Unsteady Open Channel Flow Research Articles

Hydrodynamic Characteristics of Strong, Unsteady Open-Channel Flow

Due to climate change, the intensity of extreme rainfall has been observed to increase with a shorter duration, causing flash floods (strong, unsteady flow) that lead to serious loss of life and economic damage all over the world. In this study, by repeating the same flume experiments twenty times over a bare bed or with a submerged vane installed, the hydrodynamic characteristics of a strong, unsteady open-channel flow were investigated. Acoustic Doppler velocimetry (ADV) was used to measure the instantaneous three-dimensional velocity, and the ensemble average method was then adopted to obtain the time-varying mean flow velocities. Reynolds decomposition was applied to disintegrate the instantaneous velocity to time-varying average velocity and fluctuating velocity. Turbulence characteristics such as turbulent intensity, turbulent bursting, and power spectral density (PSD) were analyzed against water depth variations. The results show that the loop pattern of the streamwise velocity against the water depth variations could significantly affect the turbulence characteristics of unsteady flow. Near the bed, the peaks of the turbulent intensity and the TKE lag behind the peak of the water depth. The PSD revealed that the turbulent energy increases at the rising and falling stages were due to the generation of small-scale turbulence vortices or eddies. As a submerged vane was present, the increase in the angle of attack caused the increase in the turbulent intensity and TKE, which means that the induced vortex became stronger and the wake region was larger. When the angle of attack was equal to 20°, the TKE abruptly enlarged in the falling stages, implying the breaking-up of the induced vortex. The PSD of the transverse fluctuation velocity showed multiple spikes at the high-frequency part, possibly denoting the shedding frequency from the induced vortex. Further downstream, behind the submerged vane, the peak frequencies of the PSD became imperceptible, likely because of the induced vortex decays or other factors such as the turbulence generated from the free surface or the channel bed mixing with the turbulence from the induced vortex.

Investigating the performance of the modified two-component pressure approach in capturing column separation with large vapour cavities

This paper investigates the performance of the modified two-component pressure approach (proposed model), recently proposed by the authors, in capturing column separation with large vapour cavities. The proposed model calculates both the cavitating and waterhammer flow regions using a single set of equations that governs unsteady flow in open channels. Comparing the results with experimental data, and the results obtained from the discrete gas cavity model (counterpart model) reveals that: (1) the proposed model can accurately capture the evolution of large vapour cavities in pipe systems; (2) the model provides more accurate results compared to the counterpart model; (3) under some circumstances the counterpart provides inaccurate results as it fails to account for the energy loss in the cavitating flow zone while the proposed model was shown to resolve the issue. Finally, the results exhibit the superiority of the proposed model over the counterpart model and the conventional open-channel-based models.

Numerical Simulation of Sediment Transport in Unsteady Open Channel Flow

This paper presented a two-dimensional, well-balanced hydrodynamic and sediment transport model based on the solutions of variable-density shallow-water equations (VDSWEs) and the Exner equation for bed change for simulating sediment transport in unsteady flows. Those equations are solved in a coupled way by the first-order Godunov-type finite volume method. The Harten–Lax–van Leer–Contact (HLLC) Riemann solver is extended to find the local Riemann fluxes to maintain the exact balance between the momentum term and the bed slope term. A well-balanced scheme is superior to an unbalanced scheme to minimize numerical dispersion as demonstrated by the synthetic standing contact-discontinuity test case. Following this, the model is employed to simulate two laboratory experiments and a field case, the 1996 Lake Ha! Ha! flood event in Canada. The results of water surface elevations and bed surface profiles agree well with the measurements. The accuracy and robustness of the numerical schemes make the model a good candidate for practical engineering applications.

Reconstruction of the 3-D fields with a depth-averaged model for open-channel flows in the smooth turbulent case

A two-dimensional depth-averaged model is derived for open-channel flows in the smooth turbulent case. The derivation is consistently obtained with a method of matched asymptotic expansions in the outer and inner layers using a mixing length model of turbulence including the free-surface reduction of the eddy viscosity. The shear effects are taken into account by an extra tensor variable of the model called the enstrophy tensor. The friction coefficient is an explicit expression of the water depth. The three-dimensional velocity field and the friction velocity can be reconstructed from the values of the depth-averaged quantities. Numerical simulations show that the enstrophy can be used to evaluate the development of the turbulent boundary layer. In the case of subcritical unsteady flows, the reconstructed velocity can be described with a logarithmic law modified by Coles’ wake function with apparent von Kármán constant, integration constant and wake-strength parameter, which differ from their values in steady flows. In the viscous sublayer the steady-state relation between the velocity and the vertical coordinate, in the inner scaling, is not valid for unsteady flows. Large errors on the calculation of the von Kármán constant can be made if the validity of the steady-state relation is assumed for unsteady flows. The comparisons of the reconstructed velocity profiles in the case of one-dimensional unsteady open-channel flows and two-dimensional wide trapezoidal channels show a good agreement with experiments.

Random-walk-path solution of unsteady flow equations for general channel networks

The random walk path method (RWP) is a stochastic model that transforms the solution of the unsteady open channel flow equations into a probability problem. In this paper, this method is introduced to solve the discretized Saint-Venant equations, with probabilistic representation of the water levels at junctions. Different boundary conditions, which specify either water levels or discharge hydrographs, can be implemented by specifying how walkers react when arriving at the boundaries. The pointwise solution at river junctions can be obtained via random walk simulations. The efficacy of the model was verified against the following test cases: (1) a real-world looped channel example documented in the manual of the software Hec-Ras and (2) a hypothetical channel system consisting of dendritic and divergent networks. The number of random simulations is an important aspect of the RWP method and needs to be chosen by considering the trade-off between precision and computational efficiency. The impact of the boundary water level or discharge on the water levels at internal nodes can be quantitatively evaluated by adopting the terminal weights, which present a distinct advantage of the RWP method. This assessment of water levels can serve as a guide in the operation of hydraulic controls, such as dams, sluices and pumps, to effectively regulate the flow in mitigating flood risks.

Estimated Zones of Saint-Venant Equations for Flood Routing with Over Bank Unsteady Flow in Open Channel

In this paper, we learn about the control of open channel water glide under the flood routing conditions. Generally, for flood routing in rivers, the Saint-Venant equations will be used which can be solved by finite distinction method. Saint-Venant equations will be converted into nonlinear equations and will be solved using the Preissmann scheme in the finite difference method. Using the Newton Raphson method, the set of equations will be changed into linear equations and will be solved by the space method. Our aims are to the estimated zones of Saint-Venant equations for flood routing by using the finite difference method with over bank unsteady flow in an open channel. The effectiveness of this method to optimize the choice of finite difference method is more accurate than other methods having adequate space and time steps.

Mathematical Modeling of water surface at Unsteady Flow in Al-Msharah River

This study is concerned with unsteady flow in open channels in boundary conditions that may exist in river or channel system. A software Hydrologic Engineering Center-River Analysis System (HEC-RAS) was used for a case study. The software contains a hydrodynamic model for simulating the unsteady flow in open channels based on a one-dimensional form of the Saint-Venant equations, by formulating a discrete form using the implicit finite difference scheme, then solving using the Newton-Raphson iteration procedure and the modified Gaussian elimination technique. The case study adopted with the help of HEC-RAS application was applied to a reach of Al Msharah River (channel) between AL Msharah Barrage (upstream)and Al Malah Bridge (Downstream), for a length of (49 Km) divided to (52) cross sections. Model runs and the comparison of results with actual field data indicate that the accuracy of the results obtained for(θ=1 and ∆t=5 minute) is quite acceptable, where θ is a weighting parameter and ∆t is the time interval. The paper covers the case of normal depth of flow and observed water stage data within the period ( from January 2006 to July 2006) . The results demonstrate that the area at Al Am'arah city at distance 17.5 Km from upstream(cross section 30)could be subjected to flooding at High Flow, therefore, it is recommended to adjust cross sections to prevent the flooding in this area. At last , Calibration of the hydrodynamic model is achieved in a study reach using the observed data (water stage) along AL Msharah River and show that a good agreement.

A Comparative Study of Two Numerical Methods for Regulating Unsteady Flow in Open Channels.(Dept.C)

Unsteady flow in open channels can be simulated by solving the Saint Venant equations to predict the discharge and water level in the channel during the future time series under the given condition. On the other hand, the operation problem is used to compute the inflow at the upstream section of the channel or a schedule of operation for the regulating structures of the delivery system to get a predefined water demand at the downstream end of the channel. This type of problem is also known as the inverse computation of open channel flow. The finite difference Preissmann implicit model was used to solve the operation-type problems in open channels, based on the saint Venant equations. The final linear system of equations was solved by the double-sweep algorithm. The computed results using the finite difference implicit model showed more accuracy and less oscillation than that obtained by an explicit model for all the numerical tests. The finite difference Preissmann implicit model was successfully applied to a nonprismatic reach of El-Mansouria canal between Meit-Ghamer and Sanayet regulator.

Numerical modelling for leak oil flow in tunnel of the cave storage

Abstract Oil storage in cavern tank is one of the important ways of oil storage. More than 80% of the oil tanks in the cave storage have been used for around 30 years in China and leakage accidents are likely to occur. When it occurs, the motor closing facilities preventing leakage spreading should be set up. The leaking oil flow characteristic in tunnel must be known for setting up the motor closing facilities. Based on the open-channel unsteady flow theory, a numerical model of tank leakage coupling the oil flow in tunnel was developed, and the characteristic line method was used to solve the model under the given initial condition and boundary condition. The variation of the depth and flow rate with time of oil flowing in the tunnel can be obtained. In order to verify the accuracy of model, the measurement data of high accuracy experimental system for unsteady open-channel flow were conducted to compare with the calculation. It fully confirms the high precision of the numerical model and reflects the variation characteristics of each hydraulic parameter in the numerical model of unsteady open channel flow truly. And then, the leaking oil flow characteristic in tunnel was obtained. It is that the farther from the leaking tank, the lower depth and flow rate of the oil in the tunnel revealed. Furthermore, the effect of the size of oil tank crack on the depth and flow rate was investigated. The greater size of crack makes the oil flow rate higher and the oil deeper in the tunnel and flow faster.

Positive surge propagating in an asymmetrical canal

A positive surge is an unsteady open channel flow motion characterised by a sudden rise in water surface elevation. The literature is focused primarily on rectangular channels. Herein, the free-surface features of positive surges propagating upstream were investigated in an asymmetrical canal. The surge propagation was a three-dimensional unsteady flow motion, resulting in a complicated transient secondary motion compared to positive surge propagating in rectangular canals. The present results may be relevant to surge propagation in man-made trapezoidal canals and natural irregular-shaped channels.

Numerical evaluation of the general flow hydraulics and estimation of the river plain by solving the Saint–Venant equation

Continuity and momentum equations govern the movement of unsteady flows in open channels, known as Saint–Venant equations. Numerical models of flood routing are classified according to the simplifications made in these equations. Among the flood routing methods, one can refer to dynamic wave and kinematic models and the Att-Kin and Muskingum hydrological models. The kinematic-wave model solves the Saint–Venant equations approximately by considering the gravity and the frictional force terms. This method assumes that the slopes of the energy gradient and the bed slope are equal. The full dynamic wave model uses all the terms in the Saint–Venant equation to simulate unsteady flows. Besides, given the historical records of the past floods and based on the flow continuity equation, the Att-Kin and Muskingum hydrological models deal with flood routing operations. This study describes the principles of Saint–Venant equations and numerical models of dynamic and kinematic-waves, as well as Att-Kin and Muskingum models, the results of which are compared. For this purpose, the Mehrian River range was used. The maximum flood discharge, the maximum area of the river flood plain, and the maximum flood flow width were estimated by presenting a new mathematical relation. The results showed that the MIKE 11 dynamic wave model yields better results than other models by solving partial differential equations. Moreover, among the hydrological methods, the mathematical flood routing and zoning indicate the closeness of Muskingum hydrological results using the least-squares model and the least error with the numerical values of the software for the observed flood.

Timescale interpolation and no-neighbour discretization for a 1D finite-volume Saint-Venant solver

ABSTRACT A new finite-volume numerical method for the one-dimensional (1D) Saint-Venant equations for unsteady open-channel flow is developed and tested. The model uses a recently-developed conservative finite-volume formulation that is inherently well-balanced for natural channels. A new timescale interpolation approach provides transition between 1st-order upwind and 2nd-order central interpolation schemes for supercritical and subcritical flow, respectively. This interpolation meets a proposed “no-neighbour” criterion for simplicity in future parallel implementation. Tests with a highly-resolved transitional flow and a coarsely-resolved natural channel show that the method is stable and accurate when applied with a flowrate damping algorithm that limits propagation of energy down to subgrid scales.

Effect of wall normal velocity on velocity distribution in unsteady flow

The effect of wall normal velocity on velocity distribution in unsteady flow has been investigated. We established a modified log-law model by adding the wake-term by Coles and deviation term together to depict the velocity deviation caused from classic log-low model. An empirical formula for deviation-correction factor α has been determined based on the present experimental data, and it yields a good agreement with other published experimental data. Experimental measurements of velocity distributions in unsteady open channel flows have been performed to prove this model.The cause of this addition was discussed and explained, and the theoretical velocity images from the improved model have been compared with the present data, they are well correlated.

Turbulence characteristics in a rough open channel under unsteady flow conditions

ABSTRACT Most open channel flows are generally unsteady in nature. Very few studies have been carried out in the past to investigate the turbulence characteristics in unsteady open channel flows over a rough bed at different test reaches. In the present study, experiments are conducted for two different flow depths over a rough bed and hydrographs are passed at the inlet of a rectangular flume. With consideration of laboratory constraints, the present study characterizes the unsteady flow as high catchment with long duration flood. The behavior of velocities in both the rising limb and falling limb for same flow depth is observed. Classical hysteresis effect of stage-discharge (h ~ Q) rating curve is illustrated. The present study investigates the turbulent intensities and Reynolds stresses under unsteady flow conditions. The turbulence intensities and Reynolds stresses in the rising limb are higher than that of the falling limb at same flow depth. Turbulence characteristics, i.e. turbulent kinetic energy (TKE), mean kinetic energy (MKE) and turbulent kinetic dissipation rate are also analyzed. It is found that the turbulence intensities, Reynolds stresses, and Turbulence characteristics are gradually decreased towards the downstream reaches.

Data Assimilation of Mobile Sensors in Hydrological Models of Unsteady Flow

Abstract In this paper, the estimation of the spatio-temporal variation of water bodies for state variables, velocity (m/s) and water surface elevation (m) for unsteady flows in open channels has been investigated. For data assimilation, average velocity measurements are obtained from mobile sensors such as Lagrangian sensors which have the ability to float passively in water bodies and provide their GPS location. One dimensional Saint-Venant equations are used for a system model linearized by a Taylor series expansion. To obtain a discrete-time state-space model, the coupled PDEs are discretized by Lax diffusive method in time and space. For state estimation of the open channel, a Kalman filter is set up with suitable filtering parameters for the channel’s model. Eulerian (fixed) sensors present at the head and tail of the canal provide the minimally required boundary conditions to run the model. The system is simulated using HEC-RAS simulation software. Water velocity profiles are used to predict the movement of the float, providing measurements for the Kalman Filter which is run in MATLAB. The estimated states are compared with actual values.

Characteristics of Positive Surges in a Rectangular Channel

A positive surge is an unsteady open channel flow motion characterized by an increase of flow depth. In previous experimental studies, a positive surge was typically induced by either a sudden increase of discharge in a channel or by the rapid closure of a downstream sluice gate, thus leading to a steep initial profile. However, in many instances, the evolution of a positive surge is of a progressive manner (e.g., in the downstream navigation canal during the emptying operation of lock chambers). In the present work, the inception and development of a positive surge induced by a progressive increase of discharge was investigated in a rectangular channel with a smooth bed. Both undular and breaking surges were studied. The results demonstrate that the maximum wave height at the first wave crest of an undular surge is in very close agreement with the McCowan theory. Additionally, the wave amplitude essentially shows a linearly increasing trend with an increasing surge Froude number up to Fr0 = 1.26 to 1.28, whereas it tends to suggest a power law reduction for larger surge Froude numbers. Moreover, the dispersion of undular surges is consistent with the linear wave theory only for surge Froude numbers close to unity. Overall, the present study demonstrates the unique features of positive surges induced by a progressive increase of discharge.

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 1: theoretical development

Abstract. The Saint-Venant equations are commonly used as the governing equations to solve for modeling the spatially varied unsteady flow in open channels. The presence of uncertainties in the channel or flow parameters renders these equations stochastic, thus requiring their solution in a stochastic framework in order to quantify the ensemble behavior and the variability of the process. While the Monte Carlo approach can be used for such a solution, its computational expense and its large number of simulations act to its disadvantage. This study proposes, explains, and derives a new methodology for solving the stochastic Saint-Venant equations in only one shot, without the need for a large number of simulations. The proposed methodology is derived by developing the nonlocal Lagrangian–Eulerian Fokker–Planck equation of the characteristic form of the stochastic Saint-Venant equations for an open-channel flow process, with an uncertain roughness coefficient. A numerical method for its solution is subsequently devised. The application and validation of this methodology are provided in a companion paper, in which the statistical results computed by the proposed methodology are compared against the results obtained by the Monte Carlo approach.

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 2: numerical application

Abstract. The characteristic form of the Saint-Venant equations is solved in a stochastic setting by using a newly proposed Fokker–Planck Equation (FPE) methodology. This methodology computes the ensemble behavior and variability of the unsteady flow in open channels by directly solving for the flow variables' time–space evolutionary probability distribution. The new methodology is tested on a stochastic unsteady open-channel flow problem, with an uncertainty arising from the channel's roughness coefficient. The computed statistical descriptions of the flow variables are compared to the results obtained through Monte Carlo (MC) simulations in order to evaluate the performance of the FPE methodology. The comparisons show that the proposed methodology can adequately predict the results of the considered stochastic flow problem, including the ensemble averages, variances, and probability density functions in time and space. Unlike the large number of simulations performed by the MC approach, only one simulation is required by the FPE methodology. Moreover, the total computational time of the FPE methodology is smaller than that of the MC approach, which could prove to be a particularly crucial advantage in systems with a large number of uncertain parameters. As such, the results obtained in this study indicate that the proposed FPE methodology is a powerful and time-efficient approach for predicting the ensemble average and variance behavior, in both space and time, for an open-channel flow process under an uncertain roughness coefficient.

Depth-averaged velocity and bed shear stress in unsteady open channel flow over rough bed

Very few studies have been carried out in the past in estimating depth-averaged velocity and bed shear stress in unsteady flow over rough beds. An experiment is thus conducted to investigate the vertical and lateral velocity profiles under unsteady flow conditions in a rough open channel for various flow depths. One hydrogram is repeatedly passed through the rectangular flume with a fixed rigid grass bed. Using micro Pitot tube and Acoustic Doppler Velocimeter (ADV), the flow patterns are investigated at both lateral and longitudinal positions over different cross-sections. For two typical flow depths, the velocities in both the rising limb and falling limb are observed. Hysteresis effect between stage-discharge (h ~ Q) rating curve between rising and falling limbs is illustrated. Lateral distribution of depth-averaged velocity and bed shear stress are plotted at three different cross sections and compared with the steady flow conditions. In falling limb of an unsteady flow case, both depth-averaged velocity and bed shear stress distribution in the central region is higher than that of steady flow case. However, in the rising limb, the bed shear stress of unsteady flow is less than that of steady flow case. Further, in an unsteady flow, the magnitude of depth-averaged velocity is found to increase towards the downstream sections. Along the downstream positions, bed shear stress values increase for lower flow depths and decrease for higher flow depth cases.

Interpolation of water surface profile in unsteady open channel flow using the adjoint method based on 2D shallow water equations

Real-time interpolation of high-water stage data obtained at separate water gauges was investigated to provide visual information for evacuation decision-making by residents near a river. The adjoint method was applied to a set of shallow water equations in a generalized coordinate system in which the hydraulic effect of tree communities is included as semi-permeable blocks. The cost function to be minimized was assumed as the summation of four misfits for water level and discharge from observation and for Manning's roughness (n) and tree community permeability (K) from standard values. The Lagrange multipliers method was adopted to determine the uncertain quantities, upstream discharge and spatial distributions of n and K, for minimizing the total cost under the constraint of the shallow water equations. The method was applied to a flood event observed in 2011 over a 20-km reach of the Tone River, where four water gauges are installed. The computation results successfully explained the time series of river discharge observed at stations located in the river reach, as well as the longitudinal profile of the trace water levels measured at 500-m intervals just after the flood event.