New theoretical model for effects of Reynolds number on laminar boundary layer along ribbed surfaces

Sharks are famously fast swimmers, thanks in part to their hydrodynamically-optimized skin covered with scaly textures called denticles. While numerous studies have noted that small-scale periodic textures on surfaces can reduce drag at flow boundaries, a single theory to account for these observations has remained elusive.

The new paper presents a new canonical theory for how laminar boundary layer flows behave over specific types of ribbed surfaces at high Reynolds numbers. By combining an extension of the boundary layer theory with the mathematical approach of conformal mapping, Raayai-Ardakani and McKinley were able to develop a model for predicting the flow dynamics of ribbed surfaces.

After scaling the Navier-Stokes equation to develop an extended version of the boundary layer equations, the researchers were able to use coordinate transformation to solve these simplified asymptotic equations. This enabled them to calculate how the frictional drag force on V-grooved plates changes as a function of the Reynolds number of the flow and the geometry of the ribs.

Using this asymptotic model, the researchers showed that the microscopic ribs on shark scales systematically modify viscous skin friction in high Reynolds number boundary layer flow. Comparison with published experimental observations reveals that shark scales are close to the optimal geometric size for reducing frictional drag.

According to the authors, this is the first successful attempt to create a mathematical foundation for understanding the effects of geometry on laminar boundary layer flows for riblet-textured surfaces. They hope that such a priori methods will be useful in the future for guiding the design and manufacturing of hydrodynamically-optimized textured surfaces for industrial applications.

Source: “Geometric optimization of riblet-textured surfaces for drag reduction in laminar boundary layer flows,” by Shabnam Raayai-Ardakani and Gareth H. McKinley, Physics of Fluids (2019). The article can be accessed at https://doi.org/10.1063/1.5090881 .