On the virtual origin determined from momentum equation analysis using experimental data within the roughness sublayer

The centroid of distributed drag acting on a roughness element, which is interpreted as the origin of the distance from the wall, is discussed for a wall-bounded turbulent flow over a rough surface with square ribs arranged with a roughness pitch ratio, PR, of 4. The first and second moments of fluctuating velocity components were measured by one-dimensional laser Doppler velocimetry to estimate spatially averaged quantities in the roughness sublayer. The spatially averaged profiles of the streamwise mean velocity and Reynolds shear stress can be scaled with the representative scales of the mixing layer observed behind a roughness element; these are $u_{\text{p}}=\sqrt{D_{\text{p}}/\left(\rho b\right)}$ and b, where u_p is the velocity scale of the driven force for the flow in a cavity, D_p is the form drag acting on a roughness element, $\rho$ is the fluid density, and b is the cavity width. The analytical solutions of the spatially averaged profiles can be formulated in exponential form, and are in good agreement with available experimental data. The displacement height, which is the distance from the centroid of the distributed drag to the roughness crest, can be approximately estimated by the spatially averaged Reynolds shear stress profile. The prediction of the displacement height well represents that of the experimental and direct numerical simulation data for 4 ≤ PR ≤ 8.

Graphical abstract

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The spatially averaged Reynolds shear stress profiles in the cavity of turbulent boundary layers over square-ribbed rough surfaces can be scaled with the representative scales of the mixing layer behind a roughness element for 4 ≤ PR ≤ 8, where PR is the roughness pitch ratio. The displacement height, which corresponds to the origin of the law of the wall, can be predicted using the scaled spatially averaged Reynolds shear stress profile and increases with the PR as the mixing layer develops over the cavity.