Minimum-dissipation transport enhancement by flow destabilization: Reynolds’ analogy revisited

Abstract

A classical transport enhancement problem is concerned with increasing the heat transfer in a system while minimizing penalties associated with shear stress, pressure drop, and viscous dissipation. It is shown by Reynolds' analogy that viscous dissipation in a wide class of flows scales linearly with the Nusselt number and quadratically with the Reynolds number. It thus follows that transport enhancement optimization is equivalent to a problem in hydrodynamic stability theory; a more unstable flow will achieve the same Nusselt number at a lower Reynolds number, and therefore at a fraction of the dissipative cost. This transport-stability theory is illustrated in a numerical study of supercritical (unsteady) two-dimensional flow in an eddy-promoter channel comprising a plane channel with an infinite periodic array of cylindrical obstructions.It is shown that the addition of small cylinders to a plane channel results in stability modes that are little changed in form or frequency from plane-channel Tollmien-Schlichting waves. However, eddy-promoter flows are dramatically less stable than their plane-channel counterparts owing to cylinder-induced shear-layer instability (with critical Reynolds numbers on the order of hundreds rather than thousands), and thus these flows yield heat transfer rates commensurate with those of a plane-channel turbulent flow but at much lower Reynolds number. Small-cylinder supercritical eddy-promoter flows are shown to roughly preserve the convective-diffusive Reynolds analogy, and it thus follows from the transport-stability theory that eddy-promoter flows achieve the same heat transfer rates as plane-channel turbulent flows while incurring significantly less dissipation.