Laminar stability and heat transport in high aspect ratio planar confined impinging flows

Abstract

We consider two-dimensional flow and heat transfer in a high aspect ratio (e=10) laminar confined impinging flow and show that unsteadiness occurs as a result of the bifurcation of the steady flow to an unstable branch of solutions. Our calculations reveal that the steady flow field displays hysteresis over a wide range of Reynolds numbers, ΔRe∼40, wherein the location of the reattachment point RP of the primary separation vortex is multi-valued. Linear stability analysis of the bifurcation branches leads to the conclusion that the upper branch corresponding to the larger value of RP is unstable. These steady flow bifurcation and stability results allow us to describe the details of the transition to unsteady flow previously observed for Re>130. We use the bifurcation results to quantify the associated hysteresis in steady wall jet heat transfer at the impinging surface. For two Reynolds numbers on either side of the bifurcation point corresponding to the jump from the stable to the unstable branch we compute the leading stability eigenmodes and the associated temperature modes. Due to the loss of flow stability on the upper branch the structure of the temperature modes is significantly different on the lower and upper branches. Consequently, perturbation Nusselt number profiles corresponding to the temperature modes on the two branches also show structural differences between these two Reynolds numbers. The comparison highlights the antisymmetric character of the temperature mode and perturbation Nusselt number corresponding to the unstable velocity mode on the upper branch, and the consequent asymmetry of the perturbed Nusselt number profile.