Energy growth of disturbances in a non-Fourier fluid

Natural convection of non-Fourier fluid of the single-phase-lagging (SPL) type between two horizontal walls (Rayleigh-Benard) has been investigated. These fluids possess a relaxation time, reflecting the delay in the response of the heat flux and the temperature gradient with respect to one another. By invoking the role of the eigenvectors to detect and quantify short-time behavior, transient growth of energy of disturbances has been illustrated. The energy of the perturbations is introduced in terms of the primary variables as a disturbance measure in order to quantify the size of the disturbance. In contrast to linear stability analysis, one does not assume exponential time dependence, but monitor the evolution of initial conditions in the pre- and post-critical ranges of Rayleigh numbers. Different growth functions for different levels of non-Fourier effects have been found, which should be thought of as the envelope of the energy evolution of individual initial conditions. Also, it is found that nonlinearities are not required for the energy growth. Energy growth can occur if non-orthogonal eigenfunctions are available. A fundamental implication of the non-normality is that there can be significant energy growth in the energy of perturbations even if the flow is stable.