Department of Civil & Infrastructure Engineering

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IIT Jodhpur

The mode of sediment transport where the sediment particles slide, roll, or travel in succession of low jumps close to the bed is known as bed-load transport. In this chapter, theories of bed-load and formulations to predict the bed-load transport rate are presented. The pioneering attempt to predict the bed-load transport rate was due to MP du Boys in 1879, who expressed bed-load transport rate as a function of excess bed shear stress, that is the bed shear stress exceeding the threshold bed shear stress. Thereafter, number of researchers suggested du Boys type equations making use of the excess bed shear stress in different forms and coefficients. Other concepts to predict the bed-load transport rate are the discharge concept (Schoklitsch type), the velocity concept, the bedform concept, the probabilistic concept (Einstein type), the deterministic concept (Bagnold type), and the equal mobility concept. The additional features of this chapter are the discussion on particle saltation, sediment sorting, streambed armoring, and sediment entrainment probability to bed load. The effects of bed load on velocity distribution, length scales of turbulence, and von Kármán constant are also discussed in details. The method of computation of bed-load transport is illustrated through worked out examples. © 2014, Springer-Verlag Berlin Heidelberg.

GeoPlanet: Earth and Planetary Sciences

Bed-load transport

Gravity waves propagating over free-surface flows with shallow depth are well-known phenomena. The small-amplitude sinusoidal wave and Korteweg-de Vries (KdV) equations are based on potential-flow theory being widely used to describe water-wave propagation. The KdV equation leads to two basic flow patterns, as follows: (1) cnoidal waves, and (2) solitary waves. However, in case of a real Newtonian fluid, the bed resistance and the rapid motion of fluid generate turbulence (eddies) in the medium. The effects of turbulence are taken into account in this paper by using the equations for the surface elevation η and depth-averaged flow velocity U developed previously. These equations are based on the Reynolds-averaged Navier-Stokes (RANS) equations for turbulent flow in open channels. The wave profile can be approximated by a form a cos kξ̂/(1 -b cos kξ̂), where a and b are constant amplitudes; k = wave number; ξ̂ = dimensionless horizontal distance given by (x-ct)/h; x = horizontal distance; c = wave velocity; t = time; and h = undisturbed flow depth. Such a profile has the characteristic that the peaks are narrower but higher compared to wider but shallower troughs. The effects of streamflow on wave propagations are also considered. If the waves travel in the direction of the streamflow, there is a lengthening effect on the peaks and troughs, whereas if the waves travel against the direction of streamflow, they become shorter. © 2014 American Society of Civil Engineers.

World University root node

Journal of Hydraulic Engineering

Gravity waves on turbulent shear flow: Reynolds averaged approach

Dimensional analysis is a powerful tool in designing, ordering, and analyzing the experiment results and also synthesizing them. One of the important theorems in dimensional analysis is known as the Buckingham Π theorem, so called since it involves non-dimensional groups of the products of the quantities. In this chapter, Buckingham Π theorem and its uses are thoroughly discussed. Physical models for hydraulic structures or river courses are usually built to carry out experimental studies under controlled laboratory conditions. The main purposes of physical models are to replicate a small-scale hydraulic structure or flow phenomenon in a river and to investigate the model performance under different flow and sediment conditions. The concept of similitude is commonly used so that the measurements made in a laboratory model study can be used to describe the characteristics of similar systems in the practical field situations. This chapter describes hydraulic similitude in terms of geometric, kinematic, and dynamic similitude. Two categories of hydraulic models are discussed: Immobile bed models and mobile bed models. The analysis leads to the definition of model-scale ratios. A number of illustrative examples are presented. © 2014, Springer-Verlag Berlin Heidelberg.

Dimensional analysis and similitude

Water research

Straining of polyelectrolyte-stabilized nanoscale zero valent iron particles during transport through granular porous media

The flow condition that is just adequate to initiate the motion of sediment particles at the bed surface is called sediment threshold. Albert Frank Shields carried out his doctoral research study on sediment transport at the Technischen Hochschule Berlin, Germany. He is well known for proposing a useful diagram, known as Shields diagram. This diagram provides the criterion for the sediment threshold, 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 for the beginning of sediment motion. In this chapter, the important experimental and theoretical studies on sediment threshold under steady stream flow are furnished, highlighting the empirical formulations and semitheoretical analyses. Both deterministic and probabilistic models of sediment threshold are discussed. The special feature of this chapter is a discussion on the influence of turbulent bursting on threshold of sediment entrainment. Latest experimental findings evidenced that the mechanism of sediment entrainment is governed by the sweep events. The concept of sediment threshold is applied to determine the stable-ideal section of a channel, known as threshold channel. It has a bank profile for which the sediment particles along the wetted perimeter are in a state of incipient motion. The design of threshold channel is demonstrated through numerical examples. © 2014, Springer-Verlag Berlin Heidelberg.

Sediment threshold

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.

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.

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.