Universal skin friction laws for turbulent flow in curved tubes

Delving into a century of turbulence research, we have long concentrated on skin friction laws by Blasius and Strickler, especially for straight-tube flows. Yet, a persistent question remains: Does skin friction in curved-tube flows possess universality? Addressing this enduring challenge, we present a phenomenological model unveiling universal laws governing turbulent skin friction, applicable to both rough and smooth curved-tube flows. We find that the skin friction coefficients, denoted by f, follow the inverse three-fourths law, f/α1/2 ∼ k α − 3 / 4 , and the inverse four-fifths law, f/α1/2 ∼ I−4/5, for rough and smooth flows, respectively, beyond certain threshold values. Here, α ≡ D/(2Rc) is the curvature ratio, D is the tube diameter, Rc is the radius of curvature, kα ≡ [(ks/D)−2α3]1/6 is the roughness-curvature number, ks is the roughness height, I ≡ (Reα2)1/4 is the Ishigaki number, and Re is the flow Reynolds number. Below their respective threshold limits, they recover the familiar skin friction laws for rough and smooth straight-tube flows. Our findings are primarily validated with an extensive dataset for smooth flow because of the data scarcity in rough flow. Supported by compelling skin friction data from smooth flow across various geometries—plane curved tube, helical tube, and toroid—gathered through experiments and simulations, our model serves as a potential bridge, connecting the theoretical and experimental realms of curved-tube turbulence.