Mathematical modeling of the transportation competency of an ice-covered flow

A two-dimensional model was developed to study the effect of ice cover on the transportation competence of ice-covered flows. The model is based on the equations of motion, impurity transfer, turbulence energy, and the ratio of turbulent energy to turbulent kinetic energy with allowance made to the effect of turbulent energy bursts on solid surfaces bounding the flow. The turbulent bursts on solid boundaries of open and ice-covered flows are shown to have no effect on the structure of flows, characterized by their aver-aged characteristics, but considerably change the distribution of suspended sediment concentrations.     

Effect of ice cover on bed deformations in lower pools

The effect of ice cover on bed deformations in the lower pools of hydraulic structures was examined by using data of laboratory and numerical experiments. Data of numerical experiments were used to parameterize bed deformations as a function of the release volume and duration, the length of ice hole, ice roughness coefficient, soil particle size, porosity, and density. The results of numerical experiments were compared with data of field and laboratory measurements.     

Suspended load in flows on erodible bed

Steady state suspended-load of sediment transported in flow over erodible beds usually is treated by the advection-diffusion approach, though in recent years, it is being treated as a two-phase flow phenomenon incorporating kinetics of sediment particles. Among the advection-diffusion approaches, Rouse＇s equation is the well-known, although a number of researchers in later periods have attempted to improve it by modifying the mixing length concept taking into account other aspects. In this paper, the advection-diffusion approach and associated logarithmic law of flow velocity are revisited. It is concluded from the logarithmic law that the Reynolds shear stress is a linear function of height above the bed, which reduces to bed shear stress in the case of a long horizontal channel. As a consequence, it is shown that the volumetric concentration of sediment is best approximated by the sum of two power laws of height above the bed. An equation is derived for the suspended-load transport rate in terms of elementary functions.     

Mathematical modeling of PCB bioaccumulation in Perna viridis

In the present work, we built a mathematical model of polychlorinated biphenyl (PCB) bioaccumulation in Perna viridis, namely, a one-compartment model with a time dependent incorporation rate R (μg g⁻¹ lipid per ppb water per day), with positive substrate cooperativity as the underlying physical mechanism. The temporal change of the PCB concentration Q (μg g⁻¹ lipid) in the soft tissues of the mussel depends on the competition of the input rate RW and the output rate kQ, where W is the concentration of PCB in water (ppb water) and k is the elimination rate (per day). From our experimental data, k=0.181±0.017 d⁻¹. The critical concentration in water W_c for positive substrate cooperativity was found to be 2.4 ppb. Below W_c, R is a constant. For a water concentration of 0.5 ppb Aroclor 1254, R=24.0±2.4 μg g⁻¹ lipid ppb⁻¹ d⁻¹. Above W_c, positive substrate cooperativity comes into effect and R becomes a function of time and dependent on the concentration Q in a form R=γQ/(Q+δ). This is the case for a water concentration of 5 ppb Aroclor 1254, where γ=15.1 μg g⁻¹ lipid ppb⁻¹ d¹ and δ≈200 μg g⁻¹ lipid. From this model, the uptake is exponentially increasing when the PCB concentration in the mussel is small compared to 200 μg g⁻¹ lipid, and hyperbolically increasing when the concentration is large compared to 200 μg g⁻¹ lipid, which are consistent with the experimental data. The model is useful for understanding the true processes taking place during the bioaccumulation and for risk assessment with higher confidence. Future experimental data which challenge the present model are anticipated and in fact desirable for improvement and perfection of the model.     

Sediment transport in ice-covered channels

The existence of ice cover has important effects on sediment transport and channel morphology for rivers in areas with an annual occurrence of an ice season. The interaction of sediment transport and s...     

Experimental study of depth-limited open-channel flows over a gravel bed

Laboratory experiments of depth-limited open-channel flows over a gravel bed were conducted in the study.Two gravel patches with identical individual element size and different lengths(3.81 m and 7.5 m)were tested.The depth-limited uniform flow regime with relative submergence S_r(= D/k_s) ranging from2.68 to 5.94 was produced by adjusting the tailgate weir.The velocity profiles were measured by using both an ultra-sound velocity profiler(UVP) and an acoustic Doppler velocimeter(ADV).The conventional methods used to determine the zero-plane displacement and estimate the bed shear velocity were then reviewed and compared.The measured double-averaged(DA) velocity profiles were found to fit well with the log law and defect law with a non-universal Karman constant κ./κ-value remains nearly constant and in the range from 0.2 to 0.3 for the long patch(LP) cases and κ-values are scattered within a wider range from 0.3 to 0.5 for the short patch(SP) cases.While the Br-value in log law remains constant and equal to 8.5 for LP cases,the Br-value was found to decrease with the increase of the dimensionless roughness height k_s~+ for SP cases.The streamwise turbulence intensity distributions were found to be independent on the patch length and agree well with the available experimental data in the intermediate region and wall region.The Manning resistance coefficient and Darcy-Weisbach friction factor were analyzed.The κ-value decreases to 0.22 for the fitting of the logarithmic flow resistance law under small relative submergence.The value of the integration constant Ar in the logarithmic law falls within the normal range between 3.25 and 6.25.     

Probabilistic approach to modeling of velocity distributions in fluid flows

This paper presents a synthesis of a probability-based approach and the underpinning, mathematical and philosophical foundations that have evolved to date, as well as applications in modeling of vertical and two-dimensional velocity distributions that have direct implications to measurements and estimation of transport of mass, momentum and energy in fluid flows. The approach draws inferences from a probability law identified by maximizing the information entropy under the constraints imposed by the available information. It gives predictions considered to be the most probable or objective on the basis of the available information. The probabilistic approach complements the deterministic approach of hydrodynamics. The difference in the point of view between the two approaches creates a different view about the available information. Some information, such as the location and magnitude of maximum velocity, the ratio of mean and maximum velocities, that may not appear to have direct use to the deterministic approach in flow predictions become important and useful to the probabilistic approach.     

Incompressible SPH scour model for movable bed dam break flows

In this study an incompressible smoothed particle hydrodynamics (ISPH) approach coupled with the sediment erosion model is developed to investigate the sediment bed scour and grain movement under the dam break flows. Two-phase formulations are used in the ISPH numerical algorithms to examine the free surface and bed evolution profiles, in which the entrained sediments are treated as a different fluid component as compared with the water. The sediment bed erosion model is based on the concept of pick-up flow velocity and the sediment is initiated when the local flow velocity exceeds a critical value. The proposed model is used to reproduce the sediment erosion and follow-on entrainment process under an instantaneous dam break flow and the results are compared with those from the weakly compressible moving particle semi-implicit (WCMPS) method as well as the experimental data. It has been demonstrated that the two-phase ISPH model performed well with the experimental data. The study shows that the ISPH modelling approach can accurately predict the dynamic sediment scouring process without the need to use empirical sediment transport formulas.     

Enhanced bed load sediment transport by unsteady flows in a degrading channel

Laboratory flume experiments were done to investigate bed load sediment transport by both steady and unsteady flows in a degrading channel. The bed, respectively composed of uniform sand, uniform gravel, or sand-gravel mixtures, always undergoes bulk degradation. It is found that both uniform and non-uniform bed load transport is enhanced greatly by unsteady flows as compared to their volume-equivalent steady flows. This enhancement effect is evaluated by means of an enhancement factor, which is shown to be larger with a coarser bed and lower discharges. Also, the fractional transport rates of gravel and sand in non-uniform sand-gravel mixtures are compared with their uniform counterparts under both steady and unsteady flows. The sand is found to be able to greatly promote the transport of gravel, whilst the gravel considerably hinders the transport of sand. Particularly, the promoting and hindering impacts are more pronounced at lower discharges and tend to be weakened by flow unsteadiness.     

Mathematical modeling of PCB bioaccumulation in Perna viridis

In the present work, we built a mathematical model of polychlorinated biphenyl (PCB) bioaccumulation in Perna viridis, namely, a one-compartment model with a time dependent incorporation rate R (μg g⁻¹ lipid per ppb water per day), with positive substrate cooperativity as the underlying physical mechanism. The temporal change of the PCB concentration Q (μg g⁻¹ lipid) in the soft tissues of the mussel depends on the competition of the input rate RW and the output rate kQ, where W is the concentration of PCB in water (ppb water) and k is the elimination rate (per day). From our experimental data, k=0.181±0.017 d⁻¹. The critical concentration in water W_c for positive substrate cooperativity was found to be 2.4 ppb. Below W_c, R is a constant. For a water concentration of 0.5 ppb Aroclor 1254, R=24.0±2.4 μg g⁻¹ lipid ppb⁻¹ d⁻¹. Above W_c, positive substrate cooperativity comes into effect and R becomes a function of time and dependent on the concentration Q in a form R=γQ/(Q+δ). This is the case for a water concentration of 5 ppb Aroclor 1254, where γ=15.1 μg g⁻¹ lipid ppb⁻¹ d¹ and δ≈200 μg g⁻¹ lipid. From this model, the uptake is exponentially increasing when the PCB concentration in the mussel is small compared to 200 μg g⁻¹ lipid, and hyperbolically increasing when the concentration is large compared to 200 μg g⁻¹ lipid, which are consistent with the experimental data. The model is useful for understanding the true processes taking place during the bioaccumulation and for risk assessment with higher confidence. Future experimental data which challenge the present model are anticipated and in fact desirable for improvement and perfection of the model.     

Evaluation of a gravel transport sensor for bed load measurements in natural flows

A recent acoustic instrument （Gravel Transport Sensor, GTS） was tested for predicting sediment transport rate （bed load rate） in gravel bed streams. The GTS operation is based on the particle collision theory of submerged obstacles in fluids. When particles collide with the GTS cylinder their momentum is recorded in the form of ping rates. The GTS is attractive for further consideration here because of its potential to provide continuous unattended local bed load measurements, especially in areas found in streams that access may be difficult under extreme conditions. Laboratory experiments coupled with numerical simulations for the same flow conditions were performed in order to determine the conditions under which particles of different size will hit the GTS cylinder and be able to register a ping rate. The GTS was able to detect the number of particles with diameter in the range of 15.9 to 25.4 mm, with reasonable accuracy, if the applied Shields effective stress τ＊e = τ＊ - τ＊cr was in the range of 0.006 to 0.015. A drawback of the tested prototype GTS, however, was that it exerted increased resistance on the incoming particles. The added drag effects increased the overall resistance that was exerted by the flow on particles and thus increased the likelihood that particles will rest in the ambient region of the cylinder instead of hitting it. Numerical simulation of the flow around the GTS cylinder revealed that changing the prototype geometry from cylindrical to ellipsoid or rhomboid will increase the likelihood of the particles hitting the instrument under the same flow conditions failed by the original tested GTS cylinder.     

Three dimensional hydrodynamic modeling over bed forms in open channels

Three dimensional numerical modeling of idealized sand dunes was used to assess the capability of various modeling formulations to capture the flow structure and resistance introduced by bed forms which are similar to those in the Lower Mississippi River. The selected models were: ECOMSED (HydroQual), MIKE 3 [Danish Hydraulic Institute (DHI)] and H3D (Hayco). The study revealed that the hydrostatic versions of models did not capture the flow separation at the crest of the dunes; however, they did respond to the presence of bed forms and gave a total resistance similar to the non-hydrostatic models.     

Motion of water in an ice-covered shallow lake

The results of analysis of data collected by direct measurements of currents in shallow Vendyurskoe Lake in winter are given. The horizontal motions of water in the lake are shown to exist throughout the freeze up period. Dominating frequencies are identified in variations in water motion with periods close to those of the longitudinal and transverse seiches of the lake. Experiments are used to show that oscillatory motions of ice can serve a source of energy for the development of water mass movement in ice-covered lakes. Wind is shown to be the most likely factor generating oscillatory motions of water in ice-covered lakes. It is shown that the velocities of residual currents existing in a lake decrease in winter and their maximum values occur above the deepwater part of the lake within the first and second months of freeze-up.     

Mathematical simulation of stable and equilibrium river bed profiles and slopes

Two approaches to the simulation of stable and equilibrium longitudinal profiles and slopes are considered. The first one deals with a solution to the equation of solid matter (sediment) continuity in the equilibrium case and the second approach deals with the application of variational principles. The approaches considered and results obtained can be used in hydroengineering construction for designing stable and equilibrium longitudinal profiles of water streams and slopes.     

A worldwide correlation for exponential bed particle size variation in subaerial aqueous flows

The particle size of the bed sediments in or on many natural streams, alluvial fans, laboratory flumes, irrigation canals and mine waste deltas varies exponentially with distance along the stream. A plot of the available worldwide exponential bed particle size diminution coefficient data against stream length is presented which shows that all the data lie within a single narrow band extending over virtually the whole range of stream lengths and bed sediment particle sizes found on Earth. This correlation applies to both natural and artificial flows with both sand and gravel beds, irrespective of either the solids concentration or whether normal or reverse sorting occurs. This strongly suggests that there are common mechanisms underlying the exponential diminution of bed particles in subaerial aqueous flows of all kinds. Thus existing models of sorting and abrasion applicable to some such flows may be applicable to others. A comparison of exponential laboratory abrasion and field diminution coefficients suggests that abrasion is unlikely to be significant in gravel and sand bed streams shorter than about 10 km to 100 km, and about 500 km, respectively. Copyright © 1999 John Wiley & Sons, Ltd.     

Mathematical modeling of flooding due to river bank failure

Modeling of flooding events resulting from bank overflooding and levee breaching is of relevant social and environmental interest. Two-dimensional (2D) hydrodynamic models integrating the shallow water equations turn out to be very effective tools for the purpose at hand. Many of the available models also use 1D channel elements, fully coupled to the 2D model, to simulate the flow of small channels dissecting the urban and rural areas, and 1D elements, referred to as 1D-links, to efficiently model the flow over levees, road and rail embankments, bunds, the flow through control gates, either free or submerged, and the operation of other hydraulic structures. In this work we propose a physically-based 1D-link to model breach formation and evolution in fluvial levees, and levee failure due to either piping or overtopping. The proposed 1D-link is then embedded in a 1D–2D hydrodynamic model, thus accounting for critical feedbacks between breach formation and changes in the hydrodynamic flow field. The breach model also includes the possibility of simulating breach closure, an important feature particularly in the view of hydraulic risk assessment and management of the emergency. The model is applied to five different case studies and the results of the numerical simulations compare favorably with field observations displaying a good agreement in terms of urban and rural flooded areas, water levels within the channel, final breach widths, and water volumes flowed through the breach.     

Vertical distribution of a pollutant in river flow: Mathematical modeling

A mathematical model has been developed to assess the role of the shape of channel cross-section, the position of pollutant source on the bank slope, and the sedimentation rate of pollutant particles in the process of pollutant transport in water flow. The procedure of numerical experiments and the obtained results and conclusions can be of use in the simulations and studies of pollution propagation from sources on riverbanks in permafrost zone, which are subject to thermal erosion.     

Fully coupled mathematical modeling of turbidity currents over erodible bed

Turbidity currents may feature active sediment transport and rapid bed deformation, such as those responsible for the erosion of many submarine canyons. Yet previous mathematical models are built upon simplified governing equations and involve steady flow and weak sediment transport assumptions, which are not in complete accordance with rigorous conservation laws. It so far remains unknown if these could have considerable impacts on the evolution of turbidity currents. Here a fully coupled modeling study is presented to gain new insights into the evolution of turbidity currents. The recent analysis of the multiple time scales of subaerial sediment-laden flows over erodible bed [Cao Z, Li Y, Yue Z. Multiple time scales of alluvial rivers carrying suspended sediment and their implications for mathematical modeling. Adv Water Resour 2007;30(4):715–29] is extended to subaqueous turbidity currents to complement the fully coupled modeling. Results from numerical simulations show the ability of the present coupled model to reproduce self-accelerating turbidity currents. Comparison among the fully and partially coupled and decoupled models along with the analysis of the relative time scale of bed deformation explicitly demonstrate that fully coupled modeling is essential for refined resolution of those turbidity currents featuring active sediment transport and rapid bed deformation, and existing models based on simplified conservation laws need to be reformulated.