An assessment of effective slope as a parameter for turbulent drag prediction over multi-scaled roughness

The streamwise effective slope ( $${\rm ES}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ES</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> ), which is the mean absolute streamwise gradient of the roughness, is considered to be a key parameter in predicting the drag penalty of rough-wall turbulent boundary-layers. However, many real-world rough surfaces are multi-scaled. For such surfaces, $${\rm ES}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ES</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> can be unbounded and its value can be dominated by scales of the topography that are invisible to the flow. To illustrate this, a campaign of drag balance measurements was conducted with a set of machined surfaces. A baseline surface is prepared with fine machining parameters. By coarsening the machining precision, artefacts called ‘scallops’ are introduced which increases the ‘measured’ $${\rm ES}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ES</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> without changing other geometrical statistics. The drag of the ‘scalloped’ surfaces is higher than the baseline surface, with the relative drag increase scaling with the viscous scaled scallop height, but only when their height exceeds $$2\sim 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>∼</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> times the viscous length scale. Further, the drag of one of the ‘scalloped’ cases, even when the scallop height is $$\mathcal {O}(10)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>10</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> viscous units, is seen to be much lower ( $$\sim 28$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>28</mml:mn> </mml:mrow> </mml:math> %) than a case with matched $${\rm ES}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ES</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> , but where the $${\rm ES}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ES</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> results from larger scale features. These findings confirm that for multi-scaled surfaces, $${\rm ES}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ES</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> may be a misleading topographical metric for drag (or $$k_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> ) prediction and one must consider which scales contribute to the average slope of a surface.