature gradient. There- fore by analyzing the turbulent budgets in a vertical channel this study may contribute to a sound basis of an universal turbulence model for natural convection. To the authors knowledge the last comprehensive vertical channel study with available data for the turbulent budgets, essen- tial for turbulence model design, was performed by Vers- teegh and Nieuwstadt [16]. Since then, computational power increased considerably so that today computations provide improved accuracy for even higher Grashof num- bers at less computational cost. It is the intention of this paper to extend the limits of available benchmark cases for natural convection in ver- tical channels. Furthermore we address the assessment of assumptions in the underlying mathematical model by considering the entropy generation. Although a broad range of Prandtl numbers may be of practical interest (e.g. water, liquid metals) we limit ourselves to gases with Pr = 0.71. P. Kis ˇ  H. Herwig (&) Institute of Thermo-Fluid Dynamics, Hamburg University of Technology, Denickestr. 17, 21073 Hamburg, Germany e-mail: h.herwig@tuhh.de; h.herwig@tu-harburg.de URL: http://www.tu-harburg.de/tt 123 Heat Mass Transfer DOI 10.1007/s00231-014-1305-5