Instability of the flow over a porous surface: Linear stability analysis

Direct numerical simulations of the fully developed turbulent flow through a porous channel show the appearance of short span-wise rollers at the fluid-porous interface along with the streaky structures of wall-bounded turbulence near the upper wall of the channel. The scope of the present research project is to investigate the instabilities in porous channel flow by means of local linear stability analysis in order to elucidate their physical origin and underlying mechanisms. The numerical approach for the problem consists of framing an eigenvalue problem using primitive variable approach to model the porous channel flows. The Volume Averaged Navier-Stokes (VANS) equations are used for modeling the flow through a porous channel with a porosity distribution function varying along the wall-normal direction separating the homogeneous fluid region and the homogeneous porous region by an interface region with smooth transition of porosity. The linearized VANS equations are discretized using 4th order dispersion-relation preserving finite-differences schemes, in order to reduce the computational resources in terms of memory and time. The modal stability analysis is performed to identify the linearly unstable region based on the critical flow parameters (critical Reynolds number and critical wavenumber). Reynold-Orr analysis is carried out to identify the contribution of different terms of the perturbation kinetic energy in the flow. A scaling approach is done to identify and distinguish between the different physical mechanisms governing the instabilities in the flow at different ranges of porosity. Non-modal analysis is carried out to understand the mechanism responsible for optimal energy amplification in the sub-critical regime and the wavenumbers over which maximum energy growth takes place. Non-modal analysis for turbulent porous channel flow is also carried out to get better insight into the underlying mechanisms governing the instabilities in the flow. It is seen that the 4th order dispersion relation preserving scheme is well able to capture the physics of the flow through porous channel. Initial analysis about the linearly unstable region shows that the Squire transformation holds for porous channel flows and hence the modal stability analysis is performed using two dimensional perturbations. The eigenspectrum and the shape of the eigenmode shows the evolution of viscid instability from fluid region at low values of porosity. At high values of porosity, the eigenmode shows strong flow reversal at the interface due to weakening of wall-blocking and wall induced viscous effects. Energy analysis shows the dominance of viscous dissipation at lower values of porosity while the porous dissipation becomes significant at higher values of porosity. The physical origin of the instability is identified by the scaling analysis. It is seen that at lower values of porosity, the instability stems from the viscous fluid region (TS instability) while the primary source of instability at higher values of porosity is a TS instability wave modified by the drag from porous region. Non-modal analysis shows the optimal mechanism of energy amplification over different ranges of porosity is the Lift-up effect due to the streaky structures. However the non-normal energy growth becomes insignificant at extremely high values of porosity. The turbulent flow analysis provides a better understanding of the perturbation kinetic energy gain as seen from the direct numerical simulations. The optimal mechanism responsible for the energy growth is due to the streaky structures for different values of porosity. However at extremely high values of porosity there is energy gain due to the span-wise rollers as well but energy gain due to streaks still dominates.Mechanical, Maritime and Materials EngineeringProcess and Energ