Analysis of polymer turbulent drag reduction in flow past a flat plate

Abstract

According to the well-established phenomenology for flow in a tube (reviewed by Virk [AIChE J. 21 (1975) 625]), polymer turbulent drag reduction begins once a critical shear stress τ w ∗ is exceeded, and then is manifested as an increased slope on a Prandtl–Karman (PK) plot. The critical shear stress depends on polymer molecular weight, and the increase in slope Δ of the Prandtl–Karman (PK) plot is controlled by polymer concentration and molecular weight. Combining this phenomenology for flow in a tube with a PK-type boundary layer analysis for flow along a flat plate with zero pressure gradient, we find the thicknesses of the laminar, buffer, and turbulent boundary layers as functions of dimensionless position Re x along the plate, as well as the drag on the plate for drag-reducing polymer as functions of τ w ∗ and Δ. At high Reynolds number ( Re x ), around 10 9, a factor of six reduction in drag is the maximum reduction possible, corresponding to the maximum drag reduction (mdr) asymptote for pipe flow. In the case in which the polymer enters the solvent through injection from the flat plate, we estimate the mass flux of polymer required to sustain the polymer concentration in the buffer layer at a level necessary to maintain drag reduction.