Direct numerical simulation of transition in unstably stratified poiseuille flow

Abstract

In the present study, we have implemented a three-dimensional direct numerical simulation (DNS) algorithm for the purpose of studying instability and transition in a Poiseuille flow with imposed unstable stratification. We are specifically interested in studying the algebraic instability mechanism as a route to transition and the effect of stratification on the same. The DNS uses a mixed finite-difference-spectral algorithm that embodies a sixth order central difference scheme in the wall normal direction and spectral fourier method in the spanwise and streamwise directions, which are taken to be periodic. We first establish the validity of the DNS code in the present context by verification of the results obtained from the DNS with that obtained from linear theory, both in the case of modal theory and transient growth. In the former case we show that the DNS captures the growth rate and frequency of an input disturbance, which is in the form of the most unstable eigenfunction, quite accurately. We then study the evolution of optimal perturbations of small amplitudes and show that the energy evolution curves, in comparison with linear theory, are captured well by the DNS, as is the evolution of the actual flow field. Our next step is to consider the evolution of a finite amplitude optimal perturbation in three dimensions which are in the form of streamwise vortices, with and without the presence of unstable stratification. We find that the optimal perturbations redistribute mean shear to form streaks which are organized regions of high and low speed fluid relative to the background mean profile. A similar mechanism is shown to redistribute the mean temperature profile to form high and low intensity temperature streaks. We first consider the evolution of the energy of the perturbation and find that the energy evolves in two different growth spurts before decaying eventually. By analyzing the temperature and flow fields, which are presented at various times in considerable detail, we are able to observe the basic mechanism of the streak instability proces. We then compare the nature of streak instability process with and without the presence of stratification and consider possible reasons - both physical and computational, for the observations.