Applied Scientific Research 47: 195-220, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands. 195 On the stability of the natural convection flow in a square cavity heated from the side R.A.W.M. HENKES & C.J. HOOGENDOORN Delft University of Technology, Department of Applied Physics, P.O. Box 5046, 2600 GA Delft, The Netherlands Received 5 April 1989; accepted in revised form 28 August 1989 Abstract. The stability of the steady laminar natural-convection flow of air (Prandtl number 0.71) and water (Prandtl number 7.0) in a square cavity is calculated by numerically solving the unsteady, two-dimensional Navier Stokes equations. The cavity has a hot and cold vertical wall and either conducting or adiabatic horizontal walls. The flow looses its stability at a lower Rayleigh number in the case of conducting horizontal walls than in the case of adiabatic horizontal walls. The flow of water is more stable than the flow of air. Directly above the critical Rayleigh number the unsteady flow shows a single-frequency oscillation. Air in the case of adiabatic horizontal walls is an exception and shows two frequencies. The instabilities in the cavity seem to be related to well-known elementary instability mechanisms. In the case of conducting and adiabatic horizontal walls the instability seems to be related to a Rayleigh/ B6nard and a Tollmien-Schlichting instability respectively. The second instability for air in the case of adiabatic horizontal walls seems to be related to an instability after a hydraulic jump. 1. Introduction The natural-convection flow in an enclosure undergoes a transition from a laminar to a turbulent state when the temperature difference (or dimension- less: the Rayleigh number) between the two vertical walls exceeds a critical value. This transition can be calculated by numerically solving the unsteady, three-dimensional Navier-Stokes equations. Because of the appearance of very fine eddy structures in the final phase of the transition, this calculation would require an enormous computational effort. Therefore in the present study we restrict ourselves to the calculation of the initial phase of the transition: we calculate how the steady, two-dimensional laminar flow looses its stability by solving the unsteady, two-dimensional Navier-Stokes equations. The stability can be examined by following the evolution of disturbances on a steady laminar flow. Beyond a critical value these disturbances no longer die out and a bifurcation to an unsteady flow is found. For very simple geometries some classical results exist, referring to elementary instability