On the Features of Thermal Convection in a Compressible Gas.

The fully nonlinear equations of gas dynamics are solved in the framework of a numerical approach in order to study the stability of the steady mode of Rayleigh-Bénard convection in compressible, viscous and heat-conducting gases encapsulated in containers with no-slip boundaries and isothermal top and bottom walls. An initial linear temperature profile is assumed. A map of the possible convective modes is presented assuming the height of the region and the value of the temperature gradient as influential parameters. For a relatively small height, isobaric convection is found to take place, which is taken over by an adiabatic mode when the height exceeds the critical value, or by a super-adiabatic mode in case of a relatively high temperature gradient. In the adiabatic mode, convective flow develops due to adiabatic processes given a stable initial stratification. An analytic formula for the critical height of the region is derived taking into account and neglecting the dependence of the gas viscosity on the temperature. Moreover, an analytic formula is obtained for the upper boundary of the region of applicability of the Boussinesq approximation for incompressible gases. These models for compressible gases are relevant to practical situations such as the study of convective flows in spatially extended gas mixtures when dealing with safety issues related to hydrocarbons stored in gas stations. A dangerous situation arises when the tank is almost empty but some hydrocarbon is left at the bottom of the tank. In the presence of convective flows, the vaporized fuel is mixed with the oxidizer (air) forming a gas-vapor medium. However, if the volumetric concentration of fuel vapor (hydrocarbon) is in the interval between the lower and upper concentration limits of ignition, then the gas-vapor mixture becomes explosive and any accidental spark is sufficient to cause an emergency. [ABSTRACT FROM AUTHOR]