Drag reduction in rough pipes

Abstract

Drag reduction caused by dilute, distilled water solutions of four polyethylene-oxides and one polyacrylamide, molecular weights respectively 0·1 × 10 6 to 8 × 10 6 and 13 × 10 6 , was studied experimentally in one smooth and three sand-roughened pipes, relative roughnesses ( R / k ) 15, 23, 35, all of about 0·34, in inside diameter. The onset of drag reduction in the rough pipes occurred at the same wall shear stress as in smooth; the onset wall shear stress was essentially independent of polymer concentration, varied inversely as the square of polymer radius of gyration and was unaffected by the flow régime, hydraulically smooth, transitional or fully rough, during which onset occurred. Following onset a flow régime was observed wherein the fractional slip, i.e. fractional increase in mean velocity relative to solvent at a given friction velocity, obtained with a given polymer solution in a rough pipe was the same as the fractional slip in the smooth pipe despite marked differences in the respective rough and smooth friction factors. This ‘effectively smooth’ régime prevailed for values of non-dimensional roughness k +* < k + < k + es from onset, k +* , to an upper limit given by k + es ∼ 50 for all of the present experiments. For k + < k + es , the fractional slip in the rough pipes was always less than that corresponding to smooth and was a function of relative roughness as well as flow and polymeric parameters. The maximum drag reduction possible in the rough pipes was limited by an asymptote which was independent of polymeric parameters. Under asymptotic conditions, friction factors in all the rough pipes identically obeyed the smooth pipe friction factor relation for k + < 12; the onset of roughness at k + ∼ 12 indicated that the maximum viscous sublayer thickness attained during drag reduction was approximately 2½ times Newtonian.