J. Fluid Mech. (2009), vol. 636, pp. 41–57. c  Cambridge University Press 2009 doi:10.1017/S0022112009007757 Printed in the United Kingdom 41 Turbulent natural convection along a vertical plate immersed in a stably stratified fluid EVGENI FEDOROVICH† AND ALAN SHAPIRO School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd, Norman, OK 73072-7307, USA (Received 20 August 2007; revised 17 April 2009; accepted 17 April 2009) The paper considers the moderately turbulent natural convection flow of a stably stratified fluid along an infinite vertical plate (wall). Attention is restricted to statistically stationary flow driven by constant surface forcing (heating), with Prandtl number of unity. The flow is controlled by the surface energy production rate F s , molecular viscosity/diffusivity ν and ambient stratification in terms of the Brunt– V¨ ais¨ al¨ a (buoyancy) frequency N . Following the transition from a laminar to a turbulent regime, the simulated flow enters a quasi-stationary oscillatory phase. In this phase, turbulent fluctuations gradually fade out with distance from the wall, while periodic laminar oscillations persist over much larger distances before they fade out. The scaled mean velocity, scaled mean buoyancy and scaled second-order turbulence statistics display a universal behaviour as functions of distance from the wall for given value of dimensionless combination F s /(νN 2 ) that may be interpreted as an integral Reynolds number. In the conducted numerical experiments, this number varied in the range from 2000 to 5000. 1. Introduction Unsteady natural convection flows abound in nature and technology. Such flows are notoriously difficult to analyse theoretically because of the intrinsic coupling between the temperature and velocity fields. The case of unsteady laminar one-dimensional natural convection along an infinite vertical plate (sometimes referred to as a double- infinite plate because no leading or trailing edges are considered) provides one of the few scenarios where the Boussinesq equations of motion and thermodynamic energy may be solved analytically (Gebhart et al. 1988). Analytical solutions for unsteady one-dimensional natural convection along an infinite vertical plate were obtained in the 1950s and 1960s for a variety of surface forcings, though with a restriction to unstratified environments. The stability of these unstratified flows was analysed by Armfield & Patterson (1992) and Daniels & Patterson (1997, 2001). The extension of the one-dimensional convection framework to include ambient stratification is a relatively recent development (Park & Hyun 1998; Park 2001; Shapiro & Fedorovich 2004a, b, 2006). Shapiro & Fedorovich (2004b ) considered unsteady laminar natural convection in a stratified flow adjacent to a single infinite vertical plate (wall). Analytical solutions were obtained for a Prandtl number of unity for the cases of impulsive (step) change in plate perturbation temperature, sudden application of a plate heat flux and for † Email address for correspondence: fedorovich@ou.edu