The Pursuit of the Equivalent Roughness Length for Turbulent Boundary Layer Flow over Streamwise and Spanwise Roughness Patches

There is continued interest in determination of an equivalent roughness length (z_0,eq) that can represent the combined drag from successive roughness patches arranged in various directions to incoming boundary layer flow. We first revisit the case of elongated patches arranged parallel to the flow from previous work and propose a generalized z_0,eq model assuming separate drag forces can be added. However, patches aligned perpendicular to the flow are more challenging as the evolution of the downstream patch depends on the incoming state and evolution of the layer from the preceding patch. Such an internal boundary layer (IBL) has primarily been studied for a single developing layer rather than for multiple interacting streamwise IBLs. Here we consider two IBLs, each with their own evolution based on separate von Karman momentum integral equations. We also revisit the Elliot (1958) model to simplify the otherwise challenging analytical expressions that result from attempting to solve the integral equations simultaneously. The results are used to develop improved modified power-law fits as functions of normalized fetch length and roughness ratio between the layers. Using these improved fits, we develop a methodology to find the z_0,eq representing both patches, with additional patches modeled using a telescoping approach. The resulting z_0,eq model is tested against large eddy simulation data gathered from several equilibrium wall modeled LES over several patches with different roughness lengths.