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The turbulence in a fluid flow is characterized by irregular and chaotic motion of fluid particles. It is a complex phenomenon. In this chapter, the turbulence characteristics are discussed with reference to flow over a sediment bed. An application of Reynolds decomposition and time-averaging to the Navier–Stokes equations yields the Reynolds-averaged Navier–Stokes (RANS) equations, containing terms of Reynolds stresses. The RANS equations along with the time-averaged continuity equation are the main equations to analyze turbulent flow. The classical turbulence theories were proposed by Prandtl and von Kármán. Prandtl simulated the momentum exchange on a macro-scale to explain the mixing phenomenon in a turbulent flow establishing the mixing length theory, while von Kármán’s relationship for the mixing length is based on the similarity hypothesis. The velocity distribution in open-channel flow follows the linear law in viscous sublayer, the logarithmic law in turbulent wall shear layer, and the wake law in the outer layer. The determination of bed shear stress is always a challenging task. Different methods for the determination of bed shear stress are discussed. Flow in a narrow channel exhibits strong turbulence-induced secondary currents, and as a result, the maximum velocity appears below the free surface, known as dip phenomenon. Isotropic turbulence theory deals with the turbulent kinetic energy (TKE) transfer from the large-scale motions to smaller-scale motions until attaining an adequately small length scale so that the fluid molecular viscosity can dissipate the TKE into heat. Anisotropy in turbulence is analyzed by the anisotropic invariant mapping (AIM) and the anisotropy invariant function to quantify the degree of the departure from isotropy. Higher-order correlations are given by skewness and kurtosis of velocity fluctuations, TKE flux, and budget. This chapter also includes most of the modern development of turbulent phenomena, such as coherent structures and burst phenomena and double-averaging of heterogeneous flow over gravel beds. © 2014, Springer-Verlag Berlin Heidelberg.

Department of Civil & Infrastructure Engineering

GeoPlanet: Earth and Planetary Sciences

Turbulence in open-channel flows

River configurations in plan view are highly variable. Under specific environmental and hydraulic conditions, the type of planform geometry of a river is controlled by the sediment transport and its capacity of the river. Alluvial river configurations are in general categorized as straight, meandering, and braided rivers. While long, straight rivers seldom occur in nature; meandering and braided rivers are common. This chapter focuses on the characteristics of meandering and braided rivers. Mathematical models of meandering rivers are presented. © 2014, Springer-Verlag Berlin Heidelberg.

Fluvial processes: Meandering and braiding

One-dimensional (1D) hydraulic models for phreatic aquifers are commonly based on Dupuit's approximation for near-horizontal flows. In the past, improved 1D models were developed by inclusion of streamline curvature in the governing equations. However, the models developed for steady flow are quite different from those developed for unsteady curvilinear flow. In this paper, a single mathematical model is proposed for application to both steady and unsteady curvilinear phreatic flows. Unsteady flow equations for phreatic aquifers are presented by adopting the shallow flow approximation. The resulting model converges to that originating from perturbation methods, thereby providing a generalized result. Limitations of the second-order shallow flow theory and Dupuit's approximation are investigated for steady flow through a dam and unsteady flow in the bank storage problems. The results obtained from the present model are compared with two-dimensional data, indicating that Dupuit's theory is valid under a wider range of conditions than previously found. © 2014 American Society of Civil Engineers.

World University root node

Journal of Hydraulic Engineering

Second-order shallow-flow theory and dupuit approximation for phreatic aquifers

The proposed study examined and reviewed the published experimental results related to clear water scour below a cylinder across the flow. It also highlighted the limitations of existing methods for estimating the scour depth below a submerged cylinder. In the present study, experiments were performed for 50% and 75% submergences of a 70 mm diameter cylinder in the free surface flow over a uniform sand bed with d50=0.98 mm downstream of an apron. Based on the experimental results, an empirical equation was proposed to estimate the amount of gap flow between the cylinder and the bed for an equilibrium scour for a given flow depth and sediment properties. Measured scour profile consisted of a scour hole and immediately followed by a dune. However, no general sediment transport was occurring away from the cylinder due to the undisturbed bed shear stress less than or equal to the critical shear stress required for the sediment entrainment. Different submergence ratios of the cylinder resulted in different longitudinal and vertical extensions of the scour hole and the dune. The maximum equilibrium scour depth occurred when the cylinder is fully submerged in the unidirectional flow with water depth equals to the cylinder diameter. The non-dimensional measured scour profiles were found to be similar. The characteristic lengths of the scour hole and the dune were computed analytically by approximating the measured scour profile by third degree polynomials. The computed non-dimensional scour profiles compared satisfactorily with the measured profiles. It was found that analytical non-dimensional scour profiles were identical for a given diameter of a cylinder with different submergences for the same flow conditions. © 2014, Indian Academy of Sciences.

Fulltext

Sadhana - Academy Proceedings in Engineering Sciences

Experimental studies of local scour in the pressurized OCF below a wooden log across the flow

A notable universal feature of wall-bounded turbulent flows is the universal logarithmic law of the wall deduced by Theodore von Kármán. This law of the wall describes how time-averaged streamwise velocity changes with distance from the wall. Despite the law of the wall having a universal von Kármán constant κ = 0.41 that governs the slope of the log-law velocity profile, as commonly known over a period of about 80 years, in fluvial streams there are a number of instances of the non-universality of κ. To be specific, it behaves as a variable in flows with low relative submergence, or where there is bed-load and/or suspended-load sediment transport. This article focuses on the aspect of non-universality of κ by inviting various open questions relating to future research directions.© Springer-Verlag Berlin Heidelberg 2013.

Evidence of Non-Universality of von Kármán's κ

When an unsteady free surface flow encounters an adverse slope, it results in a decelerating flow up the adverse slope. The time dependent turbulent flow is treated here by appropriately reducing the two-dimensional Reynolds averaged Navier-Stokes equation along with the equation of continuity considering turbulence closure. With suitable choice of parameters, the resulting differential equations are numerically solved to compute free surface and streamwise velocity profiles with time. It is found that initially the advancing free surface is convex upwards for a short time, followed by a jump of the free surface with a negative streamwise velocity that is a backwater rolling breaker due to deceleration of flow. At later time, however, the velocity becomes positive, that is, the breakers roll forward. This dual feature of motion, that is a surge followed by rolling breakers, is repeated for sometime before the jumps stop. The theoretical analysis presented here is motivated by tidal bores propagating upstream in an estuarine river. © 2012 Versita Warsaw and Springer-Verlag Wien.

Acta Geophysica

Turbulent unsteady flow profiles over an adverse slope

In this study, the hydrodynamics of three types of steady undular free surface flow problems are theoretically analyzed. These flows are governed by the steady-state equations of conservation of mass and momentum developed by Bose and Dey in 2007 (J Hydraul Eng 133:1074-1079) and 2009 (Phys Rev E 80:036304). The first type of flow is an undular hydraulic jump on a plane smooth boundary when the approach flow Froude number marginally exceeds its critical value of unity. Here, it is established that the undular hydraulic jump phenomenon can be treated as an instability of the free surface flow being described by the instability principle. The results reveal that the threshold of an undular hydraulic jump is represented by a monotonic increase of approach flow Froude number with boundary inclination. The elevation of the undular free surface increases, as the boundary inclination increases. However, the amplitude of the free surface waves decreases with downstream distance. The second type of flow is that over a submerged hemi-cylindrical boss (inverted semi-circular section) placed on the channel bottom with its horizontal diameter lateral to the flow direction. The steady-state flow analysis shows that there is a drop in the free surface elevation on the downstream end of the cylinder with an undular free surface profile. The third type of flow is that over a continuous sinusoidal boundary of a channel. A numerical experiment shows that there exists a phase lag between the free surface profile and the boundary profile. Also, it is revealed that if the flow depth is reduced, an accumulation of heaved wave occurs indicating an onset of chocking.© Springer-Verlag Berlin Heidelberg 2013.

Hydrodynamics of Undular Free Surface Flows

Submerged sluice gates may be used as discharge measuring structures in irrigation channels. One commonly used model to predict discharge under submerged gates is based on the simultaneous solution of the energy and momentum equations (EM method). The conventional assumption for the solution is that the roller forming downstream of the gate is a static bubble of water with zero net flux of both momentum and kinetic energy. Energy losses along the submerged gate flow are commonly neglected. In this study, rational analysis of the EM method shows that the roller momentum flux and the energy loss may be considerable. A novel formulation of the EM method accounting for these two phenomena is proposed. Although it yields results similar to those of the conventional method, the new method introduces rationality in the application of the EM equations for submerged gate flow and helped to critically revise the assumptions in which the conventional EM method is based. © 2013 American Society of Civil Engineers.

Journal of Irrigation and Drainage Engineering

Revisiting the energy-momentum method for rating vertical sluice gates under submerged flow conditions

A one-dimensional depth-averaged velocity deformation model is developed to study the characteristics of the unsteady flows in open channels. The vertical profile of the streamwise velocity is expressed in a power series, and the coefficients of the velocity are evaluated by using the unsteady equation of motion. A concise form of the friction velocity for unsteady nonuniform flow is then proposed by utilizing the coefficients of the power series. The applicability of the model is validated with experimental data, producing reasonably good agreement between the model results and the observed data. Finally, the deformation of velocity at the surface is assured by comparing velocity distribution of numerical model with the computational results from a previous model. © 2013 American Society of Civil Engineers.

Velocity deformation model for unsteady open-channel flows over smooth and rough beds

Scour at piers embedded in a bed of cohesive sediment mixture is investigated to quantify the effects of bed compaction. Experiments were conducted in a large rectangular laboratory flume with natural clay sediment collected from a stream located at the Chilean coastal range. Nonintrusive and high-resolution topographic measurements of scoured bed in the vicinity of piers were performed during the experimental runs with an installation of a laser distance sensor (LDS) that was traversed by the precision step motors. Measurements were taken by the LDS in different azimuthal half-planes to study the spatiotemporal variation of the topography of evolving scour holes. Experiments were conducted over 40 h, until an advanced stage of scour was reached, with the approaching flow having a bed shear stress equaling 95% of the Shields critical condition for the initiation of motion of sand in a sand-clay mixture. Results show that the bed is successively scoured in three different forms, chunks of aggregates, aggregate by aggregate, and particle by particle, when increasing the ratio of actual to Proctor's optimum molding water content, w̌ .Maximum scour depth was mainly observed at the wake region (downstream of the pier) and ranged from 10 to 58% of that in the sand alone case for mixtures with the same size of the sand fraction. For a given compaction energy, scour depths first decrease with w̌ , up to a value of w̌ 1/4 2.5, as fine clay particles are flocculated and increasingly dispersed with water content. For values of w̌ >2.5, the sediment matrix behaves as a fluid mud. In the latter case, scour depths increase with water content as the sediment bed becomes increasingly liquid. ©2013 American Society of Civil Engineers.

Effects of bed compaction on scour at piers in sand-clay mixtures

This study examines the probability for sediment entrainment to bed load and the probability for the threshold condition of sediment to be in suspension. The theoretical analysis is based on a simple one-sided exponential distribution of probability function. The probability distributions are derived from a truncated universal Gram-Charlier series expansion based on the exponential or Laplace-type distributions for turbulent velocity fluctuations. The key criterion of sediment entrainment is the hydrodynamic lift acting on a solitary particle to exceed submerged weight of the particle. In this way, a simple probability function for sediment entrainment to bed load in terms of Shields parameter containing the lift coefficient is obtained. It was found that the value of lift coefficient as 0.15 satisfactorily fitted the probability function versus Shields parameter curve with the experimental data. On the other hand, the key criterion of the threshold of sediment suspension is the fluctuations of the vertical velocity component to exceed terminal fall velocity of the particle. The probability function for the threshold of a sediment particle to be in suspension is obtained in terms of Shields parameter as a function of shear Reynolds number. Curves for different values of probabilities are drawn in respect to a Shields diagram. For the value of probability 0.05, the threshold of sediment suspension is indicated. The prediction curves for the threshold of sediment suspension are proposed in terms of Rouse number versus Shields parameter and also Shields parameter versus shear Reynolds number. © 2013 American Society of Civil Engineers.

Sediment entrainment probability and threshold of sediment suspension: Exponential-based approach

This study focuses on the double-averaged (DA) turbulence characteristics and the local flow characterization in the flow zones over and within the flow-gravel-bed interface. The cumulative running averaged (CRA) value of a turbulence quantity for a given interval (in this study, half of the gravel size) of flow measurement is the prerequisite to determine a complete DA value of that quantity. The CRA turbulence quantities within the interfacial sublayer require a longer distance to become spatially independent. A method is proposed to fit the log law for the DAvelocity over the bed and a polynomial law within the interfacial sublayer. The form-induced shear stress is 40% of the DA Reynolds shear stress occurring at the virtual bed level. The spatial velocity fluctuations diminish more rapidly than the temporal velocity fluctuations below the virtual bed level. In the inertial subrange of turbulent kinetic energy (TKE), the eddy size decreases significantly near the virtual bed level, and in the dissipation range, it increases slightly. The TKE fluxes directed downward and downstream compose the sweep events that are the prevailing mechanism within the interfacial sublayer. It is also confirmed by the quadrant analysis done for the bursting events. The form-induced energy fluxes below the crest level are comparable with the TKE fluxes. In the TKE budget, the forminduced diffusion rate is significant within the interfacial sublayer. The quantities constituting the TKE budget attain their peaks above the virtual bed level and reduce sharply below it. However, the pressure energy diffusion rate induces suction near the crest. The local flow characterization is done by separating the local velocity profiles into log, S-shaped, and accelerated classes. Unlike the velocity profiles per class, the profiles of turbulence quantities per class are almost indistinguishable within the interfacial sublayer. A comparison of the results of this study with those of previous studies shows more dissimilarities than similarities. © 2012 American Society of Civil Engineers.

Gravel-bed hydrodynamics: Double-averaging approach

An undular hydraulic jump on a smooth boundary occurs when the approaching flow Froude number marginally exceeds its critical value of unity. The free surface profiles of undular hydraulic jumps are studied theoretically by using the steady-state flow equation. It is established that the undular hydraulic jump phenomenon takes place because of the instability of the flow and can be described by the instability principle. The threshold of an undular hydraulic jump can be represented by a monotonic increasing curve of approaching flow Froude number versus boundary inclination. The elevation of the undular free surface increases with an increase in boundary inclination, but the amplitude of the free surface waves decreases with downstream distance. The theoretical results are well comparable with the experimental data. © 2012 American Society of Civil Engineers.

Free surface profiles of undular hydraulic jumps

Is the von Krmn constant affected by sediment suspension? The presence of suspended sediment in channels and fluvial streams has been known for decades to affect turbulence transfer mechanism in sediment-laden flows, and, therefore, the transport and fate of sediments that determine the bathymetry of natural water courses. This study explores the density stratification effects on the turbulent velocity profile and its impact on the transport of sediment. There is as yet no consensus in the scientific community on the effect of sediment suspension on the von Krmn parameter, . Two different theories based on the empirical log-wake velocity profile are currently under debate: One supports a universal value of =0.41 and a strength of the wake, , that is affected by suspended sediment. The other suggests that both and could vary with suspended sediment. These different theories result in a conceptual problem regarding the effect of suspended sediment on , which has divided the research area. In this study, a new mixing length theory is proposed to describe theoretically the turbulent velocity profile. The analytical approach provides added insight defining as a turbulent parameter which varies with the distance to the bed in sediment-laden flows. The theory is compared with previous experimental data and simulations using a k-turbulence closure to the Reynolds averaged Navier Stokes equations model. The mixing length model indicates that the two contradictory theories incorporate the stratified flow effect into a different component of the log-wake law. The results of this work show that the log-wake fit with a reduced is the physically coherent approximation. © 2012. American Geophysical Union. All Rights Reserved.

Journal of Geophysical Research: Earth Surface

Is the von Kármán constant affected by sediment suspension?

This study is devoted to quantify the near-bed turbulence parameters in mobile-bed flows with bed-load transport. A reduction in near-bed velocity fluctuations due to the decrease of flow velocity relative to particle velocity of the transporting particles results in an excessive near-bed damping in Reynolds shear stress (RSS) distributions. The bed particles are associated with the momentum provided from the flow to maintain their motion overcoming the bed resistance. It leads to a reduction in RSS magnitude over the entire flow depth. In the logarithmic law, the von Kármán coefficient decreases in presence of bed-load transport. The turbulent kinetic energy budget reveals that for the bed-load transport, the pressure energy diffusion rate near the bed changes sharply to a negative magnitude, implying a gain in turbulence production. According to the quadrant analysis, sweep events in mobile-bed flows are the principal mechanism of bed-load transport. The universal probability density functions for turbulence parameters given by Bose and Dey have been successfully applied in mobile-bed flows. © 2012 Versita Warsaw and Springer-Verlag Wien.

Turbulence in mobile-bed streams

A theory of turbulent shear flow over a sand bed is developed, addressing the instability principle of the fluid-granular bed interface leading to the formation of ripples. The Reynolds-averaged Navier-Stokes (RANS) equations and the time-averaged continuity equation are analyzed using a 1/7-power law of the time-averaged streamwise velocity and treating the curvilinear streamlines by the Boussinesq approximation. The integration of the RANS equations leads to a governing dynamical equation of flow over a mobile bed. A near-bed flow layer of 3.5 times the ripple height is considered being affected by the ripples. The dynamical equation of the mobile sand bed is based on the Exner's sediment continuity equation in conjunction with the Meyer-Peter and Müller bed-load transport formula as modified to account for the effect of local bed slope attributable to bed forms. The coupled dynamical equations are then analyzed to estimate the parameters for the instability that results in the formation of ripples on the bed. The nondimensional ripple length (ratio of ripple length to sand size) increases with an increase in Shields parameter. The theoretical results have an agreement with the experimental data. © 2012 American Society of Civil Engineers.

Instability theory of sand ripples formed by turbulent shear flows

Marine sediments often contain sand-clay mixtures in widely varying proportions. This study presents the results of equilibrium scour and time variation of scour depths at circular piles embedded vertically in clay alone and sand-clay mixed beds under waves. Experiments were conducted in a wave flume with different proportions of sand-clay mixtures as bed sediments. Test results for the cases of steady current and sand alone under waves are used as references. The equilibrium scour depth reduces with an increase in clay proportion n (by weight) in a sand-clay mixture. Interestingly, the scour depth reductions for n=0.3 and 1 are almost equal, suggesting that when the clay proportion in a sand-clay mixture becomes 0.3, the sand-clay mixture behaves as if it were a clay-alone mixture, in relation to scour depths. The dependency of equilibrium scour depth on the Keulegan-Carpenter number for different n-values can be expressed by an exponential law, using different coefficients and exponents. For time variation of scour, the nondimensional timescale increases with an increase in Keulegan-Carpenter number and clay proportion. However, unlike the near-equal scour depths, timescales are quite different for n=0.3 and 1. © 2011 American Society of Civil Engineers.

Journal of Waterway, Port, Coastal and Ocean Engineering

Scour at Vertical Piles in Sand-Clay Mixtures under Waves

The time-averaged characteristics of turbulent wall-wake flows downstream of a sphere placed on a rough wall are studied. The profiles of the defect of streamwise velocity, Reynolds shear stress, and turbulence intensities exhibit some degree of similarities when they are scaled by their respective peak defect values. For the velocity defect profiles, the vertical distances are scaled by the height of the location of the half-peak velocity defect. However, for the defect profiles of the Reynolds shear stress and the turbulence intensities, the vertical distances are scaled by the height of the location of the half-peak Reynolds shear stress defect. The magnitudes of the peak defect of all the quantities diminish with the distance downstream of the sphere characterizing the recovery of their undisturbed profiles. Additionally, the theoretical similarity solution for the velocity defect profiles is obtained. The third-order correlations imply that in the inner layer of wall wakes, a streamwise acceleration is prevalent and associated with a downward flux, suggesting sweeps. In contrast, in the outer layer, a streamwise deceleration exists and is associated with an upward flux, suggesting ejections. The profiles of the energy budget show that the turbulent and pressure energy diffusions oppose each other. The turbulent production has a positive peak, and the pressure energy diffusion has a negative peak, indicating a large gain in turbulence production in the wall-wake flows. The quadrant analysis confirms that in wall-wake flows, sweeps are the governing mechanism resulting from an inrush of fluid streaks. The bursting events have shorter duration, but they are more frequent than those in upstream. © 2011 American Society of Civil Engineers.

Wall-Wake Flows Downstream of a Sphere Placed on a Plane Rough Wall

Results of an experimental study on the effects of bed injection (upward seepage from the permeable bed) and suction (downward seepage) on the double-averaged (DA) turbulent flow characteristics over and within the interface of flow gravel beds are presented. The DA Reynolds shear stress within the interfacial sublayer undergoes a higher damping with suction and a lesser damping with injection, as compared to that with no seepage. The form-induced stress within the interfacial sublayer has a decelerating effect with no seepage and suction and an accelerating effect with injection. The quadrant analysis suggests that within the interfacial sublayer the sweep events in flows with injection and the ejection events with suction are the primary contributors toward the Reynolds shear stress production, while both the events in flows with no seepage are equally prevalent. In the roughness sublayer (form-induced and interfacial sublayers together), the mean duration of bursting increases with suction and decreases with injection, but the bursting events become less frequent with suction and more frequent with injection. The analysis of DA higher-order moments confirms that the seepage affects the Reynolds normal stress flux, the turbulent kinetic energy (TKE) flux, and the TKE budget. A downward-upstream flux of TKE is developed within the roughness sublayer with injection; while an upward-downstream flux of TKE is prevalent over the entire flow depth with suction. The significance of this study lies on the hydrodynamic process in such a physical system that may result in modifying the flow resistance, sediment entrainment, and morphological characteristics of a streambed. Copyright 2011 by the American Geophysical Union.

Double-averaging turbulence characteristics in seeping rough-bed streams

A. F. Shields carried out his doctoral research study on sediment transport in the Technischen Hochschule Berlin. He is well known for proposing a useful diagram, known as Shields diagram, that provides the criterion for the threshold of sediment entrainment, which is an essential requirement for the determination of sediment motion in a loose boundary stream. His diagram becomes famous and is most frequently referred in the literature. It has provided an enormous inspiration to initiate a sizable number of researches over last seven decades. Since his pioneering work, numerous attempts have so far been made to quantify the required flow condition (threshold boundary shear stress) for the beginning of sediment movement. The important laboratory experimental and theoretical studies on threshold of sediment entrainment under steady stream flows are briefly summarized, as a perspective review, highlighting the empirical formulations and semitheoretical analyses. A special attention is given towards the influence of the turbulent bursting on sediment entrainment. Latest experiments evidence the mechanism of sediment entrainment is governed by the sweep events. © Springer-Verlag Berlin Heidelberg 2011.