Natural convection of water in a rectangular cavity including density inversion Wei Tong* and Jean N. Koster Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO, USA Two-dimensional natural convection in water with density inversion is studied numerically in a rectangular cavity. The non-Boussinesq parabolic density-temperature relationship is incorporated in a finite element model. Numerical results are obtained for Rayleigh numbers up to 10s. The evolution of the temperature field and flow pattern show that density inversion and initial location of the maximum density surface within the liquid have a determining effect on convection character. The investigation of aspect ratio on flow character iselso presented. It is found that interactive convection across the density inversion surface is dependent on aspect ratio and Rayleigh number. Keyworde: natural convection; non-Boussinesq fluids; density inversion Introduction In studies of buoyancy-induced flows, Boussinesq approxima- tion is commonly used to simplify flow models. This approximation consists of two parts: (1) varying thermo- physical properties are neglected in the governing equations for density in the momentum equation for the vertical direction. (2) The density is a linear function of temperature. Conditions are non-Boussinesq if one of these conditions is violated. It is known that some fluids exhibit density inversion behavior in specific temperature ranges. Close to these density maxima the density changes in a nonlinear parabolic fashion. These systems develop convective flow for any direction of the temperature gradient. A common example is water which possesses a maximum density of 3.98°C under standard conditions. Other examples are liquid helium, which has a density inversion at about 2.18 K (Walden and Ahlers, 1981), and the pseudohinary electronic alloy Hgl_~CdxTe with an inversion at 1,028 K (Chandra and Holland 1983). The studies of density inversion have been associated with two topics: (1) Rayleigh-B6nard instability and (2) natural convection. In the first class of problems a fluid is subjected to vertical temperature gradients. At a critical Rayleigh number, convection (often referred to as "penetrative convection") develops in the unstable lower layer and extends into the stable upper layer (e.g., Merker et al. 1973; Moore and Weiss 1973; Mnsman 1968; Robillard and Vasseur 1981; Veronis, 1963). In the second class, a fluid is contained within an enclosure where two adjacent vertical walls are at different temperature. Natural convection is thus generated at the cold and hot vertical walls. This study concentrates on the second class of problems. Watson (1972) analyzed the effect of density inversion on the fluid flow and heat transfer in a square vessel. In his study, the Rayleigh number was restricted to Ra < 2 x 104. The results showed that the inversion effect is maximized when AT = 8°C. The effect of temperature-dependent viscosity was also investigated. It was found that though the fluid viscosity can change by 20 percent at the temperature range of 0-8°C, the influence of variable viscosity on the flow character is rather small. Seki et al. (1978) investigated natural convection both numerically and experimentally in rectangular vessels. The cold vertical wall was maintained at 0°C, and the hot wall temperature was varied from 1-12°C. Their experimental and numerical results of flow pattern and temperature distribution were found to be in good agreement for AT < 8°C and A = 5, and in fair agreement for AT = 10°C. More recently, Lin and Nansteel (1987) investigated natural convection in a square enclosure containing water near its density maximum. In their study the multicellular flow structures were observed for certain ranges of the density distribution parameter, which is independent of the value of Rayleigh number. The present work expands on the previous studies of natural convection in water with density inversion confined to a rectangular two-dimensional (2-D) cavity. By setting the temperature of the left wall to 0°C and varying the temperature at the right wall higher than 3.98°C, the vertical plane of the maximum density surface occurs inside the fiqnid volume. A parabolic density profile is incorporated in the finite element model. Two vertically separated liquid layers are created: the left layer with a positive density gradient in horizontal direction, and the right-side layer with a negative density gradient. Natural convection rolls of opposite vorticity develop in both layers, when for linear density profile only one roll develops. With a parabolic density correlation the governing equations become highly nonlinear. The case studies range from Rayleigh numbers 103-106. Address reprint requests to Professor Koster at the Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309, USA. * Current address: RensselaerPolytechnic Institute, Department of Mechanical Engineering, Troy, NY 12180-3590, USA Received 16 December1992; accepted 29 June 1993 © 1993 Butterworth-Heinemann Mathematical formulation The configuration of interest is illustrated in Figure 1. A rectangular cavity of aspect ratio A = H/L is filled with water and differentially heated from two vertical sides. The top and bottom surfaces are adiabatic. The flow is assumed to be 366 Int. J. Heat and Fluid Flow, Vol. 14, No. 4, December1993