Defining an equivalent homogeneous roughness length for turbulent boundary layers developing over patchy or heterogeneous surfaces

Abstract

A new approach based on the power mean is suggested for defining an equivalent homogeneous roughness length k_(ehr) which takes into account patchiness or heterogeneous distribution of roughness on ship hulls and can be readily incorporated into existing full-scale drag prediction methods. In the limit where patch sizes are much greater than the boundary layer thickness, it is readily shown that the relationship between drag coefficient and roughness length is non-linear, highlighting an obvious source of error with current approaches that attempt to define an equivalent homogeneous roughness through an area-weighed arithmetic mean. The degree of error is dependent on the roughness distribution, but is estimated to exceed 16% for highly skewed beta heterogeneous distributions. For fully-rough models, the power-mean approach returns errors of < 1% for the distributions tested here. The efficacy of the power-mean approach is also evaluated in the transitional regime and with different transitional roughness models (Nikuradse and Colebrook) and retains accuracy for most realistic operating scenarios.

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