Simple treatment of non‐aligned boundaries in a Cartesian grid shallow flow model

A simple method is proposed for treating curved or irregular boundaries in Cartesian grid shallow flow models. It directly evaluates fictional values in ‘ghost’ cells adjacent to boundary cells and requires no interpolation or generation of cut cells. The boundary treatment is implemented in a dynamically adaptive quadtree grid‐based solver of the hyperbolic shallow water equations and validated against several test cases with analytical or alternative numerical solutions. The method is easy to code, accurate, and demonstrably effective in dealing with irregular computational domains in shallow flow simulations. Results are presented for still water in a basin of complicated geometry, steady hydraulic jump in an open channel with a converging sidewall, wind‐induced circulation in a circular shallow lake, and shock wave diffraction in a channel containing a contraction and expansion. Copyright © 2007 John Wiley & Sons, Ltd.     

Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results

The hydraulic jump is the sudden transition from a high-velocity open channel flow regime to a subcritical flow motion. The flow properties may be solved using continuity and momentum considerations. In this review paper, recent advances in turbulent hydraulic jumps are developed: the non-breaking undular hydraulic jump, the positive surge and tidal bore, and the air bubble entrainment in hydraulic jumps with roller. The review paper demonstrates that the hydraulic jump is a fascinating turbulent flow motion and the present knowledge is insufficient, especially at the scales of environmental and geophysical flows.     

Flow division at a channel crossing with subcritical or supercritical flow

The passage of an extreme storm over an urban area can lead to the flooding of the streets if the rainfall intensity exceeds the design value and/or the drainage system is not functional. The study of flow distribution in street networks thus is important for the design of flood protection measures. The flow distribution is affected by the junction flow characteristics, inflow discharges and downstream water depths. To reduce the degree of empiricism, a 3D Reynolds‐averaged Navier–Stokes equations model has been implemented in this study to investigate the flow phenomena in a cross junction. The Spalart–Allmaras model is used for turbulence closure. The numerical model utilizes the split‐operator approach, in which the advection, diffusion and pressure propagations are solved separately. The numerical model predicts accurately the flow distribution in a channel crossing under different subcritical flow conditions, for which experimental data are available. Recirculation zones exist at both the downstream channels and the associated contraction coefficient varies linearly with the ratio of the discharges at the two inlets. Secondary currents are apparent for the flow with strong asymmetric outlet conditions. Under supercritical inflow conditions, the model reproduces the hydraulic jump and hydraulic drop phenomena and predicts accurately the relationship between the input power ratio and the outflow discharge ratio of the street crossing. Copyright © 2009 John Wiley & Sons, Ltd.     

Flux‐corrected transport technique for open channel flow

In modeling flow in open channels, the traditional finite difference/finite volume schemes become inefficient and warrant special numerical treatment in the presence of shocks and discontinuities. The numerical oscillations that arise by making use of a second‐ and higher‐order schemes require some additional smoothing mechanism. A characteristic feature of high‐resolution schemes lies in smooth capturing of the shock fronts. This paper provides a general formulation for a flux‐corrected transport algorithm to the one‐dimensional open channel flow equations. The preliminary results presented show that the present algorithm is an efficient, conservative and robust tool that can be easily coded. To demonstrate the robustness of the present formulation, results are compared with other published numerical results, experimental data and analytical solutions when available. In particular, a comprehensive study on the effect of the source term, dry bed, variable width channel, steep sloping channel and flow with mixed flow conditions (as in a hydraulic jump) has been carried out to test the efficacy of the present algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

明渠气二相流的双流体模型

从解决平均二相流基本方程的封闭问题出发,分析了水利工程中气水二相流的特点,介绍了据此提出的可用于水利工程稀疏气泡流计算的双流体模型,并对明渠气水二相流进行了算例验证。     

齿墩与消力池联合消能机理分析

齿墩与消力池联合消能可用附加射流水跃理论分析 ,用跃后水深、水跃长度或单宽水跃容积减少的百分数 ,估计新增消能设施的经济效益 ,并通过模型试验验证消能的理论分析 ,计算新增设施的消能量、跃后水深、水跃长度或单宽水跃容积减少的百分数。结果表明 ,采用齿墩设施 ,可增进消力池的消能作用 ,并取得显著的经济效益 ,达到附加射流水跃理论所预期的效果     

Numerical analysis of turbulent structure in compound meandering open channel by algebraic Reynolds stress model

The turbulent flow in a compound meandering channel with a rectangular cross section is one of the most complicated turbulent flows, because the flow behaviour is influenced by several kinds of forces, including centrifugal forces, pressure‐driven forces and shear stresses generated by momentum transfer between the main channel and the flood plain. Numerical analysis has been performed for the fully developed turbulent flow in a compound meandering open‐channel flow using an algebraic Reynolds stress model. The boundary‐fitted coordinate system is introduced as a method for coordinate transformation in order to set the boundary conditions along the complicated shape of the meandering open channel. The turbulence model consists of transport equations for turbulent energy and dissipation, in conjunction with an algebraic stress model based on the Reynolds stress transport equations. With reference to the pressure–strain term, we have made use of a modified pressure–strain term. The boundary condition of the fluctuating vertical velocity is set to zero not only for the free surface, but also for computational grid points next to the free surface, because experimental results have shown that the fluctuating vertical velocity approaches zero near the free surface. In order to examine the validity of the present numerical method and the turbulent model, the calculated results are compared with experimental data measured by laser Doppler anemometer. In addition, the compound meandering open channel is clarified somewhat based on the calculated results. As a result of the analysis, the present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent flow in a compound meandering open channel. Copyright © 2005 John Wiley & Sons, Ltd.     

Weighted averaged equations for modeling velocity profiles of 1D steady open channel flows

A new mathematical algorithm is proposed to address the essential details of vertical distributions of horizontal velocity for one‐dimensional steady open‐channel flow. This new algorithm comprises a system of weighted averaged equations developed from corresponding Reynolds equations by performing weighted average operations instead of conventional depth average operations. It is the system of weighted averaged equations, instead of the vertical grids, that allows for more hydraulic coefficients identifiable. It can be thought of as an extension of the St. Venant equations to address the vertical distributions of horizontal velocities, as well as the water surface profiles. To avoid the difficult expansion of governing partial differential equations in high order, an indirect scheme is proposed to solve hydraulic variables through their weighted average values. The governing partial differential equations are generated by using a variety of weight functions, and the weighted averages of relevant hydraulic variables are taken as the unknown independent variables to be solved first. Then, on the basis of the values and polynomial expansions of these weighted averaged velocities, a system of linear algebraic equations is generated and the unknown hydraulic variables or their coefficients are easily solved. Note that the new model is not proposed to compete with any three‐dimensional models in modeling accuracy or accommodation ability to all conditions. It just provides a valuable option to study the vertical structure of flow in open channels where only essential detail and reasonable accuracy of vertical distributions are required, and the data availability and other conditions limit the application of fully three‐dimensional models. The performance of the model is evaluated with experimental data of flows in two different flumes. It is shown that the model well predicted the velocity profiles of sections along the centerlines of these flumes with reasonable accuracy and essential details of vertical distributions of horizontal velocity. Copyright © 2013 John Wiley & Sons, Ltd.     

Central schemes for open‐channel flow

The resolution of the Saint‐Venant equations for modelling shock phenomena in open‐channel flow by using the second‐order central schemes of Nessyahu and Tadmor (NT) and Kurganov and Tadmor (KT) is presented. The performances of the two schemes that we have extended to the non‐homogeneous case and that of the classical first‐order Lax–Friedrichs (LF) scheme in predicting dam‐break and hydraulic jumps in rectangular open channels are investigated on the basis of different numerical and physical conditions. The efficiency and robustness of the schemes are tested by comparing model results with analytical or experimental solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

Free-surface fluctuations in hydraulic jumps: Experimental observations

A hydraulic jump is the rapid and sudden transition from a high-velocity supercritical open channel flow to a subcritical flow. It is characterised by the dynamic interactions of the large-scale eddies with the free-surface. New series of experimental measurements were conducted in hydraulic jumps with Froude numbers between 3.1 and 8.5 to investigate these interactions. The dynamic free surface measurements were performed with a non-intrusive technique while the two-phase flow properties were recorded with a phase-detection probe. The shape of the mean free surface profile was well defined and the turbulent fluctuation profiles highlighted a distinct peak of turbulent intensity in the first part of the jump roller, with free-surface fluctuation levels increasing with increasing Froude number. The dominant free-surface fluctuation frequencies were typically between 1 and 4 Hz. A comparison between the acoustic sensor signals and conductivity probe data suggested that the air–water “free-surface” detected by the acoustic sensor corresponded to about the boundary between the turbulent shear layer and the upper free-surface layer. Simultaneous measurements of free surface and bubbly flow fluctuations for Fr = 5.1 indicated that the frequency ranges of both sensors were similar (F < 5 Hz) whatever the position downstream of the toe. The present results highlighted that the dynamic free-surface measurements can be conducted successfully using acoustic displacement meters, and the time-averaged depth measurements was a physical measure of the free-surface location in hydraulic jumps.     

Development of a generalized multi‐layer model for 3‐D simulation of free surface flows

A mathematical model was developed for three‐dimensional (3‐D) simulation of free surface flows. In this model, the flow depth is divided into a number of layers and shallow water equations are integrated in each layer to derive the hydrodynamic equations. To give a general form to this model, each layer is assumed to be non‐horizontal with varying thickness in the flow domain. A non‐orthogonal curvilinear coordinate system is employed in the model, to allow for flexibility in dealing with the irregular geometry of natural watercourses. Due to the similarity in governing equations, two‐dimensional (2‐D) depth averaged programs can be developed into a multi‐layer model. The development for a depth averaged program and its numerical scheme is described in this paper. Experimental data and semi‐analytical solutions are used to evaluate the performance of the model. Three different cases of open channel flow are tested: 1‐flow in a straight open channel, 2‐the flow development region in a channel, and 3‐flow in a meandering channel. It is shown that the model has the capability to predict velocity distribution and secondary flows in complex 3‐D flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

A fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes

Simulation of 1D steady flow covers a wide range of practical applications, such as rivers, pipes and hydraulic structures. Various flow patterns coexist in such situations: free surface flows (supercritical, subcritical and hydraulic jump), pressurized flows as well as mixed flows. As a result, development of a unified 1D model for all the situations of interest in civil engineering remains challenging. In this paper, a fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes is set up and assessed. Developments are initiated from an original unified mathematical model using the Saint‐Venant equations. Application of these equations, originally dedicated to free‐surface flow, is extended to pressurized flow by means of the Preissmann slot model. In particular, an original negative slot is developed in order to handle sub‐atmospheric pressurized flow. Next, the full unsteady model is simplified under the assumption of steadiness and reformulated into a single pseudo‐unsteady differential equation. The derived pseudo‐unsteady formulation aims at keeping the hyperbolic feature of the equation. Stability analysis of the differential equation suggests a unique splitting for the finite volume scheme whatever the flow conditions. The numerical scheme obtained is a universal Flux Vector Splitting which shows robustness and simplicity. Accuracy and performance of the new methodology is assessed by comparison with analytical and experimental results. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional numerical simulation of compound meandering open channel flow by the Reynolds stress model

Turbulent flow in a compound meandering open channel with seminatural cross sections is one of the most complicated turbulent flows as the flow pattern is influenced by the combined action of various forces, such as centrifugal force, pressure, and shear stresses. In this paper, a three‐dimensional (3D) Reynolds stress model (RSM) is adopted to simulate the compound meandering channel flows. Governing equations of the flow are solved numerically with finite‐volume method. The velocity fields, wall shear stresses, and Reynolds stresses are calculated for a range of input conditions. Good agreement between the simulated results and measurements indicates that RSM can successfully predict the complicated flow phenomenon. Copyright © 2008 John Wiley & Sons, Ltd.     

A confrontation of 1D and 2D RKDG numerical simulation of transitional flow at open‐channel junction

In this study, a comparison between the 1D and 2D numerical simulation of transitional flow in open‐channel networks is presented and completely described allowing for a full comprehension of the modeling water flow. For flow in an open‐channel network, mutual effects exist among the channel branches joining at a junction. Therefore, for the 1D study, the whole system (branches and junction) cannot be treated individually. The 1D Saint Venant equations calculating the flow in the branches are then supplemented by various equations used at the junction: a discharge flow conservation equation between the branches arriving and leaving the junction, and a momentum or energy conservation equation. The disadvantages of the 1D study are that the equations used at the junction are of empirical nature due to certain parameters given by experimental results and moreover they often present a reduced field of validity. On the contrary, for the 2D study, the entire network is considered as a single unit and the flow in all the branches and junctions is solved simultaneously. Therefore, we simply apply the 2D Saint Venant equations, which are solved by a second‐order Runge–Kutta discontinuous Galerkin method. Finally, the experimental results obtained by Hager are used to validate and to compare the two approaches 1D and 2D. Copyright © 2009 John Wiley & Sons, Ltd.     

Bubbly flow measurements in hydraulic jumps with small inflow Froude numbers

The transition from supercritical to subcritical open channel flow is characterised by a strong dissipative mechanism called a hydraulic jump. A hydraulic jump is turbulent and associated with the development of large-scale turbulence and air entrainment. In the present study, some new physical experiments were conducted to characterise the bubbly flow region of hydraulic jumps with relatively small Froude numbers (2.4 < Fr₁ < 5.1) and relatively large Reynolds numbers (6.6 × 10⁴ < Re < 1.3 × 10⁵). The shape of the time-averaged free-surface profiles was well defined and the longitudinal profiles were in agreement with visual observations. The turbulent free-surface fluctuation profiles exhibited a peak of maximum intensity in the first half of the hydraulic jump roller, and the fluctuations exhibited some characteristic frequencies typically below 3 Hz. The air–water flow properties showed two characteristic regions: the shear layer region in the lower part of the flow and an upper free-surface region above. The air–water shear layer region was characterised by local maxima in terms of void fraction and bubble count rate. Other air–water flow characteristics were documented including the distributions of interfacial velocity and turbulence intensity. The probability distribution functions (PDF) of bubble chord time showed that the bubble chord times exhibited a broad spectrum, with a majority of bubble chord times between 0.5 and 2 ms. An analysis of the longitudinal air–water structure highlighted a significant proportion of bubbles travelling within a cluster structure.     

Turbulent free surface flow simulation using a multilayer model

Based on the steady hydrodynamic equations, a multilayer (ML) model has been formulated for simulating turbulent flow in open channels. The model is imposed on a general curvilinear co-ordinate system with non-staggered finite volume discretization. The turbulent quantities in the model are described by the layer-averaged K-ε turbulence model with standard coefficients. Assuming a vertical hydrostatic pressure distribution, a depth correction scheme, originating in the Rhie and Chow approach for confined flows, is incorporated into the SIMPLE procedure to compute the water surface. Using the multilayer model, flows in a 180° channel bend, near a groin, and in straight open channels are computed. The results are compared with experimental data and with calculations of a depth-averaged model (DAV) having three-dimensional effect corrections. The comparisons show that the predictions of the ML model on mean flow values are in good agreement with the available data and are better than those of the DAV model. The vertical distribution of the turbulent energy dissipation rate is also shown to agree well with the open-channel measurements.     

The study on three-dimensional mathematical model of river bed erosion for water-sediment two-phase flow

Based on the tensor analysis of water-sediment two-phase flow, the basic model equations for clear water flow and sediment-laden flow are deduced in the general curve coordinates for natural water variable-density turbulent flow. Furthermore, corresponding boundary conditions are also presented in connection with the composition and movement of non-uniform bed material. The theoretical results are applied to the calculation of the float open caisson in the construction period and good results are obtained.     

SPH modelling of depth‐limited turbulent open channel flows over rough boundaries

A numerical model based on the smoothed particle hydrodynamics method is developed to simulate depth‐limited turbulent open channel flows over hydraulically rough beds. The 2D Lagrangian form of the Navier–Stokes equations is solved, in which a drag‐based formulation is used based on an effective roughness zone near the bed to account for the roughness effect of bed spheres and an improved sub‐particle‐scale model is applied to account for the effect of turbulence. The sub‐particle‐scale model is constructed based on the mixing‐length assumption rather than the standard Smagorinsky approach to compute the eddy‐viscosity. A robust in/out‐flow boundary technique is also proposed to achieve stable uniform flow conditions at the inlet and outlet boundaries where the flow characteristics are unknown. The model is applied to simulate uniform open channel flows over a rough bed composed of regular spheres and validated by experimental velocity data. To investigate the influence of the bed roughness on different flow conditions, data from 12 experimental tests with different bed slopes and uniform water depths are simulated, and a good agreement has been observed between the model and experimental results of the streamwise velocity and turbulent shear stress. This shows that both the roughness effect and flow turbulence should be addressed in order to simulate the correct mechanisms of turbulent flow over a rough bed boundary and that the presented smoothed particle hydrodynamics model accomplishes this successfully. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd     

Quantitative flow visualization using the hydraulic analogy

The current work describes the development of a non-intrusive optical method for the quantitative determination of water heights along a hydraulic jump in shooting water flows on a water table. The technique involves optically superimposing a series of alternating dark and clear fringes on the water flow. It is proposed that the fringe deviations seen under a hydraulic jump can be simulated using a series of optical prisms oriented along the direction of the hydraulic jump. The height of each prism gives the local maximum water height at the fringe location. Three types of theoretical prism configurations (isosceles flat-topped prism, scalene flat-topped prism and rounded-topped prism models) have been studied for two flow systems: shooting flow around a wedge and around a cylinder. Equations relating the physical characteristics of the deviated fringes to the height of the theoretical prism and hence the local water height are presented. The variation in water height along a hydraulic jump for flow around a wedge obtained using the optical technique has been compared with heights obtained using a depth gauge. The results were in good agreement for the range of Froude numbers studied (Fr=1.9−3.6). The rounded-topped prism model led to the best agreement with the physical measurements, within 11% throughout the range of conditions studied. The uncertainty associated with the water height determination using the optical technique is ±10%. Received: 15 September 1998/Accepted: 16 April 1999     

Analytical solutions for transverse distributions of stream-wise velocity in turbulent flow in rectangular channel with partial vegetation

The theory of poroelasticity is introduced to study the hydraulic properties of the steady uniform turbulent flow in a partially vegetated rectangular channel. Plants are assumed as immovable media. The resistance caused by vegetation is expressed by the theory of poroelasticity. Considering the influence of a secondary flow, the momentum equation can be simplified. The momentum equation is nondimensionalized to obtain a smooth solution for the lateral distribution of the longitudinal velocity. To verify the model, an acoustic Doppler velocimeter (ADV) is used to measure the velocity field in a rectangular open channel partially with emergent artificial rigid vegetation. Comparisons between the measured data and the computed results show that the method can predict the transverse distributions of stream-wise velocities in turbulent flows in a rectangular channel with partial vegetation.