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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. [ABSTRACT FROM AUTHOR]

Quantifying turbulent friction holds significant importance, not only for understanding the fundamental flow physics but also for enriching system performance across a wide range of engineering applications. This vision article presents the state of the science of the turbulent friction in canonical flows, shedding light on its current status through a combination of theoretical developments and experimental observations. First, the article discusses the law of the wall, including the scaling behavior, the possible origin of the logarithmic law, and the effects of wall roughness. Then, it provides an overview of roughness height and its connection with the wall topography. The scaling behaviors of the logarithmic and power laws of turbulent friction are thoroughly appraised, offering insights into their implications. Additionally, the phenomenological models of turbulent friction based on the spectral and co-spectral budget theories are furnished. The behavior of turbulent friction for extremely large Reynolds number flows is examined, based on theoretical models and experimental data. The semiempirical finite Reynolds number model for turbulent friction is reviewed, emphasizing the pertinent scaling laws in various forms. The scaling laws of turbulent friction in curved-pipe and axisymmetric boundary layer flows are discussed. Finally, future research directions are outlined, highlighting the key challenges to be addressed. [ABSTRACT FROM AUTHOR]

Turbulent stream approaching a cylinder embedded in an erodible bed triggers significant erosion to form a scour hole. We present a generalized scaling framework for the equilibrium scour depth at a circular cylinder by means of the similarity principle and discover that the similarity exponent can be uniquely determined through the turbulence phenomenology. We assess the veracity of our approach aided by the experimental and field data covering wide ranges of key parameters. The scaling law offers insight into how the dimensionless scour depth relates to the inertia, drag, shear, and threshold shear forces. [ABSTRACT FROM AUTHOR]

This study examines the instability of sand waves (dunes and antidunes) from both linear and weakly nonlinear perspectives. The linear and weakly nonlinear analyses use the standard linearization and the center manifold-projection technique, respectively. The mathematical framework includes the depth-averaged continuity and momentum equations, the advection–diffusion equation for suspended sediment concentration, and Exner's equation for bed evolution. The streamline curvature is treated using the Boussinesq approximation. The model considers the departure of the pressure distribution from the hydrostatic law. Both modes of sediment transport, as bedload and as suspended load, are taken into consideration. The perturbations characterize the maximum growth rate for a selected wave number, called the resonant wave number, which is the most favorable wave number for the formation of sand waves. As the flow Froude number and the relative roughness number increase, the dimensionless resonant wave number decreases. The dimensionless amplitude of sand waves increases as the flow Froude number and the relative roughness number increase to achieve a maximum, and subsequently it decreases. The predicted wave number and amplitude of sand waves satisfactorily match the available experimental data. [ABSTRACT FROM AUTHOR]

Understanding scour at bridge piers is crucial for safeguarding public safety, ensuring infrastructure resilience, and planning effective maintenance. Despite over six decades of extensive studies aiming to develop predictive formulas for the equilibrium scour depth at bridge piers, more than 20 000 highway bridges in the United States have been spotted "scour critical." The traditional reliance on existing empirical formulas has posed a severe challenge for researchers, hindering to achieve a unified relation for the equilibrium scour depth from a fundamental scientific tenet. This perspective article presents a breakthrough—a universal law governing the equilibrium scour depth at a circular pier embedded in a sediment bed, specifically in clear-water scour condition. Derived from a phenomenological model, the universal law reveals that the equilibrium scour depth to pier diameter ratio obeys a consistent two-fifths scaling law with the introduction of a newly coined pier-scour number. This number accounts for all the key parameters involved in a local scour phenomenon, including the approach mean flow velocity, threshold shear velocity for sediment grain motion, approach flow depth, pier diameter, and sediment grain size. Importantly, the scaling law contains an additional term involving the drag coefficient raised to the power of 2/5, addressing the impact of the pier shape on the equilibrium scour depth. The derived universal law undergoes the validation through an extensive dataset of experimental measurements on circular pier scour. [ABSTRACT FROM AUTHOR]

This study presents an analytical model of a steady turbulent flow over a two-dimensional downstream-skewed wavy bed of small amplitude. The mathematical framework rests on the time-averaged continuity and Reynolds-averaged Navier–Stokes (RANS) equations. The streamwise velocity profile is considered to follow a self-similar power law, whereas the Reynolds normal stresses are founded on the turbulent diffusivity hypothesis. The curvilinearity in flow streamlines induced by the wavy bed is introduced into the analysis via Boussinesq approximation. The analysis provides solutions to the free-surface, bed shear stress, and Reynolds shear stress profiles. As the flow Froude number varies, the free-surface and bed shear stress profiles change their phases. The phase shift of the free-surface profile with respect to the bed profile decreases with an increase in skewness factor, whereas the phase shift of the bed shear stress profile with respect to the bed profile increases with an increase in skewness factor attaining a constant value. The convex and concave shapes of the Reynolds shear stress profiles on the downslope and upslope of the bed profile are attributed to the decelerated and accelerated flows, respectively. The implementation of the key findings of this work to study the hydrodynamics of fluvial bedforms is discussed from the standpoint of sediment transport, as a future scope of research. [ABSTRACT FROM AUTHOR]

In this study, at first, we analyse the linear stability of a straight river. We find that the natural perturbation modes maintain an equilibrium state by confining themselves to a threshold wavenumber band. The effects of river aspect ratio, Shields number and relative roughness number on the wavenumber band are studied. Then, we present a phenomenological concept to probe the initiation of meandering of a straight river, which is governed by the counter-rotational motion of neighbouring large-scale eddies in succession to create the processes of alternating erosion and deposition of sediment grains of the riverbed. This concept is deemed to have adequately explained by a mathematical framework stemming from the turbulence phenomenology to obtain a quantitative insight. It is revealed that at the initiation of meandering of a river, the longitudinal riverbed slope obeys a universal scaling law with the river width, flow discharge and sediment grain size forming the riverbed. This universal scaling law is validated by the experimental data obtained from the natural and model rivers.

In this paper, we seek the scaling laws of sediment transport under a turbulent flow by applying the phenomenological theory of turbulence. The results show that at the threshold of sediment motion, the densimetric Froude number follows a “(1+σ)/4” scaling law with the relative roughness number (ratio of particle size to flow depth), where σ is the spectral exponent. For the bedload transport, the bedload transport intensity follows a “3/2” and “(1+σ)/4” scaling laws with the transport stage function and the relative roughness, respectively. For the scour in a contracted stream, the dimensionless scour depth follows a “4/(3–σ)”, “– 4/(3–σ)” and “–(1+σ)/(3–σ)” scaling laws with the densimetric Froude number, the channel contraction ratio and the relative roughness, respectively.