| In the first part, the weakly nonlinear theory, based on the combined amplitude-multiple timescale expansion, is developed for the flow of an arbitrary fluid governed by the low-Mach-number equations. The approach if shown to be different from the one conventionally used for Boussinesq flows. The range of validity of the applied analysis is discussed and shown to be sufficiently large. Results are presented for the natural convection flow of air inside a closed differentially heated tall vertical cavity for a range of temperature differences far beyond the region of validity of the Boussinesq approximation. The issue of possible resonances of two different types is noted. The character of bifurcations for the shear- and buoyancy-driven instabilities and their interaction is investigated in detail. The energy transfer mechanisms are analyzed in supercritical regimes.; In the second part, the weakly nonlinear stability analysis of non-Boussinesq mixed convection flow of air in a vertical channel is performed for a wide range of temperature differences between the walls, Grashof and Reynolds numbers. It is shown that the constant mass flux and constant pressure gradient formulations result in quantitatively different, but qualitatively similar results. The physical nature of the distinct shear and buoyancy disturbances is investigated and detailed mean flow and energy analyses are presented. The complex generalized Ginzburg-Landau (GGL) equation is derived to described the dynamics of disturbance wave packets. The influence of spatial modulation of plane disturbance waves is investigated both in subcritical and supercritical regimes as well as away from the maximum linear amplification rate surface in wavenumber space. The stability analysis is modified substantially when the imaginary part of the disturbance wave packet group velocity is taken into account, and concepts of absolute and convective instabilities are generalized for this case. Subsequently, it is found that the ranges of instability are much wider than predicted by the standard linear stability analysis. Methodological aspects of extension of weakly nonlinear theory away from the marginal stability surface are given thorough consideration, and solutions of the derived GGL equation are used to justify the analysis by comparing them with results of the experimental investigation of the subcritical Poiseuille flow. |
