Applicability of Kinematic, Noninertia, and Quasi-Steady Dynamic Wave Models to Unsteady Flow Routing

Propagation of flood waves in an open channel can be mathematically approximated by the Saint-Venant equations (dynamic wave) or by their simplifications including the kinematic wave, noninertia wave, gravity wave, and quasi-steady dynamic wave models. All of these wave approximations differ not only in the physical propagation mechanism, but also in the degree of complexity involved in computation. In order to efficiently implement the approximate wave models for flood routing, their criteria of applicability should be developed. The applicability of the kinematic wave, noninertia wave, and quasi-steady dynamic wave approximations to the full dynamic wave equations for unsteady flow routing is examined by comparing the propagation characteristics of a sinusoidal perturbation to the steady gradually varying flow for different simplified wave models. Development of the applicability criteria provides a guideline for selecting an appropriate wave model for unsteady flow modeling, thus enabling an assessment of the capabilities and limitations of different simplified wave models. By using the linear stability analysis, the derived criteria can be expressed in terms of dimensionless physical parameters that represent the unsteadiness of the wave disturbance, characteristics of the downstream boundary condition (backwater effect), and the location along the channel. The developed criteria are for a specific point and time, thereby providing a more refined indication than the integrated criteria based on the testing for a hydrograph found commonly in the literature. In this study, we have justified whether the simplified wave models such as the kinematic, noninertia, or gravity wave models would be appropriate and reliable approximations to the full Saint-Venant equations with a comparable accuracy for a given flow condition. The downstream backwater effect has been taken into consideration in the developed criteria for broader engineering applications. One hypothetical example is presented for illustration.     

Deflector Ski Jump Hydraulics

Ski jumps are a standard element of dam spillways for an efficient energy dissipation if takeoff velocities are large, and stilling basins cannot be applied. This laboratory study investigates the hydraulic performance of a triangular-shaped, rather than the conventional circular-shaped, bucket placed at the takeoff of ski jumps. The following items were addressed: (1) pressure head maximum and pressure distribution along the triangular-shaped bucket; (2) takeoff characteristics as a function of the bucket deflector angle and the relative bucket height including the lower and the upper jet trajectories; (3) jet impact characteristics in a prismatic tailwater channel including the shock wave formation and the height of recirculation depth below the jet cavity; (4) energy dissipation across the ski jump, from the approach flow channel to downstream of jet impact; and (5) choking flow conditions of the flip bucket. A significant effect of the approach flow Froude number, the relative bucket height, and the deflector angle is found. A comparison with previous results for the circular-shaped bucket geometry indicates a favorable behavior of the novel bucket design.     

Finite-Element Analysis of Double-Free-Surface Flow through Slit in Dam

The fluid loads on some hydraulic structures and the free-surface profiles of the flow need to be determined for design purposes. This is a difficult task because the governing equations have nonlinear boundary conditions. The goal of the present work is to develop a suitable and accurate numerical procedure for the computation of free-surface profiles, velocity and pressure distributions, and flow rate for a 2D gravity fluid flow through a conduit in the pattern of a free jet. The problem involves two highly curved unknown free surfaces and arbitrary curve-shaped boundaries. These features make the problem more complicated than the flow under a sluice gate or over a weir. A combination of a variable domain and a fixed domain finite-element method is used to solve the problem. The results of the calculations show good agreement with previous flow solutions for the water surface profiles and pressure distributions throughout the flow domain and on the gate. Results are also confirmed by conducting a hydraulic model test.     

Ski Jump Hydraulics

Ski jumps are a major element of each dam spillway because these are the only structures able to accomplish satisfactory energy dissipation for takeoff velocities in excess of some 20?m/s. This research aims to add to several hydraulic problems with ski jumps that have not yet been systematically solved so far. Based on an experimental campaign, the following problems were addressed: (1) pressure head maximum and pressure distribution along a circular-shaped flip bucket; (2) takeoff characteristics for a certain bucket deflection and a relative bucket curvature including the jet trajectories of both the lower and the upper nappes; (3) impact characteristics in a prismatic tailwater channel with details of shock wave formation and height of recirculation depth; (4) energy dissipation across the ski jump, from the upstream channel to downstream of jet impact; and (5) choking flow conditions by the flip bucket. These results demonstrated the significant effect of the approach Froude number, the relative bucket curvature and the bucket angle. The results allow immediate application to the design of ski jumps in hydraulic engineering.     

Numerical Investigation of Wave–Current–Vegetation Interaction

A fully three-dimensional numerical model has been developed to simulate the wave–current–vegetation interaction phenomenon. Physical experiments have also been carried out to provide data for the verification of the model. The numerical model utilizes the split-operator approach, in which the advection, diffusion, and pressure propagation are solved separately. Vegetation is modeled as a sink of momentum. The unsteady fluid force on vegetation is split into a time-dependent inertia component and a drag component. The model has been applied to simulate vegetation under pure waves, pure current, as well as wave current. Compared to available experimental data, the model is capable of reproducing the turbulence and velocity profiles induced by vegetation–current interaction. The wave attenuation due to vegetation is simulated correctly with a proper value of drag coefficient. Both the physical experiments and numerical simulations show that the interaction of waves and current leads to a greater attenuation of waves in the presence of vegetation, which can be explained by the nonlinear nature of the resistance force induced by the vegetation.     

Effect of Gravity and Initial Stress on Torsional Surface Waves in Dry Sandy Medium

This problem studies the effect of gravity and initial stress on the propagation of torsional surface waves in dry sandy medium. The mathematical analysis of the problem has been dealt with the Whittaker function. Assuming the expansion of the Whittaker function up to linear term, it is concluded that the gravity field will always allow torsional waves to propagate. The expansion of the Whittaker function up to quadratic terms shows that two such wave fronts may exist in the medium. Finally, it is concluded that the sandy medium without support of a gravity field cannot allow the propagation of torsional surface waves, where as the presence of a gravity field always supports the propagation of torsional surface waves regardless of whether the medium is elastic or dry sandy.     

Local Scour Caused by Submerged Square Jets under Model Ice Cover

An experimental study was carried out to understand the local scour process of noncohesive sand beds caused by submerged three-dimensional square jets under model ice covered conditions. The characteristics of asymptotic scour have been investigated by varying the tailwater depth, the densimetric Froude number, the grain size of the bed material, and three different types of water surface conditions (open-water flow, smooth ice covered flow, and rough ice covered flow). Results show that the local scour process under covered conditions is different from that under open water, especially at lower tailwater depths. Further, for the range of test conditions, the effect of the ice cover is reduced when the bed is composed of finer sand particles or when the densimetric Froude number is small.     

VOF Model for Simulation of a Free Overfall in Trapezoidal Channels

In the past, solutions to open channel flow problems involving free surfaces were generally found on the basis of experimental data or through the development of theoretical expressions using simplified assumptions. The volume of fluid (VOF) turbulence model is applied to obtain characteristics of three-dimensional open channel flows involving free surfaces. In particular, the VOF model is used to determine the pressure head distributions, velocity distributions, and water surface profiles for the free overfall in a trapezoidal open channel. The predictions of the proposed model are validated using existing experimental data.     

The effect of gravity on the combustion synthesis of metal-ceramic composites

The effects of gravity on the combustion characteristics and microstructure of metal-ceramic composites (HfB₂/Al and Ni₃Ti/TiB₂ systems) were studied under both normal and low gravity conditions. Under normal gravity conditions, pellets were ignited in three orientations relative to the gravity vector. Low gravity combustion synthesis (SHS) was carried out on a DC-9 aircraft at the NASA-Lewis Research Center. It was found that under normal gravity conditions, both the combustion temperature and wave velocity were highest when the pellet was ignited from the bottom orientation; i.e., the wave propagation direction was directly opposed to the gravitational force. The SHS of 70 vol pct Al (in the Al-HfB₂ system) was changed from unstable, slow, and incomplete when ignited from the top to unstable, faster, and complete combustion when ignited from the bottom. The hydrostatic force (height × density × gravity) in the liquid aluminum was thought to be the cause of formation of aluminum nodules at the surface of the pellet. The aluminum nodules that were observed on the surface of the pellet when reacted under normal gravity were totally absent for reactions conducted under low gravity. Buoyancy of the TiB₂ particles and sedimentation of the Ni₃Ti phase were observed for the Ni₃Ti/TiB₂ system. The possibility of liquid convective flow at the combustion front was also discussed. Under low gravity conditions, both the combustion temperature and wave velocity were lower than those under normal gravity. The distribution of the ceramic phase, i.e., TiB₂ or HfB₂, in the intermetallic (Ni₃Ti) or reactive (Al) matrix was more uniform.     

Simulation of a 3D Numerical Viscous Wave Tank

A numerical scheme was developed to solve the unsteady three-dimensional (3D) Navier–Stokes equations and the fully nonlinear free surface boundary conditions for simulating a 3D numerical viscous wave tank. The finite-analytic method was used to discretize the partial differential equations, and the marker-and-cell method was extended to treat the 3D free surfaces. A piston-type wave generator was incorporated in the computational domain to generate the desired incident waves. This wave tank model was applied to simulate the generation and propagation of a solitary wave in the wave tank and the diffraction of periodic waves by a semiinfinite breakwater. The computation was carried out by a PC cluster established by connecting several personal computers. The message passing interface (MPI) parallel language and MPICH software were used to write the computer code for parallel computing. High consistency between the numerical results and the theoretical solutions for the wave and velocity profiles confirms the accuracy of the proposed wave tank model.     

Undular Hydraulic Jumps in Circular Conduits

Undular hydraulic jumps in circular conduits are considered with an experimental approach. Based on previous findings in rectangular channels, this research indicates differences in terms of shape effects. All present results depend on the filling ratio of the upstream conduit flow in addition to the upstream Froude number. The results include information on the wave crests and troughs, wave lengths, and generalized axial surface profiles. The wall surface profile is shown to be similar to the axial wave profile, but with smaller wave extrema and a wave shift. The design of conduits containing undular jumps should be avoided because of unstable flow. It is also demonstrated that conduits may choke in the presence of undular jumps, with a previously established choking number relating to a design limit. For flows with choking numbers in excess of 1, choking occurs associated with a transition from the free surface to the pressurized conduit flow.     

Fully Nonhydrostatic Modeling of Surface Waves

A fully nonhydrostatic model is tested by simulating a range of surface-wave motions, including linear dispersive waves, nonlinear Stokes waves, wave propagation over bottom topographies, and wave–current interaction. The model uses an efficient implicit method to solve the unsteady, three-dimensional, Navier-Stokes equations and the fully nonlinear free-surface boundary conditions. A new top-layer pressure treatment is incorporated to fully include the nonhydrostatic pressure effect. The model results are verified against either analytical solutions or experimental data. It is found that the model using a small number of vertical layers is capable of accurately simulating both the free-surface elevation and vertical flow structure. By further examining the model’s performance of resolving wave dispersion and nonlinearity, the model’s efficiency and accuracy are demonstrated.     

RANS∕Laplace Calculations of Nonlinear Waves Induced by Surface-Piercing Bodies

An interactive zonal numerical method has been developed for the prediction of free surface flows around surface-piercing bodies, including both viscous and nonlinear wave effects. In this study, a Laplace solver for potential flow body-wave problems is used in conjunction with a Reynolds-averaged Navier-Stokes (RANS) method for accurate resolution of viscous, nonlinear free surface flows around a vertical strut and a series 60 ship hull. The Laplace equation for potential flow is solved in the far field to provide the nonlinear waves generated by the body. The RANS method is used in the near field to resolve the turbulent boundary layers, wakes, and nonlinear waves around the body. Both the kinematic and dynamic boundary conditions are satisfied on the exact free surface to ensure accurate resolution of the divergent and transverse waves. The viscous-inviscid interaction between the potential flow and viscous flow regions is captured through a direct matching of the velocity and pressure fields in an overlapping RANS and potential flow computational region. The numerical results demonstrate the capability of an interactive RANS∕Laplace coupling method for accurate and efficient resolution of the body boundary layer, the viscous wake, and the nonlinear waves induced by surface-piercing bodies.     

Effect of Soil Locking on the Cylindrical Shock Wave’s Peak Pressure Attenuation

The paper investigates the characteristics of propagating shock waves in soil, resulting from an explosion of a cylindrical line charge. The main purpose of this investigation is to study the full locking parameter’s effect on the qualitative and quantitative behavior of the peak pressure attenuation. The soil is modeled as a bulk irreversible compressible elastic plastic medium, including full bulk locking and dependence of the current deviatoric yield stress on the pressure. The Lagrange approach and the modified variational-difference methods are used to simulate the process. A study of the characteristics of the equation of the state of soil was carried out for the case of a cylindrical line charge explosion in an infinite soil medium. The computed results were compared with experimental data. It was found that the full locking model somewhat overpredicts the measured stresses. The dependence of the peak stress attenuation, during the shock wave propagation, on the full locking parameter of the equation of state was studied. When the pressure-volumetric strain relationship beyond the point of full compaction is not steep, the attenuation may be described by a power law that may be expressed as a linear relationship on a logarithmic scale. When this relationship is relatively steep however, the logarithmic dependence of the peak stress on the shock wave coordinate is close to a bilinear function. It was also shown that the shock wave peak stress parameters of this power law for the high pressure range (i.e., short distance of the wave front from the explosive charge) are linearly dependent on the full locking parameter. Also it was shown that the full compaction density affects the exponent of stress-distance power relationship only at the earlier stage of the wave propagation and only when the range of nonlinear elastic compaction is relatively narrow.     

Hydrodynamic Pressures on Arch Dam during Earthquakes

A complete three-dimensional finite difference scheme has been developed and used to analyze the nonlinear hydrodynamic pressures on arch dam during earthquakes. Both free-surface waves and nonlinear convective acceleration are included in the analysis. Various dam shapes and reservoir banks were studied and the characteristic of nonlinear contribution due to curved geometry and free-surface waves are discussed. Numerical experiments have been made to determine the desirable mesh size arrangements and time increments. The effects of surface wave and convective acceleration on hydrodynamic pressure could augment the hydrodynamic pressure to 10% larger than that of a linear analysis. For an earthquake with large ground displacement, a large rise of the water surface could probably happen and be a severe threat to human safety in recreational areas and even cause overflow to affect the safe operation of the arch dam. An empirical formula is given to predict approximate hydrodynamic force.     

Modeling 3D Supercritical Flow with Extended Shallow-Water Approach

Despite the three-dimensional (3D) nature of the flow, the classical shallow-water equations are often used to simulate supercritical flow in channel transitions. A closer comparison with experimental data, however, often shows large discrepancies in the height and pattern of the shock waves that increase with the Froude number. An extension to the classical shallow-water approach is derived considering higher-order distribution functions for pressure and horizontal and vertical velocities, therefore taking nonhydrostatic pressure distribution and vertical momentum into account. The approach is applied to highly supercritical flow in a channel contraction (F0 = 4.0), a channel junction (F0 ≈ 4.5 for both branches), and a channel expansion (F0 = 8.0). Specific problems of such flows—wetting and drying of computational cells and wave breaking due to steep free-surface gradients—are discussed and solved numerically. The solutions with the extended approach are compared both with experimental data and classical shallow-water computations, and the influence of the additional terms considering the 3D nature of such flows is illustrated.     

Analysis of Dynamic Wave Model for Unsteady Flow in an Open Channel

The models for flood propagation in an open channel are governed by Saint-Venant’s equations or by their simplified forms. Assuming the full form of hyperbolic type nonlinear expressions, the complete or dynamic wave model is obtained. Hence, after first-order linearization procedure, the dispersion relation is obtained by using the classical Fourier analysis. From this expression, the phase and group speed and the variations of the amplitude of the waves are defined and investigated. Adopting Manning’s resistance formula, the effects of the variations of the Froude number, Courant number, and friction parameter are examined in the wave number domain for progressive and regressive waves. For small and high wave numbers, the simplified kinematic and gravity wave models are recovered, respectively. Moreover, the analysis confirms, according to the Vedernikov criterion, the Froude number value corresponding to the stability condition to contrast the development of roll waves. In addition, for stable flow on the group speed versus wave number curves, the results show critical points, maximum and minimum for progressive and regressive waves, respectively.     

Conservative Formulation for Natural Open Channels and Finite-Element Implementation

This investigation considers an approximate formulation of the St. Venant equations for natural channels, in which the fully conservative form is developed by revising the boundary pressure term accounting for the topographic variation in the momentum equation. As such a formulation has the potential to enhance the performance of existing models used in practice, the accuracy implications for this approximate formulation are examined using an error analysis for a simplified case. Further, an energy calculation is performed which illustrates that an earlier formulation actually results in energy gain for some cases. A more general formula for the constant water surface elevation that corrects this is introduced and tested. It is found that the refined formulation presented here is accurate for hydraulic jumps, steep surge waves, and flood wave propagation in natural channels. The shock capturing capability of the approximate formulation is illustrated for both steady- and unsteady-flow situations using the finite-element method, for which this approximate equation formulation adapts naturally. Using the characteristic-dissipative-Galerkin finite-element scheme, good results are obtained for the case of a hydraulic jump in a diverging rectangular channel, with the maximum percent error associated with the approximate formulation determined to be only 0.34%. For the case of dam break wave propagation in a converging and diverging rectangular channel, the model performs similarly well, with the maximum error only 0.0064%. Further, the approximate formulation is used to simulate the flood routing in a natural channel, the Oldman River in southern Alberta. The computational results are in good agreement with the observed data. The arrival time of peak flow is 5?h earlier and the magnitude of peak discharge is only 3.8% lower than the observed value.     

Flow through Side Sluice Gate

The hydraulic characteristics of a side sluice gate were studied experimentally. It was found that the specific energy remains constant along the side sluice gate. The coefficient of discharge for the side sluice gate is related to the main channel Froude number and the ratio of upstream depth of flow to sluice gate opening for free flow. It also depends on an additional parameter: the ratio of tailwater depth to the gate opening for submerged flow. Suitable equations for discharge coefficient are obtained.     

Supercritical Flow across Sewer Manholes

The hydraulics of supercritical flow across manholes in sewers is explored using systematic experimentation. Due to the expansion at the manhole entrance an in-manhole wave is generated. Further, at the downstream manhole end, flows with a sufficiently large filling ratio impinge on the wall and lead to a so-called swell. In addition, due to shock wave generation in the downstream sewer, a sewer wave is generated. The heights and locations of these three waves were determined in terms of basic hydraulic quantities. More importantly, the capacity of the manhole and the downstream sewer under wave action was quantified. It was found that in order for free surface flow to be maintained the common design standard for sewers with a supercritical approach flow have to be revised. These implications have to be accounted for in future designs.