Multiple Solutions in Fluid Mechanics: Uncertainty of Convection
  The nonlinear development of finite amplitude disturbances in mixed convection flow in a heated vertical annulus is studied by direct numerical simulation. The unsteady Navier Stokes equations are solved numerically by a spectral method for different initial conditions. The results indicate that the equilibrium state of the flow is not unique, but depends on the amplitude and wavenumber of the initial disturbance. In all cases, the equilibrium state consists of a single dominant mode with the wavenumber kf, and its superharmonics. The range of equilibrium wavenumbers kf was found to be narrower than the span of the neutral curve from linear theory. Flows with wavenumbers outside this range, but within the unstable region of linear theory are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that linearly stable long and short waves can also excite a wave inside this narrow band through nonlinear wave interaction. The results suggest that the selection of the equilibrium wavenumber kf is due to a nonlinear energy transfer process which is sensitive to initial conditions. 
L.S. Yao and S. Ghosh Moulic