Problem 7.3

Nikuradse used uniform sand grains of diameter € to characterize the impact of surface texture on velocity profiles. Based on his experiments the length scale €, now called the equivalent sand grain diameter, has become a standard for describing roughness of any shape. The 'equivalency' is interpreted as providing the same drag. The equivalent roughness is estimated by measuring the velocity profile and fitting it to a logarithmic profile to determine y_o. If the flow is Rough Turbulent, then y_o = €/30. This tells us that the roughness provides the same drag to the flow as uniform sand grains of diameter 30 y_o.

Consider the roughness elements depicted below. Although the shapes have the same physical scale, d, they will not necessarily offer the same roughness to the flow, that is they will not necessarily produce the same y_o. Consider two cases, d << _s, and d >> _s, and order the three cases from highest to lowest y_o.

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