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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]

This study presents a comprehensive dataset comprising multiple data packages derived from laboratory experiments on steady and unsteady hydraulic jumps interacting with a large-scale Gaussian-shaped bed obstacle in an open-channel flume. The primary objective was to accurately measure the impact of hydraulic jump on the free surface and the bed pressure along the obstacle, ensuring the transferability of the results. A multi-process method was followed: designed experiments were recorded, images were postprocessed, and water level data were digitalized. For steady conditions, the bed pressure along the obstacle were measured by piezometers. The repository data are organized and provided in a single package, supplemented by a second package containing panoramas for each experimental time instant and graphical representations of the data, facilitating rapid evaluation of the outcomes. This study provides versatile data that can be utilized in various ways, particularly for fluvial model validation and studying turbulence-driven phenomena in open-channel flows. The detailed methodology presented herein can contribute to the advancement of enhanced laboratory techniques to study similar flow problems. [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]

We report an unprecedented existence of the zeroth law of helicity spectrum (i.e., the helicity spectrum becomes independent of the wavenumber) in the transition from production range to inertial range, herein termed the pre-inertial range, of wall turbulence. The zeroth law is explained by the superposition effect of the forward joint cascade of energy and helicity caused by twisting and stretching of wall-attached superstructures in an equilibrium layer. The phenomenological model perfectly predicts the zeroth law in the pre-inertial range. Experimental data support the existence of the zeroth law. [ABSTRACT FROM AUTHOR]

In this paper, we explore the hydrodynamic instability of free river bars driven by a weakly varying turbulent flow in a straight alluvial channel with erodible bed and non-erodible banks. We employ linear stability analysis in the framework of depth-averaged formulations for the hydrodynamics and the sediment transport. A significant fraction of the sediment flux is considered to be in suspension. The analysis is performed for the alternate pattern of river bars at the leading order followed by the next order, covering the effects of flow regime. We find that the unstable region bounded by a marginal stability curve depends significantly on the shear Reynolds number, which demarcates different flow regimes, and the Shields number and the relative roughness (particle size to flow depth ratio). The results at the next order stabilize the bars with longer wavenumbers. The variations of threshold aspect ratio with Shields number and relative roughness are studied for different flow regimes. In addition, for a given Shields number and relative roughness, the diagram of threshold aspect ratio vs shear Reynolds number is explained. Unlike the conventional theories of bar instability, the analysis reveals limiting values of Shields number and relative roughness beyond which the theoretical results at the next order produce infeasible regions of instability. The limiting values of Shields number and relative roughness appear to reduce, as the shear Reynolds number increases. [ABSTRACT FROM AUTHOR]

The law of the wall, regarded as one of the very few pieces of turbulence hypothesis, predicts the mean-velocity profile (MVP) in a wall-bound flow. For about nine decades, the underlying physics of the law is deemed to be governed by an ad hoc mixing-length hypothesis. Here, we seek the origin of the law, for the first time, with the aid of a new hypothesis, which we call the mixing-instability hypothesis. The hypothesis unveils the previously unknown universal scaling behavior for the amplitude of turbulent ripples or waves (that cause spontaneous stretching and shrinking of turbulent eddies) within the overlap layer and accurately maps the experimental data of the MVPs for moderate to extremely large Reynolds numbers. This study offers a new mechanism of the momentum transfer in a turbulent wall-bound flow, calling for a revision of the conventional mixing-length hypothesis, which has persisted in standard textbooks on turbulence for many decades. [ABSTRACT FROM AUTHOR]

This state-of-the-art review describes the progress in analytical modeling of suspended sediment transport by turbulent wall shear flow. The mechanics of the suspended sediment transport by turbulent flow represented by various concepts, such as the diffusion concept, the energy concept and the stochastic concept, is appraised. The effects of sediment suspension on velocity distribution, von Kármán constant value and turbulence characteristics are highlighted. As the latest development of the subject, this paper explains the scaling law of suspended sediment concentration within the wall shear layer stemming from the energetics of turbulent eddies. This scaling law elucidates that within the wall shear layer, the suspended sediment concentration obeys a unique scaling law with the vertical distance from the bed. Finally, the modeling challenges are outlined as a future scope of research. [ABSTRACT FROM AUTHOR]

In this study, a mathematical model for the Reynolds stress profiles in different regions of a submerged turbulent plane offset jet was developed. The two-dimensional Reynolds-averaged Navier-Stokes and the continuity equations were solved for the determination of the velocity and the Reynolds stress profiles within the jet layer in different regions of a submerged offset jet. The theoretical derivations involved a boundary layer approximation. The velocity profiles in the preattachment region obeyed an exponential function, whereas those in the impingement and wall jet regions followed a combination of a power function for the inner layer and an exponential function for the outer layer of the jet. The computed velocity and the Reynolds stress profiles corresponded well with the previous experimental results of the flow in submerged turbulent plane offset jets. [ABSTRACT FROM AUTHOR]

The phenomenological theory of turbulence (PTT) remains a long-standing and fascinating theory in turbulence research. In this review article, we highlight the state-of-the-science of the impact of the PTT on the pragmatic approach to fluvial hydraulics, explored over recent decades, discussing the salient and the subtle roles that the turbulence plays in governing many physical processes. To acquire a theoretical explanation of this pragmatic approach necessitates an intuitive thought that can bring together the background mechanisms of all the physical processes under one law—a thought that is capable of finding their inextricable links with the turbulent energy spectrum. We begin here with emphasizing the spectral and the co-spectral origin of the well-recognized laws of the wall, the resistance equation, and the turbulence intensities by portraying the typical momentum transfer mechanism of eddies in a turbulent flow. Next, we focus on the scaling laws of key fluvial processes derived from the perspective of the PTT, enlightening their physical insight and ability to judge how far the so-called empirical formulas can be used with confidence. The PTT has been able to disclose the origin of several primeval empirical formulas that have been used over many years without having any theoretical clarification and confirmation. Finally, we make an effort to describe some unsolved issues to be resolved as a future scope of research. [ABSTRACT FROM AUTHOR]

This study examines the turbulence in wall-wake flow downstream of an isolated dunal bedform. The streamwise flow velocity and Reynolds shear stress profiles at the upstream and various streamwise distances downstream of the dune were obtained. The results reveal that in the wall-wake flow, the third-order moments change their signs below the dune crest, whereas their signs remain unaltered above the crest. The near-wake flow is featured by sweep events, whereas the far-wake flow is controlled by the ejection events. Downstream of the dune, the turbulent kinetic energy production and dissipation rates, in the near-bed flow zone, are positive. However, they reduce as the vertical distance increases up to the lower-half of the dune height and beyond that, they increase with an increase in vertical distance, attaining their peaks at the crest. The turbulent kinetic energy diffusion and pressure energy diffusion rates, in the near-bed flow zone, are negative, whereas they attain their positive peaks at the crest. The anisotropy invariant maps indicate that the data plots in the wall-wake flow form a looping trend. Below the crest, the turbulence has an affinity to a two-dimensional isotropy, whereas above the crest, the anisotropy tends to reduce to a quasi-three-dimensional isotropy. [ABSTRACT FROM AUTHOR]