Finite-amplitude instability of mixed convection

The finite-amplitude instability of mixed convection of air in a vertical concentric annulus with each cylinder maintained at a different temperature is studied by use of weakly nonlinear instability theory and by direct numerical simulation. A strictly shear instability and two thermally induced instabilities exist in the parameter space of Reynolds and Grashof numbers. The first thermal instability occurs at low Reynolds numbers as the rate of heating increases, and is called a thermal-shear instability because it is a shear-driven instability induced by thermal effects. The second thermal instability occurs at larger Reynolds number as heating increases, and is also a thermally induced shear instability called the interactive instability. The weakly nonlinear results demonstrate that the thermal-shear instability is supercritical at all wavenumbers. With the shear and interactive instabilities, however, both subcritical and supercritical branches appear on the neutral curves. The validity of the weakly nonlinear calculations are verified by comparison with a direct simulation. The results for subcritical instabilities show that the weakly nonlinear calculations are accurate when the magnitude of the amplification rate is small, but the accuracy deteriorates for large amplification rates. However, the trends predicted by the weakly nonlinear theory agree with those predicted by the direct simulations for a large portion of the parameter space. Analyses of the energy sources for the disturbance show that subcritical instability of the shear and interactive modes occurs at larger wavenumbers because of increased gradient production of disturbance kinetic energy. This is because, at shorter wavelengths, the growth of the wave causes the shape of the fundamental disturbance to change from that predicted by linear instability theory to a shape more favourable for shear-energy production. The results also show that many possibly unstable modes may be present simultaneously. Consequently, all of these modes, as well as all of the possible wave interactions among the modes, must be considered to obtain a complete picture of mixed-convection instability.