Prediction of an effective hydraulic length scale for surfaces with heterogeneous roughness

Practical engineering systems which include either external or internal wall-bounded turbulent flow are routinely affected by surface roughness. This causes performance degradation in the form of increased drag or head loss. Often times this roughness exhibits significant spatial heterogenenity. In this case, it not clear how one 'averages' a number of local roughness conditions to generate a single effective hydraulic length scale for the entire surface. In the present work, this question is explored. Boundary layer similarity law analysis of frictional drag is used to predict the influence of roughness variations in both the spanwise and streamwise directions. It is shown that the arithmetic mean does a poor job of predicting an effective hydraulic length scale for the entire surface due to the nonlinear relationship between the frictional drag increment and the hydraulic length scale. Alternative averaging approaches are offered which do a much better job.